[Paper Review] Convergence of Bayesian Nash Equilibrium in Infinite Bayesian Games under Discretization

Linan Huang, Quanyan Zhu|arXiv (Cornell University)|Feb 24, 2021

Game Theory and Applications18 references4 citations

TL;DR

This paper establishes the existence of Bayesian Nash Equilibrium (BNE) in general-sum infinite Bayesian games with continuous types and finite actions, under continuity assumptions on utility functions and prior distributions. It proves that BNE strategies of the infinite game can be approximated through weak convergence of BNE strategies from a sequence of discretized finite games, enabling a constructive algorithm to compute $\varepsilon$-BNE via discretization and solution of finite approximations.

ABSTRACT

We prove the existence of Bayesian Nash Equilibrium (BNE) of general-sum Bayesian games with continuous types and finite actions under the conditions that the utility functions and the prior type distributions are continuous concerning the players' types. Moreover, there exists a sequence of discretized Bayesian games whose BNE strategies converge weakly to a BNE strategy of the infinite Bayesian game. Our proof establishes a connection between the equilibria of the infinite Bayesian game and those of finite approximations, which leads to an algorithm to construct $\varepsilon$-BNE of infinite Bayesian games by discretizing players' type spaces.

Motivation & Objective

• To establish the existence of Bayesian Nash Equilibrium (BNE) in general-sum Bayesian games with continuous types and finite actions.
• To demonstrate that BNE strategies of the infinite game can be approximated by BNE strategies from a sequence of discretized finite games.
• To provide a constructive algorithm for computing $\varepsilon$-BNE of infinite Bayesian games using discretization and finite game solution.
• To extend the theoretical convergence result to $N$-player games and higher-dimensional compact type spaces via cardinality equivalence with $[0,1]$.

Proposed method

• Reformulate the BNE condition in distributional form to enable weak convergence analysis.
• Use the fact that any uncountable compact metric space of cardinality of the continuum is homeomorphic to $[0,1]$ to generalize results beyond $[0,1]$.
• Construct a sequence of finite discretizations of the continuous type space using uniform or adaptive partitioning.
• Prove weak convergence of BNE strategies from the discretized games to a BNE strategy of the original infinite game under continuity of utility functions and prior distributions.
• Develop Algorithm 1 to iteratively discretize the game, solve the finite approximation, and check for $\varepsilon$-BNE using a performance verification condition.
• Reformulate the finite game solution as a bilinear program $C^K$, and under specific conditions, reduce it to a linear program via appropriate choice of weights $\alpha_i(\theta_i)$.

Experimental results

Research questions

• RQ1 Does a Bayesian Nash Equilibrium exist in general-sum infinite Bayesian games with continuous types and finite actions under continuity of utility and prior distributions?
• RQ2 Can the BNE of an infinite Bayesian game be approximated by BNE strategies from a sequence of finite discretized games?
• RQ3 Does the sequence of BNE strategies from finite approximations converge weakly to a BNE of the original infinite game?
• RQ4 Can an algorithm be constructed to compute an $\varepsilon$-BNE of an infinite Bayesian game via discretization and finite game solution?
• RQ5 Under what conditions can the bilinear program for finite game solution be reduced to a linear program?

Key findings

• The existence of a Bayesian Nash Equilibrium is proven for general-sum infinite Bayesian games with continuous types and finite actions, assuming continuity of utility functions and prior distributions.
• There exists a sequence of discretized finite Bayesian games such that the BNE strategies of these games converge weakly to a BNE strategy of the original infinite game.
• The convergence result extends to $N$-player general-sum games and higher-dimensional compact type spaces via homeomorphism to $[0,1]$.
• An algorithm is proposed to compute an $\varepsilon$-BNE of the infinite game by iteratively discretizing the type space and solving the finite approximations.
• Under a specific condition where $m_2(\theta_2)\bar{u}^{x,y}(\theta_1,\theta_2) = -m_1(\theta_1)\bar{v}^{x,y}(\theta_1,\theta_2)$, the bilinear program $C^K$ can be reformulated as a linear program by choosing $\alpha_i(\theta_i) = \bar{b}_i(\theta_i)/m_i(\theta_i)$.
• The algorithm provides a finite-step verification for $\varepsilon$-BNE, avoiding misleading results from finite approximations by checking equilibrium conditions at each discretization level.