# Solving zero-sum extensive-form games with arbitrary payoff uncertainty models

## TL;DR

This paper introduces Harsanyi-Counterfactual Regret Minimization, a novel method to solve two-player zero-sum extensive-form games with arbitrary continuous payoff distributions, extending AI techniques to more complex uncertainty models.

## Contribution

It proposes a new approach combining Harsanyi transformation with counterfactual regret minimization to handle arbitrary payoff uncertainties in extensive-form games.

## Key findings

- Successfully applied to a published problem example.
- Demonstrated effectiveness in handling continuous payoff distributions.
- Extended AI game-solving techniques to broader uncertainty models.

## Abstract

Modeling strategic conflict from a game theoretical perspective involves dealing with epistemic uncertainty. Payoff uncertainty models are typically restricted to simple probability models due to computational restrictions. Recent breakthroughs Artificial Intelligence (AI) research applied to Poker have resulted in novel approximation approaches such as counterfactual regret minimization, that can successfully deal with large-scale imperfect games. By drawing from these ideas, this work addresses the problem of arbitrary continuous payoff distributions. We propose a method, Harsanyi-Counterfactual Regret Minimization, to solve two-player zero-sum extensive-form games with arbitrary payoff distribution models. Given a game $\Gamma$, using a Harsanyi transformation we generate a new game $\Gamma^\#$ to which we later apply Counterfactual Regret Minimization to obtain $\varepsilon$-Nash equilibria. We include numerical experiments showing how the method can be applied to a previously published problem.