The origin of cooperators and defectors
by Christoph Hauert, Version 1.0, March 2004.
- » Origin of copperators and defectors
Traditionally in both classical as well as evolutionary game theory a finite set of discrete strategies is assumed a priori. When discussing the evolution of cooperation in social dilemmas, the simplest set of strategies would consist of cooperators and defectors. Cooperators provide a benefit to other individuals at some cost, while defectors attempt to exploit such common resources. This leads to a classic conflict of interest between the individual's and the community performance - and hence the dilemma. The game dynamics then determines the relative frequencies of the different strategies in a population.
From an evolutionary perspective, however, it is essential to understand the emergence of discrete strategies and particularly of cooperators and defectors in order to justify these a priori assumptions. Since evolution is a gradual process, it is important to consider continuous variants of game theoretical metaphors for cooperative interactions such as the Prisoner's Dilemma or the Snowdrift game. The essential difference between the two games is that in the Prisoner's Dilemma the benefits of cooperation accrue exclusively to the other individuals whereas in the Snowdrift game the act of cooperation also provides some benefits to the cooperator itself.
Even though the Snowdrift game has received relatively little attention in the literature it appears to be highly relevant in many biologically scenarios: Consider for example Saccharomyces cerevisiae a single-celled yeast species. The yeast cells secrete an enzyme which 'digests' their immediate surroundings (invertase hydrolyzes sucrose) and turns it into a food resource. Since this food resource is commonly available, it represents a public good that can be easily exploited by defectors that produce little or no enzyme (but rather reproduce faster). However, if no one produces the enzyme, everybody will die of starvation. Therefore, the costs and benefits of cooperative investments crucially depend on the present environment.
Let us assume two individuals with strategies x and y. The payoff of x interacting with y could then be written as P(x,y) = B(x+y)-C(x) where the benefit B(x+y) depends on the accumulated investments made (i.e. the total amount of enzyme produced) while the incurring costs C(x) obviously only depend on the investment x.
In the following we highlight the rich and intriguing dynamics arising from this simple model. In particular, we demonstrate that this scenario can easily lead to evolutionary branching which gives rise to the spontaneous emergence of discrete strategies, i.e. cooperators and defectors. The two strategies co-exist in a stable equilibrium even under strictly mixing conditions and in absence of assortative interactions.
Different scenarios - an overview
In the simplest case with B(x+y) = b2(x+y)2+b1(x+y) and C(x) = c2 x2+c1 x the system can be fully analyzed and the dynamics is characterized by five different scenarios. Note that quadratic functions are the simplest interesting ones - with B(x+y) and C(x) linear, the dynamics is rather uninteresting and dull.
Cost & benefit functions
Quadratic cost (dotted) and benefit (dashed) functions, constrained to the trait interval [0,1], together with the monomorphic population payoff (solid). The two vertical lines highlight that the evolutionary dynamics is quite unrelated to maximizing the population payoff (dash-dotted) as opposed to the branching point (dashed). Also note that there is no easy way to predict the evolutionary outcome simply by looking at the shape of the cost and benefit functions. This allows to rule out some scenarios but in general it requires a detailed analysis of the slopes and curvature of the two functions. The parameter values correspond to the branching scenario.
Complex scenarios - examples
The only condition required by adaptive dynamics to allow for evolutionary branching are (locally) saturating cost and benefit functions. Two examples with more complicated functions which induce complex dynamics are given below.
The applet below illustrates the different components. Along the bottom there are several buttons to control the execution and the speed of the simulations. Of particular importance are the Param button and the data views pop-up list on top. The former opens a panel that allows to set and change various parameters concerning the game as well as the population structure, while the latter displays the simulation data in different ways.
Note: The shades of grey of the payoff scale are augmented by blueish and reddish shades indicating payoffs for mutual cooperation and defection, respectively.
|Params||Pop up panel to set various parameters.|
|Views||Pop up list of different data presentations.|
|Slider||Idle time between updates. On the right your CPU clock determines the update speed while on the left updates are made roughly once per second.|
|Mouse||Mouse clicks on the graphics panels start, resume or stop the simulations.|
|Structure - Strategy||Snapshot of the spatial arrangement of strategies.|
|Mean frequency||Time evolution of the strategy frequencies.|
|Histogram Strategy||Snapshot of strategy distribution in population|
|Distribution Strategy||Time evolution of the strategy distribution|
|Structure - Fitness||Snapshot of the spatial distribution of payoffs.|
|Mean Fitness||Time evolution of average population payoff bounded by the minimum and maximum individual payoff.|
|Histogram - Fitness||Snapshot of payoff distribution in population.|
The list below describes only the few parameters related to the continuous snowdrift game. Follow the link for a complete list and descriptions of all other parameters e.g. referring to update mechanisms of players and the population.
- Benefit/Cost Functions:
- A variety of different combinations of cost and benefit functions can be selected.
- Benefit b0, b1:
- Two parameters for the benefit function. Note that not all functions require both.
- Cost c0, c1:
- Two parameters for the cost function. Note that not all functions require both.
- Mean invest:
- Mean trait value of initial population.
- Sdev invest:
- Standard deviation of initial population. If set to negative values, the population will be initialized with uniform distributed traits.
Press & News
- When Laziness Pays: Math explains how cooperation and cheating evolve (2005) ScienceNews 167 (3) 35 by Erika Klarreich (January 15).
- Egoismo e altruismo, sempre in equilibrio (2004) Il Corriere della Sera (October 31) by Massimo Piattelli Palmarini.
This article 'Egoisms and altruisms, always in equilibrium' appeared in an Italian national newspaper.
- Warum einige immer abräumen und andere ständig draufzahlen (2004) Die Welt (November 22) by Ranty Islam.
This article 'Why some always scoop the market and others persistently overpay' appeared in a German national newspaper.
These fascinating research results would have never been achieved without the inspiring discussions and advice from Michael Doebeli and Timothy Killingback. The origin of all virtuallabs is easily traced back to the encouragement, scientific inspiration and advice of Karl Sigmund. To conquer the tricky subtleties of Java the patience and competence of Urs Bill was invaluable. Financial support of the Swiss National Science Foundation is gratefully acknowledged.