Multi-group Bayesian Games

TL;DR

This work introduces multi-group Bayesian games ($MBG$) to capture intra-group cooperation or noncooperation under incomplete information, and proposes a multi-group ex-ante agent game ($MEAG$) transformation that converts MBGs into normal-form games. It establishes a one-to-one correspondence between MBG equilibria and MEAG equilibria and proves that (strong) potentiality is preserved by the transformation, enabling equilibrium computation via potential equations. The paper provides explicit algebraic forms for payoffs and potential functions, derives potential equations (e.g., $\Lambda \xi = \Psi P^{T}$ and $\Lambda \xi = \Theta P^{T}$), and offers algorithms to find (strongly) MBNE by solving the MEAG’s equilibria, demonstrated on a first-price auction example. This framework allows tractable analysis of cooperative versus noncooperative behavior in group-structured Bayesian settings with broad applications in routing, auctions, and multi-agent systems.

Abstract

This paper presents a model of multi-group Bayesian games (MBGs) to describe the group behavior in Bayesian games, and gives methods to find (strongly) multi-group Bayesian Nash equilibria (MBNE) of this model with a proposed transformation. MBNE represent the optimal strategy \textit{profiles} under the situation where players within a group play a cooperative game, while strongly MBNE characterize the optimal strategy \textit{profiles} under the situation where players within a group play a noncooperative game. Firstly, we propose a model of MBGs and give a transformation to convert any MBG into a multi-group ex-ante agent game (MEAG) which is a normal-form game. Secondly, we give a sufficient and necessary condition for a MBG's MEAG to be (strongly) potential. If it is (strongly) potential, all its (strongly) Nash equilibria can be found, and then all (strongly) MBNE of the MBG can be obtained by leveraging the transformation's good properties. Finally, we provide algorithms for finding (strongly) MBNE of a MBG whose MEAG is (strongly) potential and use an illustrative example to verify the correctness of our results.

Figures (4)

• Figure 1: Compare the differences between ① a Bayesian game, ② a multi-group Bayesian game where players in a group play a cooperative game, and ③ a multi-group Bayesian game where players in a group play a noncooperative game by using an auction problem.
• Figure 2: The traffic network.
• Figure 3: The properties of multi-group ex-ante agent transformation.
• Figure 4: A potential function and all MBNE of the MBG.

Theorems & Definitions (33)

• Definition 1
• Lemma 1
• Lemma 2
• Corollary 1
• Definition 2
• Definition 3
• Example 1
• Remark 1
• Definition 4
• Example 2