On Game-Theoretic Risk Management (Part Two) -- Algorithms to Compute Nash-Equilibria in Games with Distributions as Payoffs

TL;DR

This paper develops algorithms for computing Nash-Equilibria in games where payoffs are probability distributions, addressing computational challenges with empirical data and proposing modified fictitious play and linear programming methods.

Contribution

It introduces a novel approach to compute equilibria in distribution-based games, including modifications to fictitious play and an exact linear programming method.

Findings

Fictitious play may fail to converge with certain distribution payoffs.

Tail approximation improves convergence in distribution-based games.

Linear programming provides an exact solution method.

Abstract

The game-theoretic risk management framework put forth in the precursor work "Towards a Theory of Games with Payoffs that are Probability-Distributions" (arXiv:1506.07368 [q-fin.EC]) is herein extended by algorithmic details on how to compute equilibria in games where the payoffs are probability distributions. Our approach is "data driven" in the sense that we assume empirical data (measurements, simulation, etc.) to be available that can be compiled into distribution models, which are suitable for efficient decisions about preferences, and setting up and solving games using these as payoffs. While preferences among distributions turn out to be quite simple if nonparametric methods (kernel density estimates) are used, computing Nash-equilibria in games using such models is discovered as inefficient (if not impossible). In fact, we give a counterexample in which fictitious play fails to converge for the (specifically unfortunate) choice of payoff distributions in the game, and introduce a suitable tail approximation of the payoff densities to tackle the issue. The overall procedure is essentially a modified version of fictitious play, and is herein described for standard and multicriteria games, to iteratively deliver an (approximate) Nash-equilibrium. An exact method using linear programming is also given.