Abstract
We consider interactions between players in groups of size \(d\ge 2\) with payoffs that not only depend on the strategies used in the group but also fluctuate at random over time. An individual can adopt either cooperation or defection as strategy and the population is updated from one time step to the next by a birth-death event according to a Moran model. Assuming recurrent symmetric mutation and payoffs to cooperators and defectors according to the composition of the group whose expected values, variances, and covariances are of the same small order, we derive a first-order approximation for the average abundance of cooperation in the selection-mutation equilibrium. In general, we show that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation. As for the effect of the covariance between any payoff for cooperation and any payoff for defection, we show that it depends on the number of cooperators in the group associated with these payoffs. We study in particular the public goods game, the stag hunt game, and the snowdrift game, all social dilemmas based on random benefit b and random cost c for cooperation, which lead to correlated payoffs to cooperators and defectors within groups. We show that a decrease in the scaled variance of b or c, or an increase in their scaled covariance, makes it easier for weak selection to favor the abundance of cooperation in the stag hunt game and the snowdrift game. The same conclusion holds for the public goods game except that the variance of b has no effect on the average abundance of C. Moreover, while the mutation rate has little effect on which strategy is more abundant at equilibrium, the group size may change it at least in the stag hunt game with a larger group size making it more difficult for cooperation to be more abundant than defection under weak selection.
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References
Antal T, Nowak MA, Traulsen A (2009) Strategy abundance in \(2\times 2\) games for arbitrary mutation rates. J Theor Biol 257:340–344
Andrews GE, Askey R, Roy R (1999) Special Functions. Cambridge University Press, New York
Archetti M, Scheuring I (2012) Review: Game theory of public goods in one-shot social dilemmas without assortment. J Theor Biol 299:9–20
Balakrishnan N, Nevzorov VB (2003) A Primer on Statistical Distributions. John Wiley and Sons, Hoboken, New Jersey
Christiansen FB (1991) On conditions for evolutionary stability for a continuous varying character. Am Nat 138:37–50
Conway JH, Guy RK (1995) The Book of Numbers. Copernicus Press, New York
Eshel I (1983) Evolutionary and continuous stability. J Theor Biol 103:99–111
Evans SN, Hening A, Schreiber SJ (2015) Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments. J Math Biol 71:325–359
Ewens WJ (1972) The sampling theory of selectively neutral alleles. Theor Popul Biol 3:87–112
Ewens WJ (1990) Population genetics theory - The past and the future. In: Lessard S (ed) Mathematical and Statistical Developments of Evolutionary Theory, NATO ASI Series C: Mathematical and Physical Sciences, vol 299. Kluwer Academic Publishers, Dordrecht, The Netherlands, pp 177–227
Ewens WJ (2004) Mathematical Population Genetics 1: Theoretical Introduction. Springer-Verlag, New York
Fox J, Guyer M (1978) Public choice and cooperation in \(N\)-person prisoner’s dilemma. J Conflict Resolut 22:469–481
Frank SA, Slatkin M (1990) Evolution in a variable environment. Am Nat 136:244–260
Fudenberg D, Harris C (1992) Evolutionary dynamics with aggregate shocks. Journal of Economic Theory 57:420–441
Fudenberg D, Imhof LA (2006) Imitation processes with small mutations. Journal of Economic Theory 131:251–262
Gillespie JH (1973) Natural selection with varying selection coefficients - a haploid model. Genet Res 21:115–120
Gillespie JH (1974) Natural selection for within-generation variance in offspring number. Genetics 76:601–606
Gokhale C, Traulsen A (2011) Strategy abundance in evolutionary many-player games with multiple strategies. J Theor Biol 283:180–191
Griffiths RC (1980) Lines of descent in the diffusion approximation of neutral Wright-Fisher models. Theor Popul Biol 17:37–50
Griffiths RC, Lessard S (2005) Ewens’ sampling formula and related formulae: combinatorial proofs, extensions to variable population size and applications to ages of alleles. Theor Popul Biol 68:167–177
Hamburger H (1973) \(N\)-person prisoner’s dilemma. J Math Sociol 3:27–48
Hofbauer J, Sigmund K (1988) The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge
Hoppe FM (1984) Polya-like urns and the Ewens’ sampling formula. J Math Biol 20:91–94
Imhof L, Nowak MA (2006) Evolutionary game dynamics in a Wright-Fisher process. J Math Biol 52:667–681
Imhof LA (2005) The long-run behavior of the stochastic replicator dynamics. Annals of Applied Probability 15:1019–1045
Joyce P, Tavaré S (1987) Cycles, permutations and the structures of the Yule process with immigration. Stochastic Processes and its Applications 25:309–314
Kaplan H, Hill K, Hurtado AM (1990) Risk, foraging and food sharing among the Ache. In: Cashdan E (ed) Risk and Uncertainty in Tribal and Peasant Economies. Westview Press, Boulder, Colorado, pp 107–144
Karlin S, Levikson B (1974) Temporal fluctuations in selection intensities: case of small population size. Theor Popul Biol 6:383–412
Karlin S, Liberman U (1974) Random temporal variation in selection intensities: case of large population size. Theor Popul Biol 6:355–382
Karlin S, Taylor HM (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York
Kingman JFC (1982) The coalescent. Stochastic Processes and its Applications 13:235–248
Kroumi D, Lessard S (2015) Evolution of cooperation in a multidimensional phenotype space. Theor Popul Biol 102:60–75
Kroumi D, Lessard S (2015) Strong migration limit for games in structured populations: Applications to dominance hierarchy and set structure. Games 6:318–346
Kroumi D, Lessard S (2021) The effect of variability in payoffs on average abundance in two-player linear games under symmetric mutation. J Theor Biol 513:110569
Kroumi D, Martin É, Li C, Lessard S (2021) The effect of variability in payoffs on conditions for the evolution of cooperation in a small population. Dynamic Games and Applications 11:803–834
Kurokawa S, Ihara Y (2009) Emergence of cooperation in public goods games. Proc R Soc B 276:1379–1384
Lande R, Engen S, Saether BE (2003) Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press, Oxford
Lambert A (2006) Probability of fixation under weak selection: a branching process unifying approach. Theor Popul Biol 69:419–441
Li C, Lessard S (2020) Randomized matrix game in a finite population: effect of stochastic fluctuations in the payoffs on the evolution of cooperation. Theor Popul Biol 134:77–91
Lynch M, Walsh B (1998) Genetics and Analysis of Quantitative Traits, 1st edn. Sinauer Associates, Sunderland, Massachusetts
May RM (1973) Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton
Maynard Smith J (1982) Evolution and the Theory of Games. Cambridge University Press, Cambridge
Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15–18
Moran PAP (1958) Random processes in genetics. Math Proc Cambridge Philos Soc 54:60–71
Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–650
Otto SP, Whitlock MC (1997) The probability of fixation in populations of changing size. Genetics 146:723–733
Pacheco JM, Santos FC, Souza MO, Skyrms B (2009) Evolutionary dynamics of collective action in \(N\)-person stag hunt dilemmas. Proc R Soc B 276:1655
Parsons TL, Quince C (2007) Fixation in haploid populations exhibiting density dependence II: the non-neutral case. Theor Popul Biol 72:121–135
Parsons TL, Quince C (2007) Fixation in haploid populations exhibiting density dependence II: the quasi-neutral case. Theor Popul Biol 72:468–479
Rice SH, Papadopoulos A (2009) Evolution with stochastic fitness and stochastic migration. PLoS ONE 4:e7130
Rychtar J, Taylor DT (2019) Moran process and Wright-Fisher process favor low variability. Discrete and Continuous Dynamical Systems B. https://doi.org/10.3934/dcdsb.2020242
Santos FC, Pacheco JM (2011) Risk of collective failure provides an escape from the tragedy of the commons. Proceedings of the National Academy of Sciences USA 108:10421–10425
Schreiber SJ (2012) The evolution of patch selection in stochastic environments. Am Nat 180:17–34
Schreiber SJ (2015) Unifying within- and between-generation bet-hedging theories: An ode to J. H. Gillespie. Am Nat 186:792–796
Skyrms B (2004) The Stag Hunt and Evolution of Social Structure. Cambridge University Press, Cambridge
Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under \(N\)-person snowdrift games. J Theor Biol 260:581–588
Starrfelt J, Kokko H (2012) Bet-hedging-a triple trade-off between means, variances and correlations. Biol Rev Camb Philos Soc 87:742–55
Tavaré S (1984) Line-of-descent and genealogical processes, and their application in population genetics models. Theor Popul Biol 26:119–164
Taylor P, Jonker L (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145–156
Taylor P (1989) Evolutionary stability in one-parameter models under weak selection. Theor Popul Biol 36:125–143
Uecker H, Hermisson J (2011) On the fixation process of a beneficial mutation in a variable environment. Genetics 188:915–930
Zeeman RC (1980) Populations dynamics from game theory. In: Nitecki ZH, Robinson RC (eds) Global Theory of Dynamical Systems, Lecture Notes in Mathematics, Vol 819, Springer, New York
Zheng DF, Yin H, Chan CH, Hui PM (2007) Cooperative behavior in a model of evolutionary snowdrift games with \(N\)-person interactions. Europhys Lett 80:18002
Zheng XD, Li C, Lessard S, Tao Y (2017) Evolutionary stability concepts in a stochastic environment. Phys Rev E 96:032414
Zheng XD, Li C, Lessard S, Tao Y (2018) Environmental noise could promote stochastic local stability of behavioral diversity evolution. Phys Rev Lett 120:218101
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S. Lessard is supported by NSERC of Canada, grant no. 8833.
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Appendices
Appendix A: Conditional expected frequency change
For \(i/N=x\), we have
Therefore, in the limit of a large population size N, the average payoffs to C and D in (3) can be written as
Using (1), the first two moments of the average payoff to C in (3) when the frequency of C is x are given by
and similarly the first two moments of \(P_D(x)\) by
Moreover, we have
We are interested in
where \({\bar{P}}(x)=xP_C(x)+(1-x)P_D(x)\) is the average payoff in the population. We expand the last expression by the delta-method (Lynch and Walsh 1998; Rice and Papadopoulos 2009), which gives
for two random variables Y and Z with
and
for \(k \ge 1\). For \(Y=P_C(x)-P_D(x)\) and \(Z=1+{\bar{P}}(x)\), using the facts that
for \(k\ge 2\) and
for \(k\ge 1\), we obtain
Note that
and \(\left( 1+E[{\bar{P}}(x)]\right) ^{2}=1+O(\delta )\), so that
where
Appendix B: Moments of the Dirichlet distribution
The ancestry of a random sample of n individuals is described backward in time by a coalescent tree with every pair of lines at any given time back coalescing at rate 1 independently of all the others (Kingman 1982). Moreover, mutations occur independently on each lines of the coalescent tree according to a Poisson process of intensity \(\theta >0\). When there is mutation, the mutant type is 1 or 2 with probability 1/2 for each type independently of the parental type. A line is said to be ancestral to the sample as long as no mutation has occurred on it. We are interested in the probability distribution of the sample configuration, more precisely the probability for k labeled individuals to be of type 1 and the \(n-k\) others to be of type 2, denoted by \(\psi _n^k\), for \(0\le k\le n\).
In order to determine the sample probability distribution, we will extend an approach used in Griffiths and Lessard (2005) to show the Ewens sampling formula (Ewens 1972) in the case where the mutant type is always a novel type. See also Hoppe (1984) and Joyce and Tavaré (1987) for related approaches based on urn models and cycles in permutations.
Note first that the number of ancestral lines of a sample of n individuals is a death process backward in time, where ancestral lines are lost by either mutation or coalescence. This death process was studied in Griffiths (1980) and Tavaré (1984), and the events in this death process called defining events in Ewens (1990).
Label the n sampled individuals and list them in the order in which their ancestral lines are lost backward in time, following either a mutation or a coalescence. In the case of coalescence, one of the two lines involved is chosen at random to be the one that is lost, the other one being the continuing line. There are n! different orders.
Let us first consider the probability for the n sampled individuals in a given order to be all of type 1. Note that this event occurs if and only if all ancestral lines lost by mutation lead to type 1. Now let us look at the probability of each defining event. When i ancestral lines remain, the total rate of mutation is \(i\theta \) and the total rate of coalescence is \(i(i-1)/2\), while the rate of mutation leading to type 1 on any particular ancestral line is \(\theta /2\) and the rate of coalescence involving any particular ancestral line and leading to its loss is \((i-1)/2\). Therefore, the probability that a particular ancestral line is the next one lost and that it is lost by mutation leading to type 1 is \((\theta /2)/[(i\theta +i(i-1)/2]\) for \(i\ge 1\). Similarly, the probability that a particular ancestral line is the next one lost and that it is lost by coalescence is \([(i-1)/2]/[(i\theta +i(i-1)/2]\) for \(i\ge 1\). Summing these probabilities, multiplying the sums for \(i=n, n-1, \ldots , 1\), and considering all possible orders, we get
as probability for the n sampled individuals to be of type 1.
Now let us look at the general case of k labeled individuals of type 1 and \(n-k\) of type 2. When i ancestral lines of individuals of type 1 and j ancestral lines of individuals of type 2 remain, the total rate of mutation is \((i+j)\theta \) and the total rate of coalescence is \((i+j)(i+j-1)/2\), while any particular ancestral line of an individual is lost by mutation to the type of the individual, whose rate is \(\theta /2\), or by coalescence with ancestral lines of individuals of the same type, whose rate is \((i-1)/2\) for type 1 and \((j-1)/2\) for type 2. Considering i and j decreasing from k and \(n-k\) to 0 or 1 with \(i+j=n, n-1, \ldots , 1\), we get
as probability for k labeled individuals to be of type 1 and the \(n-k\) others to be of type 2 in a random sample of size n. Here, we use \(\Gamma (\beta +1 )= \beta \Gamma (\beta )\) for \(\beta >0\). Note that \(\psi _n^k=\psi _{n-k}^k\).
Applying the same approach as above for K types with rate of mutation to type k given by \(\alpha _k/2 >0\) for \(k=1, \ldots , K\) with \(\alpha _1 + \cdots + \alpha _K=\alpha \), the probability for \(n_k\) labeled individuals to be of type k for \(k=1, \ldots , K\) in a random sample of size \(n=n_1 + \cdots + n_K\) is given by
These are the moments of the Dirichlet distribution (see, e.g., Balakrishnan and Nevzorov 2003).
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Kroumi, D., Lessard, S. Average abundancy of cooperation in multi-player games with random payoffs. J. Math. Biol. 85, 27 (2022). https://doi.org/10.1007/s00285-022-01789-1
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DOI: https://doi.org/10.1007/s00285-022-01789-1
Keywords
- Moran model
- Mutation-selection equilibrium
- Evolution of cooperation
- Dirichlet distribution
- Public goods game
- Stag hunt game
- Snowdrift game