The evolution of cooperation among non-related 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. Thus, voluntary participation offers an escape hatch out of some social traps. Cooperation can subsist in sizeable groups even if interactions are not repeated, defectors remain anonymous, players have no memory and assortement is purely random.

This tutorial illustrates several scientific articles and press releases co-authored with Karl Sigmund, Josef Hofbauer, Silvia DeMonte and Györgi Szabó. It provides interactive Java applets to visualize and explore the systems' dynamic for parameter settings of your choice.

Voluntary participation

Most theoretical and experimental studies on public goods games or the related prisoner's dilemma have tacitly built on the fact that the participants are actually prisoners, i.e. they are trapped in the dilemma. In nature, however, animals and humans often have, at least to a certain extend, the freedom to decide whether to participate in a public enterprise. In particular usually they would refuse to participate in any public enterprise that is doomed from the beginning.

For this reason we extend the public goods game to allow for voluntary participation. Individuals unwilling to participate are termed loners. They prefer autarky and rather rely on some small but fixed payoff. The loner strategy is thus risk averse. But, the option to withdraw from social or economic enterprises efficiently avoids deadlocks in states of mutual defection and economic stalemate.

In the following we thus consider three strategical types: (i) cooperators and (ii) defectors both willing to join the public goods game, with different intentions though, and (iii) the risk averse loners that refuse to participate and rather rely on some small but fixed income. These strategies lead to a rock-scissors-paper-type dynamics with cyclic dominance: if cooperators abound, they can be exploited by defectors, but if defectors prevail it is best to abstain and if no one participates, small groups can form and it pays to return to cooperation.

If such small groups are formed on voluntary terms, they are known to be efficient for difficult tasks. Military people must have been aware of this ages ago. Marbot, an officer of Napoleon, writes in his memoirs:

To face immense perils, volunteers are infinitely preferable to a body of men under orders.

Population structure


Well-mixed populations

In well-mixed populations groups of individuals that interact in public goods games (or refuse to participate) are randomly formed. The cyclic dominance of cooperators, defectors and loners promotes oscillations in the frequencies of the three strategies. Depending on the underlying dynamics, the amplitude of the oscillations increases and only one strategy survives, the amplitude decreases and all three co-exist or periodic oscillations persist.


Structured populations: group interactions

In structured populations interactions of individuals are limited to a local neighborhood. In spatially extended systems, for example if individuals are arranged on regular lattices, voluntary participation in the public goods interaction greatly enhances the parameter range where cooperators can survive because in the spatial setting, loners provide some protection against exploitation by defectors. Whenever all three strategies co-exist, the cyclic dominance drives fascinating spatio-temporal patterns such as traveling waves or even evolutionary kaleidoscopes.

The classical setting of the compulsory public goods game is extended by allowing for a third strategic type: the loners, that are risk averse and unwilling to participate in the public enterprise. Instead, they prefer to rely on a small but fixed income. Thus, the payoff for loners PL is simply a constant:

PL = σ c,

measured in units of c, i.e. the cost of cooperation. The loners payoff must satisfy 0 < σ < r-1 such that loner earn more than groups of defectors but less than groups of cooperators. Recall, that the payoffs for cooperators PC and defectors PD in a group of N interacting individuals were given by:

PD = (r nc c)/N
PC = PD - c,

where r is the multiplication factor of the public good, c the cost of cooperation and nc the number of cooperators in the group. Because of the the option to drop out, the number of actual participants S in the public goods game varies and can be smaller than N such that

PD = (r nc c)/S.

Whenever only a single individual is willing to participate, it is forced to (temporarily) act as a loner. In populations of interacting individuals the cyclic dominance leads to oscillations in the frequency of the three strategies: if cooperators abound it pays to switch to defection and if defectors previal it is better to abstain but if loners dominate, then small groups can form and cooperators can thrive again. This last change occurs whenever the typical interaction group size S becomes smaller than the multiplication factor r which relaxes the social dilemma such that cooperation temporarily becomes dominant.

This system can be fully analyzed for the replicator dynamics despite the non-linear payoff terms and, moreover, the periodic oscillations of the different strategies have been experimentally confirmed (Semmann et al. Nature 2003).

This work was first published in
Hauert, Ch., De Monte, S., Hofbauer, J. & Sigmund, K. (2002) Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games, Science 296, 1129-1132 (click on title to access the article online).

Further publications on volunteering in public goods games:

Experimental verification:

Press & News


For the development of these pages help and advice of the following two people was of particular importance: First, my thanks go to Karl Sigmund for helpful comments on the game theoretical parts and second, my thanks go to Urs Bill for introducing me into the Java language and for his patience and competence in answering my many technical questions. Financial support of the Swiss National Science Foundation is gratefully acknowledged.