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Theory and Decision

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18.08.2015

The emergence of reciprocally beneficial cooperation

verfasst von: Sergio Beraldo, Robert Sugden

Erschienen in: Theory and Decision | Ausgabe 4/2016

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Abstract

We offer a new and robust model of the emergence and persistence of cooperation when interactions are anonymous, the population is well mixed, and evolution selects strategies according to material payoffs. The model has a Prisoner’s Dilemma structure, but with an outside option of non-participation. The payoff to mutual cooperation is stochastic; with positive probability, it exceeds that from cheating against a cooperator. Under mild conditions, mutually beneficial cooperation occurs in equilibrium. This is possible because the non-participation option holds down the equilibrium frequency of cheating. Dynamic properties of the model are investigated theoretically and through simulations based on replicator dynamics.

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In parallel with attempts to explain human cooperation in terms of self-interest, there is a large literature in economics in which cooperation is explained as a product of non-selfish motivations, such as altruism, preferences for equality, fairness or welfare-maximisation, team reasoning, or intrinsic desires to cooperate. Without denying the existence of such motivations, we think it is useful to investigate the explanatory power of the self-interest assumption.

In some simplified models of the Prisoner’s Dilemma, each player makes an independent decision about whether to incur a fixed cost to confer a larger fixed benefit on the other player; payoffs in the four possible outcomes are defined as sums of these costs and benefits. Because we want to distinguish between cheating and non-participation, our model does not have this additivity property.

The game would be very different (and rather trivial) if all payoffs were common knowledge. In that case, there would be a Nash equilibrium in which there was no cheating and in which each player cooperated if and only if both players gained more from mutual cooperation than from cheating against a cooperator.

Provided that this best response structure is maintained, the main implications of the model are preserved. It is not essential that the payoff from playing C against C is stochastic and that all other payoffs are not; but there must be some random variation in the payoffs, such that the best reply to C is sometimes C and sometimes D.

In our game, unlike the ‘additive’ Prisoner’s Dilemma games described in Footnote 2, the sign of \(a-b\) imposes no constraints on whether the payoff from playing C against C is greater or less than that from playing D against D.

In this case, it is possible that the CM locus intersects the line \(\pi = 1\), creating boundary equilibria. However, this would require a high proportion of the probability mass of f(X) to be between a and \(x_{\mathrm {max}}\), despite the fact that \(x_{\mathrm {max}}\le \) ab/c.

The equilibrium value of \(\pi \) can be derived from (2) using the fact that, in an interior equilibrium, \(V_{D}= 0\).

This comparative-static property is compatible with evidence that in both human and non-human interaction, the level of cooperation is greater, the higher the payoffs to cooperation (Clutton-Brock 2002; Capraro et al. 2014).

In a preliminary version of this paper, we presented simulations based on a stochastic evolutionary process in which selection acted directly on \(\beta \) and \(\pi \). However, as a referee suggested, the main evolutionary properties of our model can be conveyed more simply through replicator dynamics.

The main difference between the two ways of modelling the dynamics relates to states in which at least one of \(p_{N}, p_{C}\) and \(p_{D}\) is equal to zero (i.e. where \(\beta =x_{\mathrm {min}}\), \(\beta =x_{\mathrm {max}}\), \(\pi = 0\) or \(\pi =1\)). In replicator dynamics, a strategy that is not played in any one period is never played, irrespective of its payoffs. Since our main concern in this section is with interior equilibria, this difference is not particularly significant.

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Titel: The emergence of reciprocally beneficial cooperation
verfasst von: Sergio Beraldo
Robert Sugden
Publikationsdatum: 18.08.2015
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2016
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-015-9514-z

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