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Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games
Abstract
The evolution of cooperation among nonrelated individuals is one of the fundamental problems in biology and social sciences. Reciprocal altruism fails to provide a solution if interactions are not repeated often enough or groups are too large. Punishment and reward can be very effective but require that defectors can be traced and identified. Here we present a simple but effective mechanism operating under full anonymity. Optional participation can foil exploiters and overcome the social dilemma. In voluntary public goods interactions, cooperators and defectors will coexist. We show that this result holds under very diverse assumptions on population structure and adaptation mechanisms, leading usually not to an equilibrium but to an unending cycle of adjustments (a Red Queen type of evolution). Thus, voluntary participation offers an escape hatch out of some social traps. Cooperation can subsist in sizable groups even if interactions are not repeated, defectors remain anonymous, players have no memory, and assortment is purely random.
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As noted in (34), most analyses of the prisoner's dilemma have tacitly built on the fact that the two partners in the original story are prisoners, whereas in most real-life examples, individuals do have the freedom to choose between playing and not playing. In the few examples where the option of not playing the game was explicitly offered to test persons playing the prisoner's dilemma game, the social welfare increased.
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If xc, xd, and xl specify the frequencies of cooperators, defectors, and loners, respectively (with xc + xd + xl = 1), then their average payoff values are As an example, consider two groups A and B, with N = 10 and r = 5. Assume that A consists of eight cooperators and two defectors. The cooperators obtain $3 and the defectors $4. Group B contains two cooperators and eight defectors. Cooperators get nothing and defectors $1. In both groups, defectors earn $1 more than cooperators. Yet on average, defectors get only $1.6 whereas cooperators earn $2.4. Whenever the payoff values allow Simpson's paradox to operate in the small groups made possible through the loner's option, rock-scissors-paper dynamics can be expected. The assumption that payoff is linear in the number nc of cooperators is used only for the sake of simplicity, and because this is the traditional way to model public goods games. The relevance of Simpson's paradox in the evolution of cooperation has been pointed out in (35).
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To simulate and visualize the evolution in well-mixed populations as well as in populations arranged on rigid regular lattices, we provide interactive virtual laboratories at www.univie.ac.at/virtuallabs/PublicGoods. These Java Applets allow one to change various parameters, including the geometry of the lattice, and observe the resulting dynamics.
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In spatial public goods games, we assume that the players invited to participate in the public goods game are, for example, those in a chess king's neighborhood of a given site (in which case N = 9). In each “generation,” every site of the lattice is the center of one game, such that each player is invited to N games. Subsequently, players update synchronously, either by adopting the strategy of the most successful neighbor, or by adopting the strategy of a more successful neighbor with a probability proportional to the difference in accumulated payoff. We note that an individual's update depends on the strategies in a 7 by 7 neighborhood.
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The threshold value rc is determined by geometrical configurations. If players imitate the most successful neighbor (including itself), rc is close to 3; one can check that r > 3 corresponds to the condition that a half-plane of cooperators can advance along a straight front into the defectors' region.
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The same holds for asynchronous updating of the strategies. A related behavior occurs in the case of the prisoner's dilemma (with two-player interactions) if sites are allowed to go empty (36).
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C.H. acknowledges support of the Swiss National Science Foundation; K.S. acknowledges support of the Wissenschaftskolleg WK W008 “Differential Equation Models in Science and Engineering.”
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Published In
Science
Volume 296 | Issue 5570
10 May 2002
10 May 2002
Submission history
Received: 5 February 2002
Accepted: 28 March 2002
Published in print: 10 May 2002
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- Christoph Hauert et al.
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