Cooperation in alternating interactions with memory constraints

Park, Peter S.; Nowak, Martin A.; Hilbe, Christian

doi:10.1038/s41467-022-28336-2

Cited by 26 publications

(13 citation statements)

References 70 publications

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“…An unconditional player will imitate a memory-1 strategy’s average cooperation probability. Similarly, a reactive player will infer effective values of his strategy components p and q , which can also be calculated from the invariant distribution [ 53 ]. We find that this variation leads to a weakening of the memory dilemma.…”

Section: Resultsmentioning

confidence: 99%

Direct reciprocity between individuals that use different strategy spaces

et al. 2022

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In repeated interactions, players can use strategies that respond to the outcome of previous rounds. Much of the existing literature on direct reciprocity assumes that all competing individuals use the same strategy space. Here, we study both learning and evolutionary dynamics of players that differ in the strategy space they explore. We focus on the infinitely repeated donation game and compare three natural strategy spaces: memory-1 strategies, which consider the last moves of both players, reactive strategies, which respond to the last move of the co-player, and unconditional strategies. These three strategy spaces differ in the memory capacity that is needed. We compute the long term average payoff that is achieved in a pairwise learning process. We find that smaller strategy spaces can dominate larger ones. For weak selection, unconditional players dominate both reactive and memory-1 players. For intermediate selection, reactive players dominate memory-1 players. Only for strong selection and low cost-to-benefit ratio, memory-1 players dominate the others. We observe that the supergame between strategy spaces can be a social dilemma: maximum payoff is achieved if both players explore a larger strategy space, but smaller strategy spaces dominate.

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Section: Resultsmentioning

confidence: 99%

Direct reciprocity between individuals that use different strategy spaces

et al. 2022

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“…In the infinitely repeated donation game, two players decide simultaneously in each round whether to cooperate (C) or to defect (D). (For the asynchronous donation game or Prisoner's Dilemma, see [43][44][45][46][47].) Cooperation incurs a cost c while conferring a benefit b on the other player [17,[48][49][50][51][52].…”

Section: Methodsmentioning

confidence: 99%

A geometric process of evolutionary game dynamics

LaPorte,

Nowak

2023

J. R. Soc. Interface.

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Many evolutionary processes occur in phenotype spaces which are continuous. It is therefore of interest to explore how selection operates in continuous spaces. One approach is adaptive dynamics, which assumes that mutants are local. Here we study a different process which also allows non-local mutants. We assume that a resident population is challenged by an invader who uses a strategy chosen from a random distribution on the space of all strategies. We study the repeated donation game of direct reciprocity. We consider reactive strategies given by two probabilities, denoting respectively the probability to cooperate after the co-player has cooperated or defected. The strategy space is the unit square. We derive analytic formulae for the stationary distribution of evolutionary dynamics and for the average cooperation rate as function of the cost-to-benefit ratio. For positive reactive strategies, we prove that cooperation is more abundant than defection if the area of the cooperative region is greater than 1/2 which is equivalent to benefit, b , divided by cost, c , exceeding 2 + 2 . We introduce the concept of strategies that are stable with probability one. We also study an extended process and discuss other games.

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“…When both players adopt memory-1 strategies, there is an explicit formula to derive their average payoffs (as described in the next section). Based on this formula, it is possible to characterize all Nash equilibria among the memory-1 strategies [32][33][34][35][36][37]. In general, however the payoff formula yields a complex expression in the players' conditional cooperation probabilities p ij .…”

Section: Introductionmentioning

confidence: 99%

Adaptive dynamics of memory-1 strategies in the repeated donation game

LaPorte

Hilbe

Nowak

2023

Preprint

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Social interactions often take the form of a social dilemma: collectively, individuals fare best if everybody cooperates, yet each single individual is tempted to free ride. Social dilemmas can be resolved when individuals interact repeatedly. Repetition allows individuals to adopt reciprocal strategies which incentivize cooperation. The most basic model to study reciprocity is the repeated donation game, a variant of the repeated prisoner’s dilemma. Two players interact over many rounds, in which they repeatedly decide whether to cooperate or to defect. To make their decisions, they need a strategy that tells them what to do depending on the history of previous play. Memory-1 strategies depend on the previous round only. Even though memory-1 strategies are among the most elementary strategies of reciprocity, their evolutionary dynamics has been difficult to study analytically. As a result, most previous work relies on simulations. Here, we derive and analyze their adaptive dynamics. We show that the four-dimensional space of memory-1 strategies has an invariant three-dimensional subspace, generated by the memory-1 counting strategies. Counting strategies record how many players cooperated in the previous round, without considering who cooperated. We give a partial characterization of adaptive dynamics for memory-1 strategies and a full characterization for memory-1 counting strategies.Author summaryDirect reciprocity is a mechanism for evolution of cooperation based on the repeated interaction of the same players. In the most basic setting, we consider a game between two players and in each round they choose between cooperation and defection. Hence, there are four possible outcomes: (i) both cooperate; (ii) I cooperate, you defect; (ii) I defect, you cooperate; (iv) both defect. A memory-1 strategy for playing this game is characterized by four quantities which specify the probabilities to cooperate in the next round depending on the outcome of the current round. We study evolutionary dynamics in the space of all memory-1 strategies. We assume that mutant strategies are generated in close proximity to the existing strategies, and therefore we can use the framework of adaptive dynamics, which is deterministic.

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Cooperation in alternating interactions with memory constraints

Cited by 26 publications

References 70 publications

Direct reciprocity between individuals that use different strategy spaces

Direct reciprocity between individuals that use different strategy spaces

A geometric process of evolutionary game dynamics

Adaptive dynamics of memory-1 strategies in the repeated donation game

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