Characterizing group strategy-proof rules in the object allocation problem with money

Abstract

We study the object allocation problem with money. The owner possesses a single object, and each agent has preferences that are not necessarily quasi-linear. We examine various group incentive properties, which are classified according to the following criteria: (i) whether the group incentive property is weak or strong, (ii) the maximum number of agents who may form a coalition, (iii) whether only self-enforcing manipulations are considered, (iv) whether agents in a coalition can reallocate the object among themselves after misrepresenting their preferences, and (v) whether agents in a coalition can arrange side payments among themselves. We characterize the classes of rules that satisfy various group incentive properties, along with a mild property of non-imposition. Furthermore, we establish the revelation principles for rules that satisfy the group incentive properties.

1 Introduction

1.1 Purpose

We study the object allocation problem with money. The owner possesses a single object. A bundle specifies the consumption of the object along with a payment. Each agent has preferences over the set of bundles, which are not necessarily quasi-linear. Non-quasi-linear preferences reflect agents’ income effects or liquidity constraints in borrowing money from financial markets. An allocation specifies a bundle for each agent. An (allocation) rule (or a direct mechanism) is a function that associates an allocation with each preference profile.

One of the most important examples of the object allocation problem with money is auctions. A significant concern in practical auction design is preventing agents from colluding. As Klemperer (2002) states, “a first major set of concerns for practical auction design involves the risk that participants may explicitly or tacitly collude to avoid bidding up prices.” Collusion among bidders is pervasive in public procurement auctions, particularly in Japanese procurement auctions for public construction (McMillan 1991, 2003). One of the most notorious recent collusion incidents involved bid-rigging for the public procurement of the test events for the Tokyo 2020 Olympics. The total winning bids under collusion amounted to about 285 million dollars.¹

The requirement to prevent collusion in auctions corresponds to the group incentive properties of rules in the object allocation problem with money. A group incentive property typically requires that coalitional misbehavior should never be beneficial to the coalition. Arguably, the most standard group incentive property in the literature is (weak or strong) group strategy-proofness, which requires that no coalition should ever benefit from misrepresenting their preferences.

Although group strategy-proofness is a reasonable group incentive property, it overlooks several important aspects of collusion in practical auctions as follows.

• Sizes of coalitions. Group strategy-proofness considers coalitions of all possible sizes. However, in practice, organizing large coalitions may be challenging, making group strategy-proofness unnecessarily demanding in terms of coalition size (e.g, Schummer 2000; Serizawa 2006). Therefore, it may be reasonable to consider group incentive properties that consider only coalitions of limited sizes.

• Self-enforcing manipulation. Group strategy-proofness requires that all profitable coalitional misrepresentations should be avoided, but some misrepresentations may be impractical. Certain manipulations may not be self-enforcing in the sense that an agent in a coalition might benefit from deviating from the manipulation (Serizawa 2006; Alva 2017), which makes such manipulations unsustainable. Thus, from a practical standpoint, it may suffice to prevent specific types of manipulations, such as self-enforcing ones.

• Other types of manipulation. Group strategy-proofness only prevents coalitional misrepresentations of preferences, but in practical auctions agents may engage in other forms of misbehavior. For example, a coalition of agents might reallocate the object after misrepresenting preferences (Hagen 2023), or arrange side payments among themselves (Schummer 2000; Bu 2016; Hagen 2023).² Therefore, it is important to consider group incentive properties that prevent not only coalitional misrepresentations of preferences but also other coalitional misbehavior.

The purpose of this paper is to examine the classes of rules that satisfy various group incentive properties, taking these aspects into account.

1.2 Main results

In this paper, we examine various group incentive properties. The most fundamental group incentive properties are weak group strategy-proofness and strong group strategy-proofness. Weak group strategy-proofness requires that no misrepresentations by a coalition should make each agent in the coalition better off. In contrast, strong group strategy-proofness requires that no misrepresentations by a coalition should make some agent in the coalition better off without making any other agent in the coalition worse off. Clearly, strong group strategy-proofness implies weak group strategy-proofness. All the group incentive properties considered in this paper are defined as extensions of either of these properties.

Specifically, the group incentive properties that we will study are classified as follows.

• l-group incentive property. For a given $l{\ }{\ge }{\ }2$, l-group incentive property requires that no coalition of size at most l should benefit from manipulations.

• Effective group incentive property. An effective group incentive property requires that no coalition should benefit from self-enforcing manipulations, where no single agent in the coalition benefits from deviating from the manipulation.

• Group incentive property with reallocation. A group incentive property with reallocation requires that no coalition should benefit from misrepresenting their preferences and then reallocating the object among themselves after misrepresentation.

• Group incentive property with side payments. A group incentive property with side payments requires that no coalition should benefit from misrepresenting their preferences and arranging side payments among themselves.

• Group incentive property with reallocation and side payments. A group incentive property with reallocation and side payments requires that no coalition should benefit from misrepresenting their preferences, reallocating the object among themselves, and arranging side payments.

We define group incentive properties by extending either weak or strong group strategy-proofness according to the above classification. For example, effective l-strong group strategy-proofness with reallocation is an extension of strong group strategy-proofness to ensure effectiveness, to restrict to coalitions of size $l{\ }{\ge }{\ }2$, and to allow the possibility of reallocation of the object. All the other extensions of either weak group strategy-proofness or strong group strategy-proofness are defined in an analogous way.

When side payments are allowed, a weak group incentive property becomes equivalent to its strong counterpart. Therefore, we do not distinguish between weak and strong group incentive properties with side payments. For example, effective weak group strategy-proofness with reallocation and side payments is equivalent to its strong counterpart, so we refer to it simply as effective group strategy-proofness with reallocation and side payments.

We clarify the relationships between the group incentive properties.

First, any strong group incentive property implies its weak counterpart. For example, strong group strategy-proofness implies weak group strategy-proofness.

Second, any l-group incentive property that considers only coalitions of size at most l gets stronger as l increases. For example, l-weak group strategy-proofness implies $l'$-weak group strategy-proofness for $l{\ }{>}{\ }l'$.

Third, any group incentive property implies its effective counterpart. For example, weak group strategy-proofness implies effective weak group strategy-proofness.

Fourth, any group incentive property with reallocation implies its counterpart without reallocation. For example, weak group strategy-proofness with reallocation implies weak group strategy-proofness.

Fifth, any group incentive property with side payments implies its counterpart without side payments. For example, group strategy-proofness with side payments implies strong (or weak) group strategy-proofness.

Finally, any group incentive property with reallocation and side payments implies its counterparts without reallocation or side payments. For example, group strategy-proofness with reallocation and side payments implies group strategy-proofness with side payments and strong (or weak) group strategy-proofness with reallocation.

Individual incentive properties have been widely studied as well as group incentive properties. One of the most important individual incentive properties is strategy-proofness. A rule satisfies strategy-proofness if no single agent should benefit from misrepresenting his preferences. Note that all the incentive properties considered in this paper imply strategy-proofness.

In our results, we will require an additional, mild property of rules. A rule satisfies non-imposition (Sakai 2008) if the welfare of an agent who is not interested in the object remains unaffected by the rule. Note that almost all standard rules satisfy non-imposition.

We introduce the two classes of rules that we will study.

First, a rule is a variable threshold-price rule (Nisan 2007; Sprumont 2013, etc.) if each agent faces a variable threshold-price, which may depend on the preferences of the other agents. An agent receives the object only if his willingness to pay for the object is greater than or equal to his variable threshold-price. Note that the class of variable threshold-price rules is so broad that it includes almost all standard rules satisfying strategy-proofness.

Second, a rule is a priority rule (Juarez 2013) if there are a priority ordering over the set of participants and a fixed price for each participant. The participant with the highest priority among those who are willing to pay their own prices receives the object and pays his own price. A rule is a priority rule with positive prices if it is a priority rule such that each participant faces a positive price. It is a dictatorial rule if there is a single participant. It is the no-trade rule if no agent participates in the rule.

The main results of this paper are characterizations of the classes of rules satisfying group incentive properties together with non-imposition as follows.

• For a given $l{\ }{\ge }{\ }2$, the variable threshold-price rules are the only rules satisfying non-imposition and any of strategy-proofness, effective l-weak group strategy-proofness, or l-weak group strategy-proofness (Theorem 1).

• For a given $l{\ }{\ge }{\ }2$, the priority rules are the only rules satisfying non-imposition and either effective l-strong group strategy-proofness or l-strong group strategy-proofness (Theorem 2).

• For a given $l{\ }{\ge }{\ }2$, the variable threshold-price rules are the only rules satisfying non-imposition and either effective l-weak group strategy-proofness with reallocation or l-weak group strategy-proofness with reallocation (Theorem 3).

• For a given $l{\ }{\ge }{\ }2$, the priority rules with positive prices are the only rules satisfying non-imposition and either effective l-strong group strategy-proofness with reallocation or l-strong group strategy-proofness with reallocation (Theorem 4).³

• For a given $l{\ }{\ge }{\ }2$, the dictatorial rules and the no-trade rule are the only rules satisfying non-imposition and l-group strategy-proofness with side payments (Theorem 5).

• For a given $l{\ }{\ge }{\ }2$, the priority rules are the only rules satisfying non-imposition and effective l-group strategy-proofness with side payments (Theorem 6).

• For a given $l{\ }{\ge }{\ }2$, the no-trade rule is the only rule satisfying non-imposition and either effective l-group strategy-proofness with side payments and reallocation or l-group strategy-proofness with side payments and reallocation (Theorem 7).

Several interesting observations are derived from our characterization results.

First, the weak group incentive properties basically characterize the variable threshold-price rules (Theorems 1 and 3).

Second, the strong group incentive properties characterize subclasses of priority rules (Theorems 2, 4, 5, 6, and 7).

Third, the size of coalitions does not affect any of our characterization results.

Fourth, under non-imposition, there is equivalence between strategy-proofness, l-weak group strategy-proofness, and l-weak group strategy-proofness with reallocation for a given $l \ge 2$ (Theorems 1 and 3).

Finally, for the class of rules satisfying strong group strategy-proofness, the effects of allowing side payments, which may restrict the resulting class of rules (Theorem 5), and the effects of focusing solely on self-enforcing manipulations, which may expand the resulting class of rules (Theorem 6), offset each other. Therefore, under non-imposition, the class of rules satisfying effective group strategy-proofness with side payments coincides with that satisfying group strategy-proofness, both of which are equivalent to the class of priority rules (Theorems 2 and 6).

From the perspective of implementation theory or mechanism design, our results can be interpreted as characterizing the classes of rules implementable by direct mechanisms in the respective group equilibrium concepts corresponding to the group incentive properties. A natural question is whether a broader class of rules can be implemented if we allow for indirect mechanisms.

For implementation in dominant strategy equilibrium, it is well known by the celebrated revelation principle (Gibbard 1973; Dasgupta et al. 1979, etc.) that restricting our attention to strategy-proof rules (or direct mechanisms) is without loss of generality. Therefore, allowing for indirect mechanisms does not expand the set of implementable rules in dominant strategy equilibrium.

We extend this insight to implementation in group equilibrium concepts by establishing the revelation principles for rules satisfying the group incentive properties (Propositions 2 and 3). Thus, allowing for indirect mechanisms does not expand the set of implementable rules in the group equilibrium concepts. In other words, it is also without loss to focus on rules (or direct mechanisms) satisfying the group incentive properties when considering implementation in the corresponding group equilibrium concepts. To the best of our knowledge, we are the first to establish revelation principles for rules satisfying group incentive properties.

1.3 Organization

The remainder of this paper is organized as follows. Section 2 introduces the model. Section 3 introduces the variable threshold-price rules and the priority rules. Section 4 provides the main results. Second Sect. 5 discusses the implementation via indirect mechanisms, and establishes the revelation principles for rules satisfying the group incentive properties. Section 6 discusses the contributions of this paper relative to the literature. Section 7 concludes. All the proofs are relegated to the Appendix.

2 Model

We study the object allocation problem with money. There are $n{\ }{\ge }{\ }2$ agents. Let $N=\{1,{\dots },n \}$ denote the set of agents. The owner possesses a single object. Let $M=\{0,1\}$. The consumption set of an agent $i{\ }{\in }{\ }N$ is $M{\ }{\times }{\ }{\mathbb {R}}$. A (consumption) bundle of an agent $i{\ }{\in }{\ }N$ is a pair $z_{i}=(x_{i},t_{i}){\ }{\in }{\ }M{\ }{\times }{\ }{\mathbb {R}}$.

2.1 Preferences

An agent $i{\ }{\in }{\ }N$ has a preference $R_{i}$ over $M{\ }{\times }{\ }{\mathbb {R}}$. As usual, the indifference and strict relations associated with $R_{i}$ are denoted by $I_{i}$ and $P_{i}$, respectively. We assume that preferences satisfy the following properties.

Weak desirability of the object. For each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$, $(1,t_{i}){\ }R_{i}{\ }(0,t_{i})$.

Money monotonicity. For each $x_{i}{\ }{\in }{\ }M$ and each pair $t_{i},t'_{i}{\ }{\in }{\ }{\mathbb {R}}$ with $t_{i}{\ }{<}{\ }t'_{i}$, $(x_{i},t_{i}){\ }P_{i}{\ }(x_{i},t'_{i})$.

Finiteness. For each $z_{i}{\ }{\in }{\ }M{\ }{\times }{\ }{\mathbb {R}}$ and each $x_{i}{\ }{\in }{\ }M$, there is $t_{i}{\ }{\in }{\ }{\mathbb {R}}$ such that $(x_{i},t_{i}){\ }I_{i}{\ }z_{i}$.

Our generic notation for a class of preferences satisfying the above three properties is ${\mathcal {R}}$. We call ${\mathcal {R}}$ a domain.

Given $R_{i}{\ }{\in }{\ }{\mathcal {R}}$, $z_{i}{\ }{\in }{\ }M{\ }{\times }{\ }{\mathbb {R}}$, and $x_{i}{\ }{\in }{\ }M$, let $V(x_{i},z_{i};R_{i}){\ }{\in }{\ }{\mathbb {R}}$ denote a payment such that $(x_{i},V(x_{i},z_{i};R_{i})){\ }I_{i}{\ }z_{i}$. The existence of such a payment is guaranteed by finiteness, and it is unique by money monotonicity. We call $v(t_{i};R_{i})=V(1,(0,t_{i});R_{i})-t_{i}$ the willingness to pay of $R_{i}$ at $t_{i}$. By weak desirability of the object, for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$, $v(t_{i};R_{i}){\ }{\ge }{\ }0$. Let $v(R_{i})=v(0;R_{i})$ denote the willingness to pay of $R_{i}$ at 0, and we call $v(R_{i})$ the willingness to pay of $R_{i}$. Then, $v(R_{i})$ solves $(1,v(R_{i})){\ }I_{i}{\ }(0,0)$.

A domain ${\mathcal {R}}$ is rich if (i) there is $R_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(t_{i};R_{i})=0$ for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$, and (ii) for each pair ${\overline{v}},{\underline{v}}{\ }{\in }{\ }{\mathbb {R}}_{+}$ with ${\overline{v}}{\ }{>}{\ }{\underline{v}}$, there is $R_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that ${\underline{v}}{\ }{<}{\ }v(R_{i}){\ }{<}{\ }{\overline{v}}$. Throughout the paper, we focus on rich domains.

The following example illustrates several rich domains. As it demonstrates, many domains of interest are indeed rich domains.

Example 1

The following are examples of rich domains.

• A preference $R_{i}$ is quasi-linear if it is represented by a utility function $u((x_{i},t_{i});v_{i})=v_{i}{\cdot }x_{i}-t_{i}$, where $v_{i}{\ }{\in }{\ }{\mathbb {R}}_{+}$. Note that if $R_{i}$ is quasi-linear, then for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$, $v(t_{i};R_{i})=v_{i}$. Let ${\mathcal {R}}^{Q}$ denote the class of quasi-linear preferences. Then, ${\mathcal {R}}^{Q}$ is rich.

• A preference $R_{i}$ exhibits positive (resp. non-negative) income effects if for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$ with $v(t_{i};R_{i}){\ }{>}{\ }0$ and for each $t'_{i}{\ }{\in }{\ }{\mathbb {R}}$ with $t'_{i}{\ }{<}{\ }t_{i}$, we have $v(t'_{i};R_{i}){\ }{>}{\ }v(t_{i};R_{i})$ (resp. $v(t'_{i};R_{i}){\ }{\ge }{\ }v(t_{i};R_{i})$). Note that we do not take an agent’s income into account explicitly, but the zero payment can be regarded as the initial income of an agent. Then, a payment level corresponds to a relative income level of an agent. In words, the positive (resp. non-negative) income effects state that as an agent’s relative income increases (i.e., his payment decreases), the willingness to pay increases (resp. does not decrease). Let ${\mathcal {R}}^{++}$ denote the class of preferences exhibiting positive income effects, and ${\mathcal {R}}^{+}$ the class of preferences exhibiting non-negative income effects. Then, ${\mathcal {R}}^{++}$ and ${\mathcal {R}}^{+}$ are both rich.

• A preference $R_{i}$ exhibits negative (resp. non-positive) income effects if for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$ with $v(t_{i};R_{i}){\ }{>}{\ }0$ and for each $t'_{i}{\ }{\in }{\ }{\mathbb {R}}$ with $t'_{i}{\ }{<}{\ }t_{i}$, we have $v(t'_{i};R_{i}){\ }{<}{\ }v(t_{i};R_{i})$ (resp. $v(t'_{i};R_{i}){\ }{\le }{\ }v(t_{i};R_{i})$). In words, the negative (resp. non-positive) income effects state that as an agent’s relative income increases, the willingness to pay decreases (resp. does not increase). Let ${\mathcal {R}}^{--}$ denote the class of preferences exhibiting negative income effects, and ${\mathcal {R}}^{-}$ the class of preferences exhibiting non-positive income effects. Then, ${\mathcal {R}}^{--}$ and ${\mathcal {R}}^{-}$ are both rich.

• Given an interest rate $r{\ }{\in }{\ }{\mathbb {R}}_{+}$ and a wealth $w_{i}{\ }{\in }{\ }{\mathbb {R}}_{+}{\ }{\cup }{\ }\{{\infty }\}$, a preference $R_{i}$ faces soft budget constraints at $(r,w_{i})$ if it is represented by a utility function

  $$\begin{aligned} u((x_{i},t_{i});v_{i},r,w_{i})={\left\{ \begin{array}{ll} v_{i}{\cdot }x_{i}-t_{i} & \text {if } \,t_{i}{\ }{\le }{\ }w_{i}, \\ v_{i}{\cdot }x_{i}-w_{i}-(1+r)(t_{i}-w_{i}) & \text {if } \,t_{i}{\ }{>}{\ }w_{i}, \end{array}\right. } \end{aligned}$$

  where $v_{i}{\ }{\in }{\ }{\mathbb {R}}_{+}$. Given a pair $(r,w_{i})$ of an interest rate and a wealth, let ${\mathcal {R}}^{SB}(r,w_{i})$ denote the class of preferences facing soft budget constraints at $(r,w_{i})$, and ${\mathcal {R}}^{SB}(r)$ the class of preferences facing soft budget constraints at $(r,w'_{i})$ for some wealth $w'_{i}$. Note that ${\mathcal {R}}^{SB}(r,w_{i})$ corresponds to the case of public budget $w_{i}$, and ${\mathcal {R}}^{SB}(r)$ corresponds to the case of private budgets. Then, for each pair $(r,w_{i})$, ${\mathcal {R}}^{SB}(r,w_{i})$ and ${\mathcal {R}}^{SB}(r)$ are both rich.

2.2 Allocations and rules

An n-tuple $z=(z_{i})_{i{\in }N}=(x_{i},t_{i})_{i{\in }N}{\ }{\in }{\ }(M{\ }{\times }{\ }{\mathbb {R}})^{n}$ is a (feasible) allocation if it satisfies $\sum _{i{\in }N} x_{i}{\ }{\le }{\ }1$. Let Z denote the set of allocations. Note that we focus on deterministic allocations.

A preference profile is an n-tuple $R=(R_{i})_{i{\in }N}{\ }{\in }{\ }{\mathcal {R}}^{n}$. Given $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $N'{\ }{\subseteq }{\ }N$, let $R_{N'}=(R_{i})_{i{\in }N'}$ and $R_{-N'}=(R_{i})_{i{\in }N{\setminus }N'}$. For a given pair of distinct agents $i,j{\ }{\in }{\ }N$, if $N'=\{i\}$, then let $R_{-i}=R_{-N'}$; if $N'=\{i,j\}$, then let $R_{i,j}=R_{N'}$ and $R_{-i,j}=R_{-N'}$.

A rule (or a direct mechanism) is a function $f:{\mathcal {R}}^{n} \rightarrow Z$. Let $f_{i}(R)=(x^{f}_{i}(R),t^{f}_{i}(R))$ denote the outcome bundle of an agent $i{\ }{\in }{\ }N$ at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ under a rule f. Let $N^{f}_{+}=\{i{\ }{\in }{\ }N:{\exists }R{\ }{\in }{\ }{\mathcal {R}}^{n}{\ }\mathrm{{s.t.}}{\ }x^{f}_{i}(R)=1\}$ denote the set of participants of a rule f who have the chance to receive the object at some preference profile under f.

2.3 Properties of rules

We introduce the properties of rules. In particular, the group incentive properties will play a central role in this paper. A coalition $N'$ is a nonempty subset of N.

First, as a benchmark, we introduce an individual incentive property, which requires that no agent should ever benefit from misrepresenting his preferences.

Strategy-proofness. For each $R{\ }\!{\in }\!{\ }{\mathcal {R}}^{n}$, each $i{\ }\!{\in }\!{\ }N$, and each $R'_{i}{\ }\!{\in }\!{\ }{\mathcal {R}}$, $f_{i}(R){\ }R_{i}{\ }f_{i}(R'_{i},R_{-i})$.

2.3.1 Manipulation by misrepresenting preferences

We first introduce the weak group incentive properties, which require that no coalitional manipulation should make all the agents in the coalition better off. A coalition $N'$ strongly benefits from misrepresenting preferences $R'_{N}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$, $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)$.

The following property requires that no coalition of size at most l should strongly benefit from misrepresenting preferences.

l-weak group strategy-proofness. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ such that $N'$ strongly benefits from misrepresenting $R'_{N'}$ at R.

We call an n-group incentive property simply the group incentive property. For example, we call n-weak group strategy-proofness simply weak group strategy-proofness. We also call a 2-group incentive property the pairwise incentive property. For example, 2-weak group strategy-proofness is called pairwise weak strategy-proofness.

Note that any 1-group incentive property (including all the group incentive properties that we will introduce below) is equivalent to strategy-proofness. Note also that any l-group incentive property gets stronger as l increases, i.e., for any pair $l,l'$ with $l{\ }{>}{\ }l'$, l-group incentive property implies its $l'$-counterpart. Thus, any group incentive property that we introduce in this paper implies strategy-proofness.

Next, we consider the incentives for agents to unilaterally deviate from the manipulation. We say that misrepresentations $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ of preferences by a coalition $N'$ are self-enforcing at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$ and each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, $f_{i}(R'_{N'},R_{-N'}){\ }R_{i}{\ }f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'})$. That is, no agent in a coalition $N'$ benefits from unilaterally deviating from self-enforcing misrepresentations $R'_{N'}$.

The following property requires that no coalition of size at most l should strongly benefit from self-enforcing misrepresentations of preferences.

Effective l-weak group strategy-proofness. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ such that $N'$ strongly benefits from misrepresenting $R'_{N'}$ at R, and if $|N'|{\ }{\ge }{\ }2$, then $R'_{N'}$ is self-enforcing at R.

Any group incentive property that considers all possible manipulations implies its effective counterpart that only considers only self-enforcing manipulations. For example, l-weak group strategy-proofness implies effective l-weak group strategy-proofness for a given $l{\ }{\ge }{\ }2$.

Next, we introduce the strong group incentive properties, which require that no coalitional manipulation should make some agent in the coalition better off without making any agent in the coalition worse off. A coalition $N'$ weakly benefits from misrepresenting preferences $R'_{N}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$, $f_{i}(R'_{N'},R_{-N'}){\ }R_{i}{\ }f_{i}(R)$, and for some $j{\ }{\in }{\ }N'$, $f_{j}(R'_{N'},R_{-N'}){\ }P_{j}{\ }f_{j}(R)$.

The following two properties are the strong counterparts of the previous ones.

l-strong group strategy-proofness. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ such that $N'$ weakly benefits from misrepresenting $R'_{N'}$ at R.

Effective l-strong group strategy-proofness. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ such that $N'$ weakly benefits from misrepresenting $R'_{N'}$ at R, and if $|N'|{\ }{\ge }{\ }2$, then $R'_{N'}$ is self-enforcing at R.

Clearly, any strong group incentive property implies its weak counterpart. For example, l-strong group strategy-proofness implies l-weak group strategy-proofness for a given $l{\ }{\ge }{\ }2$.

2.3.2 Manipulation by misrepresenting preferences and reallocating the object

In the last subsection, we considered manipulations where agents in a coalition misrepresent their preferences. We will consider other types of manipulations in this paper.

First, we study coalitional manipulations where agents in a coalition not only misrepresent their preferences but also reallocate the object among themselves after misrepresentation. A coalition $N'$ strongly benefits from misrepresenting preferences $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ and reallocating the object $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ at R if for each $i{\ }{\in }{\ }N'$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'})){\ }P_{i}{\ }f_{i}(R)$, and $\sum _{i{\in }N'} x_{i}{\ }{\le }{\ }\sum _{i{\in }N'} x^{f}_{i}(R'_{N'},R_{-N'})$. Similarly, a coalition $N'$ weakly benefits from misrepresenting preferences $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ and reallocating the object $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ at R if for each $i{\ }{\in }{\ }N'$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'})){\ }R_{i}{\ }f_{i}(R)$, for some $j{\ }{\in }{\ }N'$, $(x_{j},t^{f}_{j}(R'_{N'},R_{-N'})){\ }P_{j}{\ }f_{j}(R)$, and $\sum _{i{\in }N'} x_{i}{\ }{\le }{\ }\sum _{i{\in }N'} x^{f}_{i}(R'_{N'},R_{-N'})$. A pair of misrepresentations $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ and a reallocation $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ by a coalition $N'$ is self-enforcing at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$ and each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'})){\ }R_{i}{\ }f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'})$.

The following four properties take into account the possibility of reallocating the object among agents in a coalition.⁴

l-weak group strategy-proofness with reallocation. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ such that $N'$ strongly benefits from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R.

Effective l-weak group strategy-proofness with reallocation. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ such that $N'$ strongly benefits from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R, and if $|N'|{\ }{\ge }{\ }2$, then $R'_{N'}$ and $(x_{i})_{i{\in }N'}$ are self-enforcing at R.

l-strong group strategy-proofness with reallocation. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ such that $N'$ weakly benefits from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R.

Effective l-strong group strategy-proofness with reallocation. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ such that $N'$ weakly benefits from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R, and if $|N'|{\ }{\ge }{\ }2$, then $R'_{N'}$ and $(x_{i})_{i{\in }N'}$ are self-enforcing at R.

Clearly, any group incentive property that takes into account the possibility of reallocating the object implies its counterpart without such reallocation. For example, for a given $l{\ }{\ge }{\ }2$, l-weak group strategy-proofness with reallocation implies l-weak group strategy-proofness.

2.3.3 Manipulation by misrepresenting preferences and coordinating side payments

Next, we study coalitional manipulations that take into account the possibility of coordinating side payments among agents in a coalition. A coalition $N'$ benefits from misrepresenting preferences $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ and arranging side payments $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$, $f_{i}(R'_{N'},R_{-N'})+(0,t_{i}){\ }P_{i}{\ }f_{i}(R)$, and $\sum _{i{\in }N'} t_{i}{\ }{\ge }{\ }0$. A pair of misrepresentations $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ and side payments $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ by a coalition $N'$ is self-enforcing at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$ and each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, $f_{i}(R'_{N'},R_{-N'})+(0,t_{i}){\ }R_{i}{\ }f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'})$.

The following two properties take into account the possibility of coordinating side payments. Note that under the possibility of side payments, a strong group incentive property is equivalent to its weak counterpart, and so we do not distinguish them.⁵

l-group strategy-proofness with side payments. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ such that $N'$ benefits from misrepresenting $R'_{N'}$ and arranging $(t_{i})_{i{\in }N'}$ at R.

Effective l-group strategy-proofness with side payments. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ such that $N'$ benefits from misrepresenting $R'_{N'}$ and arranging $(t_{i})_{i{\in }N'}$ at R, and if $|N'|{\ }{\ge }{\ }2$, then $R'_{N'}$ and $(t_{i})_{i{\in }N'}$ are self-enforcing at R.

A group incentive property that takes into account the possibility of coordinating side payments implies its counterpart without side payments. For example, for a given $l{\ }{\ge }{\ }2$, l-group strategy-proofness with side payments implies l-strong group strategy-proofness.

2.3.4 Manipulation by misrepresenting preferences, reallocating the object, and coordinating side payments

We finally introduce the group incentive properties that take into account all the types of manipulations discussed thus far. A coalition $N'$ benefits from misrepresenting preferences $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, reallocating the object $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$, and arranging side payments $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'}))+(0,t_{i}){\ }P_{i}{\ }f_{i}(R)$, $\sum _{i{\in }N'} x_{i}{\ }{\le }{\ }\sum _{i{\in }N'} x^{f}_{i}(R'_{N'},R_{-N'})$, and $\sum _{i{\in }N'} t_{i}{\ }{\ge }{\ }0$. A triple of misrepresentations $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, a reallocation $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$, and side payments $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ by a coalition $N'$ is self-enforcing at $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ if for each $i{\ }{\in }{\ }N'$ and each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'}))+(0,t_{i}){\ }R_{i}{\ }f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'})$.

l-group strategy-proofness with reallocation and side payments. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$, and $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ such that $N'$ benefits from misrepresenting $R'_{N'}$, reallocating $(x_{i})_{i{\in }N'}$, and arranging $(t_{i})_{i{\in }N'}$ at R.

Effective l-group strategy-proofness with reallocation and side payments. There are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$, and $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ such that $N'$ benefits from misrepresenting $R'_{N'}$, reallocating $(x_{i})_{i{\in }N'}$, and arranging $(t_{i})_{i{\in }N'}$ at R, and if $|N'|{\ }{\ge }{\ }2$, then $R'_{N'}$, $(x_{i})_{i{\in }N'}$, and $(t_{i})_{i{\in }N'}$ are self-enforcing at R.

Note that a group incentive property that takes into account both the possibility of reallocating the object and coordinating side payments implies its counterparts without side payments and without reallocation. For example, l-group strategy-proofness with reallocation and side payments implies both l-strong group strategy-proofness with reallocation and l-group strategy-proofness with side payments for a given $l{\ }{\ge }{\ }2$.

2.3.5 Other standard properties

In addition to the group incentive properties, we require a mild property that almost all standard rules satisfy. The following property is introduced by Sakai (2008), which requires that an agent’s welfare should be unaffected by a rule whenever he is not interested in the object.

Non-imposition. For each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and each $i{\ }{\in }{\ }N$, if $v(t_{i};R_{i})=0$ for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$, then $f_{i}(R){\ }I_{i}{\ }(0,0)$.

We also introduce four additional standard properties that will be useful for the discussion.

The next property requires that no agent should ever find an outcome bundle of a rule worse than receiving no object and making no payment.

Individual rationality. For each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and each $i{\ }{\in }{\ }N$, $f_{i}(R){\ }R_{i}{\ }(0,0)$.

The next property requires that the payment of each agent should be always non-negative.

No subsidy. For each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and each $i{\ }{\in }{\ }N$, $t^{f}_{i}(R){\ }{\ge }{\ }0$.

The next remark states that the combination of individual rationality and no subsidy implies non-imposition.

Remark 1

Let ${\mathcal {R}}$ be a domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying individual rationality and no subsidy. Then, it satisfies non-imposition.

The next property requires that, given the preferences of the other agents, each agent should have a chance to receive the object for some preference.

Strong agent sovereignty. For each $i{\ }{\in }{\ }N$ and each $R_{-i}{\ }{\in }{\ }{\mathcal {R}}^{n-1}$, there is $R_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $x^{f}_{i}(R_{i},R_{-i})=1$.

Finally, the next property requires that each agent should have a chance to receive the object for some preference profile.

Weak agent sovereignty. For each $i{\ }{\in }{\ }N$, there is $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ such that $x^{f}_{i}(R)=1$.

Clearly, strong agent sovereignty implies weak agent sovereignty.

3 Rules

In this section, we introduce the two classes of rules that we will study.

3.1 Variable threshold-price rule

In this subsection, we introduce the variable threshold-price rules (Nisan 2007; Sprumont 2013, etc.). Under a variable threshold-price rule, each agent faces a variable threshold-price which may depend on the other agents’ preferences, and he receives the object at his price only if his willingness to pay is greater than or equal to his price.

Definition 1

A rule f on ${\mathcal {R}}^{n}$ is a variable threshold-price rule if for each $i{\ }{\in }{\ }N$, there is a function ${\kappa }^{f}_{i}:{\mathcal {R}}^{n-1} \rightarrow {\mathbb {R}}_{+}{\ }{\cup }{\ }\{{\infty }\}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$,

$$\begin{aligned} f_{i}(R)={\left\{ \begin{array}{ll} (1,{\kappa }^{f}_{i}(R_{-i})) & \text {if } \, v(R_{i}){\ }{>}{\ }{\kappa }^{f}_{i}(R_{-i}), \\ (1,{\kappa }^{f}_{i}(R_{-i})){\ }\mathrm{{or}}{\ }(0,0) & \text {if }\, v(R_{i})={\kappa }^{f}_{i}(R_{-i}), \\ (0,0) & \text {if }\, v(R_{i}){\ }{<}{\ }{\kappa }^{f}_{i}(R_{-i}). \end{array}\right. } \end{aligned}$$

A number of rules of interest are examples of variable threshold-price rule. For example, any priority rule that we will introduce in the next subsection, any Vickrey rule for quasi-linear preferences (Vickrey 1961), any Vickrey rule with reserve prices for quasi-linear preferences (Myerson 1981), and any generalized Vickrey rule for non-quasi-linear preferences (Saitoh and Serizawa 2008; Sakai 2008) all belong to the class of variable threshold-price rules.

3.2 Priority rule

In this subsection, we introduce the priority rules (Juarez 2013). A priority rule specifies a priority ordering over the set of participants $N^{f}_{+}$ and sets a (personal and fixed) price for each participant. It then allocates the object to a participant with the highest priority among the participants who are willing to receive the object at their own prices.

Definition 2

A rule f on ${\mathcal {R}}^{n}$ is a (fixed-price) priority rule if there are a strict order (priority) ${\succ }^{f}$ over $N^{f}_{+}$ and prices $(p^{f}_{i})_{i{\in }N^{f}_{+}}{\ }{\in }{\ }{\mathbb {R}}^{|N^{f}_{+}|}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, the following hold.

• For each $i{\ }{\in }{\ }N^{f}_{+}$, if $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$, and $v(R_{i}){\ }{>}{\ }p^{f}_{i}$, then $x^{f}_{i}(R)=1$.

• For each $i{\ }{\in }{\ }N^{f}_{+}$, if $x^{f}_{i}(R)=1$, then $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$, and $v(R_{i}){\ }{\ge }{\ }p^{f}_{i}$.

• For each $i{\ }{\in }{\ }N^{f}_{+}$, if $x_{i}^{f}(R)=1$, then $t^{f}_{i}(R)=p^{f}_{i}$.

• For each $i{\ }{\in }{\ }N$, if $x^{f}_{i}(R)=0$, then $t^{f}_{i}(R)=0$.

If a priority rule f satisfies that for each $i{\ }{\in }{\ }N^{f}_{+}$, $p^{f}_{i}{\ }{>}{\ }0$, then it is a priority rule with positive prices. If a priority rule f satisfies $N^{f}_{+}=N$, then it is a priority rule with full participation. If a priority rule f satisfies $N^{f}_{+}=N$ and $p^{f}_{i}=0$ for each $i{\ }{\in }{\ }N$, then it is a priority rule with full participation and zero prices. If a priority rule f satisfies $|N^{f}_{+}|=1$, then it is a (fixed-price) dictatorial rule. If a priority rule f satisfies $N^{f}_{+}={\varnothing }$, then it is the no-trade rule.

Our definition of a priority rule is slightly different from that of Juarez (2013) in that we allow the set $N^{f}_{+}$ of participants to depend on a rule, whereas he requires that all agents should participate in a rule. Therefore, his definition is equivalent to our definition of a priority rule with full participation.⁶

4 Main results

In this section, we present the main results of this paper, which are characterizations of the classes of rules that satisfy group incentive properties along with non-imposition.

4.1 Weak group strategy-proofness

We begin with the weak group incentive properties. The following theorem states that the variable threshold-price rules are the only rules on a rich domain satisfying non-imposition and any of strategy-proofness, effective l-weak group strategy-proofness, or l-weak group strategy-proofness for a given $l{\ }{\ge }{\ }2$.

Theorem 1

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) f satisfies strategy-proofness.

• (ii) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-weak group strategy-proofness.

• (iii) Given $l{\ }{\ge }{\ }2$, f satisfies l-weak group strategy-proofness.

• (iv) f is a variable threshold-price rule.

An implication of Theorem 1 is that on a rich domain and under non-imposition, strategy-proofness is equivalent to effective l-weak group strategy-proofness, which in turn is equivalent to l-weak group strategy-proofness. Thus, Theorem 1 can be regarded as an equivalence result between strategy-proofness and the weak group incentive properties.

Here, we outline the proof of Theorem 1.⁷

First, we explain why a variable threshold-price rule f satisfies l-weak group strategy-proofness (i.e., (iv) implies (iii)). Suppose that a coalition $N'$ of size at most $l{\ }{\ge }{\ }2$ misrepresents $R'_{N'}$ at R. Since the owner possesses a single object, there is an agent $i{\ }{\in }{\ }N'$ who does not receive the object after misrepresenting $R'_{N'}$, i.e., $x^{f}_{i}(R'_{N'},R_{-N'})=0$. Thus, $f_{i}(R'_{N'},R_{-N'})=(0,0)$. Since any variable threshold-price rule satisfies individual rationality (Lemma 3 in Appendix A), $f_{i}(R){\ }R_{i}{\ }(0,0)=f_{i}(R'_{N'},R_{-N'})$, and so $N'$ does not strongly benefit from misrepresenting $R'_{N'}$ at R. Thus, f satisfies l-weak group strategy-proofness.

Next, we briefly explain our proof that any rule satisfying strategy-proofness and non-imposition is a variable threshold-price rule (i.e., (i) implies (iv)). It is already known that for quasi-linear preferences, any rule satisfying strategy-proofness and non-imposition is a variable threshold-price rule (Nisan 2007; Sprumont 2013, etc.). We extend their arguments from quasi-linear preferences to non-quasi-linear preferences to establish the result.

4.2 Strong group strategy-proofness

Next, we turn to the strong group incentive properties. The following theorem states that the priority rules are the only rules on a rich domain satisfying non-imposition and either effective l-strong group strategy-proofness or l-strong group strategy-proofness for a given $l{\ }{\ge }{\ }2$.

Theorem 2

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-strong group strategy-proofness.

• (ii) Given $l{\ }{\ge }{\ }2$, f satisfies l-strong group strategy-proofness.

• (iii) f is a priority rule.

In order to understand that a priority rule f satisfies l-strong group strategy-proofness (i.e., (iii) implies (ii)), consider a coalition $N'$ of size at most l and an agent $i{\ }{\in }{\ }N'$ such that $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)$. Since any priority rule satisfies individual rationality (Lemma 4 in Appendix A), he is willing to receive the object (i.e., $v(R_{i}){\ }{>}{\ }p^{f}_{i}$), receives the object after misrepresenting $R'_{N'}$, and does not receive the object before the misrepresentation. Since agent i does not receive the object before the misrepresentation even though he is willing to do so, there is another agent $j{\ }{\in }{\ }N'$ with higher priority than agent i according to ${\succ }^{f}$ who is also willing to receive the object and receives the object before the misrepresentation. Since agent i receives the object after the misrepresentation and agent j is willing to receive the object, agent j ends up worse off due to the misrepresentation. Therefore, no coalition of size at most l benefits from misrepresenting preferences, and so f satisfies l-strong group strategy-proofness.

The most involved proof in this paper is to show that any rule satisfying effective l-strong group strategy-proofness and non-imposition is a priority rule (i.e., (i) implies (iii)). We outline our proof strategy for establishing that (i) implies (iii). Let f be a rule satisfying effective l-strong group strategy-proofness and non-imposition. The proof consists of two parts.

In the first part, we show the existence of a (fixed) price $p^{f}_{i}$ for each participant $i{\ }{\in }{\ }N^{f}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, if $x^{f}_{i}(R)=1$, then $t^{f}_{i}(R)=p^{f}_{i}$. While the existence of such a price is relatively straightforward under strong group strategy-proofness, which considers coalitions of any size and all possible misrepresentations of preferences, the challenge here is to establish the existence of such a price under weaker effective l-strong group strategy-proofness, which considers only coalitions of size at most l and self-enforcing misrepresentations.

In the second part, we show the existence of a priority ${\succ }^{f}$ over $N^{f}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and each $i{\ }{\in }{\ }N^{f}_{+}$, if $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$ and $v(R_{i}){\ }{>}{\ }p^{f}_{i}$, then $f_{i}(R)=(1,p^{f}_{i})$. Our proof for this part builds on the proof of Proposition 3 of Juarez (2013), which shows the existence of such a priority under strong group strategy-proofness. Since we consider weaker effective l-strong group strategy-proofness, his proof cannot be applied directly. Our proof strategy involves establishing several lemmas concerning effective l-strong group strategy-proofness, which lead us to a position where we can apply his proof.

4.3 Group strategy-proofness with reallocation

In this subsection, we study the group incentive properties that take into account the possibility of reallocation of the object after misrepresenting preferences.

First, we study the weak group incentive properties with reallocation. The following theorem states that the variable threshold-price rules are the only rules on a rich domain satisfying non-imposition and either effective l-weak group strategy-proofness with reallocation or l-weak group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$.

Theorem 3

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-weak group strategy-proofness with reallocation.

• (ii) Given $l{\ }{\ge }{\ }2$, f satisfies l-weak group strategy-proofness with reallocation.

• (iii) f is a variable threshold-price rule.

We briefly explain why any variable threshold-price rule satisfies weak group strategy-proofness with reallocation. Consider a coalition of size at least two. Since the owner has a single object, after misrepresenting preferences and reallocating the object, some agent in the coalition does not receive the object and makes a non-negative payment. Given that any variable threshold-price rule satisfies individual rationality (Lemma 3 in Appendix A), a manipulation cannot make such agent strictly better off. Therefore, any variable threshold-price rule satisfies weak group strategy-proofness with reallocation.

Recall that Theorem 1 shows that the variable threshold-price rules are the only rules on a rich domain satisfying non-imposition and either strategy-proofness or the weak group incentive properties without reallocation. Theorem 3 shows that allowing the possibility of reallocating the object after misrepresenting preferences does not change the characterization of rules satisfying the weak group incentive properties without reallocation (Theorem 1). Therefore, Theorems 1 and 3 together imply the equivalence between the weak group incentive properties with reallocation, the weak group incentive properties without reallocation, and strategy-proofness.

Corollary 1

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) f satisfies strategy-proofness.

• (ii) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-weak group strategy-proofness.

• (iii) Given $l{\ }{\ge }{\ }2$, f satisfies l-weak group strategy-proofness.

• (iv) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-weak group strategy-proofness with reallocation.

• (v) Given $l{\ }{\ge }{\ }2$, f satisfies l-weak group strategy-proofness with reallocation.

In contrast with the weak group incentive properties, the possibility of reallocating the object puts a restriction on the class of priority rules, which are the only rules satisfying the strong group incentive properties without reallocation (Theorem 2). Indeed, the following theorem states that when there are at least three agents, the priority rules with positive prices are the only rules on a rich domain satisfying non-imposition and either effective l-strong group strategy-proofness with reallocation or l-strong group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$.

Theorem 4

Let $n{\ }{\ge }{\ }3$. Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-strong group strategy-proofness with reallocation.

• (ii) Given $l{\ }{\ge }{\ }2$, f satisfies l-strong group strategy-proofness with reallocation.

• (iii) f is a priority rule with positive prices.

We explain the role of positive prices of a priority rule in Theorem 4. Under a priority rule, a typical coalitional manipulation involving both misrepresentation of preferences and reallocation of the object is such that a participant with zero willingness to pay (the sender) ends up receiving the object after misrepresentation and reallocates it to another agent (the receiver) who has a positive willingness to pay but does not initially receive it. Such a manipulation does not make the sender worse off only when his price is zero. Therefore, if some participant’s price is zero, then a coalition that includes him as the sender and another agent with a positive willingness to pay as the receiver weakly benefits from such a manipulation. Conversely, if each participant faces a positive price, then such a coalitional manipulation makes the sender worse off, and thus the coalition cannot weakly benefit from such a manipulation.

Notice that Theorem 4 is concerned with only the case of three or more agents. The next proposition states that, for the case of two agents, a priority rule such that both agents participate in the rule and face zero prices additionally satisfies the strong group incentive properties with reallocation.⁸

Proposition 1

Let $n=2$. Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-strong group strategy-proofness with reallocation.

• (ii) Given $l{\ }{\ge }{\ }2$, f satisfies l-strong group strategy-proofness with reallocation.

• (iii) f is either a priority rule with positive prices or a priority rule with full participation and zero prices.

We explain why a priority rule with full participation and zero prices satisfies strong group strategy-proofness with reallocation in the case of two agents. Consider again the sender with zero willingness to pay and the receiver with a positive willingness to pay. Since there are only these two participants, and the sender is not interested in the object, the receiver with a positive willingness to pay receives the object before the misrepresentation. Thus, the manipulation discussed in the case of three or more agents does not occur in the case of two agents.

4.4 Group strategy-proofness with side payments

In this subsection, we study the group incentive properties with the possibility of arranging side payments. The following theorem states that the dictatorial rules and the no-trade rule are the only rules on a rich domain satisfying non-imposition and l-group strategy-proofness with side payments for a given $l{\ }{\ge }{\ }2$.

Theorem 5

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) Given $l{\ }{\ge }{\ }2$, f satisfies l-group strategy-proofness with side payments.

• (ii) f is either a dictatorial rule or the no-trade rule.

The main part of the proof of Theorem 5 is devoted to showing that any priority rule with at least two participants violates l-group strategy-proofness with side payments. To understand the intuition, consider a pair of participants: the higher priority agent (the bribee), who willing to pay his price for the object, and the lower priority agent (the briber), who has such a large willingness to pay that he can compensate the bribee for receiving the object. If all other participants are not willing to pay their prices for the object, then a priority rule gives the object to the bribee because he has higher priority than the briber. If they misrepresent their preferences so that the briber receives the object and compensates the bribee, then such a manipulation makes both agents better off. Therefore, any priority rule with at least two participants violates l-group strategy-proofness with side payments.

Notice that Theorem 5 does not consider effective l-group strategy-proofness with side payments. The following theorem states that the priority rules are the only rules on a rich domain satisfying non-imposition and effective l-group strategy-proofness with side payments for a given $l{\ }{\ge }{\ }2$.

Theorem 6

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-group strategy-proofness with side payments.

• (ii) f is a priority rule.

To understand the intuition of Theorem 6, consider again the manipulation by the bribee and the briber. If the briber unilaterally deviates from the manipulation after misrepresenting their preferences, then he receives the object but no longer has to compensate the briber, i.e., he can run away with the object. Therefore, he benefits from such a unilateral deviation from the manipulation, so that the manipulation is not self-enforcing. In this way, any profitable manipulation with side payments is not self-enforcing under any priority rule, and so any priority rule satisfies effective l-group strategy-proofness with side payments.

Recall Theorem 2 states that the priority rules are the only rules on a rich domain satisfying non-imposition and the strong group incentive properties without side payments. Theorem 5 shows that incorporating the possibility of arranging side payments restricts the possible sets of participants. In contrast, Theorem 6 shows that focusing solely on self-enforcing manipulations expands the possible sets of participants. Interestingly, the effects of allowing side payments and focusing solely on self-enforcing manipulations offset each other, making the class of rules satisfying the effective group incentive properties with side payments equivalent to those satisfying the strong group incentive properties without side payments.

4.5 Group strategy-proofness with reallocation and side payments

Finally, in this subsection, we study the group incentive properties with the possibility of reallocating the object and arranging side payments. The following theorem states that the no-trade rule is the only rule on a rich domain satisfying non-imposition and either effective l-group strategy-proofness with reallocation and side payments or l-group strategy-proofness with reallocation and side payments for a given $l{\ }{\ge }{\ }2$.

Theorem 7

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying non-imposition. The following statements are mutually equivalent.

• (i) Given $l{\ }{\ge }{\ }2$, f satisfies effective l-group strategy-proofness with reallocation and side payments.

• (ii) Given $l{\ }{\ge }{\ }2$, f satisfies l-group strategy-proofness with reallocation and side payments.

• (iii) f is the no-trade rule.

To understand the intuition of Theorem 7, suppose that there is a participant (the sender-bribee) in a priority rule. Then, we can find another agent (the receiver-briber) who is either a participant with lower priority than the sender-bribee or a non-participant. Suppose that the sender-bribee is willing to pay his price for the the object, and the receiver-briber has such a large willingness to pay that he can compensate the sender-bribee for receiving the object. If all other participants are not willing to pay their prices, then the priority rule gives the object to the sender-bribee.

Now, consider the manipulation where the sender-bribee gives the object to the receiver-briber, and the receiver-briber compensates the sender-bribee. Both agents benefit from such manipulation. Further, the manipulation is self-enforcing since the sender-bribee receives sufficient compensation from the receiver-briber, and the receiver-briber cannot receive the object by unilaterally deviating from the manipulation. Thus, any priority rule with participants violates effective l-group strategy-proofness with reallocation and side payments.

5 Implementation via indirect mechanisms

In this section, we discuss the implementation via indirect mechanisms.⁹

An (indirect) mechanism is a pair (A, g) consisting of an action space $A=\times _{i{\in }N} A_{i}$ and an outcome function $g:A \rightarrow Z$. Let $a=(a_{i})_{i{\in }N}{\ }{\in }{\ }A$ denote an action profile. Given a coalition $N'$, let $A_{N'}={\times }_{i{\in }N'} A_{i}$ and $A_{-N'}={\times }_{i{\in }N{\setminus }N'} A_{i}$. Given an action profile $a{\ }{\in }{\ }A$ and a coalition $N'$, let $a_{N'}=(a_{i})_{i{\in }N'}{\ }{\in }{\ }A_{N'}$ and $a_{-N'}=(a_{i})_{i{\in }N{\setminus }N'}{\ }{\in }{\ }A_{-N'}$.

A strategy of an agent $i{\ }{\in }{\ }N$ in a mechanism (A, g) is a function ${\sigma }_{i}:{\mathcal {R}} \rightarrow A_{i}$. A strategy profile in a mechanism (A, g) is an n-tuple ${\sigma }=({\sigma }_{i})_{i{\in }N}$. Given a coalition $N'$ and $R_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, let ${\sigma }_{N'}(R_{N'})=({\sigma }_{i}(R_{i}))_{i{\in }N'}$ and ${\sigma }_{-N'}(R_{-N'})=({\sigma }_{i}(R_{i}))_{i{\in }N{\setminus }N'}$.

A strategy profile ${\sigma }$ is a (weak) dominant strategy equilibrium in a mechanism (A, g) if for each $i{\ }{\in }{\ }N$, each $R_{i}{\ }{\in }{\ }{\mathcal {R}}$, and each $a{\ }{\in }{\ }A$, $g_{i}({\sigma }_{i}(R_{i}),a_{-i}){\ }R_{i}{\ }g_{i}(a)$. A mechanism (A, g) implements a rule f in dominant strategy equilibrium if there is a dominant strategy equilibrium ${\sigma }$ in (A, g) such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $g({\sigma }(R))=f(R)$.¹⁰

A rule (or a direct mechanism) is a special case of a mechanism (A, g) such that $A_{i}={\mathcal {R}}$ for each $i{\ }{\in }{\ }N$, i.e., the action space of each agent is a domain. Thanks to the celebrated revelation principle (Gibbard 1973; Dasgupta et al. 1979, etc.), the dominant strategy equilibrium outcomes of any mechanism can be realized by a strategy-proof rule. Therefore, if we are interested in outcomes that can be implemented by mechanisms in dominant strategy equilibrium, then it is without loss of generality to focus on strategy-proof rules.

We have focused on rules that satisfy group incentive properties in this paper. Our main results in Sect. 4 can be interpreted as characterizing the classes of rules that can be implemented by direct mechanisms in the respective group equilibrium concepts corresponding to the group incentive properties. An interesting question is whether the revelation principles hold for rules satisfying the group incentive properties.

In what follows, we study the revelation principles for rules satisfying the group incentive properties. To the best of our knowledge, the discussion and the results in this section are the first study of revelation principles for rules satisfying group incentive properties.

First, we investigate whether a revelation principle holds for l-weak group strategy-proof rules for a given $l{\ }{\ge }{\ }2$. To this end, we introduce a group equilibrium concept of a mechanism that corresponds to l-weak group strategy-proofness. Specifically, given $l{\ }{\ge }{\ }2$, a strategy profile ${\sigma }$ is an l-weak group dominant strategy equilibrium in a mechanism (A, g) if there are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, and $a{\ }{\in }{\ }A$ such that for each $i{\ }{\in }{\ }N'$, $g_{i}(a){\ }P_{i}{\ }g_{i}({\sigma }_{N'}(R_{N'}),a_{-N'})$.¹¹ Note that if a strategy profile ${\sigma }$ is an l-weak group dominant strategy equilibrium in (A, g) for a given $l{\ }{\ge }{\ }2$, then it is also a dominant strategy equilibrium in (A, g). Given $l{\ }{\ge }{\ }2$, a mechanism (A, g) implements a rule f in l-weak group dominant strategy equilibrium if there is an l-weak group dominant strategy equilibrium ${\sigma }$ in (A, g) such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $g({\sigma }(R))=f(R)$.

The following proposition provides a revelation principle for l-weak group strategy-proof rules. Since the proof follows the standard argument for the revelation principle for strategy-proof rules, we omit it.

Proposition 2

Let ${\mathcal {R}}$ be a domain. Let f be a rule on ${\mathcal {R}}^{n}$. Let $l{\ }{\ge }{\ }2$. Let (A, g) be a mechanism that implements f in l-weak group dominant strategy equilibrium. Then, f satisfies l-weak group strategy-proofness.

Note that Proposition 2 holds not only for an object allocation model with money but also for arbitrary models.

It is straightforward to extend Proposition 2 to rules satisfying any other ineffective group incentive property (including those with reallocation of the object or side payments) that considers all possible manipulations, provided that we appropriately modify the group equilibrium concept in the mechanism. Thus, the revelation principle also holds for rules satisfying any ineffective group incentive property.

Next, we investigate whether a revelation principle holds for rules satisfying the effective group incentive properties that consider only self-enforcing manipulations. Although we here focus on effective l-weak group strategy-proof rules, the discussion is applicable to any other effective group incentive property.

We propose a new group equilibrium concept in a mechanism that corresponds to effective l-weak group strategy-proofness. Given $l{\ }{\ge }{\ }2$, a strategy profile ${\sigma }$ is an effective l-weak group dominant strategy equilibrium in a mechanism (A, g) if there are no $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, a coalition $N'$ with $|N'|{\ }{\le }{\ }l$, and $a{\ }{\in }{\ }A$ such that for each $i{\ }{\in }{\ }N'$, $g_{i}(a){\ }P_{i}{\ }g_{i}({\sigma }_{N'}(R_{N'}),a_{-N'})$, and if $|N'|{\ }{\ge }{\ }2$, then for each $i{\ }{\in }{\ }N'$ and each $a'_{i}{\ }{\in }{\ }A_{i}$, $g_{i}(a){\ }R_{i}{\ }g_{i}(a'_{i},a_{-i})$. In words, under an effective l-weak group dominant strategy equilibrium, no coalition of size at most l benefits from a self-enforcing deviation, where no single agent in the coalition has an incentive to deviate further from the coalitional deviation. Given $l{\ }{\ge }{\ }2$, a mechanism (A, g) implements a rule f in effective l-weak group dominant strategy equilibrium if there is an effective l-weak group dominant strategy equilibrium ${\sigma }$ in (A, g) such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $g({\sigma }(R))=f(R)$.

It is not trivial whether a revelation principle holds for rules satisfying an effective group incentive property that considers only self-enforcing manipulations. This non-triviality arises from the mismatch between the action spaces of a rule and a mechanism. Specifically, if the action space of a mechanism is sufficiently rich compared with that of a rule (i.e., ${\mathcal {R}}^{n}$), then it may be possible that some manipulations are self-enforcing under the rule, but the corresponding deviations are not self-enforcing under the mechanism. Therefore, some rules that violate an effective group incentive property may be implemented by an indirect mechanism in the corresponding effective group equilibrium concept.

Despite the non-triviality discussed above, a revelation principle does hold for effective l-weak group strategy-proof rules.

Proposition 3

Let ${\mathcal {R}}$ be a domain. Let f be a rule on ${\mathcal {R}}^{n}$. Let $l{\ }{\ge }{\ }2$. Let (A, g) be a mechanism that implements f in effective l-weak group dominant strategy equilibrium. Then, f satisfies effective l-weak group strategy-proofness.

It is straightforward to extend Proposition 3 to an arbitrary model that is not necessarily an object allocation model with money.

To understand why a revelation principle holds for effective l-weak group strategy-proof rules, consider a mechanism that implements a rule in effective l-weak group dominant strategy equilibrium. Since an effective l-weak group dominant strategy equilibrium is a dominant strategy equilibrium, any profitable unilateral deviation under the mechanism can also be achieved by the equilibrium action. Therefore, when examining the self-enforcing property of any coalitional deviation under the mechanism, we only have to consider unilateral deviations by an agent from the coalitional deviation using his equilibrium actions. Thus, since the mechanism implements the rule in effective l-weak group dominant strategy equilibrium, the rule is effective l-weak group strategy-proofness.

Finally, we discuss the implementation of the rules that we have considered thus far by simple mechanisms.

First, we consider the implementation of the priority rules. Given $l{\ }{\ge }{\ }2$, we can define an l-strong group strategy equilibrium in the same way as an l-weak group strategy equilibrium, and the implementation in l-strong group dominant strategy equilibrium as well. Then, any priority rule f on ${\mathcal {R}}$ can be implemented in l-strong group dominant strategy equilibrium by the following simple mechanism (A, g)¹²

• For each $i{\ }{\in }{\ }N$, $A_{i}=\{v(R_{i}):R_{i}{\ }{\in }{\ }{\mathcal {R}}\}{\ }{\subseteq }{\ }{\mathbb {R}}_{+}$.

• For each $a{\ }{\in }{\ }A$, the highest priority agent in $\{i{\ }{\in }{\ }N^{f}_{+}:a_{i}{\ }{>}{\ }p^{f}_{i}\}$ according to ${\succ }^{f}$ receives the object and pays $p^{f}_{i}$, while all other agents receive no object and pay nothing.

• Ties are broken in the same way as f.

Note that any priority rule can be implemented by the same mechanism in any weaker equilibrium concept or in effective l-group dominant strategy equilibrium with side payments (Theorem 6).

Next, we consider the implementation of the variable threshold-price rules. Since the threshold-price may vary depending on the entire information of the other agents’ preferences, it may be impossible to implement a variable threshold-price rule via a simple mechanism. In other words, a mechanism that implements a variable threshold-price rule may require an infinite-dimensional action space.

Thus, we restrict our attention to the simple variable threshold-price rules such that the variable threshold-price only depends on the willingness to pay of the other agents, i.e., for each $i{\ }{\in }{\ }N$, the value of ${\kappa }^{f}_{i}$ depends only on $(v(R_{j}))_{j{\in }N{\setminus }\{i\}}$. If a domain is single-dimensional in the sense that the entire information of any preference in the domain is determined by the willingness to pay, then all variable threshold-price rules are simple. An example of a single dimensional domain is a domain ${\mathcal {R}}{\ }{\subseteq }{\ }{\mathcal {R}}^{Q}$ that includes only quasi-linear preferences.

Any simple variable threshold-price rule f on ${\mathcal {R}}^{n}$ can be implemented in l-weak group dominant strategy equilibrium by the following mechanism (A, g)¹³

• For each $i{\ }{\in }{\ }N$, $A_{i}=\{v(R_{i}):R_{i}{\ }{\in }{\ }{\mathcal {R}}\}{\ }{\subseteq }{\ }{\mathbb {R}}_{+}$.

• For each $a{\ }{\in }{\ }A$,

  $$\begin{aligned} g_{i}(a)={\left\{ \begin{array}{ll} (1,{\kappa }^{f}_{i}(a_{-i})) & \text {if }\, a_{i}{\ }{>}{\ }{\kappa }^{f}_{i}(a_{-i}), \\ (1,{\kappa }^{f}_{i}(a_{-i})){\ }\mathrm{{or}}{\ }(0,0) & \text {if }\, a_{i}={\kappa }^{f}_{i}(a_{-i}), \\ (0,0) & \text {if }\, a_{i}{\ }{<}{\ }{\kappa }^{f}_{i}(a_{-i}). \end{array}\right. } \end{aligned}$$

• Ties are broken in the same way as f.

Note that any simple variable threshold-price rule can be implemented by the above mechanism in any weaker equilibrium concept or in the weak group dominant strategy equilibrium concepts with reallocation (Theorem 3).

6 Related literature

In this section, we discuss the contributions of this paper relative to the existing literature.

6.1 Fair and group strategy-proof rules in the object allocation problem with money

In the object allocation problems with money, several authors have studied weak or strong group strategy-proof rules that satisfy the properties of fairness. Mukherjee (2014) considers the single-object model with quasi-linear preferences, and provides characterizations of weak pairwise strategy-proof rules satisfying properties of fairness such as anonymity in welfare and no envy. Hagen (2019) identifies a necessary and sufficient condition for a domain to ensure a non-trivial rule satisfies strong group strategy-proofness and equal treatment of equals in the single-object model with quasi-linear preferences. Tierney (2022) studies the multi-object model with non-quasi-linear preferences, and shows that if a rule satisfies weak pairwise strategy-proofness, anonymity in welfare, and welfare continuity, then it also satisfies no envy.

Our characterization results (Theorems 1 and 2) are different from theirs in the following two points. First, we study the effective group incentive properties that consider only self-enforcing manipulations, while they do weak or strong group strategy-proofness that considers all possible manipulations. Second, we do not assume any property of fairness in our results, while they study the properties of fairness such as anonymity in welfare, no envy, and equal treatment of equals.

6.2 Strong group strategy-proofness in the cost-sharing problem

Several papers have studied the class of rules satisfying strong group strategy-proofness in the cost-sharing problem (Moulin 1999; Moulin and Shenker 2001; Pountourakis and Vidali 2012; Juarez 2013, etc.). If we ignore the cost of providing services for a coalition, then the cost-sharing problem can be viewed as equivalent to the object allocation problem with money, where the owner may possess multiple identical objects and agents receive at most one object. Our Theorem 2 differs from this strand of research in that we focus on effective l-strong group strategy-proofness, which considers only coalitions of size at most l and self-enforcing misrepresentations of preferences, while the papers in this strand of research study strong group strategy-proofness, which considers all possible coalitions and misrepresentations.

In particular, Pountourakis and Vidali (2012) and Juarez (2013) investigate the class of strong group strategy-proof rules, abstracting away the cost function. Our model can be regarded as a special case of their cost-sharing model with the feasibility condition of a single object, where at most one agent receives the object.

Juarez (2013) characterizes the classes of rules satisfying strong group strategy-proofness and the tie-breaking properties together with individual rationality and no subsidy in the identical objects model (Theorems 1 and 2 of Juarez 2013). Since his model is more general than ours, our Theorem 2 does not imply his characterization results. Furthermore, a priority rule does not necessarily satisfy his tie-breaking properties. Thus, when focusing on the single-object model, his characterization results become characterizations of the subclasses of priority rules with particular tie-breaking properties. Therefore, his results do not imply our Theorem 2.

Proposition 3 of Juarez (2013) establishes that in the single-object model, if a rule satisfies strong group strategy-proofness, weak consumer sovereignty, individual rationality, and no subsidy, then it is a priority rules with full participation. Our Theorem 2 extends his Proposition 3 in the following four respects. First, we prove the converse of his result, i.e., any priority rule satisfies strong group strategy-proofness. This converse does not follow from any of his results. Second, we provide a characterization result without weak consumer sovereignty. As a result, the class of rules characterized in our result (i.e., the priority rules) is broader than the class characterized by him (i.e., the priority rules with full participation). Note that his result follows from ours as a corollary by imposing weak consumer sovereignty. Third, we strengthen his result by relaxing strong group strategy-proofness to effective l-strong group strategy proofness and the combination of individual rationality and no subsidy to non-imposition (Remark 1). Fourth, he studies the quasi-linear domain ${\mathcal {R}}^{Q}$, which is a special case of our rich domain. Thus, our result also extends his in terms of domains.

Pountourakis and Vidali (2012) characterize the class of rules satisfying strong group strategy-proofness, strong agent sovereignty, individual rationality, and no subsidy in the identical objects model.¹⁴ Note that all priority rules violate strong agent sovereignty. Thus, the class of rules that they characterize does not overlap with the priority rules, and so our Theorem 2 is independent from their characterization result. Furthermore, Theorem 2 implies the non-existence of a rule satisfying effective l-strong group strategy-proofness, strong agent sovereignty, and non-imposition, which highlights the strong tension between strong agent sovereignty and the feasibility condition of a single object.

6.3 Reallocation of objects and side payments

To the best of our knowledge, Schummer (2000), Bu (2016), and Hagen (2023) are the only papers that study coalitional manipulations (from the dominant strategy perspective as ours) that allows the possibility of reallocating objects or arranging side payments in the object allocation problem with money. Our Theorems 3, 4, 6, and 7 and Proposition 1 differ from their results in that we study effective group incentive properties that consider only self-enforcing manipulations, whereas they do ineffective group incentive properties that consider all possible manipulations. Furthermore, to the best of our knowledge, we are the first to study the group incentive properties with the possibility of reallocation of objects but without side payments (Theorems 3 and 4 and Proposition 1).

To be more specific, Schummer (2000) characterizes the class of rules satisfying pairwise strategy-prooness with side payments on connected quasi-linear domains.¹⁵ Among our results, Theorem 5 is most closely related to his result. Indeed, if we restrict our attention to the quasi-linear domain ${\mathcal {R}}^{Q}$, then Theorem 5 can be derived using his result. However, Theorem 5 holds not only on ${\mathcal {R}}^{Q}$ but also on any non-quasi-linear rich domain, so that it eventually does not follow from his result. Furthermore, our proof of Theorem 5 is fundamentally different from his, as it does not rely on the assumption of quasi-linear preferences that crucially underpins his approach.

Bu (2016) shows that except for the no-trade rule, the priority rules with full participation (i.e., $N^{f}_{+}=N$) and equal prices (i.e., $p^{f}_{i}=p^{f}_{j}$ for each pair $i,j{\ }{\in }{\ }N$) are the only rules satisfying group strategy-proofness with side payments, no envy, and non-imposition in the identical objects model with unit-demand agents.¹⁶ Our Theorem 5 is different from his in that we (i) study the single-object model, (ii) consider weaker l-group strategy-proofness with side payments that takes into account only coalitions of size at most l, and (iii) do not impose no envy in the characterization result. Note that our Theorem 5 implies that the no-trade rule is the only rule satisfying no envy, l-group strategy-proofness with side payments, and non-imposition, which shows the tension between no envy and the feasibility condition of a single object.

The above two papers study coalitional manipulations with side payments, while Hagen (2023) studies those with reallocation of objects and side payments in the identical objects model with multi-demand agents. He characterizes the class of rules satisfying group strategy-proofness with reallocation and side payments.¹⁷ Our Theorem 7 differs from his result in that we consider effective l-group strategy-proofness with reallocation and side payments that takes into account only coalitions of limited size and self-enforcing manipulations.

6.4 Self-enforcing manipulation

Serizawa (2006) proposes an effective group incentive property (effective pairwise strong group strategy-proofness), and characterizes the class of rules satisfying effective pairwise strong group strategy-proofness together with the other desirable properties in various models such as the public goods model with money, the pure exchange model, and the allotment model. Alva (2017) identifies a domain richness condition that guarantees the equivalence between effective pairwise strong group strategy-proofness and strong group strategy-proofness in the object allocation model without money. This paper complements these papers by examining effective group incentive properties in the object allocation problem with money, which they do not cover.

6.5 Strategy-proof rules in the single-object model

Several authors have shown in the single-object model that for quasi-linear preferences, the variable threshold-price rules are the only rules satisfying strategy-proofness and non-imposition (Nisan 2007; Sprumont 2013, etc.).¹⁸ Our Theorem 1 extends this characterization result for quasi-linear preferences in the following two respects. First, we extend their characterization results to non-quasi-linear preferences. Second, we show that any variable threshold-price rule satisfies the stronger property of weak group strategy-proofness.

6.6 Equivalence between individual and group incentive properties

The equivalence between strategy-proofness and weak group strategy-proofness has been studied in various models (Barberà et al. 2010, 2016, etc.). Recall that we establish the equivalence between strategy-proofness, weak group strategy-proofness, and weak group strategy-proofness with reallocation (Corollary 1). Our equivalence result differs from the previous studies in the following two respects. First, we establish not only the equivalence between strategy-proofness and weak group strategy-proofness, but also the equivalence between these properties and weak group strategy-proofness with reallocation (Corollary 1). Second, we provide a characterization of the class of rules satisfying strategy-proofness and weak group strategy-proofness under non-imposition (Theorem 1), whereas the previous studies have only established the equivalence.

Furthermore, our equivalence between strategy-proofness and weak group strategy-proofness does not follow from the previous equivalence results. Here, we provide a detailed discussion on why the equivalence result of Barberà et al. (2016) in an arbitrary private goods model does not imply our equivalence.¹⁹

Barberà et al. (2016) introduce three properties that together constitute a sufficient condition for the equivalence between strategy-proofness and weak group strategy-proofness. To keep the discussion concise, we here focus on one of the three properties: joint monotonicity.²⁰ Given a preference $R_{i}{\ }{\in }{\ }{\mathcal {R}}$ and a bundle $z_{i}{\ }{\in }{\ }M{\ }{\times }{\ }{\mathbb {R}}$, let $U(R_{i},z_{i})=\{z'_{i}{\ }{\in }{\ }M{\ }{\times }{\ }{\mathbb {R}}:z'_{i}{\ }P_{i}{\ }z_{i}\}$ denote the (strict) upper contour set of $R_{i}$ at $z_{i}$, and $L(R_{i},z_{i})=\{z'_{i}{\ }{\in }{\ }M{\ }{\times }{\ }{\mathbb {R}}:z_{i}{\ }P_{i}{\ }z'_{i}\}$ denote the (strict) lower contour set of $R_{i}$ at $z_{i}$. A rule f on ${\mathcal {R}}^{n}$ satisfies joint monotonicity if for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, each coalition $N'$, and each $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ such that for each $i{\ }{\in }{\ }N'$, $R'_{i}{\ }{\ne }{\ }R_{i}$, $L(R_{i},f_{i}(R)){\ }{\subseteq }{\ }L(R'_{i},f_{i}(R))$, and $U(R_{i},f_{i}(R)){\ }{\supseteq }{\ }U(R'_{i},f_{i}(R))$, we have that for each $i{\ }{\in }{\ }N'$, $f_{i}(R'_{N'},R_{-N'}){\ }R'_{i}{\ }f_{i}(R)$. The following example shows that there is a variable threshold-price rule that fails joint monotonicity.

Example 2

Let $n=2$. Let ${\mathcal {R}}$ be an arbitrary rich domain. By richness, we can choose a distinct quadruple $R_{1},R'_{1},R_{2},R'_{2}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R_{1}){\ }{>}{\ }0$, $v(R'_{1}){\ }{>}{\ }v(R_{1})$, $v(R_{2}){\ }{>}{\ }0$, and $v(R'_{2})=0$. Let f be a variable threshold-price rule such that ${\kappa }^{f}_{1}(R_{2})=v(R_{1})$, ${\kappa }^{f}_{1}(R'_{2})={\infty }$, ${\kappa }^{f}_{2}(R_{1})={\infty }$, ${\kappa }^{f}_{2}(R'_{1})={\infty }$, and $x^{f}_{1}(R)=1$. Note that by Theorem 1, f satisfies strategy-proofness. We have $f_{1}(R)=(1,v(R_{1}))$, and $f_{1}(R')=f_{2}(R)=f_{2}(R')=(0,0)$. By $f_{1}(R)=(1,v(R_{1}))$ and $v(R'_{1}){\ }{>}{\ }v(R_{1})$, we have $L(R_{1},f_{1}(R)){\ }{\subseteq }{\ }L(R'_{1},f_{1}(R))$ and $U(R_{1},f_{1}(R)){\ }{\supseteq }{\ }U(R'_{1},f_{1}(R))$. By $f_{2}(R)=(0,0)$ and $v(R'_{2})=0{\ }{<}{\ }v(R_{2})$, we have $L(R_{2},f_{2}(R)){\ }{\subseteq }{\ }L(R'_{2},f_{2}(R))$ and $U(R_{2},f_{2}(R)){\ }{\supseteq }{\ }U(R'_{2},f_{2}(R))$. However, by $v(R'_{1}){\ }{>}{\ }v(R_{1})$, $f_{1}(R)=(1,v(R_{1})){\ }P'_{1}$$(0,0)=f_{1}(R')$. Thus, f fails joint monotonicity.

The above example suggests that we cannot rely on the equivalence result established by Barberà et al. (2016) to show that any strategy-proof rule that satisfies non-imposition also satisfies weak group strategy-prooness. Therefore, the equivalence between strategy-proofness and weak group strategy-proofness established in Theorem 1 does not follow from the equivalence result by Barberà et al. (2016).

6.7 Bayesian group incentive properties

The class of rules that prevent a coalition from profitable manipulations has been extensively studied in the single-object model from the Bayesian perspective (Graham and Marshall 1987; McAfee and McMillan 1992; Pavlov 2008; Che and Kim 2009; Che et al. 2018, etc.). A well-recognized drawback of the Bayesian approach is its reliance on the common prior assumption (Wilson 1987; Bergemann and Morris 2005, etc.). In contrast, our group incentive properties are defined from the dominant strategy perspective, which does not depend on this assumption. Therefore, this paper complements this strand of research by exploring a “robust” class of rules that prevent collusion without relying on the common prior assumption.

7 Conclusion

In this paper, we have characterized the classes of rule satisfying various group incentive properties together with the mild property of non-imposition. We have also established the revelation principles for rules satisfying the group incentive properties. The results in this paper are specific to the single-object model, and a possible direction of future research is to investigate the group incentive properties in the multi-object model. Although the difficulty in characterizing the class of rules satisfying the group incentive properties without auxiliary properties in the multi-object model is known (e.g., Juarez 2013), it would be possible to study the relationships between the group incentive properties in the multi-object model.

Appendices

Appendix

Preliminaries

In this section, we provide the preliminary lemmas that will be used to show the results. The proofs of the following lemmas are straightforward, so that we omit them.

First, the following lemma states that under strategy-proofness and non-imposition, an agent who does not receive the object makes no payment.

Lemma 1

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying strategy-proofness and non-imposition. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $i{\ }{\in }{\ }N$. If $x^{f}_{i}(R)=0$, then $t^{f}_{i}(R)=0$.

Second, the following lemma states that under strategy-proofness and non-imposition, an agent who receives the object makes a non-negative payment.

Lemma 2

Let ${\mathcal {R}}$ be a rich domain. Let f be a rule on ${\mathcal {R}}^{n}$ satisfying strategy-proofness and non-imposition. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $i{\ }{\in }{\ }N$. If $x^{f}_{i}(R)=1$, then $t^{f}_{i}(R){\ }{\ge }{\ }0$.

Third, the following lemma states that any variable threshold-price rule satisfies individual rationality and no subsidy, and so by Remark 1, it also satisfies non-imposition.

Lemma 3

Let ${\mathcal {R}}$ be a domain. Let f be a variable threshold-price rule on ${\mathcal {R}}^{n}$. Then, f satisfies individual rationality, no subsidy, and non-imposition.

Finally, the following lemma states that any priority rule satisfies individual rationality, no subsidy, and non-imposition.

Lemma 4

Let ${\mathcal {R}}$ be a domain. Let f be a priority rule on ${\mathcal {R}}^{n}$. Then, f satisfies individual rationality, no subsidy, and non-imposition.

Proof of Theorem 1

In this section, we prove Theorem 1. Since l-weak group strategy-proofness implies effective l-weak group strategy-proofness, which in turn implies strategy-proofness, we have that (iii) implies (ii), which in turn implies (i). Furthermore, since l-weak group strategy-proofness with reallocation implies l-weak group strategy-proofness, it follows that (iv) implies (iii) by the implication of (iii) to (ii) in Theorem 3.²¹ Thus, it suffices to show that (i) implies (iv).

Suppose that f satisfies strategy-proofness. Let $i{\ }{\in }{\ }N$. Let $R_{-i}{\ }{\in }{\ }{\mathcal {R}}^{n-1}$. By strategy-proofness, for each pair $R_{i},R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ with $x^{f}_{i}(R_{i},R_{-i})=x^{f}_{i}(R'_{i},R_{-i})=1$, we have $t^{f}_{i}(R_{i},R_{-i})=t^{f}_{i}(R'_{i},R_{-i})$. By Lemma 2, for each $R_{i}{\ }{\in }{\ }{\mathcal {R}}$ with $x^{f}_{i}(R_{i},R_{-i})=1$, $t^{f}_{i}(R_{i},R_{-i}){\ }{\ge }{\ }0$. Thus, we can define a function ${\kappa }^{f}_{i}:{\mathcal {R}}^{n-1} \rightarrow {\mathbb {R}}_{+}{\ }{\cup }{\ }\{{\infty }\}$ such that for each $R_{-i}{\ }{\in }{\ }{\mathcal {R}}^{n-1}$,

$$\begin{aligned} {\kappa }^{f}_{i}(R_{-i})={\left\{ \begin{array}{ll} t^{f}_{i}(R_{i},R_{-i}) & \text {if } \, \text { there }\, \text { is }\, R_{i}{\ }{\in }{\ }{\mathcal {R}} \,\text { s.t. }\, x^{f}_{i}(R_{i},R_{-i})=1, \\ {\infty } & \text {otherwise.} \end{array}\right. } \end{aligned}$$

We show that f is a variable threshold-price rule associated with $({\kappa }^{f}_{i})_{i{\in }N}$. Let $i{\ }{\in }{\ }N$ and $R{\ }{\in }{\ }{\mathcal {R}}^{n}$.

First, suppose that $v(R_{i}){\ }{>}{\ }{\kappa }^{f}_{i}(R_{-i})$. We show that $x^{f}_{i}(R)=1$. Suppose by contradiction that $x^{f}_{i}(R)=0$. Then, by Lemma 1, $t^{f}_{i}(R)=0$. By finiteness, ${\kappa }^{f}_{i}(R_{-i}){\ }{<}{\ }v(R_{i}){\ }{<}{\ }{\infty }$, and so by the definition of ${\kappa }^{f}_{i}$, there is $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $x^{f}_{i}(R'_{i},R_{-i})=1$ and $t^{f}_{i}(R'_{i},R_{-i})={\kappa }^{f}_{i}(R_{-i})$. Thus, by $v(R_{i}){\ }{>}{\ }{\kappa }^{f}_{i}(R_{-i})=t^{f}_{i}(R'_{i},R_{-i})$, $f_{i}(R'_{i},R_{-i})=(1,{\kappa }^{f}_{i}(R_{-i})){\ }P_{i}{\ }(0,0)=f_{i}(R)$, which contradicts strategy-proofness. Thus, $x^{f}_{i}(R)=1$, and so by the definition of ${\kappa }^{f}_{i}$, $t^{f}_{i}(R)={\kappa }^{f}_{i}(R_{-i})$.

Second, suppose that $v(R_{i})={\kappa }^{f}_{i}(R_{-i})$. If $x^{f}_{i}(R)=1$, then by the definition of ${\kappa }^{f}_{i}$, $t^{f}_{i}(R)={\kappa }^{f}_{i}(R_{-i})$. If $x^{f}_{i}(R)=0$, then by Lemma 1, $t^{f}_{i}(R)=0$.

Finally, suppose that $v(R_{i}){\ }{<}{\ }{\kappa }^{f}_{i}(R_{-i})$. We show that $x^{f}_{i}(R)=0$. By contradiction, suppose that $x^{f}_{i}(R)=1$. Then, by the definition of ${\kappa }^{f}_{i}$, $t^{f}_{i}(R)={\kappa }^{f}_{i}(R_{-i})$. By richness, there is $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(t_{i};R'_{i})=0$ for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$. Then, by non-imposition, $t^{f}_{i}(R'_{i},R_{-i})=0$. Thus, $f_{i}(R'_{i},R_{-i}){\ }R_{i}{\ }(0,0){\ }P_{i}{\ }(1,{\kappa }^{f}_{i}(R_{-i}))=f_{i}(R)$, where the first relation follows from $t^{f}_{i}(R'_{i},R_{-i})=0$ and weak desirability of the object, and the second one from $v(R_{i}){\ }{<}{\ }{\kappa }^{f}_{i}(R_{-i})=t^{f}_{i}(R)$. However, this contradicts strategy-proofness. Thus, $x^{f}_{i}(R)=0$. By Lemma 1, $t^{f}_{i}(R)=0$.       $\blacksquare$

Proof of Theorem 2

In this section, we prove Theorem 2. Since l-strong group strategy-proofness implies effective l-strong group strategy-proofness for a given $l{\ }{\ge }{\ }2$, (ii) implies (i). Thus, it suffices to show that (iii) implies (ii) and (i) implies (iii).

1.1 (iii) implies (ii)

In this subsection, we show that (iii) implies (ii). Suppose that f is a priority rule. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'$ be a coalition with $|N'|{\ }{\le }{\ }l$, and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$. Let $i{\ }{\in }{\ }N'$ be such that $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)$.

We claim that $x^{f}_{i}(R)=0$. If $x^{f}_{i}(R)=1$, then by the definition of a priority rule, $t^{f}_{i}(R)=p^{f}_{i}$. By individual rationality of f (Lemma 4) and $f_{i}(R'_{N'},R_{-N'}){\ }{\in }{\ }\{(1,p^{f}_{i}),(0,0)\}$, $f_{i}(R)=(1,p^{f}_{i}){\ }R_{i}{\ }f_{i}(R'_{N'},R_{-N'})$, which contradicts that $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)$. Thus, we have $x^{f}_{i}(R)=1$.

Next, we claim that $x^{f}_{i}(R'_{N'},R_{-N'})=1$. If $x^{f}_{i}(R'_{N'},R_{-N'})=0$, then by the definition of a priority rule, $t^{f}_{i}(R'_{N'},R_{-N'})=0$. Thus, by individual rationality of f (Lemma 4), $f_{i}(R){\ }R_{i}{\ }(0,0)=f_{i}(R'_{N'},R_{-N'})$, which contradicts $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)$. Thus, we have $x^{f}_{i}(R'_{N'},R_{-N'})=1$.

By $x^{f}_{i}(R)=0$ and $x^{f}_{i}(R'_{N'},R_{-N'})=1$, we have $t^{f}_{i}(R)=0$ and $t^{f}_{i}(R'_{N'},R_{-N'})=p^{f}_{i}$. Thus, by $(1,p^{f}_{i})=f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)=(0,0)$, $v(R_{i}){\ }{>}{\ }p^{f}_{i}$. Thus, by $x^{f}_{i}(R)=0$, there is $j{\ }{\in }{\ }N{\setminus }\{i\}$ such that $j{\ }{\succ }^{f}{\ }i$, $x^{f}_{j}(R)=1$, and $v(R_{j}){\ }{>}{\ }p^{f}_{j}$. By $x^{f}_{i}(R'_{N'},R_{-N'})=1$, $x^{f}_{j}(R'_{N'},R_{-N'})=0$. Thus by $v(R_{j}){\ }{>}{\ }p^{f}_{j}$, $f_{j}(R)=(1,p^{f}_{j}){\ }P_{j}{\ }(0,0)=f_{j}(R'_{N'},R_{-N'})$.

We claim that $j{\ }{\in }{\ }N'$. By contradiction, suppose that $j{\ }{\not \in }{\ }N'$. By $j{\ }{\succ }^{f}{\ }i$ and $v(R_{j}){\ }{>}{\ }p^{f}_{j}$, we must have $x^{f}_{i}(R'_{N'},R_{-N'})=0$. However, this contradicts that $x^{f}_{i}(R'_{N'},R_{-N'})=1$.

Thus, we have $j{\ }{\in }{\ }N'$ and $f_{j}(R){\ }P_{j}{\ }f_{j}(R'_{N'},R_{-N'})$. Thus, $N'$ does not weakly benefit from misrepresenting $R'_{N'}$ at R.       $\blacksquare$

1.2 (i) implies (iii)

In this subsection, we show that (i) implies (iii). Suppose that f satisfies effective l-strong group strategy-proofness. Note that since effective l-strong group strategy-proofness implies strategy-proofness, f satisfies strategy-proofness. The proof consists of two parts.

Part 1. In the first part, we show the existence of a (fixed) price $p^{f}_{i}{\ }{\in }{\ }{\mathbb {R}}_{+}$ for each $i{\ }{\in }{\ }N^{f}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, if $x^{f}_{i}(R)=1$, then $t^{f}_{i}(R)=p^{f}_{i}$.

First, we establish the following two preliminary lemmas.

Lemma 5

Let $i,j{\ }{\in }{\ }N$ be a distinct pair, $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, and $R'_{j}{\ }{\in }{\ }{\mathcal {R}}$ be such that $x^{f}_{i}(R)=x^{f}_{i}(R'_{j},R_{-j})=1$. Then, $t^{f}_{i}(R)=t^{f}_{i}(R'_{j},R_{-j})$.

Proof

Suppose by contradiction that $t^{f}_{i}(R){\ }{\ne }{\ }t^{f}_{i}(R'_{j},R_{-j})$. Without loss of generality, let $t^{f}_{i}(R){\ }{<}{\ }t^{f}_{i}(R'_{j},R_{-j})$. Then, $f_{i}(R){\ }P_{i}{\ }f_{i}(R'_{j},R_{-j})$. By $x^{f}_{i}(R)=x^{f}_{i}(R'_{j},R_{-j})=1$, $x^{f}_{j}(R)=x^{f}_{j}(R'_{j},R_{-j})=0$. Thus, by Lemma 1, $f_{j}(R)=(0,0)=f_{j}(R'_{j},R_{-j})$. Thus, $N'=\{i,j\}$ weakly benefits from misrepresenting $R_{N'}$ at $(R'_{j},R_{-j})$. By strategy-proofness, for each $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$, $f_{i}(R){\ }R_{i}{\ }f_{i}(R'_{i},R_{-i})$, and for each $R''_{j}{\ }{\in }{\ }{\mathcal {R}}$, $f_{j}(R)=f_{j}(R'_{j},R_{-j}){\ }R'_{j}{\ }f_{j}(R''_{j},R_{-j})$. Thus, $R_{N'}$ is self-enforcing at $(R'_{j},R_{-j})$, which contradicts effective l-strong group strategy-proofness. $\square$

Lemma 6

Let $i,j{\ }{\in }{\ }N$ be a distinct pair, $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, and $R'_{i,j}{\ }{\in }{\ }{\mathcal {R}}^{2}$ be such that $x^{f}_{i}(R)=1$, $v(R'_{i}){\ }{>}{\ }t^{f}_{i}(R)$, and $v(R'_{j})=0$. Then, $x^{f}_{i}(R'_{i,j},R_{-i,j})=1$.

Proof

Suppose by contradiction that $x^{f}_{i}(R'_{i,j},R_{-i,j})=0$. By Lemma 1, $f_{i}(R'_{i,j},R_{-i,j})=(0,0)$. By $v(R'_{i}){\ }{>}{\ }t_{i}(R)$, $f_{i}(R){\ }P'_{i}{\ }f_{i}(R'_{i,j},R_{-i,j})$. By Lemmas 1 and 2 and $v(R'_{j})=0$, $f_{j}(R)=(0,0){\ }R'_{j}{\ }f_{j}(R'_{i,j},R_{-i,j})$. Thus, $N'=\{i,j\}$ weakly benefits from misrepresenting $R_{N'}$ at $(R'_{N'},R_{-N'})$. By Lemma 1, strategy-proofness, and $v(R'_{i}){\ }{>}{\ }t^{f}_{i}(R)$, for each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, $f_{i}(R){\ }R'_{i}{\ }f_{i}(R''_{i},R_{-i})$. By Lemmas 1 and 2 and $v(R'_{j})=0$, for each $R''_{j}{\ }{\in }{\ }{\mathcal {R}}$, $f_{j}(R)=(0,0){\ }R'_{j}{\ }f_{j}(R''_{j},R_{-j})$. Thus, $R_{N'}$ is self-enforcing at $(R'_{N'},R_{-N'})$, contradicting effective l-strong group strategy-proofness. $\square$

Then, we move on to the proof of the existence of a price for each participant in the rule f. Let $i{\ }{\in }{\ }N^{f}_{+}$. We show that there is $p^{f}_{i}{\ }{\in }{\ }{\mathbb {R}}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, if $x^{f}_{i}(R)=1$, then $t^{f}_{i}(R)=p^{f}_{i}$.

Without loss of generality, let $i=1$. By Lemma 2, for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, if $x^{f}_{1}(R)=1$, then $t^{f}_{1}(R){\ }{\ge }{\ }0$. Thus, it suffices to show that for each pair $R,R'{\ }{\in }{\ }{\mathcal {R}}^{n}$ with $x^{f}_{1}(R)=x^{f}_{1}(R')=1$, $t^{f}_{1}(R)=t^{f}_{1}(R')$. Let $R,R'{\ }{\in }{\ }{\mathcal {R}}^{n}$ be such that $x^{f}_{1}(R)=x^{f}_{1}(R')=1$. By richness, we can choose $R''_{1}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R''_{1}){\ }{>}{\ }\max \{t^{f}_{1}(R),t^{f}_{1}(R')\}$. By Lemma 1 and strategy-proofness, $f_{1}(R''_{1},R_{-1})=f_{1}(R)$ and $f_{1}(R''_{1},R'_{-1})=f_{1}(R')$.

By richness, we can choose $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R''_{2})=0$. Then, by $x^{f}_{1}(R''_{1},R_{-1})=1$ and $v(R''_{1}){\ }{>}{\ }t^{f}_{1}(R)=t^{f}_{1}(R''_{1},R_{-1})$, Lemma 6 implies $x^{f}_{1}(R''_{1,2},R_{-1,2})=x^{f}_{1}(R''_{1},R_{-1})=1$. By Lemma 5, $t^{f}_{1}(R''_{1,2},R_{-1,2})=t^{f}_{1}(R''_{1},R_{-1,2})$. By the same argument, $t^{f}_{1}(R''_{1,2},R'_{-1,2})=t^{f}_{1}(R''_{1},R'_{-1})$.

Repeating the same arguments for agents $3,{\dots },n$,

$$\begin{aligned} t^{f}_{1}(R''_{1},R_{-1})=t^{f}_{1}(R''_{1,2},R_{-1,2})={\ }{\cdots }=t^{f}_{1}(R'')={\cdots }&=t^{f}_{1}(R''_{1,2},R'_{-1,2})\\&=t^{f}_{1}(R''_{1},R'_{-1}). \end{aligned}$$

By $t^{f}_{1}(R)=t^{f}_{1}(R''_{1},R_{-1})$ and $t^{f}_{1}(R')=t^{f}_{1}(R''_{1},R'_{-1})$, we get $t^{f}_{1}(R)=t^{f}_{1}(R')$, as desired.

Part 2. In the second part, we establish the existence of a priority ${\succ }^{f}$ over $N^{f}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and each $i{\ }{\in }{\ }N^{f}_{+}$, if $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$ and $v(R_{i}){\ }{>}{\ }p^{f}_{i}$, then $f_{i}(R)=(1,p^{f}_{i})$.

We establish the following four preliminary lemmas.

Lemma 7

Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $i{\ }{\in }{\ }N^{f}_{+}$ be such that $v(R_{i}){\ }{>}{\ }p^{f}_{i}$. Let $R'_{-N^{f}_{+}}{\ }{\in }{\ }{\mathcal {R}}^{|N{\setminus }N^{f}_{+}|}$. If $x^{f}_{i}(R)=1$, then $x^{f}_{i}(R_{N^{f}_{+}},R'_{-N^{f}_{+}})=1$.

Proof

Suppose $x^{f}_{i}(R)=1$. Without loss of generality, let $i=1$ and $N{\setminus }N^{f}_{+}=\{2,{\dots },I\}$, where $I=|N{\setminus }N^{f}_{+}|+1$. We claim $x^{f}_{1}(R'_{2},R_{-2})=1$. Suppose by contradiction that $x^{f}_{1}(R'_{2},R_{-2})=0$. By Lemma 1, $f_{1}(R'_{2},R_{-2})=(0,0)$. By $v(R_{1}){\ }{>}{\ }p^{f}_{1}$ and Part 1, $f_{1}(R)=(1,p^{f}_{1}){\ }P_{1}{\ }(0,0)=f_{1}(R'_{2},R_{-2})$. By $2{\ }{\not \in }{\ }N^{f}_{+}$, $x^{f}_{2}(R)=x^{f}_{2}(R'_{2},R_{-2})=0$. By Lemma 1, $f_{2}(R)=(0,0)=f_{2}(R'_{2},R_{-2})$. Thus, $N'=\{1,2\}$ weakly benefits from misrepresenting $R_{N'}$ at $(R'_{2},R_{-2})$. By strategy-proofness, for each $R'_{1}{\ }{\in }{\ }{\mathcal {R}}$, $f_{1}(R){\ }R_{1}{\ }f_{1}(R'_{1},R_{-1})$. By Lemma 1 and $2{\ }{\not \in }{\ }N^{f}_{+}$, for each $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$, $f_{2}(R)=(0,0)=f_{2}(R''_{2},R_{-2})$. Thus, $R_{N'}$ is self-enforcing at $(R'_{2},R_{-2})$, contradicting effective l-strong group strategy-proofness. Thus, we get $x^{f}_{1}(R'_{2},R_{-2})=1$. Repeating the same arguments for agents $3,{\dots },I$, we get $x_{1}(R_{N^{f}_{+}},R'_{-N^{f}_{+}})=1$. $\square$

Lemma 8

Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'{\ }{\subseteq }{\ }N^{f}_{+}$, and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ be such that $x^{f}_{i}(R)=1$ for some $i{\ }{\in }{\ }N'$ and $v(R'_{j}){\ }{>}{\ }p^{f}_{j}$ for each $j{\ }{\in }{\ }N'$. Then, $x^{f}_{j}(R'_{N'},R_{-N'})=1$ for some $j{\ }{\in }{\ }N'$.

Proof

Without loss of generality, let $N'=\{1,{\dots },I\}$, where $I=|N'|$, and $i=1$. By $x^{f}_{1}(R)=1$ and $v(R'_{1}){\ }{>}{\ }p^{f}_{1}$, Lemma 1, Step 1, and strategy-proofness imply $x^{f}_{1}(R'_{1},R_{-1})=1$. We show $x^{f}_{i}(R'_{1,2},R_{-1,2})=1$ for some $i{\ }{\in }{\ }\{1,2\}$. Suppose by contradiction that $x^{f}_{i}(R'_{1,2},R_{-1,2})=0$ for each $i{\ }{\in }{\ }\{1,2\}$. By Lemma 1, $f_{1}(R'_{1,2},R_{-1,2})=f_{2}(R'_{1,2},R_{-1,2})=(0,0)$. By $v(R'_{1}){\ }{>}{\ }p^{f}_{1}$ and $x^{f}_{1}(R'_{1},R_{-1})=1$, Part 1 implies $f_{1}(R'_{1},R_{-1})=(1,p^{f}_{1}){\ }P'_{1}{\ }(0,0)=f_{1}(R'_{1,2},R_{-1,2})$. By $x^{f}_{1}(R'_{1},R_{-1})=1$, $x^{f}_{2}(R'_{1},R_{-1})=0$. Thus, by Lemma 1, $f_{2}(R'_{1},R_{-1})=(0,0)=f_{2}(R'_{1,2},R_{-1,2})$. Thus, $N'=\{1,2\}$ weakly benefits from misrepresenting $(R'_{1},R_{2})$ at $(R'_{N'},R_{-N'})$. By strategy-proofness, for each $R''_{1}{\ }{\in }{\ }{\mathcal {R}}$, $f_{1}(R'_{1},R_{-1}){\ }R'_{1}{\ }f_{1}(R''_{1},R_{-1})$. By strategy-proofness, $v(R'_{2}){\ }{>}{\ }p^{f}_{2}$, and $x^{f}_{2}(R'_{1,2},R_{-1,2})=0$, Lemma 1 and Step 1 imply that there is no $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$ such that $x^{f}_{2}(R'_{1},R''_{2},R_{-1,2})=1$. Thus, by Lemma 1, for each $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$, $f_{2}(R'_{1},R_{-1,2})=(0,0)=f_{2}(R'_{1},R''_{2},R_{-1,2})$. Thus, $(R'_{1},R_{2})$ is self-enforcing at $(R'_{N'},R_{-N'})$, which contradicts effective l-strong group strategy-proofness. Thus, we obtain $x^{f}_{i}(R'_{1,2},R_{-1,2})=1$ for some $i{\ }{\in }{\ }\{1,2\}$. Repeating the same arguments for agents $3,{\dots },I$ inductively, we have $x^{f}_{i}(R'_{N'},R_{-N'})=1$ for some $i{\ }{\in }{\ }N'$. $\square$

Lemma 9

Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'{\ }{\subseteq }{\ }N^{f}_{+}$, and $i{\ }{\in }{\ }N'$ be such that $x^{f}_{i}(R)=1$ and $v(R_{j}){\ }{>}{\ }p^{f}_{j}$ for each $j{\ }{\in }{\ }N'$. Then, for each $R'_{N'{\setminus }\{i\}}{\ }{\in }{\ }{\mathcal {R}}^{|N'|-1}$, $x^{f}_{i}(R'_{N'{\setminus }\{i\}},R_{-N'{\setminus }\{i\}})=1$.

Proof

Without loss of generality, let $N'=\{1,{\dots },I\}$, where $I=|N'|$, and $i=1$. Let $R'_{N'{\setminus }\{1\}}{\ }{\in }{\ }{\mathcal {R}}^{|N'|-1}$. We show $x^{f}_{1}(R'_{2},R_{-2})=1$. Suppose not. By Lemma 1, $f_{1}(R'_{2},R_{-2})=(0,0)$. By $x^{f}_{1}(R)=1$, Part 1 implies $f_{1}(R)=(1,p^{f}_{1})$. Thus, by $v(R_{1}){\ }{>}{\ }p^{f}_{1}$, $f_{1}(R){\ }P_{1}{\ }f_{1}(R'_{2},R_{-2})$. By $x^{f}_{2}(R)=0$, Lemma 1 implies $f_{2}(R)=(0,0)$. By Part 1 and $v(R_{2}){\ }{>}{\ }p^{f}_{2}$, strategy-proofness implies that there is no $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$ such that $x^{f}_{2}(R''_{2},R_{-2})=1$. Thus, $x^{f}_{2}(R'_{2},R_{-2})=0$, and so by Lemma 1, $f_{2}(R)=(0,0)=f_{2}(R'_{2},R_{-2})$. Thus, $N'=\{1,2\}$ weakly benefits from misrepresenting $R_{N'}$ at $(R'_{2},R_{-2})$. By strategy-proofness, for each $R''_{1}{\ }{\in }{\ }{\mathcal {R}}$, $f_{1}(R){\ }R_{1}{\ }f_{1}(R''_{1},R_{-1})$. Since $x^{f}_{2}(R''_{2},R_{-2})=1$ for no $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$, by Lemma 1, $f_{2}(R)=(0,0)=f_{2}(R''_{2},R_{-2})$ for each $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$. Thus, $R_{N'}$ is self-enforcing at $(R'_{2},R_{-2})$, which contradicts effective l-strong group strategy-proofness. Thus, $x^{f}_{1}(R'_{2},R_{-2})=1$. Repeating the same arguments for agents $3,{\dots },I$, we obtain $x^{f}_{i}(R'_{N'{\setminus }\{i\}},R_{-N'{\setminus }\{i\}})=1$. $\square$

Lemma 10

Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $i{\ }{\in }{\ }N^{f}_{+}$ be such that $x^{f}_{i}(R)=1$ and $v(R_{i}){\ }{>}{\ }p^{f}_{i}$. Let $N'{\ }{\subseteq }{\ }N^{f}_{+}{\setminus }\{i\}$ and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ be such that $v(R'_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\in }{\ }N'$. Then, $x^{f}_{i}(R'_{N'},R_{-N'})=1$.

Proof

Without loss of generality, let $i=1$ and $N'=\{2,{\dots },I\}$, where $I=|N'|+1$. We show that $x^{f}_{1}(R'_{2},R_{-2})=1$. Suppose by contradiction that $x^{f}_{1}(R'_{2},R_{-2})=0$. By Lemma 1, $f_{1}(R'_{2},R_{-2})=(0,0)$. By Part 1 and $x^{f}_{1}(R)=1$, $f_{1}(R)=(1,p^{f}_{1})$. By $v(R_{1}){\ }{>}{\ }p^{f}_{1}$, $f_{1}(R)=(1,p^{f}_{1}){\ }P_{1}{\ }(0,0)=f_{1}(R'_{2},R_{-2})$. By $x^{f}_{1}(R)=1$, $x^{f}_{2}(R)=0$. Thus, by Lemma 1, $f_{2}(R)=(0,0)$. By $v(R'_{2}){\ }{\le }{\ }p^{f}_{2}$, Lemma 1 and Step 1 imply $f_{2}(R)=(0,0){\ }R'_{2}{\ }f_{2}(R'_{2},R_{-2})$. Thus, $N'=\{1,2\}$ weakly benefits from misrepresenting $R_{N'}$ at $(R'_{2},R_{-2})$. By strategy-proofness, for each $R'_{1}{\ }{\in }{\ }{\mathcal {R}}$, $f_{1}(R){\ }R_{1}{\ }f_{1}(R'_{1},R_{-1})$. By $v(R'_{2}){\ }{\le }{\ }p^{f}_{2}$, Lemma 1 and Step 1 imply that for each $R''_{2}{\ }{\in }{\ }{\mathcal {R}}$, $f_{2}(R)=(0,0){\ }R'_{2}{\ }f_{2}(R''_{2},R_{-2})$. Thus, $R_{N'}$ is self-enforcing at $(R'_{2},R_{-2})$, which contradicts effective l-strong group strategy-proofness. Thus, $x^{f}_{1}(R'_{2},R_{-2})=1$. Repeating the same arguments for agents $3,{\dots },I$, we get $x^{f}_{i}(R'_{N'},R_{-N'})=1$. $\square$

Now, we are in a position to show that there is a priority ${\succ }^{f}$ over $N^{f}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and each $i{\ }{\in }{\ }N^{f}_{+}$, if $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$ and $v(R_{i}){\ }{>}{\ }p^{f}_{i}$, then $f_{i}(R)=(1,p^{f}_{i})$. As discussed in Sect. 4.2, our proof of the existence of such a priority basically follows the proof of Proposition 3 of Juarez (2013). The previous four lemmas (Lemmas 7 to 10) enable us to adopt this approach. However, we cannot follow his proof directly due to the difference between our weaker notion of effective l-strong group strategy-proofness and his stronger notion of strong group strategy-proofness. Therefore, our proof requires additional induction steps (Step 2 below) compared with his proof.

If $N^{f}_{+}={\varnothing }$, then the conclusion is vacuously true. Suppose that $N^{f}_{+}{\ }{\ne }{\ }{\varnothing }$. Without loss of generality, let $N^{f}_{+}=\{1,{\dots },I\}$, where $I=|N^{f}_{+}|{\ }{\ge }{\ }1$. The proof proceeds in three steps.

Step 1. We show that there is ${\iota }_{1}{\ }{\in }{\ }N^{f}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, if $v_{{\iota }_{1}}(R_{{\iota }_{1}}){\ }{>}{\ }p^{f}_{{\iota }_{1}}$, then $x^{f}_{{\iota }_{1}}(R)=1$. Suppose not. Then, for each $i{\ }{\in }{\ }N^{f}_{+}$, there is $R^{i}{\ }{\in }{\ }{\mathcal {R}}^{n}$ such that $v(R^{i}_{i}){\ }{>}{\ }p^{f}_{i}$ and $x^{f}_{i}(R^{i})=0$.

By $1{\ }{\in }{\ }N^{f}_{+}$, there is ${\tilde{R}}^{1}{\ }{\in }{\ }{\mathcal {R}}^{n}$ such that $x^{f}_{1}({\tilde{R}}^{1})=1$. By richness, we can choose $R_{N^{f}_{+}}{\ }{\in }{\ }{\mathcal {R}}^{|N^{f}_{+}|}$ such that $v(R_{i}){\ }{>}{\ }p^{f}_{i}$ for each $i{\ }{\in }{\ }N^{f}_{+}$. By Lemma 8, there is $i{\ }{\in }{\ }N^{f}_{+}$ such that $x^{f}_{i}(R_{N^{f}_{+}},{\tilde{R}}^{1}_{-N^{f}_{+}})=1$. By Lemma 7, $x^{f}_{i}(R_{N^{f}_{+}},R^{i}_{-N^{f}_{+}})=1$. By $v(R_{j}){\ }{>}{\ }p^{f}_{j}$ for each $j{\ }{\in }{\ }N^{f}_{+}{\setminus }\{i\}$, Lemma 9 implies $x^{f}_{i}(R_{i},R^{i}_{-i})=1$. Thus, by Lemma 1, Step 1, and $v(R^{i}_{i}){\ }{>}{\ }p^{f}_{i}$, strategy-proofness implies $x^{f}_{i}(R^{i})=1$, which contradicts $x^{f}_{i}(R^{i})=0$.

Step 2. By Step 1, there is ${\iota }_{1}{\ }{\in }{\ }N^{f}_{+}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, if $v_{{\iota }_{1}}(R_{{\iota }_{1}}){\ }{>}{\ }p^{f}_{{\iota }_{1}}$, then $x^{f}_{{\iota }_{1}}(R)=1$. Without loss of generality, let ${\iota }_{1}=1$. In this step, we show that there is ${\iota }_{2}{\ }{\in }{\ }N^{f}_{+}{\setminus }\{1\}$ such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, if $v(R_{1}){\ }{\le }{\ }p^{f}_{1}$ and $v_{{\iota }_{2}}(R_{{\iota }_{2}}){\ }{>}{\ }p^{f}_{{\iota }_{2}}$, then $x^{f}_{{\iota }_{2}}(R)=1$. Suppose not. Then, for each $i{\ }{\in }{\ }N^{f}_{+}{\setminus }\{1\}$, there is $R^{i}{\ }{\in }{\ }{\mathcal {R}}^{n}$ such that $v(R^{i}_{1}){\ }{\le }{\ }p^{f}_{1}$, $v(R^{i}_{i}){\ }{>}{\ }p^{f}_{i}$, and $x^{f}_{i}(R^{i})=0$.

By $2{\ }{\in }{\ }N^{f}_{+}$, there is ${\tilde{R}}^{2}{\ }{\in }{\ }{\mathcal {R}}^{n}$ such that $x^{f}_{2}({\tilde{R}}^{2})=1$. By richness, we can choose $R_{N^{f}_{+}{\setminus }\{1\}}{\ }{\in }{\ }{\mathcal {R}}^{|N^{f}_{+}|-1}$ such that $v(R_{i}){\ }{>}{\ }p^{f}_{i}$ for each $i{\ }{\in }{\ }N^{f}_{+}{\setminus }\{1\}$. By Lemma 8, there is $i{\ }{\in }{\ }N^{f}_{+}{\setminus }\{1\}$ such that $x^{f}_{i}(R_{N^{f}_{+}{\setminus }\{1\}},{\tilde{R}}^{2}_{-N^{f}_{+}{\setminus }\{1\}})=1$. By assumption, there is $R^{i}{\ }{\in }{\ }{\mathcal {R}}^{n}$ such that $v(R^{i}_{1}){\ }{\le }{\ }p^{f}_{1}$, $v(R^{i}_{i}){\ }{>}{\ }p^{f}_{i}$, and $x^{f}_{i}(R^{i})=0$. By $v(R^{i}_{1}){\ }{\le }{\ }p^{f}_{1}$ and $x^{f}_{i}(R_{N^{f}_{+}{\setminus }\{1\}},{\tilde{R}}^{2}_{-N^{f}_{+}{\setminus }\{1\}})=1$, Lemma 10 gives $x^{f}_{i}(R^{i}_{1},R_{N^{f}_{+}{\setminus }\{1\}},{\tilde{R}}^{2}_{-N^{f}_{+}})=1$. By Lemma 7, $x^{f}_{i}(R^{i}_{1},R_{N^{f}_{+}{\setminus }\{i\}},R^{i}_{-N^{f}_{+}})=1$. By $v(R_{j}){\ }{>}{\ }p^{f}_{j}$ for each $j{\ }{\in }{\ }N^{f}_{+}{\setminus }\{1,i\}$, Lemma 9 implies $x^{f}_{i}(R_{i},R^{i}_{-i})=1$. By Lemma 1, Step 1, and $v(R^{i}_{i}){\ }{>}{\ }p^{f}_{i}$, strategy-proofness implies that $x^{f}_{i}(R^{i})=1$, which contradicts that $x^{f}_{i}(R^{i})=0$.

Step 3. Repeating the same arguments as in Step 2 for the remaining agents in $N^{f}_{+}$ inductively, we obtain a sequence $({\iota }_{i})_{i=1}^{I}$ such that for each $i{\ }{\in }{\ }\{1,{\dots },I\}$, if $v_{{\iota }_{j}}(R_{{\iota }_{j}}){\ }{\le }{\ }p^{f}_{{\iota }_{j}}$ for each $j{\ }{<}{\ }i$ and $v_{{\iota }_{i}}(R_{{\iota }_{i}}){\ }{>}{\ }p^{f}_{{\iota }_{i}}$, then $x^{f}_{{\iota }_{i}}(R)=1$. Let ${\succ }^{f}$ be a strict order over $N^{f}_{+}$ such that

$$\begin{aligned} {\iota }_{1}{\ }{\succ }^{f}{\ }{\iota }_{2}{\ }{\succ }^{f}{\ }{\cdots }{\ }{\succ }^{f}{\ }{\iota }_{I-1}{\ }{\succ }^{f}{\ }{\iota }_{I}. \end{aligned}$$

Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $i{\ }{\in }{\ }N^{f}_{+}$ be such that $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$ and $v(R_{i}){\ }{>}{\ }p^{f}_{i}$. Then, there is $j{\ }{\in }{\ }\{1,{\dots },I\}$ such that $i={\iota }_{j}$. By $v(R_{k}){\ }{\le }{\ }p^{f}_{k}$ for each $k{\ }{\succ }^{f}{\ }i$, $v_{{\iota }_{k}}(R_{{\iota }_{k}}){\ }{\le }{\ }p^{f}_{{\iota }_{k}}$ for each $k{\ }{<}{\ }j$. Thus, by $v(R_{{\iota }_{j}})=v(R_{i}){\ }{>}{\ }p^{f}_{i}=p^{f}_{{\iota }_{j}}$, $x^{f}_{i}(R)=x^{f}_{{\iota }_{j}}(R)=1$.

We are now in a position to complete the proof. By Part 2, f satisfies the first condition of a priority rule. By Part 1, it satisfies the third condition, and by Lemma 1, it also satisfies the fourth one. Therefore, it remains to show that f satisfies the second condition.

Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $i{\ }{\in }{\ }N^{f}_{+}$. We show that if $x^{f}_{i}(R)=1$, then $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$ and $v(R_{i}){\ }{\ge }{\ }p^{f}_{i}$.

Suppose that $x^{f}_{i}(R)=1$. First, we show $v(R_{j}){\ }{\le }{\ }p^{f}_{j}$ for each $j{\ }{\succ }^{f}{\ }i$. Suppose by contradiction that there is $j{\ }{\succ }^{f}{\ }i$ such that $v(R_{j}){\ }{>}{\ }p^{f}_{j}$. Without loss of generality, let j be the highest priority agent among $\{k{\ }{\in }{\ }N^{f}_{+}:k{\ }{\succ }^{f}{\ }i{\ }\mathrm{{and}}{\ }v(R_{k}){\ }{>}{\ }p^{f}_{k}\}$ according to ${\succ }^{f}$. Then, for each $k{\ }{\succ }^{f}{\ }j$, $v(R_{k}){\ }{\le }{\ }p^{f}_{k}$. Thus, by $v(R_{j}){\ }{>}{\ }p^{f}_{j}$, Part 2 implies that $x^{f}_{j}(R)=1$, which contradicts $x^{f}_{i}(R)=1$ and $j{\ }{\ne }{\ }i$.

Next, we show $v(R_{i}){\ }{\ge }{\ }p^{f}_{i}$. If there is $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $x_{i}(R'_{i},R_{-i})=0$, then by Lemma 1, Step 1, and $x^{f}_{i}(R)=1$, strategy-proofness implies that $v(R_{i}){\ }{\ge }{\ }p^{f}_{i}$. Suppose that for each $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$, $x_{i}(R'_{i},R_{-i})=1$. By richness, there is $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(t_{i};R'_{i})=0$ for each $t_{i}{\ }{\in }{\ }{\mathbb {R}}$. Then, by Step 1 and non-imposition, $p^{f}_{i}=t_{i}(R'_{i},R_{-i})=0$. Thus, $v(R_{i}){\ }{\ge }{\ }0=p^{f}_{i}$.       $\blacksquare$

Proof of Theorem 3

In this section, we prove Theorem 3. Since l-weak group strategy-proofness with reallocation implies effective l-weak group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$, (ii) implies (i). Since effective l-weak group strategy-proofness with reallocation implies strategy-proofness for a given $l{\ }{\ge }{\ }2$, it follows that (i) implies (iii) by the implication of (i) to (iv) in Theorem 1.²² Thus, it suffices to show (iii) implies (ii).

Suppose that f is a variable threshold-price rule on ${\mathcal {R}}^{n}$. We show that it satisfies l-weak group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'$ be a coalition with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ be such that $\sum _{i{\in }N'}x_{i}{\ }{\le }{\ }\sum _{i{\in }N'} x^{f}_{i}(R'_{N'},R_{-N'})$.

First, suppose that $|N'|=1$. Let $i{\ }{\in }{\ }N'$. If $x_{i}=0$, then by individual rationality and no subsidy of f (Lemma 3), $f_{i}(R){\ }R_{i}{\ }(0,0){\ }R_{i}{\ }(x_{i},t^{f}_{i}(R'_{i},R_{-i}))$. If $x_{i}=1$, then $x^{f}_{i}(R'_{N'},R_{-N'})=1$, and so by strategy-proofness of f (Theorem 1), $f_{i}(R){\ }R_{i}{\ }f_{i}(R'_{i},R_{-i})=(x_{i},t^{f}_{i}(R'_{i},R_{-i}))$. Thus, in either case, $f_{i}(R){\ }R_{i}{\ }(x_{i},t^{f}_{i}(R'_{i},R_{-i}))$, and so $N'$ does not strongly benefit from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R.

Next, suppose that $|N'|{\ }{\ge }{\ }2$. Let $i{\ }{\in }{\ }N'$ be such that $(x_{i},t^{f}_{i}(R'_{i},R_{-i})){\ }P_{i}{\ }f_{i}(R)$. We claim that $x_{i}=1$. Suppose by contradiction that $x_{i}=0$. Then, by individual rationality and no subsidy of f (Lemma 3), $f_{i}(R){\ }R_{i}{\ }(0,0){\ }R_{i}{\ }(x_{i},t^{f}_{i}(R'_{N'},R_{-N'}))$, which contradicts $(x_{i},t^{f}_{i}(R'_{i},R_{-i})){\ }P_{i}{\ }f_{i}(R)$. Thus, $x_{i}=1$. By $|N'|{\ }{\ge }{\ }2$, there is $j{\ }{\in }{\ }N'{\setminus }\{i\}$. By $x_{i}=1$, $x_{j}=0$. Thus, by individual rationality and no subsidy of f (Lemma 3), $f_{j}(R){\ }R_{j}{\ }(0,0){\ }R_{j}{\ }(x_{j},t^{f}_{j}(R'_{N'},R_{-N'}))$. Thus, $N'$ does not strongly benefit from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R.       $\blacksquare$

Proof of Theorem 4

In this section, we prove Theorem 4. Since l-strong group strategy-proofness with reallocation implies effective l-strong group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$, (ii) implies (i). Thus, it suffices to show (iii) implies (ii) and (i) implies (iii).

1.1 (iii) implies (ii)

In this subsection, we show that (iii) implies (ii). Suppose that f is a priority rule with positive prices. We show that f satisfies l-strong group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'$ be a coalition with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ be such that $\sum _{i{\in }N'}x_{i}{\ }{\le }{\ }\sum _{i{\in }N'} x^{f}_{i}(R'_{N'},R_{-N'})$.

If $|N'|=1$, then in the same way as in the proof of Theorem 3, we can show that for $i{\ }{\in }{\ }N'$, $f_{i}(R){\ }R_{i}{\ }(x_{i},t^{f}_{i}(R'_{i},R_{-i}))$. Thus, $N'$ does not weakly benefit from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R.

Suppose that $|N'|{\ }{\ge }{\ }2$. We show that $N'$ does not weakly benefit from misrepresenting $R'_{N'}$ and reallocating $(x_{i})_{i{\in }N'}$ at R. By contradiction, suppose that for each $i{\ }{\in }{\ }N'$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'})){\ }R_{i}{\ }f_{i}(R)$, and there is $j{\ }{\in }{\ }N'$ such that $(x_{j},t^{f}_{j}(R'_{N'},R_{-N'})){\ }P_{j}{\ }f_{j}(R)$. We claim that $x_{j}=1$. If $x_{j}=0$, then by individual rationality and no subsidy of f (Lemma 4), $f_{j}(R){\ }R_{j}{\ }(0,0){\ }R_{j}{\ }(x_{j},t^{f}_{j}(R'_{N'},R_{-N'}))$, a contradiction. Thus, $x_{j}=1$. Thus, there is $i{\ }{\in }{\ }N'$ such that $x^{f}_{i}(R'_{N'},R_{-N'})=1$. Note that we have either $i=j$ or $i{\ }{\ne }{\ }j$. Thus, there are two cases.

First, suppose that $i=j$. Then, $f_{j}(R'_{N'},R_{-N'})=(x_{j},t^{f}_{j}(R'_{N'},R_{-N'})){\ }P_{j}{\ }f_{j}(R)$. Thus, we must have $x^{f}_{j}(R)=0$, and so $f_{j}(R)=(0,0)$. By $(1,p^{f}_{j})=f_{j}(R'_{N'},R_{-N'}){\ }P_{j}{\ }f_{j}(R)=(0,0)$, $v(R_{j}){\ }{>}{\ }p^{f}_{j}$. Thus, by $x^{f}_{j}(R)=0$, there is $k{\ }{\in }{\ }N^{f}_{+}{\setminus }\{j\}$ such that $x^{f}_{k}(R)=1$, $k{\ }{\succ }^{f}{\ }j$, and $v(R_{k}){\ }{>}{\ }p^{f}_{k}$. We show that $k{\ }{\not \in }{\ }N'$. Suppose by contradiction that $k{\ }{\in }{\ }N'$. Then, by $x^{f}_{j}(R'_{N'},R_{-N'})=1$, $f_{k}(R'_{N'},R_{-N'})=(0,0)$. By $x_{j}=1$, $x_{k}=0$. Thus, by $v(R_{k}){\ }{>}{\ }p^{f}_{k}$, $f_{k}(R){\ }P_{k}{\ }(0,0)=(x_{k},t^{f}_{k}(R'_{N'},R_{-N'}))$, which contradicts $(x_{k},t^{f}_{k}(R'_{N'},R_{-N'})){\ }R_{k}{\ }f_{k}(R)$. Thus, we must have $k{\ }{\not \in }{\ }N'$. However, by $k{\ }{\succ }^{f}{\ }j$ and $v(R_{k}){\ }{>}{\ }p^{f}_{k}$, we have $x^{f}_{j}(R'_{N'},R_{-N'})=0$, which contradicts $x^{f}_{j}(R'_{N'},R_{-N'})=1$.

Next, suppose that $i{\ }{\ne }{\ }j$. By $x_{j}=1$, $x_{i}=0$. Thus, by individual rationality of f (Lemma 4), $p^{f}_{i}{\ }{>}{\ }0$, and $x^{f}_{i}(R'_{N'},R_{-N'})=1$, we have $f_{i}(R){\ }R_{i}{\ }(0,0){\ }P_{i}{\ }(0,p^{f}_{i})=(x_{i},t^{f}_{i}(R'_{N'},R_{-N'}))$. However, this contradicts $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'})){\ }R_{i}{\ }f_{i}(R)$.       $\blacksquare$

1.2 (i) implies (iii)

In this subsection, we show that (i) implies (iii). Suppose that f satisfies effective l-strong group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$. Since effective l-strong group strategy-proofness with reallocation implies effective l- strong group strategy-proofness, by Theorem 2, f is a priority rule. We show that for each $i{\ }{\in }{\ }N^{f}_{+}$, $p^{f}_{i}{\ }{>}{\ }0$. By contradiction, suppose that there is $i{\ }{\in }{\ }N^{f}_{+}$ such that $p^{f}_{i}=0$. There are three cases.

Case 1. There is $j{\ }{\in }{\ }N^{f}_{+}{\setminus }\{i\}$ such that $p^{f}_{j}{\ }{>}{\ }0$.

By richness and $p^{f}_{j}{\ }{>}{\ }0$, we can choose $R_{j}{\ }{\in }{\ }{\mathcal {R}}$ such that $0{\ }{<}{\ }v(R_{j}){\ }{<}{\ }p^{f}_{j}$. Also, by richness, we can choose $R_{-j}{\ }{\in }{\ }{\mathcal {R}}^{n-1}$ such that for each $k{\ }{\in }{\ }N{\setminus }\{j\}$, $v(R_{k})=0$. Then, $t^{f}_{i}(R)=0$ and $f_{j}(R)=(0,0)$. Let $N'=\{i,j\}$. By richness, we can choose $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R'_{i}){\ }{>}{\ }p^{f}_{i}$. Let $R'_{j}=R_{j}$. Then, $f_{i}(R'_{N'},R_{-N'})=(1,p^{f}_{i})=(1,0)$ and $f_{j}(R'_{N'},R_{-N'})=(0,0)$. Let $(x_{k})_{k{\in }N'}{\ }{\in }{\ }M^{|N'|}$ be such that $x_{i}=0$ and $x_{j}=1$. Note that $\sum _{k{\in }N'} x_{k}=1=\sum _{k{\in }N'} x^{f}_{k}(R'_{N'},R_{-N'})$. By $v(R_{i})=0$, $(x_{i},t^{f}_{i}(R'_{i},R_{-i}))=(0,0){\ }I_{i}{\ }f_{i}(R)$. By $v(R_{j}){\ }{>}{\ }0$, $(x_{j},t^{f}_{j}(R'_{N'},R_{-N'}))=(1,0){\ }P_{j}{\ }(0,0)=f_{j}(R)$. Thus, $N'$ weakly benefits from misrepresenting preferences $R'_{N'}$ and reallocating the object $(x_{k})_{k{\in }N'}$ at R.

For each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, by $p^{f}_{i}=0$ and $v(R_{i})=0$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'}))=(0,0){\ }I_{i}{\ }f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'})$. For each $R''_{j}{\ }{\in }{\ }{\mathcal {R}}$, by $p^{f}_{j}{\ }{>}{\ }0$, $(x_{j},t^{f}_{j}(R'_{N'},R_{-N'}))=(1,0){\ }R_{j}{\ }f_{j}(R''_{j},R'_{N'{\setminus }\{j\}},R_{-N'})$. Thus, a pair $R'_{N'}$ and $(x_{k})_{k{\in }N'}$ is self-enforcing at R. Thus, f violates effective l-strong group strategy-proofness with reallocation, a contradiction.

Case 2. For each $j{\ }{\in }{\ }N^{f}_{+}$, $p^{f}_{j}=0$, and $N^{f}_{+}{\ }{\subsetneq }{\ }N$.

By $N^{f}_{+}{\ }{\subsetneq }{\ }N$, there is $j{\ }{\in }{\ }N{\setminus }N^{f}_{+}$. By richness, we can choose $R_{j}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R_{j}){\ }{>}{\ }0$. By richness, we can choose $R_{-j}{\ }{\in }{\ }{\mathcal {R}}^{n-1}$ such that for each $k{\ }{\in }{\ }N{\setminus }\{j\}$, $v(R_{k})=0$. Then, $t^{f}_{i}(R)=0$, and by $j{\ }{\not \in }{\ }N^{f}_{+}$, $f_{j}(R)=(0,0)$. Let $N'=\{i,j\}$. By richness, we can choose $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R'_{i}){\ }{>}{\ }p^{f}_{i}$. Let $R'_{j}=R_{j}$. Then, $f_{i}(R'_{N'},R_{-N'})=(1,p^{f}_{i})=(1,0)$ and $f_{j}(R'_{N'},R_{-N'})=(0,0)$. Let $(x_{k})_{k{\in }N'}{\ }{\in }{\ }M^{|N'|}$ be such that $x_{i}=0$ and $x_{j}=1$. By the same argument as in Case 1, we can show that $N'$ weakly benefits from misrepresenting $R'_{N'}$ and reallocating $(x_{k})_{k{\in }N'}$ at R. Also, by the same argument as in Case 1, we can show that for each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'})){\ }I_{i}{\ }f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'})$. For each $R''_{j}{\ }{\in }{\ }{\mathcal {R}}$, by $p^{f}_{j}=0$, $(x_{j},t^{f}_{j}(R'_{N'},R_{-N'}))=(1,0){\ }R_{j}{\ }f_{j}(R''_{j},R'_{N'{\setminus }\{j\}},R_{-N'})$. Thus, $R'_{N'}$ and $(x_{k})_{k{\in }N'}$ are self-enforcing at R. Thus, f violates effective l-strong group strategy-proofness with reallocation, a contradiction.

Case 3. For each $j{\ }{\in }{\ }N^{f}_{+}$, $p^{f}_{j}=0$, and $N^{f}_{+}=N$.

By $n{\ }{\ge }{\ }3$ and $N^{f}_{+}=N$, we can choose a distinct pair $j,j'{\ }{\in }{\ }N^{f}_{+}{\setminus }\{i\}$. Note that the discussions in Cases 1 and 2 do not depend on the number of agents. Without loss of generality, let $i{\ }{\succ }^{f}{\ }j{\ }{\succ }^{f}{\ }j'$. By richness, we can choose a pair $R_{j},R_{j'}{\ }{\in }{\ }{\mathcal {R}}^{2}$ such that $v(R_{j}){\ }{>}{\ }0$ and $v(R_{j'}){\ }{>}{\ }0$. Also, by richness, we can choose $R_{-j,j'}{\ }{\in }{\ }{\mathcal {R}}^{n-2}$ such that for each $k{\ }{\in }{\ }N{\setminus }\{j,j'\}$, $v_{k}(R_{k})=0$. Then, by $j{\ }{\succ }^{f}{\ }j'$, $f_{i}(R)=f_{j'}(R)=(0,0)$ and $f_{j}(R)=(1,0)$. Let $N'=\{i,j'\}$. By richness, we can choose $R'_{i}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R'_{i}){\ }{>}{\ }0$. Let $R'_{j'}=R_{j'}$. Then, by $i{\ }{\succ }^{f}{\ }j{\ }{\succ }^{f}{\ }j'$, $f_{i}(R'_{N'},R_{-N'})=(1,0)$ and $f_{j'}(R'_{N'},R_{-N'})=(0,0)$. Let $(x_{k})_{k{\in }N'}{\ }{\in }{\ }M^{|N'|}$ be such that $x_{i}=0$ and $x_{j'}=1$. By the same argument as in Case 1, we can show that $N'$ weakly benefits from misrepresenting $R'_{N'}$ and reallocating $(x_{k})_{k{\in }N'}$ at R. We can show in the same way as in Case 1 that for each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, $(x_{i},t^{f}_{i}(R'_{N'},R_{-N'})){\ }I_{i}{\ }f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'})$. Also, in the same way as in Case 2, we can show that for each $R''_{j'}{\ }{\in }{\ }{\mathcal {R}}$, $(x_{j'},t^{f}_{j'}(R'_{N'},R_{-N'})){\ }R_{j'}{\ }f_{j'}(R''_{j'},R'_{N'{\setminus }\{i\}},R_{-N'})$. Thus, $R'_{N'}$ and $(x_{k})_{k{\in }N'}$ are self-enforcing at R. Thus, f violates effective l-strong group strategy-proofness with reallocation, a contradiction.       $\blacksquare$

Proof of Proposition 1

In this section, we prove Proposition 1. As in the proof of Theorem 4, it suffices to show that (iii) implies (ii) and (i) implies (iii).

1.1 (iii) implies (ii)

In this subsection, we show that (iii) implies (ii). Suppose that f is either a priority rule with positive prices or a priority rule with full participation and zero prices. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'$ be a coalition with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(x_{i})_{i{\in }N'}{\ }{\in }{\ }M^{|N'|}$ with $\sum _{i{\in }N'} x_{i}{\ }{\le }{\ }\sum _{i{\in }N'} x^{f}_{i}(R'_{N'},R_{-N'})$. Note that the proof that (iii) implies (ii) in Theorem 4 does not depend on the number of agents, and so is valid here as well. Thus, if f is a priority rule with positive prices, then $N'$ does not weakly benefit from misrepresenting $R'_{N'}$ and reallocating $(x_{k})_{k{\in }N'}$ at R.

Suppose that f is a priority rule with full participation and zero prices, i.e., $N^{f}_{+}=N$ and $p^{f}_{i}=0$ for each $i{\ }{\in }{\ }N$. If $|N'|=1$, then by strategy-proofness, individual rationality, and no subsidy of f (Lemma 4), $f_{i}(R){\ }R_{i}{\ }(x_{i},t^{f}_{i}(R'_{i},R_{-i}))$. Thus, $N'$ does not weakly benefit from misrepresenting $R'_{N'}$ and reallocating $(x_{k})_{k{\in }N'}$ at R.

Suppose that $|N'|=2$. By $n=2$ and $N^{f}_{+}=N$, $N'=N=N^{f}_{+}$. Suppose that $(x_{i},t^{f}_{i}(R')){\ }R_{i}{\ }f_{i}(R)$ for each $i{\ }{\in }{\ }N'$. We show that $(x_{i},t^{f}_{i}(R')){\ }I_{i}{\ }f_{i}(R)$ for each $i{\ }{\in }{\ }N'$.

First, suppose that $v(R_{i})=0$ for each $i{\ }{\in }{\ }N'$. Then, for each $i{\ }{\in }{\ }N'$, by $p^{f}_{i}=0$, $(x_{i},t^{f}_{i}(R')){\ }I_{i}{\ }(0,0){\ }I_{i}{\ }f_{i}(R)$.

Next, suppose that $v(R_{i}){\ }{>}{\ }0$ for some $i{\ }{\in }{\ }N'$. By $p^{f}_{i}=0$ for each $i{\ }{\in }{\ }N'$, there is $i{\ }{\in }{\ }N'$ such that $v(R_{i}){\ }{>}{\ }0$ and $f_{i}(R)=(1,0)$. By $(x_{i},t^{f}_{i}(R')){\ }R_{i}{\ }f_{i}(R)=(1,0)$, $v(R_{i}){\ }{>}{\ }0$, and $p^{f}_{i}=0$, we must have $x_{i}=1$. Thus, $(x_{i},t^{f}_{i}(R'))=(1,0)=f_{i}(R)$. Let $j{\ }{\in }{\ }N'{\setminus }\{i\}$. By $x^{f}_{i}(R)=1$, $x^{f}_{j}(R)=0$. By $x_{i}=1$, $x_{j}=0$. Thus, $(x_{j},t^{f}_{j}(R))=(0,0)=f_{j}(R)$.       $\blacksquare$

1.2 (i) implies (iii)

In this subsection, we show that (i) implies (iii). Suppose that f satisfies effective l-strong group strategy-proofness with reallocation for a given $l{\ }{\ge }{\ }2$. Since effective l-strong group strategy-proofness with reallocation implies effective l- strong group strategy-proofness, by Theorem 2, f is a priority rule. We show that either for each $i{\ }{\in }{\ }N^{f}_{+}$, $p^{f}_{i}{\ }{>}{\ }0$, or $N^{f}_{+}=N$ and for each $i{\ }{\in }{\ }N^{f}_{+}$, $p^{f}_{i}=0$. By contradiction, suppose that for some $i{\ }{\in }{\ }N^{f}_{+}$, $p^{f}_{i}=0$, and we have either $N^{f}_{+}{\ }{\subsetneq }{\ }N$ or there is $j{\ }{\in }{\ }N^{f}_{+}$ such that $p^{f}_{j}{\ }{>}{\ }0$. If $N^{f}_{+}{\ }{\subsetneq }{\ }N$, then we can derive a contradiction to effective l-strong group strategy-proofness with reallocation in the same way as in Case 2 of the proof that (i) implies (iii) in Theorem 4. Instead, if there is $j{\ }{\in }{\ }N^{f}_{+}$ such that $p^{f}_{j}{\ }{>}{\ }0$, then we can derive a contradiction to effective l-strong group strategy-proofness with reallocation in the same way as in Case 1 of the proof that (i) implies (iii) in Theorem 4.       $\blacksquare$

Proof of Theorem 5

In this section, we prove Theorem 5. It is straightforward to show that (ii) implies (i), i.e., both a dictatorial rule and the no-trade rule satisfy l-group strategy-proofness with side payments for a given $l{\ }{\ge }{\ }2$. Thus, we here show that (i) implies (ii).

Suppose that f satisfies l-group strategy-proofness with side payments. Since l-group strategy-proofness with side payments implies l-strong group strategy-proofness, f satisfies l-strong group strategy-proofness. Thus, by Theorem 2, f is a priority rule. By contradiction, suppose that f is neither a dictatorial rule nor the no-trade rule. Then, $|N^{f}_{+}|{\ }{\ge }{\ }2$. Thus, we can choose a distinct pair $i,j{\ }{\in }{\ }N^{f}_{+}$. Without loss of generality, suppose that $i{\ }{\succ }^{f}{\ }j$. By richness, we can choose $R_{i,j}{\ }{\in }{\ }{\mathcal {R}}^{2}$ such that $v(R_{i}){\ }{>}{\ }p^{f}_{i}$ and $v(R_{j}){\ }{>}{\ }p^{f}_{j}-V(0,(1,p^{f}_{i});R_{i})$. Note that by finiteness, $V(0,(1,p^{f}_{i});R_{i}){\ }{\in }{\ }{\mathbb {R}}$, and so we can choose such $R_{j}$. By richness, we can also choose $R_{-i,j}{\ }{\in }{\ }{\mathcal {R}}^{n-2}$ such that for each $k{\ }{\in }{\ }N{\setminus }\{i,j\}$, $v(R_{k})=0$. By $i{\ }{\succ }^{f}{\ }{j}$ and $v(R_{i}){\ }{>}{\ }p^{f}_{i}$, $f_{i}(R)=(1,p^{f}_{i})$ and $f_{j}(R)=(0,0)$.

Let $N'=\{i,j\}$. By richness, we can choose $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ such that $v(R'_{i})=0$ and $v(R'_{j}){\ }{>}{\ }p^{f}_{j}$. Then, $f_{i}(R'_{N'},R_{-N'})=(0,0)$ and $f_{j}(R'_{N'},R_{-N'})=(1,p^{f}_{j})$. By $v(R_{j}){\ }{>}{\ }p^{f}_{j}-V(0,(1,p^{f}_{i});R_{i})$, we can choose ${\tau }{\ }{\in }{\ }{\mathbb {R}}$ such that $v(R_{j})-p^{f}_{j}{\ }{>}{\ }{\tau }{\ }{>}{\ }-V(0,(1,p^{f}_{i});R_{i})$. Let $(t_{k})_{k{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ be such that $t_{i}=-{\tau }$ and $t_{j}={\tau }$. Note that $\sum _{k{\in }N'} t_{k}=0$.

By ${\tau }{\ }{>}{\ }-V(0,(1,p^{f}_{i});R_{i})$ and $f_{i}(R)=(1,p^{f}_{i})$, $t_{i}=-{\tau }{\ }{<}{\ }V(0,f_{i}(R);R_{i})$. Thus,

$$\begin{aligned} f_{i}(R'_{N'},R_{-N'})+(0,t_{i})=(0,-{\tau }){\ }P_{i}{\ }(0,V(0,f_{i}(R);R_{i})){\ }I_{i}{\ }f_{i}(R). \end{aligned}$$

By $t_{j}={\tau }{\ }{<}{\ }v(R_{j})-p^{f}_{j}$ and $t^{f}_{j}(R'_{N'},R_{-N'})=p^{f}_{j}$, $t^{f}_{j}(R'_{N'},R_{-N'})+t_{j}{\ }{<}{\ }v(R_{j})$. Thus,

$$\begin{aligned} f_{j}(R'_{N'},R_{-N'})+(0,t_{j})=(1,t^{f}_{j}(R'_{N'},R_{-N'})+t_{j}){\ }P_{j}{\ }(1,v(R_{j})){\ }I_{j}{\ }(0,0)=f_{j}(R). \end{aligned}$$

Thus, $N'$ benefits from misrepresenting $R'_{N'}$ and arranging $(t_{k})_{k{\in }N'}$ at R, which contradicts l-group strategy-proofness with side payments.       $\blacksquare$

Proof of Theorem 6

In this section, we prove Theorem 6.

First, we show that (i) implies (ii). Suppose that f satisfies effective l-group strategy-proofness with side payments. Since effective l-group strategy-proofness with side payments implies effective l-group strategy-proofness, f satisfies effective l-group strategy-proofness. Thus, by Theorem 2, f is a priority rule.

Then, we show that (ii) implies (i). Suppose that f is a priority rule. We show that f satisfies effective l-group strategy-proofness with side payments. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'$ be a coalition with $|N'|{\ }{\le }{\ }l$, $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$, and $(t_{i})_{i{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ be such that $N'$ benefits from misrepresenting $R'_{N'}$ and arranging $(t_{i})_{i{\in }N'}$ at R. Note that since f satisfies strategy-proofness (Theorem 2), $|N'|{\ }{\ge }{\ }2$. We show that $R'_{N'}$ and $(t_{i})_{i{\in }N'}$ are not self-enforcing at R.

We show that there is $i{\ }{\in }{\ }N'$ such that $x^{f}_{i}(R'_{N'},R_{-N'})=1$ and $x^{f}_{i}(R)=0$. By contradiction, suppose that for each $i{\ }{\in }{\ }N'$, if $x^{f}_{i}(R'_{N'},R_{-N'})=1$, then $x^{f}_{i}(R)=1$. Let $i{\ }{\in }{\ }N'$. Since $N'$ benefits from misrepresenting $R'_{N'}$ and arranging $(t_{i})_{i{\in }N'}$ at R, $f_{i}(R'_{N'},R_{-N'})+(0,t_{i}){\ }P_{i}{\ }f_{i}(R)$.

First, suppose $x^{f}_{i}(R'_{N'},R_{-N'})=1$. Then, by the assumption, $x^{f}_{i}(R)=1$, and so $f_{i}(R'_{N'},R_{-N'})=f_{i}(R)=(1,p^{f}_{i})$. Thus, by $f_{i}(R'_{N'},R_{-N'})+(0,t_{i}){\ }P_{i}{\ }f_{i}(R)$, $t_{i}{\ }{<}{\ }0$.

Next, suppose that $x^{f}_{i}(R'_{N'},R_{-N'})=0$. Then, $f_{i}(R'_{N'},R_{-N'})=(0,0)$. Thus, by $(0,t_{i})=f_{i}(R'_{N'},R_{-N'})+(0,t_{i}){\ }P_{i}{\ }f_{i}(R)$, $t_{i}{\ }{<}{\ }V(0,f_{i}(R);R_{i}){\ }{\le }{\ }0$, where the second inequality follows from $f_{i}(R){\ }R_{i}{\ }(0,0)$ (Lemma 4).

Thus, for each $i{\ }{\in }{\ }N'$, $t_{i}{\ }{<}{\ }0$, which implies $\sum _{i{\in }N'} t_{i}{\ }{<}{\ }0$. However, since $N'$ benefits from misrepresenting $R'_{N'}$ and arranging $(t_{i})_{i{\in }N'}$ at R, we must have $\sum _{i{\in }N'} t_{i}{\ }{\ge }{\ }0$, a contradiction.

Thus, there is $i{\ }{\in }{\ }N'$ such that $x^{f}_{i}(R'_{N'},R_{-N'})=1$ and $x^{f}_{i}(R)=0$. Next, we show that for each $j{\ }{\in }{\ }N'{\setminus }\{i\}$, $t_{j}{\ }{<}{\ }0$. Let $j{\ }{\in }{\ }N'{\setminus }\{i\}$. By $x^{f}_{i}(R'_{N'},R_{-N'})=1$, $x^{f}_{j}(R'_{N'},R_{-N'})=0$. Thus, $f_{j}(R'_{N'},R_{-N'})=(0,0)$. By $(0,t_{j})=f_{j}(R'_{N'},R_{-N'})+(0,t_{j}){\ }P_{j}{\ }f_{j}(R)$, $t_{j}{\ }{<}{\ }V(0,f_{j}(R);R_{j}){\ }{\le }{\ }0$, where the second inequality follows from $f_{j}(R){\ }R_{j}{\ }(0,0)$ (Lemma 4).

Thus, for each $j{\ }{\in }{\ }N'{\setminus }\{i\}$, $t_{j}{\ }{<}{\ }0$. Thus, by $|N'|{\ }{\ge }{\ }2$, $\sum _{j{\in }N'{\setminus }\{i\}} t_{j}{\ }{<}{\ }0$. Thus, by $\sum _{j{\in }N'}t_{j}=t_{i}+\sum _{j{\in }N'{\setminus }\{i\}} t_{j}{\ }{\ge }{\ }0$, $t_{i}{\ }{>}{\ }0$. By $t_{i}{\ }{>}{\ }0$, $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R'_{N'},R_{-N'})$$+(0,t_{i})$. Thus, $R'_{N'}$ and $(t_{i})_{i{\in }N'}$ are not self-enforcing at R, and so f satisfies effective l-group strategy-proofness with side payments.       $\blacksquare$

Proof of Theorem 7

In this section, we prove Theorem 7. Since l-group strategy-proofness with reallocation and side payments implies effective l-group strategy-proofness with reallocation and side payments for a given $l{\ }{\ge }{\ }2$, (ii) implies (i). Further, it is clear that the no-trade rule satisfies l-group strategy-proofness with reallocation and side payments, i.e., (iii) implies (ii). Thus, it suffices to show that (i) implies (iii).

Suppose that f satisfies effective l-group strategy-proofness with reallocation and side payments. Then, it satisfies effective l-strong group strategy-proofness. Thus, by Theorem 2, it is a priority rule.

We show that $N^{f}_{+}={\varnothing }$. By contradiction, suppose that $N^{f}_{+}{\ }{\ne }{\ }{\varnothing }$. Then, there is $i{\ }{\in }{\ }N^{f}_{+}$. Without loss of generality, suppose that i has the highest priority according to ${\succ }^{f}$, i.e., for each $j{\ }{\in }{\ }N^{f}_{+}{\setminus }\{i\}$, $i{\ }{\succ }^{f}{\ }j$. Let $j{\ }{\in }{\ }N{\setminus }\{i\}$ be an agent such that if $N^{f}_{+}{\ }{\ne }{\ }\{i\}$, then $j{\ }{\in }{\ }N^{f}_{+}$, and otherwise (i.e., $N^{f}_{+}=\{i\}$), $j{\ }{\in }{\ }N{\setminus }N^{f}_{+}$. By richness, we can choose $R_{i,j}{\ }{\in }{\ }{\mathcal {R}}$ such that $v(R_{i}){\ }{>}{\ }p^{f}_{i}$ and $v(R_{j}){\ }{>}{\ }p^{f}_{i}-V(0,(1,p^{f}_{i});R_{i})$. Note that by finiteness, $V(0,(1,p^{f}_{i});R_{i}){\ }{\in }{\ }{\mathbb {R}}$, and so we can choose such $R_{j}$. Also, by richness, we can choose $R_{-i,j}{\ }{\in }{\ }{\mathcal {R}}^{n-2}$ such that for each $k{\ }{\in }{\ }N{\setminus }\{i,j\}$, $v(R_{k})=0$. Then, $f_{i}(R)=(1,p^{f}_{i})$ and $f_{j}(R)=(0,0)$.

Let $N'=\{i,j\}$. Let $R'_{N'}=R_{N'}$. Clearly, $f_{i}(R'_{N'},R_{-N'})=f_{i}(R)=(1,p^{f}_{i})$ and $f_{j}(R'_{N'},R_{-N'})=f_{j}(R)=(0,0)$. Let $(x_{k})_{k{\in }N'}{\ }{\in }{\ }M^{|N'|}$ be such that $x_{i}=0$ and $x_{j}=1$. Note that $\sum _{k{\in }N'} x_{k}=\sum _{k{\in }N'} x^{f}_{k}(R'_{N'},R_{-N'})$. By $v(R_{j}){\ }{>}{\ }p^{f}_{i}-V(0,(1,p^{f}_{i});R_{i})$, we can choose ${\tau }{\ }{\in }{\ }{\mathbb {R}}$ such that $p^{f}_{i}-V(0,(1,p^{f}_{i});R_{i}){\ }{<}{\ }{\tau }{\ }{<}{\ }v(R_{j})$. Let $(t_{k})_{k{\in }N'}{\ }{\in }{\ }{\mathbb {R}}^{|N'|}$ be such that $t_{i}=-{\tau }$ and $t_{j}={\tau }$. Note that $\sum _{k{\in }N'} t_{k}=0$.

By ${\tau }{\ }{>}{\ }p^{f}_{i}-V(0,(1,p^{f}_{i});R_{i})$ and $f_{i}(R)=f_{i}(R'_{N'},R_{-N'})=(1,p^{f}_{i})$,

$$\begin{aligned} (x_{i},t^{f}_{i}(R'_{N'},R_{N'}))+(0,t_{i})=(0,-{\tau }+p^{f}_{i}){\ }P_{i}{\ }(0,V(0,f_{i}(R);R_{i})){\ }I_{i}{\ }f_{i}(R). \end{aligned}$$

(1)

Also, by ${\tau }{\ }{<}{\ }v(R_{j})$ and $f_{j}(R)=f_{j}(R'_{N'},R_{-N'})=(0,0)$,

$$\begin{aligned} (x_{j},t^{f}_{j}(R'_{N'},R_{-N'}))+(0,t_{j})=(1,{\tau }){\ }P_{j}{\ }(1,v(R_{j})){\ }I_{j}{\ }(0,0)=f_{j}(R). \end{aligned}$$

(2)

Thus, $N'$ benefits from misrepresenting $R'_{N'}$, reallocating $(x_{k})_{k{\in }N'}$, and arranging $(t_{k})_{k{\in }N'}$ at R.

For each $R''_{i}{\ }{\in }{\ }{\mathcal {R}}$, by $f_{i}(R''_{i},R'_{N'{\setminus }\{i\}},R_{-N'}){\ }{\in }{\ }\{(1,p^{f}_{i}),(0,0)\}$, (1), and individual rationality of f (Lemma 4), we have $(x_{i},t^{f}_{i}(R'_{N'},R_{N'}))+(0,t_{i}){\ }P_{i}{\ }f_{i}(R''_{i},$$R'_{N'{\setminus }\{i\}},R_{-N'})$. Also, since $v(R'_{i})=v(R_{i}){\ }{>}{\ }p^{f}_{i}$ and we have either $j{\ }{\in }{\ }N^{f}_{+}$ and $i{\ }{\succ }^{f}{\ }j$, or $j{\ }{\in }{\ }N{\setminus }N^{f}_{+}$, for each $R''_{j}{\ }{\in }{\ }{\mathcal {R}}$, we have $f_{j}(R''_{j},R'_{N'{\setminus }\{j\}},R_{-N'})=(0,0)$. Thus, by (2), for each $R''_{j}{\ }{\in }{\ }{\mathcal {R}}$, we have $(x_{j},t^{f}_{j}(R'_{N'},R_{-N'}))+(0,t_{j}){\ }P_{j}{\ }(0,0)=f_{j}(R''_{j},R'_{N'{\setminus }\{j\}},R_{-N'})$. Thus, $R'_{N'}$, $(x_{k})_{k{\in }N'}$, and $(t_{k})_{k{\in }N'}$ are self-enforcing at R. Thus, f violates effective l-group strategy-proofness with reallocation and side payments, a contradiction.       $\blacksquare$

Proof of Proposition 3

In this section, we prove Proposition 3. Since (A, g) implements f in effective l-weak group dominant strategy equilibrium, there is an effective l-weak group dominant strategy equilibrium ${\sigma }$ in (A, g) such that for each $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $g({\sigma }(R))=f(R)$. We show that f satisfies effective l-weak group strategy-proofness. Let $R{\ }{\in }{\ }{\mathcal {R}}^{n}$, $N'$ be a coalition with $|N'|{\ }{\le }{\ }l$, and $R'_{N'}{\ }{\in }{\ }{\mathcal {R}}^{|N'|}$ be such that for each $i{\ }{\in }{\ }N'$, $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)$. We show that $R'_{N'}$ is not self-enforcing at R. For each $i{\ }{\in }{\ }N'$, by $g({\sigma }(R''))=f(R'')$ for each $R''{\ }{\in }{\ }{\mathcal {R}}^{n}$ and $f_{i}(R'_{N'},R_{-N'}){\ }P_{i}{\ }f_{i}(R)$, $g_{i}({\sigma }_{N'}(R'_{N'}),{\sigma }_{-N'}(R_{-N'})){\ }P_{i}{\ }g_{i}({\sigma }(R))$. Thus, since ${\sigma }$ is an effective l-weak group dominant strategy equilibrium in (A, g), there are $i{\ }{\in }{\ }N'$ and $a_{i}{\ }{\in }{\ }A_{i}$ such that $g_{i}(a_{i},{\sigma }_{N'{\setminus }\{i\}}(R'_{N'{\setminus }\{i\}}),{\sigma }_{-N'}(R_{-N'})){\ }P_{i}{\ }g_{i}({\sigma }_{N'}(R'_{N'}),{\sigma }_{-N'}(R_{-N'}))$. Since ${\sigma }$ is an effective l-weak group dominant strategy equilibrium in (A, g), it is a dominant strategy equilibrium in (A, g). Thus, we have

$$\begin{aligned} g_{i}({\sigma }_{i}(R_{i}),{\sigma }_{N'{\setminus }\{i\}}(R'_{N'{\setminus }\{i\}}),{\sigma }_{-N'}(R_{-N'})){\ }R_{i}{\ }g_{i}(a_{i},{\sigma }_{N'{\setminus }\{i\}}(R'_{N'{\setminus }\{i\}}),{\sigma }_{-N'}(R_{-N'})). \end{aligned}$$

Thus, by $g_{i}(a_{i},{\sigma }_{N'{\setminus }\{i\}}(R'_{N'{\setminus }\{i\}}),{\sigma }_{-N'}(R_{-N'})){\ }P_{i}{\ }g_{i}({\sigma }_{N'}(R'_{N'}),{\sigma }_{-N'}(R_{-N'}))$,

$$\begin{aligned} g_{i}({\sigma }_{i}(R_{i}),{\sigma }_{N'{\setminus }\{i\}}(R'_{N'{\setminus }\{i\}}),{\sigma }_{-N'}(R_{-N'})){\ }P_{i}{\ }g_{i}({\sigma }_{N'}(R'_{N'}),{\sigma }_{-N'}(R_{-N'})). \end{aligned}$$

Since $g({\sigma }(R'')){=}f(R'')$ for each $R''{\ }{\in }{\ }{\mathcal {R}}^{n}$, we have $f_{i}(R_{i},R'_{N'{\setminus }\{i\}},R_{-N'}){\ }P_{i}{\ }f_{i}(R'_{N'},$$R_{-N'})$. Thus, $R'_{N'}$ is not self-enforcing at R, and so f satisfies effective l-weak group strategy-proofness.       $\blacksquare$