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Communication games, sequential equilibrium, and mediators

Ivan Geffner, Joseph Y. Halpern

TL;DR

This work studies implementing $k$-resilient sequential equilibria without a trusted mediator in both synchronous and asynchronous distributed environments. The authors show that any $k$-resilient sequential equilibrium with a mediator in $ ext{Γ}_d$ can be implemented in cheap-talk $ ext{Γ}_{ ext{CT}}$ when $n > 3k$ (synchronous) or $n > 4k$ (asynchronous), by constructing a $k$-resilient Nash equilibrium using verifiable secret sharing and circuit computation to simulate the mediator, and then extending to a $k$-resilient sequential equilibrium via a $k$-paranoid belief system. They provide a precise characterization of SE$_k( ext{Γ}_d)$ in terms of correlated and communication equilibria for normal-form and Bayesian games, respectively, and establish the corresponding asynchronous counterparts, with tight lower bounds aligning with prior results. The methodology connects mediator-based solution concepts to cheap-talk implementations and underscores the interplay between distributed cryptographic primitives (VSS, CC, consensus) and game-theoretic notions (nash, sequential, communication equilibria). Overall, the results advance robust mechanism design for coalition-resilient strategic interactions in both synchronous and asynchronous networks, with implications for correlated and communication equilibria in distributed settings.

Abstract

We consider $k$-resilient sequential equilibria, strategy profiles where no player in a coalition of at most $k$ players believes that it can increase its utility by deviating, regardless of its local state. We prove that all $k$-resilient sequential equilibria that can be implemented with a trusted mediator can also be implemented without the mediator in a synchronous system of $n$ players if $n >3k$. In asynchronous systems, where there is no global notion of time and messages may take arbitrarily long to get to their recipient, we prove that a $k$-resilient sequential equilibrium with a mediator can be implemented without the mediator if $n > 4k$. These results match the lower bounds given by Abraham, Dolev, and Halpern (2008) and Geffner and Halpern (2023) for implementing a Nash equilibrium without a mediator (which are easily seen to apply to implementing a sequential equilibrium) and improve the results of Gerardi, who showed that, in the case that $k=1$, a sequential equilibrium can be implemented in synchronous systems if $n \ge 5$.

Communication games, sequential equilibrium, and mediators

TL;DR

This work studies implementing -resilient sequential equilibria without a trusted mediator in both synchronous and asynchronous distributed environments. The authors show that any -resilient sequential equilibrium with a mediator in can be implemented in cheap-talk when (synchronous) or (asynchronous), by constructing a -resilient Nash equilibrium using verifiable secret sharing and circuit computation to simulate the mediator, and then extending to a -resilient sequential equilibrium via a -paranoid belief system. They provide a precise characterization of SE in terms of correlated and communication equilibria for normal-form and Bayesian games, respectively, and establish the corresponding asynchronous counterparts, with tight lower bounds aligning with prior results. The methodology connects mediator-based solution concepts to cheap-talk implementations and underscores the interplay between distributed cryptographic primitives (VSS, CC, consensus) and game-theoretic notions (nash, sequential, communication equilibria). Overall, the results advance robust mechanism design for coalition-resilient strategic interactions in both synchronous and asynchronous networks, with implications for correlated and communication equilibria in distributed settings.

Abstract

We consider -resilient sequential equilibria, strategy profiles where no player in a coalition of at most players believes that it can increase its utility by deviating, regardless of its local state. We prove that all -resilient sequential equilibria that can be implemented with a trusted mediator can also be implemented without the mediator in a synchronous system of players if . In asynchronous systems, where there is no global notion of time and messages may take arbitrarily long to get to their recipient, we prove that a -resilient sequential equilibrium with a mediator can be implemented without the mediator if . These results match the lower bounds given by Abraham, Dolev, and Halpern (2008) and Geffner and Halpern (2023) for implementing a Nash equilibrium without a mediator (which are easily seen to apply to implementing a sequential equilibrium) and improve the results of Gerardi, who showed that, in the case that , a sequential equilibrium can be implemented in synchronous systems if .
Paper Structure (23 sections, 18 theorems, 19 equations)

This paper contains 23 sections, 18 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. $k$-resilient equilibrium
  3. $k$-resilient Nash, sequential, Bayesian Nash, and communication equilibrium
  4. Cheap-talk games
  5. The main result
  6. Outline of the proof
  7. Proof of Theorem \ref{['thm:main']}
  8. Constructing a $k$-resilient Nash equilibrium
  9. Constructing a $k$-resilient sequential equilibrium
  10. Characterization of $SE_k(\Gamma_{d})$ and $SE_k^S(\Gamma_{d})$
  11. The asynchronous setting
  12. Main results
  13. Proof of Theorem \ref{['thm:main2']} (outline)
  14. Characterization of $SE_k(\Gamma_{d, asyn})$ and $SE_k^S(\Gamma_{d, asyn})$
  15. Extending the set of equilibria
  16. ...and 8 more sections

Key Result

Theorem 1

If $\Gamma = (P, T, q, A,U)$ is a Bayesian game for $n$ players and $n > 3k$, then $SE_k(\Gamma_{\mathit{CT}}) = SE_k(\Gamma_{d})$ and $SE_k^S(\Gamma_{\mathit{CT}}) = SE_k^S(\Gamma_{d})$.

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Example 1
  • Theorem 1
  • Proposition 1
  • ...and 36 more