This is an interactive tutorial exploring the conditions for success and failure of cooperative behavior in two simple evolutionary games: the Prisoner's Dilemma and the Snowdrift game (the latter is also known as the Hawk-Dove or Chicken game). Example simulations implemented by Java applets illustrate and highlight the most important aspects of the evolutionary dynamics of cooperation. In addition, the applets provide ample opportunity for veryfications and further investigations of the dynamics with parameter settings of your choice.

We demonstrate that spatial structure, implemented by placing individuals on regular lattices with limited interaction ranges, has different effects on the evolution of cooperation in the two simple games. Based on extensions of the Prisoner's Dilemma to spatially structured populations it is generally believed that spatial extension promotes cooperation. However, spatial structure fails to similarly favor cooperation under the apparently less stringent conditions of the Snowdrift or Hawk-Dove game. In fact, in these games spatial structure actually tends to reduce the readiness to cooperate. Thus, our results caution against the established view that spatial structure is necessarily beneficial for cooperation.

Two aspects are emphasized below: (a) well-mixed populations with random encounters versus structured populations with limited local interactions and (b) pure strategies where individuals either cooperate or defect versus mixed strategies where individuals cooperate with some probability in each interaction. A brief summary of the background and the relevance of our results is given below.

This tutorial complements and illustrates a research article with Michael Doebeli in Nature (2004) 428 643-646.

Different scenarios

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Well-mixed populations: pure strategies

In well-mixed populations the dynamics of the Prisoner's Dilemma and Snowdrift or Hawk-Dove game can be fully analyzed. Besides, in well-mixed populations there is no distinction between pure or mixed strategies. These results provide the basis for the discussion of effects of spatial structuring in populations on the fate of cooperative behavior.

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Well-mixed populations: mixed strategies

The results for well-mixed populations do not depend on whether they refer to individuals with pure or mixed strategies. However, the interpretation of the underlying mechanisms is quite different and, in particular, the continuous strategy space of mixed strategies requires the introduction of mutations.

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Structured populations: pure strategies

In structured populations individuals are arranged on a lattice or network and interact only with their nearest neighbors. For pure strategies, where each individual is either a cooperator or defector, spatial structures promote cooperation in the Prisoner's Dilemma but fail to do so in the Snowdrift or Hawk-Dove game. In fact, this reduces cooperation as compared to well-mixed populations.

For deterministic update rules on the individual and population level fascinating spatio-temporal patterns may emerge from symmetrical initial configurations that resemble persian carpets or kaleidoscopes.

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Structured populations: mixed strategies

As above individuals are arranged on a lattice or network and interact only within a limited local neighborhood, but now each cooperates with a certain probability which leads to a continuous strategy space. In the Prisoner's Dilemma spatial structure is again beneficial for cooperation. However for the Snowdrift or Hawk-Dove game spatial structures promote cooperation only for asynchronous population updates while inhibiting it for synchronized updating.

Prisoner's Dilemma game

In the Prisoner's Dilemma two players simultaneously decide cooperate or to defect. Cooperation results in a benefit b to the recipient but incurs costs c to the donor (b > c > 0). Thus, mutual cooperation pays a net benefit of R = b - c whereas mutual defection results in P = 0. However, unilateral defection yields the highest payoff T = b and the cooperator has to bear the costs S = -c. It immediately follows that it is best to defect regardless of the opponents decision. For this reason defection is the evolutionarily stable strategy even though all individuals would be better of if all would cooperate (R > P). The characteristics of the Prisoner's Dilemma is determined by the rank ordering of the four payoff values: T > R > P > S.

Whenever the success of a strategy depends on its relative payoff, i.e. on the difference in performance as compared to other contestants, the payoffs T, R, S and P can be rescaled without loss of generality such that R = 1 and P = 0 holds. In terms of costs c and benefits b this yields T = 1 + r and S = -r where r denotes the cost-to-benefit ratio r = c / b. This convenient mapping reduces the number of parameters from four to one and ensures that the required payoff ranking is always preserved.

Snowdrift game

To illustrate the snowdrift game, imagine two drivers caught in a blizzard and trapped on either side of a snowdrift unable to communicate. They both want to get home and so they have the options to cooperate, i.e. to get out in the cold and start shovelling or to defect and stay in the cozy warmth of the car hoping the other guy would do the job. If both cooperate and shovel they have the benefit b of getting home while sharing the labor c (b > c > 0). Thus R = b - c / 2. Whereas if both stay in the car they don't get anywhere before spring arrives and therefore P = 0. However, if only one shovels, then both get home but the one that stayed in the car avoids the trouble and gets T = b whereas the diligent one is left with the whole work S = b - c. The resulting rank ordering of the payoff values is similar to the Prisoner's Dilemma except that P and S have a reverse ordering: T > R > S > P. Nevertheless this leads to fundamental changes because now the best action depends on the behavior of the opponent: defect if the other cooperates but cooperate if the other defects.

As for the Prisoner's Dilemma, the payoff values can be again conveniently rescaled such that R = 1, P = 0, T = 1 + r and S = 1 - r where r denotes a slightly different cost-to-benefit ratio r = c /(2b-c). This parametrization results in a single parameter and preserves the proper payoff ranking required for the Snowdrift game.

Hawk-Dove game

In behavioral ecology the Snowdrift game is bettern known as the Hawk-Dove game which models intraspecific competition. When competing for resources or mates hawks escalate conflicts while doves are conciliatory. If two doves meet they share the resource s and both get R = s / 2 but when facing a hawk the dove takes flight (S = 0) and the hawks gets the entire resource T = s. However, if two hawks meet they escalate the conflict until one is injured at a cost i (i > s). Escalation thus yields on average P = (s - i) / 2. Consequently it pays to escalate when facing a dove but the dove is better off when facing a hawk. It is important to note that hawk and dove may refers to behavioral patterns of a single individual which are adopted with a certain probability. Thus the Hawk-Dove game can be viewed as a mixed strategy game. Such Hawk-Dove interactions are equivalent to the Snowdrift game with b = (s + i) / 2 and c = s.

Since the Hawk-Dove game is equivalent to the Snowdrift game the payoffs can be rescaled in an analogous way which again results in R = 1, P = 0, T = 1 + r and S = 1 - r with r = 2 s / (s + i).

Summary

The emergence of cooperative behavior in human and animal societies is one of the fundamental problems in biology and social sciences. Cooperation lead to major transitions in the history of life: molecules aggregated to protocells, genes arranged in chromosomes, cells shaped complex organisms or individuals formed societies - to name only a few. All examples have one thing in common: they are apparently at odds with Darwinian selection because they are prone to exploitation by cheaters that take advantage of the favorable conditions without contributing to it.

Over the last decades game theory - which describes strategic interactions between individuals - complemented by evolutionary principles - adding selection and reproduction - has become a powerful framework to investigate the problem of cooperation. A particulalry simple evolutionary game has attracted most attention: the Prisoner's Dilemma is a mathematical metaphor for situations where community and individual performance lead to a conflict of interest. One major theoretical result that was obtained from variations of the Prisoner's Dilemma setup states that any form of associative interactions favor cooperation. In particular, cooperation can thrive in the spatial Prisoner's Dilemma where individuals are confined to lattice sites and interact only within their local neighborhood.

Despite the considerable theoretical achievements the discomfort with the Prisoner's Dilemma as the sole paradigm to discuss cooperative behavior increased because of the considerable gap between theory and experimental evidence. In field and experimental studies it is often difficult to assess the payoffs in terms of fitness for the different behavioral patterns - even the proper ranking of the payoffs is challenging. Therefore, the stringent conditions of the Prioner's Dilemma may not be satisfied in many real natural situations. An interesting and biologically viable alternative is given by the Snowdrift game which equally refers to cooperative interactions but under relaxed conditions. In well-mixed populations with random encounters cooperators and defectors co-exist in a stable equilibrium which is stark contrast to the Prisoner's Dilemma where cooperators would go extinct.

In contrast to the spatial Prisoner's Dilemma, adding spatial structure to the Snowdrift game does not benefit cooperation. In fact, spatial structure tends to reduce cooperative behavior relative to well-mixed populations. In the spatial Prisoner's Dilemma cooperators can thrive by forming compact clusters such that losses of cooperators against defectors along the boundary are outweighed by gains from interactions within the cluster. However, this mechanism does not operate in the spatial Snowdrift game. Ironically the ultimate reason for this is the maintenance of cooperative behavior in well-mixed populations.

In behavioral ecology the Snowdrift game is better known as the Hawk-Dove game which models intraspecific competition - cooperation refers to sharing some resource while defectors escalate conflicts and attempt to monopolize the resource which bears the risk of injury when facing another defector. The two behavioral patterns can be adopted by one individual with certain probabilities and hence refers to mixed strategies. Results for well-mixed populations do not discriminate between pure and mixed strategies. Therefore, cooperative behavior is expected to occur with a certain non-zero probability. Adding spatial structure to the mixed strategy case of the Hawk-Dove game leads to another counter intuitive result: asynchronous population updates favors cooperation whereas synchronous updating increases defection. This contrasts with the established view that increased stochasticity should be detrimental to cooperation.

Therefore, our results for both, pure and mixed strategies, caution against the common belief that spatial structure favors cooperation and thus may not be as universally beneficial as previously believed.

References

This work was first published in
Hauert, Ch. & Doebeli, M. (2004) Spatial structure often inhibits the evolution of cooperation in the spatial Snowdrift game. Nature (2004) 428 643-646.

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Acknowledgements

For the development of these pages help and advice of the following two people was of particular importance: First, my thanks go to Michael Doebeli 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.