Iterated Regret Minimization in Game Graphs

TL;DR

This paper extends the concept of iterated regret minimization to infinite-duration two-player quantitative games on graphs, providing algorithms for strategy computation in various graph classes with polynomial or pseudo-polynomial complexity.

Contribution

It introduces methods to compute iterated regret strategies in complex game graphs, expanding the applicability of regret minimization beyond explicit matrix games.

Findings

Polynomial-time algorithms for certain graph classes

Pseudo-polynomial algorithms for others

Strategies for trees and graphs with positive weights

Abstract

Iterated regret minimization has been introduced recently by J.Y. Halpern and R. Pass in classical strategic games. For many games of interest, this new solution concept provides solutions that are judged more reasonable than solutions offered by traditional game concepts -- such as Nash equilibrium --. Although computing iterated regret on explicit matrix game is conceptually and computationally easy, nothing is known about computing the iterated regret on games whose matrices are defined implicitly using game tree, game DAG or, more generally game graphs. In this paper, we investigate iterated regret minimization for infinite duration two-player quantitative non-zero sum games played on graphs. We consider reachability objectives that are not necessarily antagonist. Edges are weighted by integers -- one for each player --, and the payoffs are defined by the sum of the weights along the paths. Depending on the class of graphs, we give either polynomial or pseudo-polynomial time algorithms to compute a strategy that minimizes the regret for a fixed player. We finally give algorithms to compute the strategies of the two players that minimize the iterated regret for trees, and for graphs with strictly positive weights only.