Second-order productivity, second-order payoffs, and the Owen value

Abstract

We introduce the concepts of the components’ second-order productivities in cooperative games with transferable utility (TU games) with a coalition structure (CS games) and of the components’ second-order payoffs for one-point solutions for CS games as generalizations of the players’ second-order productivities in TU games and of the players’ second-order payoffs for one-point solutions for TU games (Casajus in Discrete Appl Math 304:212–219, 2021). The players’ second-order productivities are conceptualized as second-order marginal contributions, that is, how one player affects another player’s marginal contributions to coalitions containing neither of them by entering these coalitions. The players’ second-order payoffs are conceptualized as the effect of one player leaving the game on the payoff of another player. Analogously, the components’ second-order productivities are conceptualized as their second-order productivities in the game between components; the components’ second-order payoffs are conceptualized as their second-order payoffs in the game between components. We show that the Owen value is the unique efficient one-point solution for CS games that reflects the players’ and the components’ second-order productivities in terms of their second-order payoffs.

1 Introduction

A cooperative game with transferable utility for a finite player set (TU game or simply game) is given by a coalition function that assigns a worth to any coalition (subset of the player set), where the empty coalition obtains zero. (One-point) solutions for TU games assign a payoff to any player in any TU game. The Shapley value (Shapley, 1953) probably is the most eminent one-point solution concept for TU games. And its characterization by Young (1985) probably is its most important one.

Young (1985) characterizes the Shapley value by three properties of solutions: efficiency, symmetry, and marginality or strong monotonicity. Efficiency: the players’ payoffs sum up to the worth generated by the grand coalition. Symmetry: equally productive¹ players obtain the same payoff. Marginality: a player’s payoff only depends on her own productivity. Strong monotonicity: whenever a player’s productivity in a game weakly increases so does her payoff. Note that strong monotonicity implies marginality. This result indicates that the Shapley value is the efficient solution that reflects the players’ productivities by their payoffs.

The organization of players into groups can be modelled by coalition structures—partitions of the player set. Games enriched with a coalition structure are addressed as CS games and the corresponding solutions as CS solutions. Owen (1977) generalizes of the Shapley value into an efficient CS solution where the components of the coalition structure are treated like players.² Khmelnitskaya and Yanovskaya (2007) provide a characterization of the Owen value that breathes the spirit of Young ’s (1985) characterization of the Shapley value.³ This characterization uses four properties: efficiency, marginality, symmetry within components, and symmetry across components. Symmetry across components: components that are equally productive in the game between components obtain the same sum of payoffs of their members.

Recently, Casajus (2021) suggests a second-order version of Young ’s (1985) characterization of the Shapley value. This characterization is based on the notions of the players’ second-order productivities and second-order payoffs. A player’s second-order productivity with respect to another player reflects how the former affects the latter player’s marginal contribution to coalitions containing neither of them by entering these coalitions; a player’s second-order payoff with respect to another player reflects how the former affects the latter player’s payoff by leaving the game. The Shapley value is the unique efficient solution the reflects the players’ second-order productivities in terms of their second-order payoffs. More precisely, it is the unique solution that satisfies efficiency and second-order versions of symmetry and marginality. Second-order symmetry: players who are equally second-order productive with respect to a third player obtain the same second-order payoff with respect to this third player. Second-order marginality: a player’s second-order payoff with respect to another player only depends on her own second-order productivity with respect to this other player.

In this paper, we suggest a second-order version of Khmelnitskaya and Yanovskaya ’s (2007) characterization of the Owen value. In particular, we show that the Owen value is the unique CS solution that satisfies efficiency, second-order marginality, and second-order versions of symmetry within components and symmetry across components (Theorem 10). Second-order symmetry within components is just the restriction of second-order symmetry to players within the same component. Second-order symmetry across components: components that are equally second-order productive in the game between components obtain the same sum of second-order payoffs of their members. This result is partly based on three facts. Second-order marginality implies marginality (Proposition 6). Efficiency and second-order symmetry within components imply symmetry within components (Proposition 7 ). Efficiency and second-order symmetry across components imply symmetry across components (Proposition 9).

The remainder of this paper is organized as follows. In Section 2, we provide basic definitions and notation. In Section 3, we survey the characterizations of the Shapley value by Young (1985) and by Casajus (2021) . In Section 4, we first survey the characterization of the Owen value by Khmelnitskaya and Yanovskaya (2007) . Then, we provide our second-order approach to the Owen value. Some remarks conclude the paper.

2 Basic definitions and notation

Let the universe of players $\mathfrak {U}$ be a countably infinite set, and let $\mathcal {N}$ denote the set of all finite subsets of $\mathfrak {U}$. The cardinalities of $S,T,N\in \mathcal {N}$ are denoted by s,  t,  and n,  respectively. A (finite TU) game for the player set $N\in \mathcal {N}$ is given by a coalition function $v:2^{N}\rightarrow \mathbb {R}$, $v\left( \emptyset \right) =0,$ where $2^{N}\$denotes the power set of N. Subsets of N are called coalitions; $v\left( S\right)$ is called the worth of coalition S. The set of all games for N is denoted by $\mathbb {V} \left( N\right)$; the set of all games is denoted by $\mathbb {V} :=\bigcup _{N\in \mathcal {N}}\mathbb {V}\left( N\right) .$

For $N\in \mathcal {N}$, $T\subseteq N,$ and $v\in \mathbb {V}\left( N\right)$, the subgame $v|_{T}\in \mathbb {V}\left( T\right)$ is given by $v|_{T}\left( S\right) =v\left( S\right)$ for all $S\subseteq T;$ for $i\in N$ and $S\subseteq N,$ we occasionally write $v_{-i}$ and $v_{-S}$ instead of $v|_{N\setminus \left\{ i\right\} }$ and $v|_{N\setminus S},$ respectively. For $N\in \mathcal {N}$, $v,w\in \mathbb {V}\left( N\right) ,$ and $\alpha \in \mathbb {R}$, the coalition functions $v+w\in \mathbb {V}\left( N\right)$ and $\alpha \cdot v\in \mathbb {V}\left( N\right)$ are given by $\left( v+w\right) \left( S\right) =v\left( S\right) +w\left( S\right)$ and $\left( \alpha \cdot v\right) \left( S\right) =\alpha \cdot v\left( S\right)$ for all $S\subseteq N.$ For $T\subseteq N,$ $T\ne \emptyset ,$ the game $u_{T}^{N}\in \mathbb {V}$ given by $u_{T}^{N}\left( S\right) =1$ if $T\subseteq S$ and $u_{T}^{N}\left( S\right) =0$ otherwise is called a unanimity game. Any $v\in \mathbb {V}\left( N\right) ,$ $N\in \mathcal {N}$ can be uniquely represented by unanimity games. In particular, we have

$$\begin{aligned} v=\sum _{T\subseteq N:T\ne \emptyset }\lambda _{T}\left( v\right) \cdot u_{T}^{N}, \end{aligned}$$

(1)

where the coefficients $\lambda _{T}\left( v\right)$ are known as the Harsanyi dividends (Harsanyi, 1959) and can be determined recursively by

$$\begin{aligned} \lambda _{T}\left( v\right) :=v\left( T\right) -\sum _{S\subsetneq T:S\ne \emptyset }\lambda _{S}\left( v\right) . \end{aligned}$$

(2)

Players $i,j\in N$ are called symmetric in $v\in \mathbb {V}\left( N\right)$ if $v\left( S\cup \left\{ i\right\} \right) -v\left( S\right) =v\left( S\cup \left\{ j\right\} \right) -v\left( S\right)$ for all $S\subseteq N\setminus \left\{ i,j\right\} .$

A rank order of $N\in \mathcal {N}$ is a bijection $\rho :N\rightarrow \left\{ 1,2,\dots ,\left| N\right| \right\}$ with the interpretation that i is the $\rho \left( i\right)$th player in $\rho$; the set of rank orders of N is denoted by $R\left( N\right) .$ The set of players before i in $\rho$ is denoted by $B_{i}\left( \rho \right) :=\left\{ \ell \in N:\rho \left( \ell \right) <\rho \left( i\right) \right\} .$ The marginal contribution of i in $\rho$ and $v\in \mathbb {V}\left( N\right)$ is denoted by

$$\begin{aligned} MC_{i}^{v}\left( \rho \right) :=v\left( B_{i}\left( \rho \right) \cup \left\{ i\right\} \right) -v\left( B_{i}\left( \rho \right) \right) . \end{aligned}$$

(3)

A solution for $\mathbb {V}$ is an operator that assigns to any $N\in \mathcal {N}$, $v\in \mathbb {V}\left( N\right) ,$ and $i\in N$ a payoff $\varphi _{i}\left( v\right) .$ The Shapley value (Shapley, 1953) for $\mathbb {V}$, $\mathrm {Sh}$, is given by

$$\begin{aligned} \mathrm {Sh}_{i}\left( v\right) :=\sum _{T\subseteq N:i\in T}\frac{\lambda _{T}\left( v\right) }{t}=\sum _{S\subseteq N\setminus \left\{ i\right\} }\frac{v\left( S\cup \left\{ i\right\} \right) -v\left( S\right) }{n\cdot \left( {\begin{array}{c}n-1\\ s\end{array}}\right) }=\sum _{\rho \in R}\frac{1}{\left| R\left( N\right) \right| }\cdot MC_{i}^{v}\left( \rho \right) \end{aligned}$$

(4)

for all $N\in \mathcal {N},$ $v\in \mathbb {V}\left( N\right) ,$ and $i\in N.$

For $N\in \mathcal {N},$ let $\mathfrak {P}\left( N\right)$ denote the set of all partitions (coalition structures) of N;  the component of player $i\in N$ in $\mathcal {P}\in \mathfrak {P}\left( N\right)$ is denoted by $\mathcal {P}\left( i\right) .$ For $N\in \mathcal {N},$ $\mathcal {P} \in \mathfrak {P}\left( N\right) ,$ $T\subseteq N$ and $i\in N,$ let $\mathcal {P}\left( T\right) \subseteq \mathcal {P}$ be given by $\mathcal {P} \left( T\right) :=\left\{ P\in \mathcal {P}\mid T\cap P\ne \emptyset \right\}$, let $\mathcal {P}|_{T}\in \mathfrak {P}\left( T\right)$ be given by $\mathcal {P}|_{T}:=\left\{ T\cap P\mid P\in \mathcal {P}\left( T\right) \right\} ,$ let $\mathcal {P}_{-T}\in \mathfrak {P}\left( N\setminus T\right)$ be given by $\mathcal {P}_{-T}:=\mathcal {P}|_{N\setminus T},$ and let $\mathcal {P}_{-i}\in \mathfrak {P}\left( N\setminus \left\{ i\right\} \right)$ be given by $\mathcal {P}_{-i}:=\mathcal {P}_{-\left\{ i\right\} }.$

A CS game for $N\in \mathcal {N}$ is a pair $\left( v,\mathcal {P} \right)$, where $v\in \mathbb {V}\left( N\right)$ and $\mathcal {P} \in \mathfrak {P}\left( N\right) .$ Let $\mathbb{VP}\mathbb{}\left( N\right)$ denote the set of all CS games for N and let $\mathbb{VP}\mathbb{}:=\cup _{N\in \mathcal {N} }\mathbb{VP}\mathbb{}\left( N\right)$ denote the set of all CS games.

A (CS) solution for $\mathbb{VP}\mathbb{}$ is an operator $\varphi$ that assigns to any $N\in \mathcal {N}$, $i\in N$, and $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right)$ a payoff $\varphi _{i}\left( v,\mathcal {P} \right) ;$ for $P\in \mathcal {P},$ we set $\varphi _{P}\left( v,\mathcal {P} \right) =\sum _{i\in P}\varphi _{i}\left( v,\mathcal {P}\right) .$ For $N\in \mathcal {N}$ and $\mathcal {P}\in \mathfrak {P}\left( N\right) ,$ the set of all rank orders that respect $\mathcal {P}$ is denoted by

$$\begin{aligned} R\left( N,\mathcal {P}\right) :=\left\{ \rho \in R\left( N\right) \mid \text {for all }P\in \mathcal {P}\text { and }i,j\in P:\left| \rho \left( i\right) -\rho \left( j\right) \right| <\left| P\right| \right\} , \end{aligned}$$

that is, in any such rank order, the players from any component follow each other without players from other components between them. The Owen value (Owen, 1977) for $\mathbb{VP}\mathbb{}$, $\mathrm {Ow}$, is the CS solution given by

$$\begin{aligned} {\mathrm {Ow}}_{i}\left( v,\mathcal {P}\right)&\left. :=\right. \sum _{T\subseteq N:i\in T}\frac{\lambda _{T}\left( v\right) }{\left| \mathcal {P}\left( i\right) \cap T\right| \cdot \left| \mathcal {P} \left( T\right) \right| } \end{aligned}$$

(5a)

$$\begin{aligned}&=\sum _{\mathcal {C}\subseteq \mathcal {P}\setminus \left\{ \mathcal {P}\left( i\right) \right\} }\sum _{S\subseteq \mathcal {P}\left( i\right) \setminus \left\{ i\right\} }\frac{v\left( S\cup \left\{ i\right\} \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) -v\left( S\cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) }{\left| \mathcal {P}\left( i\right) \right| \cdot \left( {\begin{array}{c}\left| \mathcal {P}\left( i\right) \right| -1\\ s\end{array}}\right) \cdot \left| \mathcal {P}\right| \cdot \genfrac(){0.0pt}0{\left| {\mathcal {P}} \right| -1}{\left| \mathcal {C}\right| }} \end{aligned}$$

(5b)

$$\begin{aligned}&=\sum _{\rho \in R\left( N,\mathcal {P}\right) }\frac{1}{\left| R\right| }\cdot MC_{i}^{v}\left( \rho \right) \end{aligned}$$

(5c)

for all $N\in \mathcal {N}$, $i\in N$, and $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) .$

Fix an injection $\imath :\mathcal {N}\rightarrow \mathfrak {U},$ $N\mapsto \imath _{N}$ for $N\in \mathcal {N}.$ For any $N\in \mathcal {N}$, $\mathcal {P} \in \mathcal {P}\left( N\right) ,$ and $v\in \mathbb {V}\left( N\right) \mathfrak {,}$ set $\left[ \mathcal {P}\right] :=\left\{ \imath _{P}\mid P\in \mathcal {P}\right\}$ and let $v_{\mathcal {P}}\in \mathbb {V}\left( \left[ \mathcal {P}\right] \right)$ be given by

$$\begin{aligned} v_{\mathcal {P}}\left( \left[ \mathcal {C}\right] \right) :=v\left( {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) \qquad \text {for all }\mathcal {C}\subseteq \mathcal {P}. \end{aligned}$$

(6)

The TU game $v_{\mathcal {P}}$ is called the game between components or intermediate game for the CS game $\left( v,\mathcal {P}\right)$. For $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right)$, and $P\in \mathcal {P},$ we have

$$\begin{aligned} \mathrm {Ow}_{P}\left( v,\mathcal {P}\right) =\mathrm {Sh}_{\imath _{P}}\left( v_{\mathcal {P}}\right) . \end{aligned}$$

(7)

3 The Shapley value

This section partly follows Casajus (2021).

The marginal contributions of a player $i\in N,$ $N\in \mathcal {N}$ in the game $v\in \mathbb {V}\left( N\right)$ given as

$$\begin{aligned} v\left( S\cup \left\{ i\right\} \right) -v\left( S\right) ,\qquad S\subseteq N\setminus \left\{ i\right\} \end{aligned}$$

(8)

indicate her (individual) productivity or contribution to the generation of worth in the game v. The right-hand formula of the Shapley value in (4) indicates that the players’ Shapley value payoffs reflects their productivities in games as expressed by their own marginal contributions. Young (1985) shows that the Shapley value is the unique efficient such solution.

Efficiency, E. For all $N\in \mathcal {N}$ and $v\in \mathbb {V}\left( N\right) ,$ we have $\sum _{\ell \in N}\varphi _{\ell }\left( v\right) =v\left( N\right) .$

Symmetry, S. For all $N\in \mathcal {N}$, $v\in \mathbb {V}\left( N\right) ,$ and $i,j\in N$ such that i and j are symmetric in v,  we have $\varphi _{i}\left( v\right) =\varphi _{j}\left( v\right) .$

Marginality, M. For all $N\in \mathcal {N}$, $v,w\in \mathbb {V}\left( N\right) ,$ and $i\in N$ such that $v\left( S\cup \left\{ i\right\} \right) -v\left( S\right) =w\left( S\cup \left\{ i\right\} \right) -w\left( S\right)$ for all $S\subseteq N\setminus \left\{ i\right\} ,$ we have $\varphi _{i}\left( v\right) =\varphi _{i}\left( w\right)$.

Theorem 1

(Young 1985) The Shapley value is the unique solution for $\mathbb {V}$ that satisfies efficiency (E), symmetry (S), and marginality (M).⁵

Symmetry and marginality can be paraphrased as follows. Symmetry: players who are equally productive in a game should obtain the same payoff. Marginality: a player who is equally productive in two games should obtain the same payoff in these games. Therefore, a solution that is intended to reflect the players’ productivities should satisfy these properties.

Later on, Casajus (2021) introduces the notions of the players’ second-order productivities and second-order payoffs. Second-order productivities are conceptualized as second-order marginal contributions: the second-order marginal contributions of player $i\in N,$ $N\in \mathcal {N}$ with respect to player $j\in N\setminus \left\{ i\right\}$ in a game $v\in \mathbb {V}\left( N\right)$ are given as

$$\begin{aligned} \left[ v\left( S\cup \left\{ i,j\right\} \right) -v\left( S\cup \left\{ i\right\} \right) \right] -\left[ v\left( S\cup \left\{ j\right\} \right) -v\left( S\right) \right] ,\qquad S\subseteq N\setminus \left\{ i,j\right\} . \end{aligned}$$

(9)

These describe how player i affects the productivity of player j.⁶ The second-order payoff of player $i\in N,$ $N\in \mathcal {N}$ with respect to player $j\in N\setminus \left\{ i\right\}$ in a game $v\in \mathbb {V}\left( N\right)$ is given by

$$\begin{aligned} \varphi _{j}\left( v\right) -\varphi _{j}\left( v_{-i}\right) . \end{aligned}$$

It describes how player i affects the payoff of player j.⁷

Based on these notions, Casajus (2021) motivates natural second-order versions of symmetry and marginality. For all $N\in \mathcal {N},$ $v\in \mathbb {V}\left( N\right) ,$ and $i,j,k\in N,$ $i\ne j\ne k\ne j,$ players i and j are called second-order symmetric with respect to player k if

$$\begin{aligned}&\left[ v\left( T\cup \left\{ i,k\right\} \right) -v\left( T\cup \left\{ i\right\} \right) \right] -\left[ v\left( T\cup \left\{ k\right\} \right) -v\left( T\right) \right] \\&\left. \qquad =\right. \left[ v\left( T\cup \left\{ j,k\right\} \right) -v\left( T\cup \left\{ j\right\} \right) \right] -\left[ v\left( T\cup \left\{ k\right\} \right) -v\left( T\right) \right] \end{aligned}$$

for all $T\subseteq N\setminus \left\{ i,j,k\right\} .$

Second-order symmetry, 2S. For all $N\in \mathcal {N},$ $v\in \mathbb {V}\left( N\right)$ and $i,j,k\in N,$ $i\ne j\ne k\ne j$ such that players i and j are second-order symmetric with respect to player k,  we have

$$\begin{aligned} \varphi _{k}\left( v\right) -\varphi _{k}\left( v_{-i}\right) =\varphi _{k}\left( v\right) -\varphi _{k}\left( v_{-j}\right) . \end{aligned}$$

Second-order marginality, 2M. For all $N\in \mathcal {N},$ $v,w\in \mathbb {V}\left( N\right)$ and $i,j\in N,$ $i\ne j$ such that

$$\begin{aligned}&\left[ v\left( T\cup \left\{ i,j\right\} \right) -v\left( T\cup \left\{ i\right\} \right) \right] -\left[ v\left( T\cup \left\{ j\right\} \right) -v\left( T\right) \right] \\&\left. \qquad =\right. \left[ w\left( T\cup \left\{ i,j\right\} \right) -w\left( T\cup \left\{ i\right\} \right) \right] -\left[ w\left( T\cup \left\{ j\right\} \right) -w\left( T\right) \right] \end{aligned}$$

for all $T\subseteq N\setminus \left\{ i,j\right\} ,$ we have

$$\begin{aligned} \varphi _{j}\left( v\right) -\varphi _{j}\left( v_{-i}\right) =\varphi _{j}\left( w\right) -\varphi _{j}\left( w_{-i}\right) . \end{aligned}$$

Second-order symmetry and second-order marginality can be paraphrased as follows. Second-order symmetry: players who are equally second-order productive with respect to a third player in a game should be assigned the same second-order payoff with respect to the latter. Second-order marginality: a player who is equally second-productive with respect to a another player in two games should be assigned the same second-order payoff with respect to the latter in these games. Therefore, it seems to be plausible that a solution the second-order payoffs of which are intended to reflect the players’ second-order productivities satisfies these properties.

It turns out that the Shapley value reflects the players’ second-order productivities in terms of their second-order payoffs in the same vein as it reflects the players’ (first-order) productivities in terms of their (first-order) payoffs.

Theorem 2

(Casajus, 2021) The Shapley value is the unique solution for $\mathbb {V}$ that satisfies efficiency (E), second-order symmetry (2S), and second-order marginality (2M).

The proof of this theorem uses the fact that second-order marginality implies marginality, the proof of which is rather short.

Proposition 3

(Casajus, 2021) If a solution for $\mathbb {V}$ satisfies second-order marginality (2M), then it satisfies marginality (M).

Nevertheless, the proof of Theorem 2 is much more involved than the proof of Theorem 1 due to use of second-order symmetry instead of symmetry.

On the one hand, second-order symmetry does not imply symmetry (Casajus, 2021, Remark 3). On the other hand, the counterexamples in Casajus (2021, Remark 3) fail efficiency. As our first result, we show that the proof of Theorem 2 can be simplified substantially by providing a rather short proof that efficiency and second-order symmetry imply symmetry.

Proposition 4

If a solution for $\mathbb {V}$ satisfies second-order symmetry (2S) and efficiency (E), then it satisfies symmetry (S).

Proof

Let the solution $\varphi$ satisfy 2S and E. For $\left| N\right| =1,$ nothing is to show. Let now $\left| N\right| >1$. Let (*) $i,j\in N,$ $i\ne j,$ $N\in \mathcal {N}$ be symmetric in $v\in \mathbb {V}\left( N\right) .$ Fix $h\in \mathfrak {U} \setminus N$, set $M:=N\cup \left\{ h\right\} ,$ and let $w\in \mathbb {V} \left( M\right)$ be given by

$$\begin{aligned} w&=\sum _{T\subseteq N:T\ne \emptyset }\lambda _{T}\left( v\right) \cdot u_{T}^{M}\\&\qquad +\sum _{T\subseteq N\setminus \left\{ i,j\right\} }\left[ \lambda _{T\cup \left\{ i\right\} }\left( v\right) \cdot u_{T\cup \left\{ h\right\} }^{M}+\lambda _{T\cup \left\{ i,j\right\} }\left( v\right) \cdot u_{T\cup \left\{ i,h\right\} }^{M}+\lambda _{T\cup \left\{ i,j\right\} }\left( v\right) \cdot u_{T\cup \left\{ j,h\right\} }^{M}\right] , \end{aligned}$$

that is, a player h is added to v such that (**) i and j remain symmetric in w,  (***) h is symmetric to both i and j in w,  and (****) $w_{-h}=v.$

Since i and j are symmetric in w, they are second-order symmetric with respect to any $k\in M\setminus \left\{ i,j\right\}$ in w. Hence, we have

$$\begin{aligned} \varphi _{k}\left( w_{-i}\right) \overset{{{\textbf {2S}}}}{=}\varphi _{k}\left( w_{-j}\right) \qquad \text {for all }k\in M\setminus \left\{ i,j\right\} . \end{aligned}$$

(10)

Now, we obtain

$$\begin{aligned} \varphi _{j}\left( w_{-i}\right)&\overset{{{\textbf {E}}}}{=} w_{-i}\left( M\setminus \left\{ i\right\} \right) -\sum _{k\in N\setminus \left\{ i,j\right\} }\varphi _{k}\left( w_{-i}\right) \\&\overset{\text {(**),(10)}}{=}w_{-j}\left( M\setminus \left\{ j\right\} \right) -\sum _{k\in N\setminus \left\{ i,j\right\} }\varphi _{k}\left( w_{-j}\right) \\&\overset{{{\textbf {E}}}}{=}\varphi _{i}\left( w_{-j}\right) . \end{aligned}$$

In view of (***), we analogously obtain

$$\begin{aligned} \varphi _{j}\left( w_{-h}\right) =\varphi _{h}\left( w_{-j}\right) \qquad \text {and\qquad }\varphi _{i}\left( w_{-h}\right) =\varphi _{h}\left( w_{-i}\right) . \end{aligned}$$

(11)

Finally, we have

$$\begin{aligned} \varphi _{i}\left( v\right) \overset{{(****)}}{=}\varphi _{i}\left( w_{-h}\right) \overset{{(11)}}{=}\varphi _{h}\left( w_{-i}\right) \overset{{(10)}}{=}\varphi _{h}\left( w_{-j}\right) \overset{{(11)}}{=}\varphi _{j}\left( w_{-h}\right) \overset{{(****)}}{=}\varphi _{j}\left( v\right) , \end{aligned}$$

which concludes the proof. $\square$

4 The Owen value

In this section, we first survey the characterization of the Owen value by Khmelnitskaya and Yanovskaya (2007) . Then, we provide a second-order version of this characterization similar to the second-order characterization of the Shapley value by Casajus (2021) as surveyed in Section 3.

4.1 The (first-order) characterization by Khmelnitskaya and Yanovskaya (2007)

Khmelnitskaya and Yanovskaya (2007) generalize the characterization of the Shapley value due to Young (1985) . This characterization indicates that the Owen value is the unique efficient CS solution that reflects both the players’ and the components’ (first-order) productivities in terms of their (first-order) payoffs.⁸

Efficiency, E. For all $N\in \mathcal {N}$ and $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ we have $\sum _{\ell \in N}\varphi _{\ell }\left( v,\mathcal {P}\right) =v\left( N\right) .$

Marginality, M. For all $N\in \mathcal {N}$, $\left( v,\mathcal {P}\right) ,\left( w,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ and $i\in N$ such that $v\left( S\cup \left\{ i\right\} \right) -v\left( S\right) =w\left( S\cup \left\{ i\right\} \right) -w\left( S\right)$ for all $S\subseteq N\setminus \left\{ i\right\} ,$ we have $\varphi _{i}\left( v,\mathcal {P}\right) =\varphi _{i}\left( w,\mathcal {P}\right)$.

Symmetry within components, SwC. For all $N\in \mathcal {N}$, $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ $P\in \mathcal {P}$, and $i,j\in P$ such that i and j are symmetric in v,  we have $\varphi _{i}\left( v\right) =\varphi _{j}\left( v\right) .$

For all $N\in \mathcal {N}$ and $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ the components $P,Q\in \mathcal {P}$ are called symmetric in $\left( v,\mathcal {P}\right) ,$ if

$$\begin{aligned} v\left( P\cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) -v\left( {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) =v\left( Q\cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) -v\left( {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) \end{aligned}$$

for all $\mathcal {C}\subseteq \mathcal {P}\setminus \left\{ P,Q\right\} ,$ that is, if and only if the representatives of P and Q are symmetric in the intermediate game $v_{\mathcal {P}}.$

Symmetry across components, SaC. For all $N\in \mathcal {N}$, $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right)$, and $P,Q\in \mathcal {P}$ such that P and Q are symmetric in $\left( v,\mathcal {P}\right) ,$ we have $\varphi _{P}\left( v,\mathcal {P}\right) =\varphi _{Q}\left( v,\mathcal {P}\right) .$

Efficiency and marginality are just the CS versions of the properties for (TU) solutions with the same name and with the same interpretation. Symmetry within components is a natural relaxation of symmetry within the CS framework. Symmetry across components treats the components as players: equally productive components should obtain the same payoff as expressed by the sum of their members’ payoffs. Moreover, both symmetry within components and symmetry across component can be viewed as generalizations of symmetry. Whereas the former is equivalent to symmetry for the trivial coalition structure $\left\{ N\right\} ,$ the latter is so for the atomistic coalition structure $\left\{ \left\{ i\right\} \mid i\in N\right\} .$

Theorem 5

(Khmelnitskaya and Yanovskaya, 2007) The Owen value is the unique CS solution for $\mathbb{VP}\mathbb{}$ that satisfies efficiency (E) symmetry within components (SwC), symmetry across components (SaC), and marginality (M).

4.2 A second-order characterization

In this subsection, we simultaneously translate the second-order characterization of the Shapley value to CS solutions and the (first-order) characterization of the Owen value to the second-order framework.

Second-order marginality, 2M. For all $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) ,$ $\left( w,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ and $i,j\in N,$ $i\ne j$ such that

$$\begin{aligned}&\left[ v\left( T\cup \left\{ i,j\right\} \right) -v\left( T\cup \left\{ i\right\} \right) \right] -\left[ v\left( T\cup \left\{ j\right\} \right) -v\left( T\right) \right] \\&\left. \qquad =\right. \left[ w\left( T\cup \left\{ i,j\right\} \right) -w\left( T\cup \left\{ i\right\} \right) \right] -\left[ w\left( T\cup \left\{ j\right\} \right) -w\left( T\right) \right] \end{aligned}$$

for all $T\subseteq N\setminus \left\{ i,j\right\} ,$ we have

$$\begin{aligned} \varphi _{j}\left( v,\mathcal {P}\right) -\varphi _{j}\left( v_{-i} ,\mathcal {P}_{-i}\right) =\varphi _{j}\left( w,\mathcal {P}\right) -\varphi _{j}\left( w_{-i},\mathcal {P}_{-i}\right) . \end{aligned}$$

In essence, this property is just a restatement of second-order marginality for TU games, where the coalition structure is fixed but can be ignored otherwise. Therefore, the proof of Proposition 3 runs through smoothly within the framework of CS games and we obtain

Proposition 6

If a solution for $\mathbb{VP}\mathbb{}$ satisfies second-order marginality (2M), then it satisfies marginality (M).

Second-order symmetry within components, 2SwC. For all $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ $P\in \mathcal {P},$ $i,j\in P,$ and $k\in N\setminus P$ such that i and j are second-order symmetric with respect to k in v,  we have

$$\begin{aligned} \varphi _{k}\left( v,\mathcal {P}\right) -\varphi _{k}\left( v_{-i} ,\mathcal {P}_{-i}\right) =\varphi _{k}\left( v,\mathcal {P}\right) -\varphi _{k}\left( v_{-j},\mathcal {P}_{-j}\right) . \end{aligned}$$

This property restricts second-order symmetry for TU games to players within the same component. Yet, the coalition structure can be ignored regarding the third player to whom the second-order marginal contributions and the second-order payoffs are related. The proof of Proposition 4 essentially runs through smoothly with second-order symmetry within components instead of symmetry within components and symmetry within components instead of symmetry: one just has to put player h into the component containing players i and j. Hence, we obtain

Proposition 7

If a solution for $\mathbb {V}$ satisfies strong second-order symmetry within components (2SwC) and efficiency (E), then it satisfies symmetry within components (SwC).

In order to obtain a second-order version of symmetry across components, we first provide the notion of second-order symmetry of components. For all $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ and $A,B,C\in \mathcal {P}$ pairwise different, components A and B are called second-order symmetric with respect to component C in $\left( v,\mathcal {P}\right)$ if

$$\begin{aligned}&\left[ v\left( C\cup A\cup \bigcup _{D\in \mathcal {D}}D\right) -v\left( A\cup \bigcup _{D\in \mathcal {D}}D\right) \right] -\left[ v\left( C\cup \bigcup _{D\in \mathcal {D}}D\right) -v\left( \bigcup _{D\in \mathcal {D} }D\right) \right] \\&\left. \qquad =\right. \left[ v\left( C\cup B\cup \bigcup _{D\in \mathcal {D}}D\right) -v\left( B\cup \bigcup _{D\in \mathcal {D}}D\right) \right] -\left[ v\left( C\cup \bigcup _{D\in \mathcal {D}}D\right) -v\left( \bigcup _{D\in \mathcal {D}}D\right) \right] \end{aligned}$$

for all $\mathcal {D}\subseteq \mathcal {P}\setminus \left\{ A,B,C\right\} .$

Remark 8

Note that the components A and B are second-order symmetric with respect to component C in $\left( v,\mathcal {P}\right)$ if and only if their representatives $\imath _{A}$ and $\imath _{B}$ are second-order symmetric with respect to the representative $\imath _{C}$ of component C in the intermediate game $v_{\mathcal {P}}$.

Second-order symmetric components are equally second-order productive with respect to a third component. Therefore, if a CS solution is intended to reflect the components’ second-order productivities in terms of their second-order payoffs, it seems to be plausible that the second-order payoffs of second-order symmetric components are the same.

Second-order symmetry across components, 2SaC. For all $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ and $A,B,C\in \mathcal {P}$ pairwise different such that A and B are second-order symmetric with respect to C in v,  we have

$$\begin{aligned} \varphi _{C}\left( v,\mathcal {P}\right) -\varphi _{C}\left( v_{-A} ,\mathcal {P}_{-A}\right) =\varphi _{C}\left( v,\mathcal {P}\right) -\varphi _{C}\left( v_{-B},\mathcal {P}_{-B}\right) . \end{aligned}$$

Using the general idea of the proof of Proposition 4 one shows that second-order symmetry across components and efficiency imply symmetry across components.

Proposition 9

If a solution for $\mathbb{VP}\mathbb{}$ satisfies second-order symmetry across components (2SaC) and efficiency (E), then it satisfies symmetry across components (SaC).

Proof

Let the CS solution $\varphi$ satisfy 2SaC and E. If $\left| \mathcal {P}\right| =1,$ then nothing is to show. Let now $N\in \mathcal {N}$ and $\left( v,\mathcal {P} \right) \in \mathbb{VP}\mathbb{}\left( N\right)$ be such that $\left| \mathcal {P}\right| >1.$ Moreover, let (*) $P,Q\in \mathcal {P},$ $P\ne Q$ be symmetric in v. Fix $h\in \mathfrak {U}\setminus N$, set $M:=N\cup \left\{ h\right\}$ and $\mathcal {Q}:=\mathcal {P}\cup \left\{ \left\{ h\right\} \right\} ,$ and let $w\in \mathbb {V}\left( M\right)$ be given by

$$\begin{aligned} w&=\sum _{T\subseteq N:T\ne \emptyset }\lambda _{T}\left( v\right) \cdot u_{T}^{M}\\&\qquad +\sum _{T\subseteq N\setminus \left( P\cup Q\right) }\sum _{S\subseteq P:S\ne \emptyset }\lambda _{T\cup S}\left( v\right) \cdot u_{T\cup \left\{ h\right\} }^{M}\\&\qquad +\sum _{T\subseteq N\setminus \left( P\cup Q\right) }\sum _{S\subseteq P\cup Q:S\cap P\ne \emptyset ,S\cap Q\ne \emptyset }\lambda _{T\cup \left\{ i,j\right\} }\left( v\right) \cdot u_{T\cup \left( S\setminus Q\right) \cup \left\{ h\right\} }^{M}\\&\qquad +\sum _{T\subseteq N\setminus \left( P\cup Q\right) }\sum _{S\subseteq P\cup Q:S\cap P\ne \emptyset ,S\cap Q\ne \emptyset }\lambda _{T\cup \left\{ i,j\right\} }\left( v\right) \cdot u_{T\cup \left( S\setminus P\right) \cup \left\{ h\right\} }^{M}, \end{aligned}$$

that is, $\left\{ h\right\}$ is added to $\left( v,\mathcal {P}\right)$ such that (**) P and Q remain symmetric in $\left( w,\mathcal {Q}\right) ,$ (***) $\left\{ h\right\}$ is symmetric to both P and Q in w,  and (****) $\left( w_{-\left\{ h\right\} },\mathcal {Q}_{-\left\{ h\right\} }\right) =\left( v,\mathcal {P}\right) .$

Since P and Q are symmetric in $\left( w,\mathcal {Q}\right)$, they are second-order symmetric with respect to any $R\in \mathcal {Q}\setminus \left\{ P,Q\right\}$ in $\left( w,\mathcal {Q}\right) .$ Hence, we have

$$\begin{aligned} \varphi _{R}\left( w_{-P},\mathcal {Q}_{-P}\right) \overset{{{\textbf {2SaC}}}}{=}\varphi _{R}\left( w_{-Q},\mathcal {Q}_{-Q}\right) \qquad \text {for all }R\in \mathcal {Q}\setminus \left\{ P,Q\right\} . \end{aligned}$$

(12)

Now, we obtain

$$\begin{aligned} \varphi _{Q}\left( w_{-P},\mathcal {Q}_{-P}\right)&\overset{{{\textbf {E}}}}{=}w_{-P}\left( M\setminus P\right) -\sum _{R\in \mathcal {Q}\setminus \left\{ P,Q\right\} }\varphi _{R}\left( w_{-P} ,\mathcal {Q}_{-P}\right) \\&\overset{\text {(**),(12)}}{=}w_{-Q}\left( M\setminus Q\right) -\sum _{R\in \mathcal {Q}\setminus \left\{ P,Q\right\} }\varphi _{R}\left( w_{-Q}\right) \\&\overset{{{\textbf {E}}}}{=}\varphi _{P}\left( w_{-Q},\mathcal {Q} _{-Q}\right) . \end{aligned}$$

In view of (***), we analogously obtain

$$\begin{aligned} \varphi _{Q}\left( w_{-\left\{ h\right\} },\mathcal {Q}_{-\left\{ h\right\} }\right) =\varphi _{\left\{ h\right\} }\left( w_{-Q},\mathcal {Q} _{-Q}\right) \end{aligned}$$

(13)

and

$$\begin{aligned} \varphi _{P}\left( w_{-\left\{ h\right\} },\mathcal {Q}_{-\left\{ h\right\} }\right) =\varphi _{\left\{ h\right\} }\left( w_{-P},\mathcal {Q} _{-P}\right) . \end{aligned}$$

(14)

Finally, we have

$$\begin{aligned} \varphi _{P}\left( v,\mathcal {P}\right)&\overset{\text {(****)}}{=}\varphi _{P}\left( w_{-h},\mathcal {Q}_{-\left\{ h\right\} }\right) \\&\overset{{(14)}}{=}\varphi _{\left\{ h\right\} }\left( w_{-P},\mathcal {Q}_{-P}\right) \\&\overset{{(12)}}{=}\varphi _{\left\{ h\right\} }\left( w_{-Q},\mathcal {Q}_{-Q}\right) \\&\overset{{(13)}}{=}\varphi _{Q}\left( w_{-\left\{ h\right\} },\mathcal {Q}_{-\left\{ h\right\} }\right) \overset{\text {(****)}}{=}\varphi _{Q}\left( v,\mathcal {P}\right) , \end{aligned}$$

which concludes the proof. $\square$

Propositions 6, 7, and 9, allow us to “transfer” Theorem 5 to the second-order framework. We obtain

Theorem 10

The Owen value is the unique CS solution for $\mathbb{VP}\mathbb{}$ that satisfies efficiency (E), second-order marginality (2M), second-order symmetry within components (2SwC), and second-order symmetry across components (2SaC).

Proof

It is well-known that the Owen value satisfies E. Straightforward but tedious calculations using (5b) show the following formulas for the second-order Owen value payoffs in terms of second-order marginal contributions. Let $N\in \mathcal {N},$ $i,k\in N,$ $i\ne k,$ and $\left( v,\mathcal {P}\right) \in \mathbb {V}\left( N\right) .$ If $k\in \mathcal {P}\left( i\right) ,$ then

$$\begin{aligned}&\mathrm {Ow}_{k}\left( v,\mathcal {P}\right) -\mathrm {Ow}_{k}\left( v_{-i},\mathcal {P}_{-i}\right) \nonumber \\&\quad \left. =\right. \sum _{\mathcal {C}\subseteq \mathcal {P}\setminus \left\{ \mathcal {P}\left( k\right) \right\} }\sum _{S\subseteq \mathcal {P}\left( i\right) \setminus \left\{ i,k\right\} }\left[ \frac{v\left( S\cup \left\{ i,k\right\} \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) -v\left( S\cup \left\{ i\right\} \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) }{\left| \mathcal {P}\left( k\right) \right| \cdot \left( {\begin{array}{c}\left| \mathcal {P}\left( k\right) \right| -1\\ s+1\end{array}}\right) \cdot \left| \mathcal {P}\right| \cdot \genfrac(){0.0pt}0{\left| \mathcal {P} \right| -1}{\left| \mathcal {C}\right| }}\dots \right. \nonumber \\&\left. \qquad -\right. \left. \frac{v\left( S\cup \left\{ k\right\} \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) -v\left( S\cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) }{\left| \mathcal {P}\left( k\right) \right| \cdot \left( {\begin{array}{c}\left| \mathcal {P}\left( k\right) \right| -1\\ s+1\end{array}}\right) \cdot \left| \mathcal {P}\right| \cdot \genfrac(){0.0pt}0{\left| \mathcal {P} \right| -1}{\left| \mathcal {C}\right| }}\right] . \end{aligned}$$

(15)

If $k\in N\setminus \mathcal {P}\left( i\right) ,$ then

$$\begin{aligned}&\mathrm {Ow}_{k}\left( v,\mathcal {P}\right) -\mathrm {Ow}_{k}\left( v_{-i},\mathcal {P}_{-i}\right) \nonumber \\&\quad \left. =\right. \sum _{\mathcal {C}\subseteq \mathcal {P}\setminus \left\{ \mathcal {P}\left( k\right) ,\mathcal {P}\left( i\right) \right\} } \sum _{S\subseteq \mathcal {P}\left( k\right) \setminus \left\{ k\right\} }\left[ \frac{v\left( S\cup \left\{ k\right\} \cup \mathcal {P}\left( i\right) \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) -v\left( S\cup \mathcal {P}\left( i\right) \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) }{\left| \mathcal {P}\left( k\right) \right| \cdot \left( {\begin{array}{c}\left| \mathcal {P}\left( k\right) \right| -1\\ s\end{array}}\right) \cdot \left| \mathcal {P}\right| \cdot \genfrac(){0.0pt}0{\left| \mathcal {P} \right| -1}{\left| \mathcal {C}\right| +1}}\dots \right. \nonumber \\&\left. \qquad -\right. \left. \frac{v\left( S\cup \left\{ k\right\} \cup \left( \mathcal {P}\left( i\right) \setminus \left\{ i\right\} \right) \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) -v\left( S\cup \left( \mathcal {P}\left( i\right) \setminus \left\{ i\right\} \right) \cup {\displaystyle \bigcup \limits _{C\in \mathcal {C}}} C\right) }{\left| \mathcal {P}\left( k\right) \right| \cdot \left( {\begin{array}{c}\left| \mathcal {P}\left( k\right) \right| -1\\ s\end{array}}\right) \cdot \left| \mathcal {P}\right| \cdot \genfrac(){0.0pt}0{\left| \mathcal {P} \right| -1}{\left| \mathcal {C}\right| +1}}\right] . \end{aligned}$$

(16)

From (15) and (16) it is immediate that the Owen value satisfies 2M and 2SwC. By Remark ?? and in view of the well-known fact that the Owen values satisfies IG, it also satisfies 2SaC.

Let the CS solution $\varphi$ satisfy E, 2M, 2SwC, and 2SaC. By Propositions 6, 7, and 9, the CS solution $\varphi$ satisfies M, SwC, and SaC. By Theorem 5, we have $\varphi =\mathrm {Ow}$. $\square$

Remark 11

The characterization in Theorem 10 is non-redundant for $\left| N\right| >1$. The zero CS solution, $\mathrm {Z},$ given by $\mathrm {Z}_{i}\left( v,\mathcal {P}\right) :=0$ for all $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{}\left( N\right) ,$ and $i\in N$ satisfies all properties but efficiency. The component egalitarian CS solution, $\mathrm {CE},$ given by

$$\begin{aligned} \mathrm {CE}_{i}\left( v,\mathcal {P}\right) :=\frac{v\left( N\right) }{\left| \mathcal {P}\left( i\right) \right| \cdot \left| \mathcal {P}\right| } \end{aligned}$$

for all $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{} \left( N\right) ,$ and $i\in N$ satisfies all properties but marginality. Fix a bijection $\varrho :\mathfrak {U}\rightarrow \mathbb {N}.$ For any $N\in \mathcal {N}$ and $\mathcal {P}\in \mathfrak {P}\left( N\right) ,$ let

$$\begin{aligned}&R\left( N,\mathcal {P},\varrho \right) \\&\left. :=\right. \left\{ \rho \in R\left( N,\mathcal {P}\right) \mid \text {for all }P\in \mathcal {P}\text { and }i,j\in P:\rho \left( i\right)>\rho \left( j\right) \text { if and only if }\varrho \left( i\right) >\varrho \left( j\right) \right\} . \end{aligned}$$

The $\varrho$-Owen value, $\mathrm {Ow}^{\varrho },$ given by

$$\begin{aligned} \mathrm {Ow}_{i}^{\varrho }\left( v,\mathcal {P}\right) :=\sum _{\rho \in R\left( N,\mathcal {P},\varrho \right) }MC_{i}^{v}\left( \rho \right) \end{aligned}$$

for all $N\in \mathcal {N},$ $\left( v,\mathcal {P}\right) \in \mathbb{VP}\mathbb{} \left( N\right) ,$ and $i\in N$ satisfies all properties but second-order symmetry within components. The Shapley value for CS games ignoring the coalition structure satisfies all properties but second-order symmetry across components.

5 Concluding remarks

In this paper, we suggest a characterization of the Owen value indicating that the latter is the unique efficient CS solution that reflects the players’ and components’ second-order productivities in terms of their second-order payoffs. The natural question now arises whether this may hold true for higher-order productivities and higher-order payoffs. In view of the results of Casajus (2020, Appendix A), the Owen value should satisfy the corresponding higher-order properties, whereas not being the unique efficient CS solution to do so.

Winter (1989) generalizes the Owen value to games enriched with a level structure, that is, a finite sequence of coalition structures becoming successively finer. Khmelnitskaya and Yanovskaya (2007, Theorem 2) indicate how their characterization can be extended to this level structure value. We leave it to the reader to provide the obvious extension of our characterization of the Owen value to that level structure value.

Owen (1982) suggests an extension of the Banzhaf value (Banzhaf, 1965; Owen, 1975) to games with a coalition structure. It remains an open question which properties in place of efficiency in Theorem 10 would yield a characterization of the Banzhaf value for games with a coalition structure.⁹