Zero-Determinant strategies in finitely repeated n-player games

Abstract

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.

Zero-determinant strategies under observation errors in repeated games

“…Our results are limited to the two player RPD games. Other studies have focused on n-player games [19,26,27,56]. It is worth investigating games including observation errors and a discount factor for n-player games .…”

“…Finally, our results are limited to the two-player RPD games. Other studies have focused on n-player games [19,30,33,34]. Investigating the conditions for the existence of ZD strategies in n-player games warrant future work.…”

“…Thus, some studies have focused on a discount factor for ZD strategies [30,31,[54][55][56] and mathematically found the minimum discount factor above which the ZD strategies can exist [55].…”

Zero-Determinant Strategies in Repeated Multiplayer Social Dilemmas With Discounted Payoffs

“…In this article, we use the framework of ZD strategies in infinitely repeated multiplayer social dilemmas from [9] and extend it to the case in which future payoffs are discounted with a fixed and common discount factor, or equivalently, to a repeated game with a finite expected number of rounds. We then build upon our results in [4], in which enforceable payoff relations were characterized, by developing new theory that allows us to express threshold discount factors that determine how fast a desired linear payoff relation can be enforced in a multiplayer social dilemma game. These results extend the work of [11] to a broad class of multiplayer social dilemmas.…”

Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

“…Thus, it is quite natural to incorporate a discount factor into the model. Some studies have revealed the existence of ZD strategies with a discount factor [26,27,[35][36][37]. Hence, in this study, we incorporate a discount factor into the model and explore whether the opponent is led to an unconditional cooperation.…”