The emergence and maintainance of cooperation by natural selection is an enduring conundrum in evolutionary biology. Inspired by biological situations, different game theoretical models were proposed - most notably, of course, the Prisoner's Dilemma, the Snowdrift game and by-product mutualism for pairwise interactions, as well as Public Goods games in larger groups of interacting individuals. Introducing synergy and discounting of cooperation leads to general framework for social dilemmas in which all the traditional scenarios can be recovered as special cases. In social dilemmas cooperators provide a benefit to the group at some cost, while defectors exploit the group by reaping the benefits without bearing the costs of cooperation. In groups with more than one cooperator, the value of the accumulated benefits may be discounted or synergistically enhanced. This framework provides the first unifying approach to model cooperation in different kinds of social dilemmas.

This tutorial complements several scientific articles co-authored with Michael Doebeli, Martin Nowak and Franziska Michor. It provides interactive Java applets to visualize and experiment with the system's dynamics for parameter settings of your choice.

Social Dilemmas

Social dilemmas occur whenever conflicts of interest arise between the preferences of individuals as compared to the preferences of the community. The simplest and most general definition of a social dilemma consists of two conditions imposed on situations where cooperators produce a valuable and publicly accessible public good b at some cost c to themselves (b>c), while defectors attempt to free ride on the benefits of the common resource without bearing the costs of cooperation:

  1. Groups of cooperators outperform groups of defectors because the former profit from the public good whereas the latter forego the opportunity of mutually beneficial interactions.
  2. In every mixed group, defectors outperform cooperators because they avoid the costs of cooperation.

The first condition states that from the community perspective it is clearly advantageous to cooperate but the second condition dictates that individuals should opt for defection in order to maximize their profit. Hence the dilemma. Situations that meet these two conditions are abundant in nature and range from bacterial colonies to human interactions. For example, foraging yeast cells secrete an enzyme that lyses their environment and thereby creates a valuable, publicly accessible food resource. Naturally, this resource can be exploited by other cells that do not produce the enzyme. Other famous examples include alarm calls in merkats, predator inspection behavior in fish, blood sharing in vampire bats or public goods experiments with students.

In all these examples it is important to note that the accumulated value of cooperative benefits can be of crucial importance for the evolutionary outcome. In particular, the value of additional benefits may be discounted or synergistically enhanced - depending on the characteristics of the interactions.

Population structure


Well-mixed populations

In well-mixed populations groups are formed randomly and the dynamics recovers the four basic scenarios of evolutionary dynamics given by (i) dominating defection, (ii) co-existence of defectors and cooperators, (iii) dominating cooperation and (iv) bi-stability, in which cooperators and defectors cannot invade each other. Scenarios (i)-(iii) correspond to social dilemmas with decreasing severity, whereas in scenario (iv) the problem of cooperation becomes a coordination game. This setting allows for a fully analytic treatment for interaction groups of arbitrary size.


Structured populations

In structured populations, individuals interact and compete within their respective limited local neighborhood. This enables strategies to form clusters which leads to assortative interactions among and between the different strategic types. For example, in Prisoner's Dilemma or Public Goods type interactions where defection dominates in well-mixed populations, the ability to form clusters benefits cooperation and enables cooperators to survive in structured populations. Conversely, under the relaxed conditions of Snowdrift type interactions where cooperators and defectors can co-exist in well-mixed populations, limited local interactions often inhibit cooperation and reduce the equilibrium fraction of cooperators.

Each cooperator produces a benefit b that is equally shared among all N members of the group (including the individual itself). However, in groups containing several cooperators, the actual value of the accumulated benefits must not necessarily increase linearly with increasing numbers of cooperators. Instead, each additional benefit may be discounted or synergistically enhanced by a factor w. In a group with k cooperators, the payoffs for defectors and cooperators can be written as follows:

PD(k) = (b/N) (1+w+w2+ … +wk-1 = (b/N) (1-wk)/(1-w)
PC(k) = PD(k) - c

Hence, the first cooperator provides a benefit b which is shared by all N members of the group (including itself), the second one increases everyone's benefit by w b/N, and so on, to the last of the k cooperators in the group providing an additional benefit of wk-1 b/N. The costs of cooperation c, however, incur only to cooperators. If w = 1, then all cooperators provide the same incremental benefit b/N. This corresponds to the traditional formulation of Public Goods games with PD(k) = r k c/N (or b = r c) where r denotes the multiplication factor of the common pool. If w < 1, then the benefits are discounted and the value of the benefits provided by each additional cooperator is lower than the previous one. If w > 1, then the benefits are synergistically enhanced, and each additional cooperator provides incremental benefits of increasing value.

In natural systems, the actual value of the benefits provided by cooperators may depend on the number of cooperators in the group. For example, in the case of foraging yeast cells, the food resource provided by the first cooperator may be vital for the survival for all group members but, in particular, also for the cooperator itself. However, the value of additional food provisions decreases until further increases become essentially useless because of the cells limited capabilities of food intake. Conversely, the value of additional benefits may be synergistically enhanced. This occurs, for example, in situations where cooperators produce enzymes for enzyme-mediated reactions. The efficiency of the reaction is generally sensitive to the concentration of reactive compounds and often increases faster than linear - at least at low concentrations. Such situations can occur not only in foraging yeast and chemical reactions but essentially whenever individuals create any kind of common good, be it in the form of replication enzymes in viruses or in the form of information gained from predator inspection behavior in fish.

Note that, particularly in the economics literature, the term discounting is often used to refer to potential future benefits in repeated interactions but in the present context, neither discounting nor synergy involve temporal components and instead just characterize the value of accumulated benefits.

This work was first published in
Hauert, Ch., Michor, F., Nowak, M. A. & Doebeli, M. (2006) Synergy and discounting of cooperation in social dilemmas, J. theor. Biol. 239 195-202.

Further publications on synergy and discounting: