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Replicator Dynamics for Optional Public Good Games Christoph Hauertwz,Silviademontewy, Josef Hofbauerw and Karl Sigmundnw8

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Replicator Dynamics for Optional Public Good Games Christoph Hauertwz,Silviademontewy, Josef Hofbauerw and Karl Sigmundnw8 J. theor. Biol. (2002) 218, 187–194 doi:10.1006/yjtbi.3067, available online at http://www.idealibrary.com on Replicator Dynamics for Optional Public Good Games Christoph Hauertwz,SilviaDeMontewy, Josef Hofbauerw and Karl Sigmundnw8 w Institute for Mathematics, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria z Department of Zoology, University of British Columbia, 6270 University Boulevard, Vancouver, BC, Canada V6T1Z4 y Department of Physics, Danish Technical University, DK-2800 Kgs. Lyngby, Denmark and 811ASA, A-2361 Laxenburg, Austria (Received on 8 October 2001, Accepted in revised form on 12 April 2002) The public goods game represents a straightforward generalization of the prisoner’s dilemma to an arbitrary number of players. Since the dominant strategy is to defect, both classical and evolutionary game theory predict the asocial outcome that no player contributes to the public goods. In contrast to the compulsory public goods game, optional participation provides a natural way to avoid deadlocks in the state of mutual defection. The three resulting strategiesFcollaboration or defection in the public goods game, as well as not joining at allFare studied by means of a replicator dynamics, which can be completely analysed in spite of the fact that the payoff terms are nonlinear. If cooperation is valuable enough, the dynamics exhibits a rock-scissors-paper type of cycling between the three strategies, leading to sizeable average levels of cooperation in the population. Thus, voluntary participation makes cooperation feasible. But for each strategy, the average payoff value remains equal to the earnings of those not participating in the public goods game. r 2002 Elsevier Science Ltd. All rights reserved. Introduction In this article, we present another mechanism to Most theories on the emergence of cooperation achieve sizeable levels of cooperation in a popula- among selfish individuals are based on kin tion. The following investigation is based on the . selection (Hamilton, 1963), group selection public goods game (see Fehr& G achter, 2002; (Wilson & Sober, 1994) and reciprocal altruism Kagel & Roth, 1995) which represents a natural (Trivers, 1971). In all three models, cooperative extension of the prisoner’s dilemma to an arbitrary behavioris attained throughbasic mechanisms number of players (see Boyd & Richerson, 1988; of discrimination enabling individuals to target Dawes, 1980; Hauert & Schuster, 1997). their altruistic acts towards certain partners In a typical public goods game, an experi- only. menter gives 20 dollars to each of eight players. The players may contribute part or all of their money to some common pool. The experimenter then triples this amount and divides it equally n Corresponding author. Institute for Mathematics, Uni- among the eight players, irrespective of the versity of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria. Tel.: +43-3-4277-50612; fax: +43-1-4277-9506. amount of theirindividual contribution.If all E-mail address: [email protected] (K. Sigmund). players contribute maximally, they will end up 0022-5193/02/$35.00/0 r 2002 Elsevier Science Ltd. All rights reserved. 188 C. HAUERT ET AL. with 60 dollars each. But each individual is faced whetherto participatein the public goods game with the temptation to exploit, as a free rider, or not. [For a similar approach in the prisoner’s the contributions of the co-players. Hence, the dilemma see Batali & Kitcher(1995); Orbell& dominating strategy is to invest nothing at all. If Dawes (1993)]. Individuals unwilling to join the all players do this, they will not increase their public goods game are termed loners. These initial capital. In this sense, the ‘‘rational’’ players prefer to rely on a small but fixed payoff equilibrium solution prescribed to ‘‘homo oeco- Pl ¼ sc with nomicus’’ leads to economic stalemate. In actual experiments, players tend to invest a lot, how- 0osor À 1; ð2Þ ever: typically, in the first round, they invest 10 dollars or more (Fehr & Gachter,. 2002). such that the members in a group where all Public goods games are abundant in human cooperate are better off than loners, but loners and animal societies, and can be seen as basic are better off than members in a group of examples of economic interactions (see e.g. defectors. Binmore, 1994; Dugatkin, 1997). For the optional public goods game, there are thus three behavioral types in the population: (a) The Model the loners unwilling to join the public goods We considera largepopulation of players. game, (b) the cooperators ready to join the From time to time, N such players are chosen group and to contribute their effort, and (c) the randomly. Within such a group, players can either defectors who join, but do not contribute. contribute some fixed amount c ornothing at all. Assuming that groups form randomly, the pay- The return of the public good, i.e. the payoff to the offs for the different strategies Pc; Pd and Pl are players in the group, depends on the abundance of then determined by the relative frequencies x; y and z of the three strategies. cooperators. If nc denotes theirnumberamong the public goods players, the net payoff for coopera- tors Pc and defectors Pd is given by The Equations of Motion n Evolutionary game theory assumes that a P ¼Àc þ rc c; c N strategy’s payoff determines the growth rate of its frequency within the population. More n precisely, following (Weibull (1995), Schlag P ¼ rc c; d N (1998) and Hofbauer& Sigmund (1998), we postulate in ourmodel that playersusing where r denotes the interest rate on the common strategies i ¼ 1; y; n occasionally compare their pool. Fora public goods game deservingits name, payoff with that of a randomly chosen ‘‘model’’ we must have memberof the population, and adopt the strategy of their model with a probability 1 r N: ð1Þ o o proportional to the difference between the The first inequality states that if all do the same, model’s payoff and theirown, if this is they are better off cooperating than defecting; the positive (and with probability 0 otherwise). In second inequality states that each individual is the continuous time model, the evolution of the betteroff defecting than cooperating.Selfish frequencies xi of the strategies i is given by players will therefore always avoid the cost of X cooperation c; i.e. a collective of selfish players will x’i ¼ xixjðPi À PjÞð3Þ nevercooperate.Defection is the dominating j strategy. Hence both classical and evolutionary game theory predict that all players will defect, and with 1pi; jpn; which reduces to the replicator obtain zero payoff. equation We now extend the public goods game. In this ’ % optional public goods game, players can decide xi ¼ xiðPi À PÞ; ð4Þ OPTIONAL PUBLIC GOOD GAMES 189 P % where P ¼ xjPj is the average payoff in the game. In the case S ¼ 1 (no co-playershows up) population. we assume that the playerhas no otheroption To be precise, we consider a very large, well- than to play as a loner, and obtains payoff s: mixed population with three types of players: This happens with probability zNÀ1: Fora given loners, cooperators and defectors. From time to playerwilling to join the public goods game, the time, sample groups of N players are randomly probability of finding, among the N À 1 other chosen and offered to participate in a single players in the sample, S À 1 co-players joining public goods game. Note that in large popula- the group (S41), is tions, the probability that two players ever ! encounteragain can be neglected. Depending N À 1 ð1 À zÞSÀ1zNÀS: on their type, players either refuse to participate S À 1 orjoin the public goods game. In the lattercase, they eitherdefect, orcooperate.But, their The probability that m of these players are strategies are specified beforehand, and do not cooperators, and S À 1 À m defectors, is depend on the composition of the sampled ! group. In particular, we do not consider condi- x m y SÀ1Àm S À 1 tional strategies of the type: cooperate if and : x þ y x þ y m only if the group contains more than two cooperators, or the like. In that case, the payoff fordefectorsis rm=S: Each playeris sampled a numberof times, and Hence the expected payoff fora defectorin a obtains an average payoff which depends on his group of S players (S ¼ 2; y; N)is strategy as well as the composition of the entire ! population. This composition changes according r XSÀ1 x m y SÀ1Àm S À 1 to the replicator dynamics [see eqns (3) and (4)]. m S x þ y x þ y The intuition behind the dynamics is that m¼0 m occasionallyFand independently of the sam- r x pling of the public goods teamsFa randomly ¼ ðS À 1Þ : chosen player A compares his or her payoff with S x þ y that of anotherplayer B (also randomly chosen, Thus, within the entire population and not limited to ! the ‘‘public goods’’ groups to which A belonged), XN NÀ1 x N À 1 and adopts the strategy of B; if it yields a higher Pd ¼ sz þ r 1 À z S À 1 payoff, with a probability proportional to the S¼1 difference in their payoffs. Opportunities for 1 updating, i.e. changing the strategy, are sup- ð1 À zÞSÀ1zNÀS 1 À posed to occurmuch less frequentlythan S opportunities to play in a public goods group. x We emphasize that there are other reasonable ¼ szNÀ1 þ r game dynamics, but forthe replicatorequation 1 À z we can produce a full analysis. "# ! Forsimplicity and without loss of generality, XN N À 1 1 1 À ð1 À zÞSÀ1zNÀS we set the cost c of cooperation equal to 1: The S À 1 S payoff forlonersis then given by the constant S¼1 Pl ¼ s: N À 1 N and using ¼ S=N leads to S À 1 S In order to compute the payoff values for cooperators and defectors, we first derive the N NÀ1 x 1 À z probability that S of the N sampled individuals Pd ¼ sz þ r 1 À : ð5Þ are actually willing to join the public goods 1 À z Nð1 À zÞ 190 C.
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