The good, the bad and the lonely

In game theory, 'loners' who choose not to participate in fact promote cooperation between players. The dynamics of the game show phase transitions and complex phenomena reminiscent of statistical physics.

Hungarian-born genius John von Neumann, in a rare moment of humility, confessed that he considered himself a failure: according to his own ranking, he was only the third-best mathematician of the time. Yet von Neumann not only built one of the first computers and achieved breakthroughs in fields as diverse as operator theory, mathematical logic and hydrodynamics, he also created several fields of mathematics from scratch. Two of these fields, game theory1 and cellular automata2, are merging in the study of evolutionary dynamics3. Earlier this year, Hauert et al.4 pushed this development further by investigating the effects of voluntary participation in 'public-goods games' — players can choose whether or not to join in a game (or interaction) that may benefit them. That work is now extended by Szabó and Hauert5, writing in Physical Review Letters, to a large interacting community of players on spatial grids.

The quest for cooperation is as old as evolution itself. In the Origin of Species, Darwin noted that natural selection cannot directly promote altruistic acts where individuals reduce their own competitive ability but increase that of others. Yet cooperation is abundant in nature. The standard explanations that have been developed for this include kin selection, group selection and reciprocity6,7,8,9,10,11.

The essence of cooperation is captured by the public-goods game. Each individual of a group can decide whether or not to invest some money in a common pool. The common pool is increased by some amount and then equally distributed among all group members regardless of whether or not they made a contribution. The optimum outcome for the group occurs if everybody cooperates. But the temptation is to 'free ride': those who don't contribute (defectors) always get a higher pay-off than cooperators who do contribute. If everyone defects, however, no one will enjoy the public goods. Self-interest is self-defeating! This social dilemma threatens the success of public enterprises such as social security, the conservation of environmental resources or group defence against external threats. Notice that the well-known 'prisoner's dilemma' is a public-goods game for groups of two people.

Hauert et al.4 proposed a new and surprisingly simple mechanism to promote cooperation. Rather than allowing cooperation or defection as the only strategic alternatives, they added a third option — 'loners', who do not participate in the game but instead receive a small, independent pay-off corresponding to a modest income from some self-sufficient occupation. Those who choose to participate in the public-goods game must forgo the loner's pay-off. Equivalently, one could regard the pay-off as the avoidance of a fixed cost for participating in the public-goods game.

In this game, defectors dominate cooperators, as they do in a game without loners; but loners dominate defectors, because in a world of defectors, the public-goods game yields nothing, and loners avoid the cost of participating. Cooperators, on the other hand, dominate loners, because the public-goods game pays in the absence of defectors. The circle is closed. The outcome depends on the details of the underlying evolutionary dynamics, but there are oscillations in the frequencies of the three strategies. For example, if players always adopt the strategy providing a higher pay-off, then the oscillations of the frequencies of all three strategies are stable and robust. The updating mechanism of best-reply dynamics — switching to the best strategy, given the composition of the population at a certain point in time — leads, through damped oscillations, to a stable equilibrium among all three strategies. Hence, the social dilemma is relaxed: instead of defectors winning the world, there is coexistence among cooperators, defectors and loners.

Voluntary participation, though, is only a 'Red Queen' mechanism for cooperation. The Red Queen in Lewis Carroll's Alice's Adventures in Wonderland always has to run to stay on the same spot. Similarly, the oscillations among cooperators, defectors and loners lead to average pay-offs that are no larger than having no public-goods game at all. Volunteering does not lead to the fixation of cooperators, but it prevents the fixation of defectors in the population. It needs other mechanisms, such as punishment12,13, to achieve a stable regime of all-out cooperation. Interestingly, however, optional participation does not require individual recognition.

In spatially extended games, as investigated by Szabó and Hauert5, voluntary participation is even more successful in promoting cooperation than in well-mixed populations. In the computer simulation, players are represented by cells on a spatial grid. The size of the neighbourhood determines the number of players that interact in the public-goods game. Players with a higher pay-off have an increased probability of being imitated by their neighbours. Hence, successful strategies spread locally. Clusters of cooperators do better than clusters of defectors, but isolated defectors flourish if surrounded by cooperators. In spatial versions without volunteering, there is coexistence between cooperators and defectors for a restricted range of parameters. Adding loners greatly increases the range of parameters allowing coexistence. Hence, optional participation enables the survival of cooperation under circumstances that would normally favour the dominance of defectors.

The intriguing spatiotemporal patterns show cyclic, complex behaviour and coexistence of all three strategies (Fig. 1). Some of the cellular games are closely related to models in statistical mechanics, such as the Ising model for magnetic spins. Spatial games exhibit phase transitions and the full range of complex phenomena observed in the world of cellular automata14. Hauert has designed a beautiful web page (http://www.univie.ac.at/virtuallabs) where visitors can run simulations for their own choice of parameters and initial conditions. Sometimes, extreme parameter values or small system sizes lead to the extinction of strategies; for example, extinction of defectors in small societies leads to homogeneous populations of cooperators.

Figure 1: Evolutionary kaleidoscope.

a, Players in a public-goods game are represented as cells distributed spatially on a grid. Cooperators are shown in blue, defectors in red and loners in yellow. Players may change their strategy in each round of the game. For this simulation, all strategy updates are synchronous. Intermediate colours indicate players who have updated their strategies in the last round of the game. b, Another simulation shows the effects of asynchronous strategy updating and small amounts of noise in the system. In both cases, there is coexistence between cooperators, defectors and loners. Szabó and Hauert5 show that the dynamics of such systems have much in common with models in statistical physics.

At the dawn of life on Earth (or elsewhere), replicating molecules had to cooperate to take the first step towards increasing complexity and stability of molecular and later cellular ecosystems. Multicellularity requires cooperation among cells. Cooperation is common in animal societies, but it is often confined to interactions among related individuals. Large-scale cooperation among unrelated individuals seems to be a particularly human trait. Of course, cooperation is always threatened by defection; oscillations between 'war and peace' have been a recurring theme in the cooperation literature. The new work shows that it is optional, rather than compulsory, interactions that promote cooperation.


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Correspondence to Martin A. Nowak.

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Michor, F., Nowak, M. The good, the bad and the lonely. Nature 419, 677–679 (2002). https://doi.org/10.1038/419677a

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