Evidence reading mechanisms

Abstract

In an environment with privately informed agents who can produce evidence, we study implementation of a social choice function by reading mechanisms: mechanisms that simply apply the social choice function to a consistent interpretation of the evidence. We provide sufficient conditions on the social choice function and the evidence structure for ex post implementability by such mechanisms. If the first-best policy of a mechanism designer satisfies this condition, then its implementation by a reading mechanism does not require commitment. We show that with rich evidence structures, (1) a function that is implementable with transfers is also implementable with evidence but no transfer, (2) under private value, the efficient allocation is implementable with budget balanced and individually rational transfers, and (3) in single-object auction and bilateral trade environments with interdependent values, the efficient allocation is implementable with budget balanced and individually rational transfers.

A Appendix

This appendix presents additional definitions to provide necessary and sufficient conditions for interim and ex-post implementation.

1.1 A.1 Additional definitions

The interim belief of agent i about the types of the other agents is given by a distribution $p_i(\cdot |t_i)\in \Delta ({\mathcal {T}}_{-i})$. A messaging strategy profile $\mu :{\mathcal {T}}\rightarrow {\mathcal {M}}$ is an interim equilibrium⁶ of this game if, for every i, every $t_i$, and every $m_i\in M_i(t_i)$,

$$\begin{aligned} E(u_i(g(\mu (t));t)|t_i)\ge E(u_i(g(m_i,\mu _{-i}(t_{-i}));t)|t_i). \end{aligned}$$

A mechanism $g(\cdot )$ interim implements the social choice function $f(\cdot )$ if there exists an interim equilibrium $\mu (\cdot )$ of the game generated by $g(\cdot )$, such that $g(\mu (t))=f(t)$ for every $t\in {\mathcal {T}}$.

The interim masquerading payoff of agent i is given by the function

$$\begin{aligned} v_i(s_i|t_i)=\sum _{t_{-i}\in {\mathcal {T}}_{-i}}u_i\bigl (f(s_i,t_{-i});t_i,t_{-i}\bigr )p_i(t_{-i}|t_i). \end{aligned}$$

For the interim masquerade relation, we say that $t_i$ wants to masquerade as $s_i$, denoted by $t_i\xrightarrow {{\mathfrak {M}}} s_i$, if and only if $v_i(s_i|t_i)>v_i(t_i|t_i)$. The set of interim worst-case types is denoted by

$$\begin{aligned} {{\,\mathrm{wct}\,}}({\mathcal {S}}_i):=\bigl \{s_i\in {\mathcal {S}}_i\,|\,\not \exists \; t_i\in {\mathcal {S}}_i, t_i\xrightarrow {{\mathfrak {M}}}s_i\bigr \}. \end{aligned}$$

An interim evidence base for agent i is a set of messages ${\mathcal {E}}_i\subseteq {\mathcal {M}}_i$ such that there exists a one-to-one function $e_i:{\mathcal {T}}_i\rightarrow {\mathcal {E}}_i$ that satisfies $e_i(t_i)\in M_i(t_i)$, and $t_i\in {{\,\mathrm{wct}\,}}(M_i^{-1}(e_i(t_i)))$ for every $t_i$. If cheap talk completion of the evidence structure is allowed, the condition for an interim evidence base is that, for each $t_i$, there exists $m_i$ such that $t_i\in {{\,\mathrm{wct}\,}}(M_i^{-1}(m_i))$.

A reading is independent if for every i the reading of the evidence satisfies $\rho _i(m_i,m_{-i})=\rho _i(m_i, m_{-i}')$ for every $m_i$, $m_{-i}$ and $m_{-i}'$. It means that agent i’s evidence is interpreted independently of the evidence submitted by other agents.⁷

As in the main paper, we say that a reading mechanism implements $f(\cdot )$ with accurate readings if it reads the evidence correctly on the equilibrium path of the corresponding equilibrium.

Definition 2

(Accurate Implementation) A reading mechanism with associated reading $\rho (\cdot )$ accurately (interim or ex post) implements $f(\cdot )$ if there exists an (interim or ex post) equilibrium strategy profile $\mu (\cdot )$ such that, for every $t\in {\mathcal {T}}$, $\rho (\mu (t)) = t$.

Definition 3

(Straightforward Implementation) A reading mechanism with associated reading $\rho (\cdot )$ straightforwardly (interim or ex post) implements $f(\cdot )$ if there exists an (interim or ex post) equilibrium strategy profile $\mu (\cdot )$ such that $\rho (\mu (t)) = t$, and $\rho _{-i}(\mu _{-i}(t_{-i}), m_{i})=t_{-i}$, for every $t\in {\mathcal {T}}$, every $i\in N$, and every $m_i\in {\mathcal {M}}_i$.

Hence straightforward implementation is more restrictive than accurate implementation. It implies that, if all agents except i use their equilibrium strategy, then the type profile of these non deviators is correctly interpreted. Note also that accurate implementation by an independent reading implies straightforward implementation.

1.2 A.2 Necessary and sufficient conditions for implementation

Theorem 6

(Interim Implementation) There exists a reading mechanism that accurately interim implements $f(\cdot )$ with an independent reading if and only if the following conditions hold for every agent i:

• (i) For every message $m_i\in {\mathcal {M}}_i$, the set $M_i^{-1}(m_i)$ admits an interim worst case type.

• (ii) $M_i(\cdot )$ admits an interim evidence base.

Proof

($\Leftarrow$) By (ii), we can pick, for each agent i, a one-to-one mapping $e_i:{\mathcal {T}}_i\rightarrow {\mathcal {M}}_i$ corresponding to an evidence base of i. By (i), we can choose an independent reading $\rho (\cdot )$ such that, for every $m_i$, $\rho _i(m_i)\in {{\,\mathrm{wct}\,}}(M_i^{-1}(m_i))$ and for every $t_i$, $\rho _i(e_i(t_i))=t_i$. Suppose that every agent i adopts $e_i(\cdot )$ as her strategy in the game defined by the mechanism associated with $\rho (\cdot )$. Then for every t, the mechanism selects the outcome $f(\rho (e(t)))=f(t)$. Hence, if the strategy profile $e(\cdot )$ is an equilibrium of the game, we have succeeded in implementing $f(\cdot )$ with accurate readings. It remains to show that $e(\cdot )$ is indeed an equilibrium. Suppose then that agent i of type $t_i$ deviates with a message $m_i\ne e_i(t_i)$. Then the implemented outcome is $f(w_i,t_{-i})$, where $w_i \in {{\,\mathrm{wct}\,}}(M_i^{-1}(m_i))$. But then we know that $v_i(w_i|t_i)\le v_i(t_i|t_i)$, so the deviation is not profitable for i.

($\Rightarrow$) Let $\rho (\cdot )$ be an independent reading such that the associated mechanism implements $f(\cdot )$ with accurate readings, and let $\mu ^*(\cdot )$ be the associated equilibrium strategy profile. Then, by definition of accurate implementation, $\rho (\mu ^*(t))=t$. Consider some message $m_i$ of agent i. The equilibrium condition implies that, for every $t_i\in M_i^{-1}(m_i)$,

where first equality is a consequence of accuracy and independence. Since, by definition of a reading mechanism, $\rho _i(m_i))\in M_i^{-1}(m_i)$, this proves that $\rho _i(m_i)\in {{\,\mathrm{wct}\,}}(M_i^{-1}(m_i))$. This proves (i).

To prove (ii), consider the particular case in which $m_i=\mu _i^*(s_i)$ for some type $s_i\in {\mathcal {T}}_i$. Then $\rho _i\bigl (m_i,\mu ^*_{-i}(t_{-i})\bigr )=s_i$, by accuracy and independence, and therefore we have shown that $s_i$ is a worst case type of the set certified by $\mu _i^*(s_i)$. The accuracy property also implies that $\mu _i^*(s_i)\ne \mu _i^*(t_i)$ whenever $s_i\ne t_i$. Otherwise, we would have $t_i=\rho _i\bigl (\mu _i^*(t_i)\bigr )=\rho _i\bigl (\mu _i^*(s_i)\bigr )=s_i$. Therefore, the function $\mu _i^*:{\mathcal {T}}_i\rightarrow {\mathcal {M}}_i$ defines an evidence base for i. $\square$

It is easy to show that the existence of an evidence base for each agent is necessary for implementation with any mechanism. The worst case type condition, however, is only necessary if we require accurate implementation and independent readings. If the reading is not required to be independent, then ex post instead of interim worst case types could be used (see Theorem 7). To illustrate the importance of accuracy, the following example exhibits a social choice function that is not accurately interim implementable with independent reading, because of a missing interim worst case type, but can nevertheless be implemented by a reading mechanism with independent reading.

Example 4

(Committing to incorrect readings) There are two agents and five alternatives ${\mathcal {A}} = \{a, b, c, d, e\}$. The set of agent 1’s types is ${\mathcal {T}}_1 = \{t_1^0,t_1^1, t_1^2, t_1^3, t_1^4\}$, and the set of agent 2’s types is ${\mathcal {T}}_2 = \{t_2^1, t_2^2\}$, with a uniform prior probability distribution. Consider the following social choice function:⁸

$f(\cdot ,\cdot )$	$t_1^0$	$t_1^1$	$t_1^2$	$t_1^3$	$t_1^4$
$t_2^1$	e	b	a	d	c
$t_2^2$	e	a	b	c	d

Assume that agent 2’s utility is maximized when $f(\cdot )$ is implemented (so that he never has an incentive to deviate), and agent 1’s utility function is given by the following table, where the squares indicate the outcomes prescribed by the social choice function:

Fig. 2

Committing to incorrect readings: interim masquerade relations and evidence structures

The interim masquerade relations of the agents and the evidence structures are summarized in Fig. 2. Agent 1’s interim masquerade relation has a cycle. There is an interim evidence base for each agent, but the certifiable set $\{t_1^0, t_1^1, t_1^2, t_1^3\}$ has no interim worst case type. Hence, $f(\cdot )$ is not accurately interim implementable with an independent reading. However, it is implemented with the following independent reading and interim equilibrium strategies, where the red lines correspond to incorrect readings given the equilibrium strategies:

The intuition is that, by committing to incorrect readings, the designer can emulate the use of inconsistent punishments while remaining within the boundaries of reading mechanisms. To see that, note that, given the masquerade relation of agent 1, the key is to dissuade the use of the message $m_1^{0123}$. This cannot be done accurately because of the cycle. In the mechanism described above, $m_1^{0123}$ is interpreted as $t_1^3$, which should make $t_1^1$ willing to use this message. The trick is that the designer is voluntarily misinterpreting the equilibrium messages of agent 2, so agent 1 with type $t_1^1$, expects that the outcome implemented by the designer when she pretends to be $t_1^3$ and the true type of agent 2 is $t_2^2$ will be $f(t_1^3,t_2^2)=f(t_1^4,t_2^1)=c$. Thus, this is as if the designer attributed the message $m_1^{0123}$ to $t_1^4$, which no type in $M_1^{-1}(m^{0123})$ wants to masquerade as. The designer cannot do that directly because such a reading would not be consistent with evidence. But she can emulate that outcome by misreading evidence from agent 2 on the equilibrium path.⁹$\square$

For ex post implementation, we weaken the necessary properties of the mechanisms used in the characterization since we require straightforward implementation instead of accurate implementation with an independent reading.

Theorem 7

(Ex Post Implementation) There exists a reading mechanism that straightforwardly ex post implements $f(\cdot )$ if and only if the following conditions hold for every agent i:

• (i) For every $t_{-i}\in {\mathcal {T}}_{-i}$, and every message $m_i\in {\mathcal {M}}_i$, the set $M_i^{-1}(m_i)$ admits a worst case type given $t_{-i}$.

• (ii) $M_i(\cdot )$ admits an evidence base.

Proof

($\Leftarrow$) This part of the proof is the same as the proof of Theorem 1. The straightforwardness property is satisfied by construction of $\rho (\cdot )$.

($\Rightarrow$) Let $\rho (\cdot )$ be a reading such that the associated mechanism straightforwardly implements $f(\cdot )$, and let $\mu ^*(\cdot )$ be the associated ex post equilibrium strategy profile. Consider some message $m_i$ of agent i. The equilibrium condition implies that, for every $t_{-i}\in {\mathcal {T}}_{-i}$, and every $t_i\in M_i^{-1}(m_i)$, and

where the second line comes from the straightforward implementation property. Since, by definition of a reading mechanism, $\rho _i(m_i\mu _{-i}^*(t_{-i}))\in M_i^{-1}(m_i)$, this proves that $\rho _i(m_i,\mu _{-i}^*(t_{-i}))\in {{\,\mathrm{wct}\,}}\bigl (M_i^{-1}(m_i)|t_{-i}\bigr )$. This proves (i).

Now, consider the particular case where $m_i=\mu _i^*(s_i)$ for some type $s_i\in {\mathcal {T}}_i$. Then $\rho _i(m_i,\mu _{-i}^*(t_{-i}))=s_i$, by the straightforwardness property, and therefore we have shown that $s_i$ is a worst case type of the set certified by $\mu _i^*(s_i)$ given $t_{-i}$. The straightforwardness property also implies that $\mu _i^*(s_i)\ne \mu _i^*(t_i)$ whenever $s_i\ne t_i$. Otherwise, we would have $t_i=\rho _i(\mu _i^*(t_i),\mu _{-i}^*(t_{-i}))=\rho _i\bigl (\mu _i^*(s_i),\mu _{-i}^*(t_{-i})\bigr )=s_i$. Therefore, the function $\mu _i^*:{\mathcal {T}}_i\rightarrow {\mathcal {M}}_i$ defines an evidence base for i. $\square$

Fig. 3

Accurate interim implementation without independence or straightforwardness—ex post and interim masquerade relations of agent 1

Ex post implementability by a reading mechanism implies interim implementability by a reading mechanism. However, the reading used for ex post implementation, even if it satisfies straightforwardness, may not satisfy independence. To illustrate the relations between ex post and interim implementation by reading mechanisms, we provide an example such that the conditions of Theorems 6 and 7 are not satisfied, but accurate interim implementation by a reading mechanism is possible.

Example 5

(Accurate interim implementation without independence or straightforwardness) Consider two agents. The type sets are ${\mathcal {T}}_1=\{t_1,t_1'\}$ and ${\mathcal {T}}_2=\{t_2,t_2',t_2''\}$. The common prior is that the types of the two agents are independently and uniformly distributed over their respective supports. We assume that the only certifiable sets of agent 2 are the singletons $\{t_2\}$, $\{t_2'\}$ and $\{t_2''\}$, so that there is no need to incentivize full revelation from agent 2. The certifiable sets for agent 1 are the singletons, $\{t_1\}$ and $\{t_1'\}$, and the set $\{t_1,t_1'\}$, so that there exists an evidence base, but agent 1 needs to be incentivized to provide precise information. For simplicity, we denote the messages by the sets they certify.

The ex post masquerading relations of agent 1 and her interim masquerading relation are given in Fig. 3 with intensities. There is an ex post cycle when the type of agent 2 is $t_2$, and there is an interim cycle. Therefore the conditions of Theorems 6 and 7 are not satisfied. Accurate interim implementation is possible with the following reading:

$$\begin{aligned} \rho _1\left( \{t_1,t_1'\},\{t_2\}\right) =\rho _1\left( \{t_1,t_1'\},\{t_2'\}\right) =t_1 \quad \text {and} \quad \rho _1\left( \{t_1,t_1'\},\{t_2''\}\right) =t_1'. \end{aligned}$$

Indeed, if the type of agent 2 is $t_2''$, the uninformative message $\{t_1,t_1'\}$ of agent 1 is read as $t_1'$, which is an ex post worst case type. Hence she has no incentive to be vague conditionally on the type of agent 2 being $t_2''$. Agent 1 cannot be given ex post incentives if the type of agent 2 is $t_2$, but the designer can dissuade her from being vague by pooling this event with the event in which agent 2 has type $t_2'$. The expected masquerading gain conditional of agent 2 not being of type $t_2$ is + 6 for a $t_1$ type masquerading as a $t_1'$ type, and $-\,2$ for a $t_1'$ type masquerading as a $t_1$ type, an therefore interpreting the vague message as $t_1$ is dissuasive. $\square$