Zero-Determinant strategies in finitely repeated n-player games
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In two-player repeated games, Zero-Determinant (ZD) strategies are a class of strategies that can enable a player to unilaterally enforce a linear payoff relation between her own and her opponent's payoff irrespective of the opponent's strategy. This manipulative nature of the ZD strategies attracted significant attention from researchers due to its close connection to controlling distributively the outcome of evolutionary games in large populations. In this paper, we study the existence of ZD strategies in repeated n-player games with a finite but undetermined time horizon. Necessary and sufficient conditions are derived for a linear relation to be enforceable by a ZD strategist in n-player social dilemmas, in which the expected number of rounds can be captured by a fixed and common discount factor (0<δ<1). Thresholds exist for such a discount factor above which generous, extortionate and equalizer payoff relations can be enforced. For the first time in the studies of repeated games, ZD strategies are examined in the setting of finitely repeated n-player, two-action games. Our results show that depending on the group size and the ZD-strategist's initial probability to cooperate, for finitely repeated n-player social dilemmas, it is possible for extortionate, generous and equalizer ZD-strategies to exist. The threshold discount factors rely on the slope and baseline payoff of the desired linear relation and the variation in the "one-shot" payoffs of the n-player game. To show the utility of our general results, we apply them to a linear n-player public goods game.
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