Group Selection under the Replicator Dynamic

We consider a society where individuals get divided into diﬀerent groups, each playing a different strict Nash equilibrium of the same game. Under group selection, a group playing a Nash equilibrium with a higher payoﬀ evolves faster. As individual agents always play a Nash equilibrium, there is no conﬂict between individual selection and group selection. We deﬁne the dynamics of group selection in terms of the replicator dynamic. In the weak form of group selection, the absolute mass of all groups increase but the relative mass of all groups except the one at the Pareto eﬃcient equilibrium go to zero. In the strong form of group selection, even the absolute mass of all groups except the one at the Pareto eﬃcient equilibrium go to zero. We also provide microfoundations to the dynamics of group selection. Finally, we apply the model to the stag hunt game.


Introduction
Group selection refers to the idea that if individual behavior is beneficial to a group, then that group will acquire evolutionary advantage over other groups.The origins of this notion can be traced back to Darwin himself.In its modern form, it was first proposed by Wynne-Edwards [17,18] in a biological context as a justification for the prevalence of even such behavior that is detrimental to individuals but beneficial to the group.Applied to human populations, it may be invoked to explain the prevalence of "other-regarding" preferences incorporating features like altruism or kindness which are beneficial to the society at large but may end up reducing individual payoff.
Such group selection arguments have, however, been sharply criticised on the ground that it is individuals and not groups who are selected by evolution (Williams [16], Dawkins [4]).Individuals who behave altruistically for the benefit of the group at the cost of their own payoff would be at an evolutionary disadvantage against selfish individuals and would, therefore, be eliminated.Hence, individual selection would triumph over group selection.
Formalizing Wynne-Edwards' [17,18] notion of group selection in game theoretic terms would require, for example, that individual agents play the Pareto optimal outcome in the prisoner's dilemma even though that outcome is in strictly dominated strategies.The well-known haystack model of Maynard Smith [9] is one such attempt and it illustrates that group selection can prevail over individual selection in the prisoner's dilemma only under some very restrictive assumptions.
Due to such difficulties, the naive view of where individuals may sacrifice themselves for the group is no longer considered a strong explanation of group selection.Instead, the alternative view of group selection that is more credible is one where there is no conflict between individual and group selection.Individual's actions respect their self-interest.Within that constraint, if members of a particular group behave in a way that ensures greater payoff, then that group is selected by evolution.In game theoretic terms, this means different groups play different Nash equilibria of a game.The group that plays the Nash equilibrium with the highest payoffs gets selected.Hence, such an interpretation of group selection would be untenable in the prisoner's dilemma with its unique Nash equilibrium. 1he present paper focuses on the second and, arguably, more robust view of group selection.
Boyd and Richerson [1,2] pioneered this approach to group selection.These models have a biological motivation with payoffs in the game being interpreted as reproductive fitness.As individuals within each group continue to play a Nash equilibrium, there is no contradiction between individual selection and group selection in these models.Individuals are not behaving against their selfinterest to benefit their group.Instead, group selection emerges as a mechanism to select among different Nash equilibria.The Pareto efficient Nash equilibrium always gets selected.
We seek to contribute to this literature on group selection in two additional ways.First, we provide formal dynamical models of group selection by specifying evolutionary dynamics that determine the change of the mass of each group.Second, we also provide microeconomic foundations to our model of group selection.This is important for making the model amenable for applications involving not just biological but human populations.Like in Boyd and Richerson [1,2], our model also involves a game with a number of strict Nash equilibria which can be Pareto ranked.The entire society of agents gets split up into several groups, each playing one of the strict Nash equilibria.But Boyd and Richerson [1,2] do not present any formal model of an evolutionary dynamic to track group selection.Instead, their argument is that occasionally, groups face extinction with the rate of extinction being higher for groups which play a Pareto inferior equilibrium.The empty habitat of the extinct groups is then occupied by members of other groups.Since groups playing the Pareto superior equilibrium have a higher average fitness, there is a greater chance that those empty habitats would be occupied by such groups leading to their evolutionary selection.
As our first main contribution, we provide two dynamical models for the absolute mass of different groups playing different equilibria.In one model, the rate of growth of the absolute mass of a group equals its equilibrium payoff.The motivation is purely biological with the payoff representing reproductive fitness.In the second model, the rate of growth is the equilibrium payoff minus the average payoff of all groups.This formulation can also be given a biological justification with the main difference being that, unlike the first dynamic, it allows for the absolute mass of a group to also fall.Under both formulations, it turns out that the relative mass of a group, which is the proportion of agents in the entire society that is in that group, is given by the well-known replicator dynamic (Taylor and Jonker [15]).The replicator dynamic has the well-known property of eliminating strictly dominated strategies (Samuelson and Zhang [12]).By definition, the payoff of the Pareto efficient equilibrium strictly dominates that of all other equilibria.Therefore, the relative mass of all other groups except the one playing the Pareto efficient equilibrium gets driven to zero under both the dynamics of absolute mass.
The implications on the absolute mass, however, differ.Under the first dynamic, the absolute mass of each group grows exponentially.But the mass of the group playing the Pareto efficient equilibrium grows so much faster that it overwhelms the growth of the mass of the other groups.We refer to this phenomenon as a weak form of group selection.No group becomes extinct in an absolute sense but in a relative sense, the one at the Pareto optimal equilibrium becomes predominant.The second dynamic, however, generates a strong form of group selection.Under this dynamic, the total absolute mass of agents remains constant.Moreover, all other groups except the one at the Pareto optimal equilibrium becomes extinct in terms of absolute mass.Only the one at the Pareto optimum survives and it occupies the entire initial mass of agents in the society.As far as we know, this is the first analysis of such different forms of group selection that can emerge under the replicator dynamic or its variants.
Our second main contribution is the provision of microeconomic foundations to the second dynamic, which makes it suitable for not just biological applications but also an economic one.We generate this dynamic using a revision protocol, which is a standard method of deriving dynamics in economic applications of evolutionary game theory (Lahkar and Sandholm [8], Sandholm [13]).
A revision protocol describes the rate at which an agent may consciously abandon a strategy and adopt a new one.This is an important concept for economic applications of evolutionary game theory, where we would like strategy revision to be motivated by some conscious deliberation instead of simple birth and death processes which may be more suitable for biological applications.
The revision protocols we consider involve migration of agents from one group to another.For example, an agent may randomly observe a member of another group.If that member obtains a higher equilibrium payoff, the first agent chooses to migrate to the new group and obtain the higher payoff.Such revision protocols are, in fact, a reinterpretation, in the context of group selection, of imitative revision protocols used to generate the replicator dynamic (Hofbauer [5], Björnerstedt and Weibull [3], Schlag [14]).Thus, in economic applications, the process of migration whereby agents abandon less successful groups and move to more successful groups can facilitate group selection.
More successful groups acquire more agents and, therefore, become larger while less successful one gradually becomes extinct as it loses members.Again, as far as we know, this is the first paper that interprets group selection in terms of such revision protocols.
We then apply our model to the well-known stag hunt game.This game has two Nash equilibria.
The Pareto efficient equilibrium is risky and requires greater trust among players while the strategy involved in the Pareto inferior equilibrium generates a certain payoff.The difference between the two payoffs is the cost of agents not trusting or cooperating with each other.Our general analysis implies that eventually, there will be group selection in favor of the Pareto efficient equilibrium.
But lower is the cost of not cooperating, the slower are the dynamics of group selection in the stag hunt game.Groups playing the inefficient equilibrium will tend to persist longer under such conditions.This conclusion perhaps has some relevance in explaining situations where inefficient outcomes are likely to persist for long.Suppose natural or geographic conditions are sufficiently benign that survival is not difficult even without cooperation.Group selection is likely to be slow.
On the other hand, if conditions are so harsh that unless individuals cooperate, their chances of survival are bleak.Cost of not cooperating is then high.Then, group selection would be relatively fast and groups trapped in the inefficient outcome would be eliminated fairly rapidly.
In fact, our application to the stag-hunt game provides an additional insight.It allows us to relate the speed of group selection to the well-known notion of risk dominance in two-strategy symmetric games like the stag hunt (Kandori et al. [7], Young [19]).Under certain reasonable conditions which we describe more fully in Section 5, the group playing the risk dominant equilibrium will be the larger group.But the risk dominant equilibrium may be Pareto inferior, particularly if the cost of not cooperating is sufficiently low.By increasing the initial mass of the group at the Pareto inferior equilibrium, the low cost of not cooperating can then act as a further drag on the dynamics of group selection.
The rest of the paper is as follows.Section 2 presents the preliminaries of the model.In Section 3, we analyze the dynamics of the absolute and relative mass of each group.Section 4 discuses the microfoundations of these dynamics.In Section 5, we apply our analysis to the stag hunt game.Section 6 concludes.

Preliminaries
We consider a society of N > 1 populations.Each population consists of agents of mass 1, with each agent being of measure zero.Thus, the total mass in the society of population is N .Agents within a population will be called upon to play a n− strategy symmetric normal form game with strategy set S = {1, 2, • • • , n}.The payoff matrix of the game is Thus, v ij is the payoff of a player who plays strategy i ∈ S against another player who plays j ∈ S.
Agents are born at time t = 0 with a propensity to play one of the n strategies.We describe that initial propensity within a population by the vector x ∈ R n such that x i ∈ [0, 1] is the proportion of agents in that population with the propensity to play strategy i.We refer to such a vector x as the initial state of a population and X = {x ∈ R n : x i ∈ [0, 1] for all i ∈ S} as the set of all such possible states.The initial state of every population will be in X.There will be N such initial states, one for each population in the society.
In addition, we assume that in (1), v ii > v ij , for all i, j ∈ S. Hence, in each column of (1), the diagonal element is the highest.Therefore, the normal form game (1) has n strict Nash equilibria.
The standard basis vector e i constitutes the i−th such Nash equilibria, with i ∈ {1, 2, • • • , n}. 2 We further assume that v 11 > v 22 > • • • > v nn .Hence, the n strict equilibria can be Pareto ranked, with e 1 being the Pareto optimal Nash equilibrium.
Once agents are born at t = 0 with their initial disposition, they are randomly matched within a population to play the game (1).If x ∈ X is the initial state in a population, then such random matching generates a population game F such that the payoff of an agent playing i ∈ S in that population is There would be N such population games, one for each population in the society.We reiterate that such random matching is happening entirely within a population.There is no interaction between agents of different populations at this stage.
Let BR(x) be the set of pure best responses to x ∈ X.Thus, BR(x) = {i ∈ S : i ∈ argmax j∈X F j (x)}.Further, let #BR(x) be the cardinality of BR(x).Hence, #BR(x) is the number of strategies which are a best response to x. Define X 0 = {x ∈ X : #BR(x) > 1}.Thus, X 0 ⊂ X is the set of initial states to which there are strictly more than one best response.Hence, it must be that X 0 is a measure zero set.We now assume that the initial states of all the populations are in X \ X 0 .This is an important assumption for us but not a strenuous one as the initial states do lie in a set of full measure.We then denote as B i = {x ∈ X \ X 0 : i = BR(x)}.Thus, B i is the set of such initial states to which i ∈ X is the sole best response.We denote as N i the number of populations whose initial state x ∈ B i .Thus, for N i populations, i ∈ S is the unique best response to its initial state x ∈ X ∈ X 0 .In view of our assumption that all initial states are in X \ X 0 , it must be that j∈S N j = N .Further, in view of the fact that e i is a strict Nash equilibrium for all i ∈ S, e i ∈ B i .We also assume that N i > 0 for all i ∈ S.
We now assume that after being randomly matched, all agents in a population immediately play a best response to the initial state x in their population at time t = 0.The fact that, by assumption, the best response is unique for all populations leads us to the following result about the resulting outcome in each population.The proof is in the Appendix.
Proposition 2.1 Suppose agents in a population are randomly matched in (1) at time t = 0 following which, all agents immediately play a best response in the resulting population game F defined by (2).Let x ∈ X \ X 0 be the initial state of that population.
1.If x ∈ B i , then agents in that population immediately coordinate on the strict Nash equilibrium 2. Hence, agents in every population immediately coordinate on a strict Nash equilibrium.
3. The number of populations who play the strict Nash equilibrium e i is N i .
Recall that in addition to being the number of populations, N is also the mass of agents in the society.Thus, at time 0 itself, the entire mass N of agents get divided into n groups corresponding to the n strict Nash equilibria in the normal form game (1).Henceforth, we refer to the group playing strict equilibrium e i as group i.By part 3 of Proposition 2.1, group i consists of N i populations.Hence, each group consists of multiple populations, all of whom play the same Nash equilibrium.As each of those populations has mass 1, the mass of group i at time t = 0 is also N i .
We refer to N i as the absolute mass of group i.
In addition, we also define the ratio of the mass of agents in group i to the total mass of agents, to be the relative mass of group i. Equivalently, it is the proportion of populations who play the strict equilibrium e i at time 0. As we have assumed N i > 0, it must be that m i > 0 for all i ∈ S.
We note that the vector of relative masses m

Replicator Dynamic
Our model thus far has been confined to time t = 0.As described in Section 2, agents get divided at t = 0 into n groups.Group i consists of those populations who play strict equilibrium e i .The absolute mass of group i is N i and the relative mass of group i is m i .We now allow time to proceed beyond t = 0 and examine how N i and m i evolves according to dynamics that we introduce.In doing so, we assume that a group playing the Nash equilibrium e i will keep playing that equilibrium forever after, thereby receiving payoff v ii .This is an innocuous assumption if groups remain isolated and agents play their best response.Then even if random matchings are rearranged at subsequent times, agents within the group will best respond with strategy i to e i .More care is required to justify this assumption if there is migration between groups.We will consider this point further when we discuss microfoundations to our dynamics in Section 4.
We will denote the values of these variables at time t as N i (t) and m i (t) respectively with If we find that the absolute and relative mass of some group increases at the expense of other groups, then we then we would regard that as evidence of selection in favor of that group.In our model, that would constitute group selection.
The more fundamental part of our analysis is the dynamics of N i .In view of the dependence of m i on N i , the dynamics of m i would follow from that of N i .We specify two evolutionary dynamics for N i .Such a dynamic will take the form of an ordinary differential equation (ODE) that specifies the direction and magnitude of change in N i .Recall from (1) that v ii is the payoff at the strict equilibrium e i .Hence, the first dynamic that we specify for N i is Thus, according to (4), the growth rate of group i, Ṅi N i = v ii .As we have assumed v 11 > v 22 > • • • > v nn , group 1 has the highest growth rate under (4).This is the group that plays the Pareto optimal Nash equilibrium.The motivation behind (4) is largely biological, with the equilibrium payoff v ii being the reproductive fitness of group i. 3 Higher is fitness, higher is the growth rate.We discuss this point in greater detail in Section 4. As the ODE (4) is linear, its solution is straightforward.
We present that solution in the following observation.Observation 3.1 Under the dynamic (4), the mass of group i agents at time t is where Thus, under the dynamic (4), the mass of all groups grows exponentially as specified in (5).
The mass of group i at a time t depends upon the initial mass N i of that group and, much more importantly due to its presence inside the exponential factor in (5), the payoff v ii .In fact, due to this exponential factor, even though the mass of all groups is always increasing, we would expect the eventually, the group with the highest payoff would predominate in the society.To verify this conjecture, it would be useful to derive the dynamics of the relative mass m i (t) using ( 3) and (4).We state the relevant result in the following lemma.The proof, which follows from direct calculation, is in the Appendix.
Lemma 3.2 Consider the dynamic (4) and recall that the relative mass of group i at time t is where is the average payoff of all the groups.Thus, under ( 6), the relative mass m i (t) evolves according to the replicator dynamic when the payoff of strategy k ∈ S is the constant value v kk .
In a general population game with strategy set then the payoff of a strategy i would depend upon m and would be denoted π i (m).It is then well-known that the replicator dynamic would take the form ṁi = m i π i (m) − k∈S m k π k (m) . 4The dynamic in ( 6) is of this form with the only difference being that the payoff to strategy i is the constant v ii .This is because in our model, once the segregation into different groups happen at t = 0, the group playing i ∈ S continues to receive v ii forever thereafter.Hence, by Lemma 3.2, when the absolute mass of group i changes according to (4), the relative mass changes according to the replicator dynamic (6).Like in any replicator dynamic, the relative mass m i increases when the payoff of group i, v ii , is greater than the average payoff of all the groups.We then obtain the following result about the evolution of the relative masses of the different groups.
Proposition 3.3 Let the absolute mass N i (t) of group i evolve according to (4).Recall the relative mass m i (t) = N i (t) N (t) .Then, as t → ∞, m 1 (t) → 1 and m j (t) → 0 for all j ̸ = 1.
The proof of Proposition 3.3 relies on the elimination of strictly dominated strategies under the replicator dynamic (Theorem 1, Samuelson and Zhang [12]).Formally, strategy i has the j∈S N j exp(v jj t) by ( 5).Hence, m 1 (t) → 1 implies N 1 (t) → j∈S N j (t) = N (t).After a sufficiently long period of time, almost the entire mass of agents in the society consists of group 1 agents.This is despite the fact that under the dynamic (4), the mass of every other group also increases exponentially.
It is worth noting that the dynamics (4) and ( 6) depends only on the diagonal payoffs of (1).
The off-diagonal payoffs do not matter.Hence, any two such normal form games will have the The time scale differs in the horizontal axis of the two panels.This difference is required to make N 3 (t) visible in the same figure as N 1 (t) and N 2 (t).same dynamics as long as the diagonal payoffs are identical. 5Figure 1 presents the trajectories of the absolute and relative masses under these dynamics for such a three-strategy normal form game with v 11 = 10, v 22 = 8 and v 33 = 1.Each of the absolute masses are growing exponentially but the relative masses of groups 2 and 3 are falling to zero.The relative mass m 3 (t), which obtains the worst equilibrium payoff, first drops towards zero.Once that happens, the game is effectively one with two groups, with group 2 now getting the worst payoff.The relative mass m 2 (t) then drops towards zero.With more than three strategies, we would expect the same pattern to emerge as groups keep getting eliminated successively from the worst to the second best.Observation 3.1 and Proposition 3.3 together constitute weak form of group selection in our model.In this form of group selection, the absolute mass of every group keeps growing according to (5).But as Proposition 3.3 shows, the relative mass of all other groups except group 1 goes to zero.Hence, there is group selection in favour of group 1 but only in terms of relative mass.On the other hand, a strong form of group selection in favor of group 1 would require that not only m j (t) → 0 for all j ̸ = 1 but also N j (t) → 0. Thus, even in terms of absolute mass, all other groups except the one playing the Pareto optimal Nash equilibrium should be driven to extinction.
To explore that possibility of strong group selection, we change the dynamics of N i (t) from ( 4) where v(t) is the average payoff (7) and j∈S N j (t) .Thus, under this dynamic, the growth rate of the absolute mass of group i is Ṅi (t) N i (t) = v ii − v(t). 6Like in (4), this growth rate is higher the larger is v ii .But the key difference is that unlike in (4), the growth rate in (8) can also be negative.Therefore, under (8), it is possible that the absolute mass of some can decline and, in fact, go to zero.It is this feature of (8) that opens up the possibility of strong group selection.We will discuss the microfoundations of (8) in Section 4 As the first step towards analyzing (8), we compute the resulting dynamics of the relative mass.We present the result in the following proposition.The proof is in the Appendix.The main conclusion is that despite differences in the dynamics of the absolute mass, the relative mass evolves identically under the replicator dynamic.
Proposition 3.4 Consider the dynamic (8) and recall that the relative mass of group i at time t where v(t) is the average payoff (7).This is the same replicator dynamic (6) we computed in Lemma 3.2.Therefore, under (8), m 1 (t) → 1 and m j (t) → 0 for all j ̸ = 1 as t → ∞.
In one way, it is not surprising that the evolution of the relative mass under (8) also follows the replicator dynamic.The structure of ( 8) is similar to the replicator dynamic (6) except that m i gets replaced by N i .Since it the same replicator dynamic that governs the evolution of the relative mass, Proposition 3.4 also establishes that m 1 (t) → 1 and m j (t) → 0 for j ̸ = 1.Like in Proposition 3.3, this is a consequence of the elimination of dominated strategies under the replicator dynamic.
Hence, just like under the first dynamic (4), the relative mass of all but group 1 is also driven to zero under the second dynamic (8).
We now consider the dynamics of the absolute mass N i (t) under (8).For strong group selection, we require that all groups other than group 1 should be driven to extinction in terms of absolute mass.First, we establish that the total mass of all groups remains unchanged at the initial value N under (8).The proof of the result, stated in the following lemma, is in the Appendix.
Lemma 3.5 Under the dynamic (8), j∈S Ṅj (t) = 0. Hence, the total mass of the n groups, N (t), remains unchanged at the initial value N at all time t > 0. Lemma 3.5 allows us to determine the trajectory of the absolute mass of different groups under this dynamic.The following proposition states the result.The proof, which follows from a combination of Proposition 3.4 and Lemma 3.5, is in the Appendix.Proposition 3.6 Let t → ∞.Then, under the dynamic (8), N 1 (t) → N and N j (t) → 0 for all j ̸ = 1.Proposition 3.6 is our main result on the strong form of group selection under the dynamic (8).
Recall that strong group selection requires both the relative and absolute mass of all groups other than group 1 to go to zero.Proposition 3.4 established the first of these requirements.But we know also vii − v(t).from Observation 3.1 and Proposition 3.3 that relative mass going to zero itself is not sufficient for absolute mass to also go to zero.Instead, in those results on weak group selection, we had the absolute mass of all groups growing exponentially.In the present case, though, Lemma 3.5 already established that the total mass of agents remain unchanged at N under (8).Proposition 3.6 then shows that eventually, this entire mass will consist of group 1 agents, i.e. the group that plays the Pareto optimal Nash equilibrium.All other groups will be driven to extinction in the sense that their absolute mass will go to zero.
Figure 2 illustrates the trajectories of the absolute masses under the dynamic (8) for the same parameters as in Figure 1, i.e. v 11 = 10, v 22 = 8 and v 33 = 1 and initial masses N 1 (0) = N 2 (0) = N 3 (0) = 100.Thus, the initial total mass of the groups is 300.Hence, eventually, N 1 (t) → 300 and both N 2 (t), N 3 (t) → 0. The trajectory of the relative masses, being governed by the same replicator dynamic (6), would be the same as in the right panel of Figure 1. 7

Microfoundations
Section 3 establishes the dynamics of the absolute and relative masses of the various groups in our model.The more important dynamics are those of the absolute masses as the dynamics of the relative masses arise from the former.But the earlier section simply defined those dynamics and analyzed them without providing sufficient explanation of where these dynamics arise from.The present section addresses this issue and provides microfoundations to these dynamics.
As noted in Section 3, the motivation for the first of these dynamics, which is (4), is biological.
The growth rate of group i is simply the equilibrium payoff v ii that group receives.This equilibrium payoff is the reproductive fitness of the group.An underlying assumption behind the biological motivation of ( 4) is that there is no interaction between the groups.Hence, for example, they do not directly compete or come into conflict with each other.As each group is isolated and, as assumed at the beginning of Section 3, they continue playing their strict Nash equilibrium e i forever, their reproductive fitness is captured entirely by v ii at all times t.We can extend this biological foundation to the second dynamic for N i , which is (8), by introducing some such competition or conflict between groups.As a result, even though the reproductive fitness of each individual in the group continues to be v ii , the growth rate of group i in ( 8) is v ii − v(m).Groups where the fitness is less than the average fitness experience a fall in absolute mass.Intuitively, the average payoff is the rate at which agents are eliminated or die due to conflict between different groups.Therefore, we can also interpret v ii − v(m) as the net reproductive fitness in this model.
In economic or social applications of evolutionary game theory, however, we seek microfoundations of an evolutionary dynamic that do not have a biological origin.In such applications, which involve human agents and not genotypes or phenotypes, changes in strategy distribution should be a result of conscious deliberation rather than a pure birth and death process.Accordingly, we invoke the notion of a revision protocol which describes the manner and rate at which agents consciously shift from one strategy to another.Such revision protocols have been defined to describe changes in the proportion or relative mass of agents using different strategies (Lahkar and Sandholm [8], Sandholm [13]).We can extend that definition to our context of group selection and evolution of the absolute mass of agents.This will enable us to provide microfoundations to the second of our two dynamics of absolute mass, which is (8).This was the dynamic that generated the strong form of group selection.

Denote the vector of equilibrium payoffs as
A revision protocol ρ : R n × X → R n×n formalizes the process by which an agent abandons a group and moves to a new group.This may happen, for example, through a process of migration.As time passes, agents are chosen at random and offered the opportunity to switch groups.When an agent from group i obtains such a revision opportunity, he switches to group j at the rate ρ ij (v, m(t)).Equivalently, an agent from a population constituting group i migrates to another population constituting group j at the rate ρ ij (v, m(t)).Under such a revision protocol, the rate of change in the absolute of group i would be Ṅi = j∈S The first term in the RHS of ( 9), j∈S N j ρ ji (v, m), represents the absolute mass of agents from all groups j who abandon that group and migrate to group i.The second term, N i j∈S ρ ij (v, m) is the absolute mass of agents who abandon group i and go to various other groups j.The difference between the two terms then represents the change in the absolute mass of group j.The particular form of the mean dynamic will then depend upon the form of the revision protocol ρ ij we specify.
For a deeper interpretation of revision protocols and the mean dynamic, we can follow Section 4.1.2 of Sandholm [13] and assume that each agent in the society receives revision opportunities independently of each other and according to an exponential distribution of rate R. Upon receiving a revision opportunity, a group j player migrates to group i with probability R , where R is assumed to be sufficiently large that these are sensible probability values for all i, j ∈ S. Basic properties of the exponential distribution then implies that an agent would expect to receive Rdt revision opportunities in the next dt time units. 8Hence, the expected number of revision opportunities received by group j agents in the next dt time interval is N j Rdt.As each group j agent migrates to group i with probability ρ ji R , the expected number of group j agents migrating to group i during the next dt time interval is N j ρ ji R Rdt = N j ρ ji dt.Summing over all groups, it follows that the expected change in the absolute mass of group i is then Taking the derivative dN i (t) dt then gives us the mean dynamic (9).By specifying a particular form of a revision protocol, we can then generate a dynamic such as (8).Before doing so, we should make two further assumptions to clarify the behavior and incentives that agents face as they decide to migrate from one group to another.Recall a group consists of several populations, with each population playing the same strict Nash equilibrium.At t = 0, as we have assumed, agents in each population are randomly matched and they immediately coordinate on a strict Nash equilibrium.Our first assumption is that at subsequent time periods also, such random matching continues with matches being rearranged into new pairs within each population.
However, as all agents in the population are already at the same Nash equilibrium, say e j , all agents continue to play the mutual best response j ∈ S in all such subsequent rounds of random matching.Hence, the payoff of all such agents in any population belonging to group j continues to be v jj .Our second assumption is that when a new agent migrates to a population in group j, that agent will also be randomly matched like existing members of that population.Moreover, like other agents, the new agent will also best respond.As the prevailing state is the strict equilibrium e j , the new agent will play the best response j and, therefore, obtain payoff v jj .Therefore, as noted in the beginning of Section 3, we can justify agents receiving the equilibrium payoff v jj even when migration is allowed.
We now specify a revision protocol that will, in fact, generate the dynamic (8) through the general form (9).This revision protocol is due to Schlag [14] and was initially proposed to generate the replicator dynamic.Adapted to our context, this protocol will capture the idea of agents migrating to a more successful group.This revision protocol, which is called pairwise proportional imitation, takes the form Intuitively, at time t, if an agent in group i receives a strategy revision opportunity, he picks out one member from the society at random.The agent was receiving payoff v ii in group i.The probability of choosing a member from group j is m j .Recall that once an agent migrates to group j, that agent will find it optimal to play j ∈ S and, therefore, receive payoff v jj .Therefore, the agent decides to imitate the group j member's behavior and migrate to that group provided v jj > v ii , i.e. group j has a strictly higher equilibrium payoff than group i.The following result then demonstrates the derivation of ( 8) through (10).The proof, which follows from calculation, is in the Appendix.
We now present two other revision protocols that also generate the dynamic (8).The derivation of the dynamic is quite straightforward and follows the methodology of Proposition 4.1.Hence, we skip the details of the calculation.As with (10), both these other protocols have been applied earlier to derive the replicator dynamic.We adapt them to our purpose of deriving the dynamic of absolute mass.The first, due to Björnerstedt and Weibull ( [3]), takes the form where K > v 11 .Hence, K > v ii , for all i ∈ S.This revision protocol, known as imitation driven by dissatisfaction, captures the idea that agents from every group i are always dissatisfied by their present payoff v ii .The parameter K represents an aspiration level and the difference K − v ii measures the extent of this dissatisfaction.Such an agent randomly picks out someone else from group j, which happens with probability m j , and migrates to that agent's group at a rate equal to (11).Greater is the dissatisfaction (K −v ii ) at the current payoff, the faster is the rate of migration.
The second protocol is adapted from Hofbauer [5].This protocol, known as imitation of success, takes the form where K < v nn .Therefore, K < v ii for all i ∈ S. Here, K is once again an aspiration level but is sufficiently low that the difference (v jj − K) can be interpreted as the success of the equilibrium payoff v jj against that aspiration level.Like in (10) and (11), an agent randomly picks a group j member, which happens with probability m j .That agent then imitates and migrates to group j at a rate equal to (12).The higher is the success (v jj − K), the greater is the rate of imitation.
Thus, revision protocols such as (10)-( 12) provide plausible microfoundations to evolution of the absolute mass under (18) in our model.As agents abandon less successful groups or migrate to more successful groups under any of these protocols, the rate of change of the absolute mass of any group will be given by (8).Propositions 3.4 and 3.6 have already established the asymptotic values of the relative and absolute masses of each group under this dynamic.Eventually, the entire mass N of agents will become concentrated in group 1, which plays the Pareto optimal Nash equilibrium.
The other groups will become extinct.In terms of the above revision protocols, all agents will eventually leave all other groups and migrate to group 1.
Can such revision protocols also provide microfoundations to the first of our two dynamics of absolute mass, which is ( 4)?The answer is no.Revision protocols such as ( 10)-( 12) require agents to switch from one group to another and, hence, will generate a dynamic that also depends upon payoffs in other groups.This is true for the dynamic (8) through the presence of the average payoff v(t).But the dynamic (4) is independent of payoffs in other groups.Hence, to generate that dynamic, we cannot invoke any revision protocol that requires agents to migrate from one group to another.Instead, as we have remarked at the beginning of this section, we need to assume that groups are isolated from each other.Change of strategy within the group is also not meaningful as all agents in the group play the same Nash equilibrium strategy.Hence, under such conditions, the biological motivation where we interpret equilibrium payoffs as reproductive fitness seems to be the only plausible way of deriving that dynamic.
To end this section, we address the question of why we have chosen to focus on the replicator dynamic.There are three main reasons.First, being the most well-known evolutionary dynamic, the use of the replicator dynamic is standard in evolutionary game theoretic models.Second, the replicator dynamic is amenable to both a biological and an economic interpretation.As discussed in this section, the biological interpretation involves viewing the payoffs as reproductive fitness while the economic interpretation arises from the revision protocols ( 10)- (12).Hence, the replicator dynamic provides a common dynamical model that can be applied to understand group selection in both biological and human populations.Third, the replicator dynamic has the property that it eliminates dominated strategies.This is not a property that other evolutionary dynamics necessarily have. 9As our results on group selection depend critically on the elimination of groups with the lowest equilibrium payoffs, the replicator dynamic is particularly well suited for our purpose.

Application: Stag Hunt
We now apply our general analysis to the stag hunt game.This is a two-strategy symmetric game where, for ease of reference, we refer to strategy 1 as "stag" (S) and strategy 2 as "hare" (H).As per the well-known parable of the stag-hunt game, the two players are hunters.If both hunt the stag together, they obtain a payoff of V , which is the highest possible payoff.But if only one player hunts the stag, that player's payoff is 0. Thus, hunting stag is risky as it requires the cooperation of both players.On the other hand, hunting hare is a riskless endeavor.Whether alone or jointly, hunting hare gives a payoff of V − c, where V > c > 0. Thus, that payoff is certain but is lower than that of hunting the stag jointly.The payoff matrix of this game is, therefore, In terms of the notation in (1), v 11 = V and v 22 = V − c.The reason behind presenting payoffs in this manner is that the parameter c will play an important role in our analysis of this game.
As in Section 2, we continue to assume that there are N populations, with each population having a mass 1 of players.Within each population, players are randomly matched in pairs to play the game.Since there are only two strategies in (13), we can represent an initial state in a population in this game by the scalar x ∈ [0, 1], which denotes the proportion of agents in the population with the propensity to play the strategy 1.There will be N such initial states in the society, all in [0, 1].In terms of the initial state, the payoff functions in the game are Clearly, there are two strict equilibria, x = 1 and x = 0 in this game.The former is the equilibrium where all players play strategy 1 (stag) while the latter is the one where all players play strategy 2 (hare).Accordingly, we call x = 1 as the stag equilibrium and x = 0 as the hare equilibrium.
It is evident that the stag equilibrium is Pareto optimal.In fact, we can interpret c in (13) as the cost of not coordinating on the Pareto optimal equilibrium.Higher is c, the greater is the difference between the two equilibrium payoffs V and V − c.In addition to the two pure equilibria, there is also a mixed equilibrium given by Adapting the notation of Section 2 to the stag hunt game, we denote as B S = (x * , 1] the set of initial states to which "stag" is the unique best response while B H = [0, x * ) as the set of initial states to which "hare" is the unique best response.All N initial states are either is B S or B H with at least one in both B S and B H . Thus, in keeping with the general structure of Section 2, there is no initial state in the measure zero set {x * }.For the purpose of this section, however, we also make a stronger assumption.It is that the N initial states are distributed uniformly on [0, 1], with none of them lying exactly at x * .As we will see, this assumption will allow us to relate the notion of a risk dominant equilibrium to group selection in the stag-hunt game.
By Proposition 2.1, all populations whose initial states are in B S coordinate on the stag equilibrium (x = 1) while the populations whose initial states are in B H coordinate on the hare equilibrium (x = 0).Our assumption of uniform distribution of the initial states on [0, 1], however, allows us to go further in predicting the number of populations at each of these pure equilibria.Recall that Similarly, the replicator dynamic for the relative mass m S would be ṁS We know from Propositions 3.4 and 3.6 the asymptotic behavior of ( 18) and (19).As the stag equilibrium is Pareto efficient, N S → N in (18) and m S → 1 in (19).Therefore, eventually, almost the entire society will consist of stag players and the group playing hare will become extinct.But the specific analysis of the stag hunt game provides another interesting insight.It is that lower is c, the slower is the rate of convergence under (18) or (19).Thus, a society with a lower c will take a longer time to converge to the Pareto optimal stag equilibrium than one with a higher c.
Our conclusion that a lower c reduces the speed of group section under (18) or ( 19) is independent of our assumption of the uniform distribution of the N initial states in the stag hunt game.This conclusion follows from Section 3 where no such assumption was made.The uniform distribution assumption, however, reveals another reason why a lower c will act as a further drag on group selection in the stag hunt game.Recall our conclusion following ( 16) that a majority of populations initially coordinate on the stag equilibrium if and only if that equilibrium is risk dominant, for which we require c > V 2 .Hence, if c < V 2 , the hare equilibrium risk dominant and it is on that equilibrium that a majority of populations will coordinate.The hare group is, therefore, initially in the majority if c is sufficiently low.This second factor, which arises from the uniform distribution assumption, will further increase the time required for the hare group to be driven to extinction under (18) or (19).
We can relate these observations on the effect of c on group selection to certain real world circumstances.Suppose a lower c results from benign geographical or natural conditions.Then, even if agents do not trust or cooperate with each other, they can still attain a reasonably high standard of living.In our stag-hunt game that would be the payoff V − c, which is not too different from V .Such a society would see a greater number of groups playing a socially worse equilibrium and for a longer period of time.On the other hand, in a society where natural conditions are harsh, failure to cooperate can impose severe costs (high c) and may make even survival difficult. 10Then, more groups are likely to play the Pareto superior equilibrium and the ones that find themselves trapped in the Pareto inferior equilibrium will experience a faster rate of extinction.

Conclusion
We have considered the dynamics of group selection in this paper.We view group selection as a process of selection between different strict Nash equilibria in a normal form game.The society initially gets segregated into different groups, with each group playing a different strict Nash equilibrium.We then define the dynamics of the absolute and relative masses of each group.The first dynamic of absolute mass is one where the growth rate equals the equilibrium payoff of the group while in the second dynamic, the growth rate equals the difference between the equilibrium payoff and average equilibrium payoff.In both cases, the dynamics of relative masses are given by the replicator dynamic.The first of these dynamics generate a weak form of group selection where the absolute mass of each group grows exponentially but in terms of relative mass, the group playing the Pareto optimal equilibrium takes over the society.Under the second dynamic, we obtain a strong form of group selection.The absolute mass of all groups except the one at the Pareto optimal equilibrium go to zero.Hence, all such groups become extinct and the entire society eventually consists of the group at the Pareto optimal equilibrium.Naturally, the relative mass of this group converges to one.
We can provide a biological justification for both dynamics in terms of reproductive fitness.
Higher is the equilibrium payoff of a group, higher is the fitness.Naturally, the group with the highest fitness eventually gets selected by evolution.In addition, we can also provide our second dynamic process microeconomic foundations using the notion of a revision protocol.Agents observe the equilibrium payoffs in different groups and migrate to one with a higher payoff.Eventually, all agents will migrate to the group with the highest equilibrium payoff.Finally, we apply our model to the stag hunt game.In this application, a smaller difference between the Pareto optimal stag equilibrium and the Pareto inferior hare equilibrium reduces the speed of group selection in favor of the Pareto optimal equilibrium.We end our discussion with a few general remarks on group selection and the migration based revision protocol we have considered.In our model, the replicator dynamic and its variants evolve fast.The agents are segregated into different groups at t = 0 itself and within a short span of time, the dynamics reach close to their asymptotic values.In reality, of course, the process is likely to be more drawn out.The initial separation into groups will itself take more time.In fact, a more complete model would probably involve two dynamic processes operating simultaneously.Under the first process, each population would converge to an equilibrium and under the other much slower process, agents will switch between groups.For reasons of tractability, we have simplified our model and assumed instantaneous coordination on a Nash equilibrium a the initial point of time.This does imply that our dynamics evolve fast.But we do not view that as a serious limitation as we can slow down the dynamics as much as we want by dividing them with a large enough constant.Such a rescaling will not affect our results on convergence except that the time span required for convergence will be longer.We leave the analysis of the more complicated model with a two-speed dynamic process as a task for future research.
The revision protocols we have suggested also deserve some comments.It is important that when agents migrate to a new group, it does not upset the Nash equilibrium of that group.Only then will migrants find it optimal to follow the equilibrium strategy of the new group.For that to happen, the mass of new migrants should be sufficiently low that the existing equilibrium strategy continues to be the unique best response.In our model, that is not a problem.As time is a continuous variable, the probability that two agents get revision opportunities and, hence, migrate at exactly the same time is zero.Hence, at most one migrant will enter a group at a given time and given that migrant is of measure zero, will not have any impact on the best response or the Nash equilibrium of the group.Such a migrant, when randomly matched with another agent in the new group, will find the prevailing equilibrium strategy to be the best response.More realistically, this condition requires that should not be a sudden or huge influx of migrants into a new group.
Otherwise, the best response and Nash equilibrium of that group may be disrupted.

Figure 1 :
Figure 1: The trajectories of absolute masses (left panel) and relative masses (right panel) under the dynamics (4) and (6) for a normal form game (1) with v 11 = 10, v 22 = 8 and v 33 = 1.The initial values of the absolute masses are N 1 (0) = N 2 (0) = N 3 (0) = 100 so that the initial relative masses are m 1(0) = m 2 (0) = m 3 (0) = 13 .The time scale differs in the horizontal axis of the two panels.This difference is required to make N 3 (t) visible in the same figure as N 1 (t) and N 2 (t).

Figure 2 :
Figure 2: The trajectories of absolute masses under the dynamic (8) for a normal form game (1) with v 11 = 10, v 22 = 8 and v 33 = 1.The initial values of the absolute masses are N 1 (0) = N 2 (0) = N 3 (0) = 100.The trajectories of relative masses are given by the right panel of Figure 1.