Abstract
We consider the implementation problem with socially responsible agents who want to report a socially desirable outcome to a mechanism designer. We design a simple and natural mechanism in which each agent reports an outcome. We show that if there are at least two socially responsible agents, then the mechanism implements any unanimous social choice correspondence in Nash equilibria with at least three agents.
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Notes
The mechanism might be criticized because of the use of the modulo game which may lead to new unwanted mixed strategy equilibria. We are planning to conduct an experiment to investigate whether the mechanism causes this problem.
For the formal definition of common knowledge, see Aumann (1995).
We have omitted the proof of Theorem (1) and that of (2) in the case of three agents. The proofs are provided in an earlier version of the paper (Hagiwara et al. 2016). For the proof of Theorem (2) when \(n=3\), we need to ask each agent to report an objection flag in addition to an outcome and a positive integer between 1 and n.
While in the modulo game, any agent can exactly designate the winner from all agents, in the integer game, any agent can designate the winner from either himself or an agent who announces the highest integer. Since a socially responsible agent \(s\in S\backslash \{i\}\) needs to make agent \( j\in N\backslash \{i,s\}\) the winner of the modulo game, we must use the modulo game instead of the integer game.
Formally,
$$\begin{aligned} {\mathcal {P}}_{1}= & {} \{\left\{ (abc,abc)\text {,}~(abc,acb)\right\} \text {,}~\left\{ (acb,abc)\text {,}~(acb,acb)\right\} \text {,}~\{(bca,abc)\text {,}~(bca,acb)\},\\&\left\{ (abc,bca)\right\} \text {,}~\left\{ (acb,bca)\right\} \text {,}~\left\{ (bca,bca)\right\} \},\\ {\mathcal {P}}_{2}= & {} \{\left\{ (abc,abc)\text {,}~(acb,abc)\right\} \text {,}~\left\{ (abc,acb)\text {,}~(acb,acb)\right\} \text {,}~\{(abc,bca)\text {,}~(acb,bca)\},\\&\left\{ (bca,abc)\right\} ,\left\{ (bca,acb)\right\} \text {,}~\left\{ (bca,bca)\right\} \}, \end{aligned}$$and
$$\begin{aligned} {\mathcal {P}}_{j}= & {} \{\left\{ (abc,abc)\text {,}~(acb,abc)\text {,}~(abc,acb)\text {,}~(acb,acb)\right\} \text {,}~\{(abc,bca)\text {,}~(acb,bca)\},\\&\{(bca,abc)\text {,}~(bca,acb)\}\text {,}~\left\{ (bca,bca)\right\} \} \end{aligned}$$for each \(j\in N\backslash \{1,2\}\).
If the event, \(F^{-1}(F(R))=\{R^{\prime }\in {\mathcal {R}}:F(R)=F(R^{\prime })\}\), is common knowledge among agents, then the set of socially desirable outcomes F(R) is also common knowledge among agents. We are grateful to an anonymous reviewer for pointing out this fact.
Given \(R_{i}\in {\mathcal {R}}_{i}\), let \(t(R_{i})=\{a\in A|aR_{i}b\) for each \( b\in A\}\) be the set of top outcomes in A according to \(R_{i}\). An SCC F satisfies tops-only if for each pair of preference profiles R, \( R^{\prime }\in {\mathcal {R}}\), if \(t(R_{i})=t(R_{i}^{\prime })\) for each \(i\in N\), then \(F(R)=F(R^{\prime })\).
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An earlier version of this paper was Hagiwara et al. (2016). We are indebted to an associate editor and two anonymous reviewers of the journal for remarkably insightful and detailed comments. For helpful discussions and comments, we are grateful to Luis Corchón, Biung-Ghi Ju, Ryo Kawasaki, Michele Lombardi, Nozomu Muto, Ryan Tierney, and Naoki Yoshihara. Hagiwara gratefully acknowledges the financial support by a research granted from The Murata Science Foundation. This work was partially supported by JSPS KAKENHI Grant Number JP26285045.
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Hagiwara, M., Yamamura, H. & Yamato, T. Implementation with socially responsible agents. Econ Theory Bull 6, 55–62 (2018). https://doi.org/10.1007/s40505-017-0123-6
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DOI: https://doi.org/10.1007/s40505-017-0123-6