Game Implementation: What Are the Obstructions?

TL;DR

This paper investigates the computational complexity of designing incentives in games to ensure desired strategies, revealing NP-hardness results and providing algorithms for special cases, with implications for game theory and mechanism design.

Contribution

It proves NP-hardness of game implementation even with two players and small strategy sets, and offers a corrected algorithm for specific cases.

Findings

Game implementation is NP-hard for two players.

Designing cost-free incentives is characterized by a stable core concept.

An efficient algorithm is repaired for small strategy and player sets.

Abstract

In many applications, we want to influence the decisions of independent agents by designing incentives for their actions. We revisit a fundamental problem in this area, called GAME IMPLEMENTATION: Given a game in standard form and a set of desired strategies, can we design a set of payment promises such that if the players take the payment promises into account, then all undominated strategies are desired? Furthermore, we aim to minimize the cost, that is, the worst-case amount of payments. We study the tractability of computing such payment promises and determine more closely what obstructions we may have to overcome in doing so. We show that GAME IMPLEMENTATION is NP-hard even for two players, solving in particular a long open question (Eidenbenz et al. 2011) and suggesting more restrictions are necessary to obtain tractability results. We thus study the regime in which players have only a small constant number of strategies and obtain the following. First, this case remains NP-hard even if each player's utility depends only on three others. Second, we repair a flawed efficient algorithm for the case of both small number of strategies and small number of players. Among further results, we characterize sets of desired strategies that can be implemented at zero cost as a kind of stable core of the game.