Cost Minimization for Equilibrium Transition

TL;DR

This paper investigates the computational complexity of incentivizing players to transition between Nash equilibria using monetary rewards, providing complexity results and efficient algorithms for special cases.

Contribution

It introduces the problem of cost minimization for equilibrium transition, proves NP-completeness and APX-hardness, and offers polynomial-time algorithms for fixed parameters and single-peaked utilities.

Findings

Determining zero minimum reward is NP-complete.

Computing the minimum reward is APX-hard.

Efficient algorithms exist when either k or n is fixed, or for single-peaked utilities.

Abstract

In this paper, we delve into the problem of using monetary incentives to encourage players to shift from an initial Nash equilibrium to a more favorable one within a game. Our main focus revolves around computing the minimum reward required to facilitate this equilibrium transition. The game involves a single row player who possesses $m$ strategies and $k$ column players, each endowed with $n$ strategies. Our findings reveal that determining whether the minimum reward is zero is NP-complete, and computing the minimum reward becomes APX-hard. Nonetheless, we bring some positive news, as this problem can be efficiently handled if either $k$ or $n$ is a fixed constant. Furthermore, we have devised an approximation algorithm with an additive error that runs in polynomial time. Lastly, we explore a specific case wherein the utility functions exhibit single-peaked characteristics, and we successfully demonstrate that the optimal reward can be computed in polynomial time.