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

TL;DR

This work analyzes synchronous versus asynchronous coalitions in multiplayer zero-sum games, focusing on the continuous three-player Guts Poker as a key case. It defines the coalition values $V_S$, $V_A$, and $V_N=0$, showing how $V_A$ can be nonconvex and differ from Nash outcomes, and provides a complete analytical treatment of the continuous guts game yielding $V_A=V_N=0$ with $V_S<0$, alongside a detailed examination of discrete analogs like Odds-Evens and Rock-Paper-Scissors. The authors develop and benchmark numerical methods (BFGS, SLSQP, fictitious play, and joint variants) and apply them to both random and discretized game instances, revealing typical gap distributions and offering guidance on algorithm choice. The results illuminate when asynchronous coalitions offer an advantage, establish benchmarks for future algorithmic work in nonconvex multiplayer optimization, and pose open questions for higher-dimensional and correlated settings with practical implications for game design and strategy.

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.

Figures (21)

• Figure 1: Numerical evaluation of inner loop of Maximin, illustrating oscillation between local minima; cf. Figure \ref{['negativefig']}.
• Figure 2: Blowup of graph plotting best response $\alpha_a$ and $\alpha_b$ vs. $p_1^*$. Maximin occurs at $p_1^*\approx 0.6437$, $\alpha\approx -0.0056$, at intersection of $\alpha_a$ and $\alpha_b$.
• Figure 3: Blowup of graph plotting best response $\alpha_a$ and $\alpha_b$ vs. $p_1^*$. with tangent lines at crossing points showing $\alpha_a'\approx .02$, $\alpha_b'\approx -1.4$ at maximin occurring at intersection of $\alpha_a$ and $\alpha_b$, giving probability $y\approx .875$ for strategy $(p_a^3,p_a^3)$ and $(1-y)\approx .125$ for strategy $(0,p_b^3)$, where, by our formulae, $p_3^a\approx .6764$ and $p_3^b\approx .864$.
• Figure 4: Payoff for player 1 against optimal player 2-3 strategy, plotted vs. $p_1^*$.
• Figure 5: Pyramid function, contour map for the RPS OMI maximin.

Theorems & Definitions (40)

• Proposition 2.1
• Proposition 2.2
• Theorem 3.1
• Proposition 4.1
• Remark 4.2
• Remark 4.3
• Proposition 4.4
• Remark 4.5
• Proposition 5.1: CCZ
• Lemma 5.2