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Games with Incomplete Information Played by Risk-Revising Players

Shutian Liu

Abstract

This paper introduces risk-revising players to a class of games with incomplete information. These players enter the game with ex ante risk preferences represented by coherent risk measures and develop time-consistent interim revisions of them contingent on their private information. The standard Nash equilibrium at ex ante stage and Bayesian Nash equilibrium at interim stage are extended to their risk-averse counterparts. Risk-revising Bayesian Nash equilibrium is proposed to capture behavioral outcomes resulting from interim plays based on revised risk preferences. We discuss existence results of these equilibrium concepts. When players' risk revisions correspond to their ex ante equilibrium play, connections are established between equilibria at the ex ante and interim stages. The effect of risk-aversion is analyzed using comparative statics. With the help of the dual representation of risk measures, we illustrate the role of risk-aversion in representing inconsistent beliefs. A numerical example is presented to illustrate the proposed equilibrium concepts under a specific choice of risk preference.

Games with Incomplete Information Played by Risk-Revising Players

Abstract

This paper introduces risk-revising players to a class of games with incomplete information. These players enter the game with ex ante risk preferences represented by coherent risk measures and develop time-consistent interim revisions of them contingent on their private information. The standard Nash equilibrium at ex ante stage and Bayesian Nash equilibrium at interim stage are extended to their risk-averse counterparts. Risk-revising Bayesian Nash equilibrium is proposed to capture behavioral outcomes resulting from interim plays based on revised risk preferences. We discuss existence results of these equilibrium concepts. When players' risk revisions correspond to their ex ante equilibrium play, connections are established between equilibria at the ex ante and interim stages. The effect of risk-aversion is analyzed using comparative statics. With the help of the dual representation of risk measures, we illustrate the role of risk-aversion in representing inconsistent beliefs. A numerical example is presented to illustrate the proposed equilibrium concepts under a specific choice of risk preference.
Paper Structure (16 sections, 12 theorems, 40 equations, 6 figures, 5 tables)

This paper contains 16 sections, 12 theorems, 40 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Game model
  3. Risk-averse game with incomplete information
  4. Equilibria played by risk-revising players
  5. Ex ante equilibria.
  6. Interim equilibria
  7. Connection to existing equilibrium notions.
  8. Equilibrium analysis
  9. Equilibrium existence
  10. Existence in infinite games
  11. Connections between ex ante and interim plays
  12. Effects of risk-aversion
  13. Behaviors of equilibria
  14. Commonization of inconsistent beliefs
  15. Numerical example
  16. ...and 1 more sections

Key Result

Lemma 1

(Proposition 3.1 of ruszczynski2006optimization). Suppose that $\rho:\mathcal{L}^{\infty}(T,\mathcal{F},P)\rightarrow \mathbb{R}$ satisfies properties (i) and (ii) in Definition def:CRM, then $\rho(\cdot)$ is continuous.

Figures (6)

  • Figure 1: Game tree. Players' costs are show under the leaf nodes. For instance, under types $GG$ and strategies $(Ds)$, player $1$ observes loss $11$ and player $2$ observes loss $52$. Dashed lines represent knowledge in interim play.
  • Figure 1: Ex ante risks. Risks corresponding to best responses are underlined.
  • Figure 2: Random losses and risks of player $1$ without risk revision under different types and pure strategies at the interim stage given player $2$ playing ss. The left table corresponds to type $G$; the right table corresponds to type $H$.
  • Figure 3: Random losses and risks of player $2$ without risk revision under different types and pure strategies at the interim stage given player $1$ playing DD. The left table corresponds to type $G$; the right table corresponds to type $H$.
  • Figure 4: Random losses and risks of player $1$ without risk revision under different types and pure strategies at the interim stage given player $2$ playing ds. The left table corresponds to type $G$; the right table corresponds to type $H$.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • Theorem 1
  • proof
  • ...and 17 more