Sychronous vs. asynchronous coalitions in multiplayer games, with applications to guts poker

TL;DR

This paper analyzes the differences between synchronous and asynchronous coalitions in multiplayer games, focusing on continuous Guts poker, and introduces methods for approximating optimal strategies and conditions for their behavior.

Contribution

It provides a comprehensive comparison of coalition strategies, introduces numerical methods for nonconvex optimization, and resolves an open problem regarding asynchronous coalition values in 3-player Guts.

Findings

Optimal asynchronous coalition value for 3-player Guts equals zero.

Sufficient conditions for different coalition behaviors are established.

Methods for approximating nonconvex optimization problems in game theory are discussed.

Abstract

We study the issue introduced by Buck-Lee-Platnick-Wheeler-Zumbrun of synchronous vs. asynchronous coalitions in multiplayer games, that is, the difference between coalitions with full and partial communication, with a specific interest in the context of continuous Guts poker where this problem was originally formulated. We observe for general symmetric multiplayer games, with players 2-n in coalition against player 1, that there are three values, corresponding to symmetric Nash equilibrium, optimal asynchronous, and optimal synchronous strategies, in that order, for which inequalities may for different examples be strict or nonstrict (i.e., equality) in any combination. Different from Nash equilibria and synchronous optima, which may be phrased as convex optimization problems, or classical 2-player games, determination of asynchronous optima is a nonconvex optimization problem. We discuss methods of numerical approximation of this optimum, and examine performance on 3-player rock-paper-scissors and discretized Guts poker. Finally, we present sufficient conditions guaranteeing different possibilities for behavior, based on concave/convexity properties of the payoff function. These answer in the affirmative the open problem posed by Buck-Lee-Platnick-Wheeler-Zumbrun whether the optimal asynchronous coalition value for 3-player guts is equal to the Nash equilibrium value zero. At the same time, we present a number of new results regarding synchronous coalition play for continuous $3$-player guts.