Generalized Bayesian Nash Equilibrium with Continuous Type and Action Spaces

TL;DR

This paper introduces a generalized Bayesian game model with type-dependent and action-dependent spaces, proving equilibrium existence and uniqueness under certain conditions, and proposes an approximation method for computing equilibria.

Contribution

It extends Bayesian game theory to include action spaces dependent on types and rivals' actions, providing existence, uniqueness results, and an approximation scheme.

Findings

Existence of continuous GBNE under moderate conditions

Uniqueness when action space depends only on own type

Approximation scheme converges and performs well in tests

Abstract

Bayesian game is a strategic decision-making model where each player's type parameter characterizing its own objective is private information: each player knows its own type but not its rivals' types, and Bayesian Nash equilibrium (BNE) is an outcome of this game where each player makes a strategic optimal decision according to its own type under the Nash conjecture. In this paper, we advance the literature by considering a generalized Bayesian game where each player's action space depends on its own type parameter and the rivals' actions. This reflects the fact that in practical applications, a firm's feasible action is often related to its own type (e.g. marginal cost) and the rivals' actions (e.g. common resource constraints in a competitive market). Under some moderate conditions, we demonstrate existence of continuous generalized Bayesian Nash equilibria (GBNE) and uniqueness of such an equilibrium when each player's action space is only dependent on its type. In the case that each player's action space is also dependent on rivals' actions, we give a simple example to show that uniqueness of GBNE is not guaranteed under standard monotone conditions. To compute an approximate GBNE, we restrict each player's response function to the space of polynomial functions of its type parameter and consequently convert the GBNE problem to a stochastic generalized Nash equilibrium problem (SGNE). To justify the approximation, we discuss convergence of the approximation scheme. Some preliminary numerical test results show that the approximation scheme works well.