Generalized Bayesian Nash Equilibrium with Continuous Type and Action Spaces

TL;DR

The paper develops a generalized Bayesian Nash equilibrium (GBNE) framework with continuous type and action spaces where feasible sets depend on own type and rivals’ actions, establishing existence of continuous GBNE via Schauder fixed points and detailing conditions for uniqueness when action sets depend only on type. It introduces a polynomial decision-rule approximation to compute GBNE by recasting it as a stochastic generalized Nash equilibrium (SGNE) and proves convergence of the polynomial GBNE to the true GBNE as the polynomial degree grows. The work also extends to risk-aware feasibility through CVaR constraints, and demonstrates the approach with numerical tests in a symmetric Bayesian Cournot setting, showing convergence behavior and trade-offs between ECC and CCC formulations. Collectively, the results provide both theoretical guarantees and practical computational methods for GBNE with rich type-action dependence and continuous spaces, enabling applications in rent-seeking, price competition, and production economies with shared resource constraints.

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.

Figures (4)

• Figure 1: Illustration of the proof of Case 2 in Part (ii), where notation '$\longleftrightarrow$' denotes $\mathsf {d l}(x",X^*(y,t'))$ which is larger than $\epsilon$.
• Figure 2: Illustration the proof of Part (ii), where $\mathcal{C}_d(f^d)$ denotes the polynomial feasible set defined by $f^d$.
• Figure 3: Optimal response functions at polynomial GBNE of symmetric BCG with ECC. The blue dashed curve represents the optimal response function at the true GBNE.
• Figure 4: Optimal response functions at the polynomial equilibria with degree $d=1$ for symmetric BCG-CCC model. The blue dashed curve represents the optimal response function at the true GBNE.

Theorems & Definitions (18)

• Definition 2.1: Generalized Bayesian Nash equilibrium (GBNE)
• Proposition 2.1: Equivalent formulations of the GBNE model
• Example 2.1: Rent-seeking games
• Example 2.2: The price competition in discrete choice models liu2024Bayesian
• Example 2.3: Cournot games
• Lemma 3.1: Uniform continuity of the feasible set of (\ref{['eqn:para_min_problem']})
• Theorem 3.1: Uniform continuity of the optimal solution set of (\ref{['eqn:para_min_problem']})
• Theorem 3.2: Schauder's fixed point theorem zeilder1985nonlinear
• Proposition 3.1: Arzela-Ascoli Theorem Theorem A5 in rudin1973functional
• Remark 3.1