Abstract
The outcomes of many strategic situations such as parlor games or competitive economic scenarios are rankings of the participants, with higher ranks generally at least as desirable as lower ranks. Here we define ranking games as a class of n-player normal-form games with a payoff structure reflecting the players' von Neumann–Morgenstern preferences over their individual ranks. We investigate the computational complexity of a variety of common game-theoretic solution concepts in ranking games and deliver hardness results for iterated weak dominance and mixed Nash equilibrium when there are more than two players, and for pure Nash equilibrium when the number of players is unbounded but the game is described succinctly. This dashes hope that multi-player ranking games can be solved efficiently, despite their profound structural restrictions. Based on these findings, we provide matching upper and lower bounds for three comparative ratios, each of which relates two different solution concepts: the price of cautiousness, the mediation value, and the enforcement value.
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