Autocratic strategies for iterated games with arbitrary action spaces

Abstract

The recent discovery of zero-determinant strategies for the iterated prisoner's dilemma sparked a surge of interest in the surprising fact that a player can exert unilateral control over iterated interactions. These remarkable strategies, however, are known to exist only in games in which players choose between two alternative actions such as "cooperate" and "defect." Here we introduce a broader class of autocratic strategies by extending zero-determinant strategies to iterated games with more general action spaces. We use the continuous donation game as an example, which represents an instance of the prisoner's dilemma that intuitively extends to a continuous range of cooperation levels. Surprisingly, despite the fact that the opponent has infinitely many donation levels from which to choose, a player can devise an autocratic strategy to enforce a linear relationship between his or her payoff and that of the opponent even when restricting his or her actions to merely two discrete levels of cooperation. In particular, a player can use such a strategy to extort an unfair share of the payoffs from the opponent. Therefore, although the action space of the continuous donation game dwarfs that of the classic prisoner's dilemma, players can still devise relatively simple autocratic and, in particular, extortionate strategies.cooperation | evolutionary game theory | extortion | repeated games G ame theory provides a powerful framework to study interactions between individuals ("players"). Among the most interesting types of interactions are social dilemmas, which result from conflicts of interest between individuals and groups (1, 2). Perhaps the most well-studied model of a social dilemma is the prisoner's dilemma (3). A two-player game with actions, C ("cooperate") and D ("defect"), and payoff matrix,is said to be a prisoner's dilemma if T > R > P > S (4). In a prisoner's dilemma, defection is the dominant action, yet the players can realize higher payoffs from mutual cooperation (R) than they can from mutual defection (P), resulting in a conflict of interest between the individual and the pair, which characterizes social dilemmas. Thus, in a one-shot game (i.e., a single encounter), two opponents have an incentive to defect against one another, but the outcome of mutual defection (the unique Nash equilibrium) is suboptimal for both players. One proposed mechanism for the emergence of cooperation in games such as the prisoner's dilemma is direct reciprocity (5, 6), which entails repeated encounters between players and allows for reciprocation of cooperative behaviors. In an iterated game, a player might forgo the temptation to defect in the present due to the threat of future retaliation-"the shadow of the future"-or the possibility of future rewards for cooperating (4, 7), phenomena for which there is both theoretical and empirical support (8, 9). One example of a strategy for the iterated game is to copy the action of the opponent in the previous round ("tit for tat") (4). Alternatively, a player might choose to re...

Zero-determinant strategies in finitely repeated games

“…However, Eq. (27) indicates that the payoff that an equalizer enforces on the co-player, π Y , depends on the value of p 0 . Because Eq.…”

“…The advent of the ZD strategies has spurred new lines of investigations of direct reciprocity. They include the examination and extension of ZD strategies such as their evolution [11][12][13][14][15][16][17][18][19][20][21][22], multiplayer games [20,[23][24][25][26], continuous action spaces [25][26][27][28], alternating games [28], human reactions to computerized ZD strategies [29,30], and human-human experiments [25,31].…”

Linear algebraic structure of zero-determinant strategies in repeated games

“…The expected payoff vector e = (1,ē T ) T should be given by a non-zero solution of the linear equation α2, · · · , αK). [15] One has SA = (u1, u2, · · · , uK) = 1M b T +SĀ, [16] whereS ≡ (s1, · · · , sN ). The Rouché-Capelli theorem (26) tells us that rankĀ = rank A is a necessary and sufficient condition for the linear equationē TĀ + b T = 0 T K inē to have a solution, that is, for span(Vn) n∈N ′ to be consistent.…”

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Zero-Determinant strategies in finitely repeated n-player games

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