Complexity

Volume 2018, Article ID 9836150, 14 pages

https://doi.org/10.1155/2018/9836150

## Stochasticity, Selection, and the Evolution of Cooperation in a Two-Level Moran Model of the Snowdrift Game

Correspondence should be addressed to Laurence Loewe; ude.csiw@eweol

Received 1 September 2017; Accepted 28 November 2017; Published 13 February 2018

Academic Editor: Francisco J. S. Lozano

Copyright © 2018 Brian McLoone et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Snowdrift Game, also known as the Hawk-Dove Game, is a social dilemma in which an individual can participate (cooperate) or not (defect) in producing a public good. It is relevant to a number of collective action problems in biology. In a population of individuals playing this game, traditional evolutionary models, in which the dynamics are continuous and deterministic, predict a stable, interior equilibrium frequency of cooperators. Here, we examine how finite population size and multilevel selection affect the evolution of cooperation in this game using a two-level Moran process, which involves discrete, stochastic dynamics. Our analysis has two main results. First, we find that multilevel selection in this model can yield significantly higher levels of cooperation than one finds in traditional models. Second, we identify a threshold effect for the payoff matrix in the Snowdrift Game, such that below (above) a determinate cost-to-benefit ratio, cooperation will almost surely fix (go extinct) in the population. This second result calls into question the explanatory reach of traditional continuous models and suggests a possible alternative explanation for high levels of cooperative behavior in nature.

#### 1. Introduction

Evolutionary game theory (Hofbauer and Sigmund 1998 [1]; Smith 1982 [2]; Smith and Price 1973 [3]) allows one to analyze the evolutionary dynamics of a population of individuals in which individual fitness is frequency dependent. A model in evolutionary game theory is based on a payoff matrix, which describes the payoff an individual will receive given its own behavior and the behavior of its partner(s). Payoff Matrix 1 represents a symmetric, two-player game. The entries in the matrix—, , , and —represent the payoff the row player will receive when it plays strategy* A* or* B* with its partner, the column player, who can also play either strategy* A *or* B*.

* Payoff Matrix 1.* A two-player, symmetric game is as follows:

The Prisoner’s Dilemma, in which , is perhaps the most widely studied form of social interaction, having been used to model systems ranging from microorganisms (Conlin et al. 2014; [4], Frey 2010 [5]; West et al. 2006 [6]) to human societies (Axelrod and Hamilton [7]; Bowles and Gintis [8]). However, The Prisoner’s Dilemma does not accurately represent social dilemmas in which there is no strictly dominant strategy.

The Snowdrift Game (Sugden 1986 [9]), also known as the Hawk-Dove Game (Smith 1982 [2]), is a social dilemma in which an individual can participate (“cooperate") or not (“defect") in producing a public good. In this game, two individuals, each in her own car, want to get home, but their cars are blocked by a large pile of snow. For either to get home, at least one person must get out of her car to shovel the snow. But there is a cost to doing so, creating a strategic dilemma in which . The game is relevant to a number of phenomena in biology, such as collective defense and resource extraction among microorganisms (Gore et al. 2009 [10]; Conlin et al. 2014 [4]), behavioral contests (Smith 1982 [2]), distributive justice in humans (Sugden 1986 [9]), behavioral diversification (Doebeli et al. 2004 [11]), and branching events (Wakano and Lehmann 2014 [12]).

In what we might think of as a “standard" model in evolutionary game theory, there is an infinitely large, well-mixed population within which individuals either cooperate or defect in one-time, pairwise games with each other. One generally assumes that individuals reproduce asexually and that the number of offspring each individual has is proportional to its fitness. This makes it easy to track how the frequencies of cooperators and defectors change in the population over time. The rate of change of a strategy is given by the replicator dynamics (Hofbauer and Sigmund 1998 [1]; Hofbauer et al. 1979 [13]; Taylor and Jonker 1978 [14]), an ordinary differential equation.

If a population of individuals is playing this game, then traditional evolutionary models, in which the dynamics are continuous and deterministic, predict a stable, interior equilibrium frequency of cooperators (Doebeli and Hauert 2005 [15]; Doebeli et al. 2004 [11]; Hauert and Doebeli 2004 [16]) (see (7)). Thus, the standard, deterministic model of the Snowdrift Game describes a scenario in which, even in the absence of facultative trait expression or heterozygote superiority, a stable polymorphism of behaviors can emerge.

However, all biological communities are finite, and many are small and organized into groups that together form a metapopulation (Gilpin 2012 [17]; Hanski 1999 [18]). This was certainly the case for ancestral hominins, which has plausibly influenced the evolution of human cooperation (Bowles and Gintis 2011 [8]), and it also appears to characterize many other taxa, including bacteria (Lieberman et al. 2016 [19]). It is therefore important to understand how a model with finite population size and metapopulation structure can change the predictions generated from models that assume a single population whose size tends to infinity.

Here we explore a discretization of the standard model of the Snowdrift Game. We consider a metapopulation composed of a finite set of discrete, nonintermixing groups, which are themselves composed of a finite set of discrete individuals who either cooperate or defect in the Snowdrift Game. The evolutionary dynamics both between and within groups are governed by a discrete time Moran process (Moran 1958 [20], 1962 [21]), a special case of a discrete time Markov chain.

Our analysis has two main results. First, we show that the combination of within-group stochasticity and group selection can promote the evolution of cooperation in the metapopulation and can even result in cooperation’s fixation. This would be an impossible result were the evolutionary dynamics deterministic.

Second, we describe a phase transition for the fixation and extinction probabilities of cooperation in any finite group of a constant size whose members play the Snowdrift Game. Letting* r *stand for the cost-to-benefit ratio in a given Snowdrift Game, this threshold quantity, which we call , is approximately equal to If , the probability cooperators will fix tends to 1 as group sizes go to infinity. If , the probability cooperators will go extinct tends to 1 as group size goes to infinity. This is true so long as the starting frequency of cooperators is strictly between 0 and 1. (As we detail, more complicated dynamics occur when .) The existence of this threshold quantity allows us to state a sufficient condition for cooperators to fix in a metapopulation. Moreover, while this threshold result comes about from taking the limit of group size, we in fact show that even when group size is fairly small (e.g., 100), the value of effectively determines whether cooperators will fix or go extinct within a group. This threshold has no analogue in a deterministic model of the Snowdrift Game, and it makes the important differences between discrete and continuous models of evolution salient.

As we discuss, our results provide insight into the evolution of cooperation, particularly from a multilevel perspective (Luo 2014 [22]; Okasha 2006 [23]; Simon et al. 2013 [24]; Traulsen and Nowak 2006 [25]), evolving games (Akçay and Roughgarden 2011 [26]; Hashimoto and Kumagai 2003 [27]; Smead 2014 [28]), and the relationship between discrete and continuous models of evolutionary dynamics (Traulsen et al. 2005 [29]).

#### 2. The Model

Here, we describe the within- and between-group Moran processes in our model.

Payoff Matrix 2 provides the payoff matrix for the two-player Snowdrift Game we will assume throughout this paper. Let* b *refer to the* benefit* of getting home and let* c *refer to the* cost* of shoveling snow, where . Each driver can either get out of her car to shovel ( for “cooperate”) or stay in her car ( for “defect"). If both drivers cooperate, then each receives a net payoff of , since both receive the benefit of going home, while the cost of shoveling is divided in half. If one driver cooperates and the other defects, then both get to go home, but the cooperator must pay the full cost of shoveling snow, receiving a net payoff of , while the defector pays no cost, receiving a net payoff of . If each driver stays in her car, neither pays the cost of shoveling, but neither gets to go home, so neither receives any reward. Following Zheng et al. (2007 [30]), we let stand for the ratio of the cost of cooperating in the Snowdrift Game to the benefit of doing so (). We can thereby speak of the “-value" of an instantiation of the game.

*Payoff Matrix 2.* The Snowdrift Game is as follows ():

Note that “cooperation” in this context refers to the shoveling snow behavior—that is, strategy* C* in Payoff Matrix 2. This strategy is not a form of altruism since cooperation can be in the interest of the actor, depending on the behavior of the other player. Strategy* C* in Payoff Matrix 2 coincides with a technical definition of cooperation (West et al. 2007 [31]): cooperative behaviors (i) increase the payoff to others and (ii) carry a benefit (or cost) to the actor contingent on the behavior of others.

##### 2.1. Within-Group Moran Process

Suppose there is a metapopulation composed of a finite number of discrete groups, indexed by , so that . The size of a given group () is finite and constant, and each group in the metapopulation is of the same size. We assume the individuals in each group are hard-wired to either cooperate or defect in the Snowdrift Game. The evolutionary dynamics within each group are governed by a discrete time Moran process (Moran 1958 [20], 1962 [21]). At each time step, an individual in a group* j *is chosen to reproduce and have one offspring. The probability that a given individual is chosen to reproduce is proportional to its fitness relative to the average fitness of the individuals in its group. The behavior of an offspring is always identical to the behavior of its parent—that is, cooperators always beget cooperators, while defectors always beget defectors. At the moment it is born, an offspring replaces uniformly at random some individual in* j*, perhaps its parent, but not itself. The probability an individual will be replaced is unaffected by that individual’s phenotype or fitness and is always . For our purposes here, we assume there is no migration between groups and no mutations during reproduction.

Since we are interested in modeling interactions that generate public goods that can be used by all, we will assume individuals play the Snowdrift Game with all of the members of their respective groups, including themselves, simultaneously. This is equivalent to using the expected fitness of random pairwise interactions among members of the group allowing for self-interaction.

Formally, we can represent the fitness of cooperators and defectors in a group* j* as follows. Let and index the parameters* b *and* c *for a given group* j*, let stand for the frequency of cooperators in* j*, and let stand for the frequency of defectors in* j*. The fitness of a cooperator and a defector in group* j* for a given value of are given, respectively, byThe average individual fitness of the members of a group* j *is given by

The composition of a group can change in one of two ways: a cooperator can replace a defector, or a defector can replace a cooperator. Letting stand for the number of cooperators in a group* j*, we can represent the first transition as and the second as . As described elsewhere (Fudenberg et al. 2006 [32]; Nowak 2006 [33]; Taylor et al. 2004 [34]), we can use the fitness given in (3)-(4) to calculate the probability, , of each of these two transitions:

Since no other changes within a group are possible, the probability that the state of the group will not change is given by,

Were each group well-mixed and infinitely large, each group* j* would converge to a stable, internal equilibrium frequency of cooperators (), which is given by the following (Hauert and Doebeli 2004 [16]):

However, because group size is finite in our model and there are no mutations, the only truly stable states of a group are the two absorbing states, in which cooperators fix or go extinct . Nevertheless, the frequency of cooperators in a group in our model will often temporarily oscillate around what its internal equilibrium value would be were group size to be infinite; for a finite population, this is sometimes called its “quasi-equilibrium” (Shpak et al. 2013 [35]).

##### 2.2. Between-Group Moran Process

Our model also involves a discrete time Moran process that occurs between groups. Within the metapopulation, a “parent" group is chosen to replicate, thereby producing a “daughter" group, which replaces uniformly at random some group in the metapopulation, perhaps its parent, but not itself. A daughter group of some group* j* that is created at time* t* has the same frequency of cooperators as* j* at* t* and has the same payoff matrix as* j*. (See Figure 1.)