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Information Compression in Dynamic Games

Dengwang Tang, Vijay Subramanian, Demosthenis Teneketzis

TL;DR

This work tackles the challenge of computing equilibria in stochastic dynamic games with asymmetric information by introducing two strategy-independent information-state concepts: Mutually Sufficient Information (MSI) and Unilaterally Sufficient Information (USI). The authors prove that MSI guarantees the existence of Bayes-Nash Equilibria and Sequential Equilibria, while USI ensures that the sets of BNE and SE payoffs coincide with those under full information; however, USI can reduce weak Perfect Bayesian Equilibria payoffs in some cases. The paper also shows that certain strategy-dependent compression maps may fail to preserve equilibrium existence or payoff sets, and it discusses MSI/USI realizations in known models along with open problems surrounding strategy-dependent maps. Overall, MSI/USI provide a principled, computationally tractable framework to analyze and design equilibria in complex dynamic games, with potential extensions to broader classes and applications.

Abstract

One of the reasons why stochastic dynamic games with an underlying dynamic system are challenging is since strategic players have access to enormous amount of information which leads to the use of extremely complex strategies at equilibrium. One approach to resolve this challenge is to simplify players' strategies by identifying appropriate compression of information maps so that the players can make decisions solely based on the compressed version of information, called the information state. For finite dynamic games with asymmetric information, inspired by the notion of information state for single-agent control problems, we propose two notions of information states, namely mutually sufficient information (MSI) and unilaterally sufficient information (USI). Both these information states are obtained with information compression maps independent of the strategy profile. We show that Bayes-Nash Equilibria (BNE) and Sequential Equilibria (SE) exist when all players use MSI-based strategies. We prove that when all players employ USI-based strategies the resulting sets of BNE and SE payoff profiles are the same as the sets of BNE and SE payoff profiles resulting when all players use full information-based strategies. We prove that when all players use USI-based strategies the resulting set of weak Perfect Bayesian Equilibrium (wPBE) payoff profiles can be a proper subset of all wPBE payoff profiles. We identify MSI and USI in specific models of dynamic games in the literature. We end by presenting an open problem: Do there exist strategy-dependent information compression maps that guarantee the existence of at least one equilibrium or maintain all equilibria that exist under perfect recall? We show, by a counterexample, that a well-known strategy-dependent information compression map used in the literature does not possess any of the properties of MSI or USI.

Information Compression in Dynamic Games

TL;DR

This work tackles the challenge of computing equilibria in stochastic dynamic games with asymmetric information by introducing two strategy-independent information-state concepts: Mutually Sufficient Information (MSI) and Unilaterally Sufficient Information (USI). The authors prove that MSI guarantees the existence of Bayes-Nash Equilibria and Sequential Equilibria, while USI ensures that the sets of BNE and SE payoffs coincide with those under full information; however, USI can reduce weak Perfect Bayesian Equilibria payoffs in some cases. The paper also shows that certain strategy-dependent compression maps may fail to preserve equilibrium existence or payoff sets, and it discusses MSI/USI realizations in known models along with open problems surrounding strategy-dependent maps. Overall, MSI/USI provide a principled, computationally tractable framework to analyze and design equilibria in complex dynamic games, with potential extensions to broader classes and applications.

Abstract

One of the reasons why stochastic dynamic games with an underlying dynamic system are challenging is since strategic players have access to enormous amount of information which leads to the use of extremely complex strategies at equilibrium. One approach to resolve this challenge is to simplify players' strategies by identifying appropriate compression of information maps so that the players can make decisions solely based on the compressed version of information, called the information state. For finite dynamic games with asymmetric information, inspired by the notion of information state for single-agent control problems, we propose two notions of information states, namely mutually sufficient information (MSI) and unilaterally sufficient information (USI). Both these information states are obtained with information compression maps independent of the strategy profile. We show that Bayes-Nash Equilibria (BNE) and Sequential Equilibria (SE) exist when all players use MSI-based strategies. We prove that when all players employ USI-based strategies the resulting sets of BNE and SE payoff profiles are the same as the sets of BNE and SE payoff profiles resulting when all players use full information-based strategies. We prove that when all players use USI-based strategies the resulting set of weak Perfect Bayesian Equilibrium (wPBE) payoff profiles can be a proper subset of all wPBE payoff profiles. We identify MSI and USI in specific models of dynamic games in the literature. We end by presenting an open problem: Do there exist strategy-dependent information compression maps that guarantee the existence of at least one equilibrium or maintain all equilibria that exist under perfect recall? We show, by a counterexample, that a well-known strategy-dependent information compression map used in the literature does not possess any of the properties of MSI or USI.
Paper Structure (31 sections, 28 theorems, 129 equations, 2 figures)

This paper contains 31 sections, 28 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Contributions
  4. Notation
  5. Game Model and Objectives
  6. Game Model
  7. Objectives
  8. Two Definitions of Information State
  9. Mutually Sufficient Information
  10. Unilaterally Sufficient Information
  11. Comparison
  12. Information-State Based Equilibrium
  13. Information-State Based Bayes--Nash Equilibrium
  14. Information-State Based Sequential Equilibrium
  15. Discussion
  16. ...and 16 more sections

Key Result

Lemma 1

If for all $i\in\mathcal{I}$ and all $K^{-i}$-based strategy profiles $\rho^{-i}$, there exist functions $(\Phi_t^{i, \rho^{-i}})_{t\in\mathcal{T}}$ where $\Phi_t^{i, \rho^{-i}}\colon\mathcal{K}_t^i \mapsto \Delta(\mathcal{X}_t\times \mathcal{K}_t^{-i})$ such that for all behavioral strategies $g^i$, all $t\in\mathcal{T}$, and all $h_t^i$ admissible under $(g^i, \rho^{-i})$, then $K=(K^i)_{i\in\m

Figures (2)

  • Figure 2: Extensive form of the game in Example \ref{['ex:zerosumobsac']}.
  • Figure 3: The pieces (polygons) for which $J^*(\alpha)$ is linear on. The extreme points of the pieces are labeled.

Theorems & Definitions (79)

  • Remark 1
  • Definition 1: Bayes-Nash Equilibrium
  • Definition 2: Sequential Equilibrium
  • Remark 2
  • Remark 3
  • Definition 3
  • Definition 4: Mutually Sufficient Information
  • Lemma 1
  • proof
  • Definition 5: Unilaterally Sufficient Information
  • ...and 69 more