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Dynamic Games and Applications

20-02-2024

Group Selection Under the Replicator Dynamic

Author: Ratul Lahkar

Published in: Dynamic Games and Applications

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Abstract

We consider a society where individuals get divided into different groups. All groups play the same normal form game, and each group is in the basin of attraction of a particular strict Nash equilibrium of the game. Group selection refers to the evolution of the mass of each group under the replicator dynamic. We provide microfoundations to the replicator dynamic using revision protocols, wherein agents migrate between groups based on average payoffs of the groups. Individual selection of strategies within each group happens in two possible ways. Either all agents instantaneously coordinate on a strict Nash equilibrium or all agents change their strategies under the logit dynamic. In either case, the group playing the Pareto efficient Nash equilibrium gets selected. Individual selection under the logit dynamic may slow down the process and introduce non-monotonicity. We then apply the model to the stag hunt game.

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Appendix
Available only for authorised users
Footnotes
1
For a review of this view on group selection, we refer the reader to Wilson and Sober [29], Robson [21], Wilson and Wilson [30] and Robson and Samuelson [22].
 
2
The average payoff of the entire society is the average of the group average payoffs.
 
3
See Lahkar [18], Heifetz et al. [14] and Bandhu and Lahkar [1] for applications of the replicator dynamic in indirect evolution.
 
4
Sandholm [25] presents an extensive analysis of this dynamic and of the more general class of perturbed best response dynamics to which it belongs.
 
5
The standard basis vector \(e_s\in {\textbf{R}}^n\) is the vector with 1 in the s-th place and 0 otherwise.
 
6
This is the typical way to describe population games, i.e., games with a large number of agents (Section 2.1, Sandholm [25]). Intuitively, we can think of each agent as a point in the interval [0, 1]. Thus, the total mass of 1 is the length of the interval and each agent, being a point, is of size zero.
 
7
As is standard in evolutionary game theory, we interpret (6) as the current payoff, i.e., the payoff based on the current state and mass of the group. It is liable to change when the state or mass changes under evolutionary dynamics.
 
8
Here, \(\sum _{k\in S}m^k\pi _k(m)\) is the average payoff of the strategies.
 
9
If revision opportunities arrive according to an exponential distribution with rate R, then the number of revision opportunities that arrive in the time interval \([\tau _1,\tau _2]\) follows a Poisson distribution with rate \(R(\tau _2-\tau _1)\). Hence, the expected number of revision opportunities in the time interval \([0,d\tau ]\) is \(Rd\tau \). A stochastic process satisfying such conditions is, therefore, called a Poisson process. See Section 10.C.2 in Sandholm [25] for further details.
 
10
The same is true in the two other revision protocols, (12) and (13), that follow.
 
11
The best response to \(\frac{m^ie_i}{m^i}=e_i\) is i. Hence, the initial state being \(m^ie_i\) is consistent with our assumption that the best response for every agent in the group to \(\frac{x^i(0)}{m^i}\) is strategy i.
 
12
The off-diagonal payoffs will, however, matter for the size of the sets \(B_i\) as defined in (4).
 
13
A more general reason behind such convergence is that the logit dynamic converges in potential games (Sandholm [24], Hofbauer and Sandholm [16]). Pure coordination games are the prototypical examples of potential games. Only if (1) is a two-strategy game can we relax the assumption that it is a coordination game. The logit dynamic must converge to an approximation of a strict Nash equilibrium even if (1) is not a coordination game. Section 6 presents such an example.
 
14
One reason is that we seek to track the entire trajectory \(m^i(\tau )\) in order to check whether \(m^1(\tau )\) increases monotonically, i.e., whether selection of group 1 is monotonic.
 
15
To clarify interpretation, we note that of the total mass of 0.5 in each group in (19), the mass of agents playing strategy 1 is 0.1717 in group 1 and 0.005 in group 2.
 
16
Thus, by \(\tau =5\), \(m^1(\tau )\) is nearly equal to 1 in the right panel, which the not the case in the left panel.
 
17
In fact, the same pattern of monotonic convergence of \(m^1(\tau )\) is also observed in Fig. 1.
 
18
This happens because (17) is a two-strategy game with two strict equilibria.
 
19
Thus, as noted in the last paragraph before Sect. 5.1, with multilevel selection, it is no longer the case that a single group always has the highest average payoff. Hence, an argument based on elimination of dominated strategies as in Proposition 4.3 is no longer applicable.
 
20
With instantaneous coordination, the average payoff of group 1 would immediately be 2 and that of group 2 would be 1.
 
21
In a two-strategy game with two strict Nash equilibria, this would be true even if the game is not a coordination game. See also footnote 13 in this connection.
 
22
The stag hunt game is clearly not a coordination game. But recall from footnote 13 that we can apply multilevel selection to two-strategy symmetric games even if they are not coordination games.
 
23
With two strategies, the state space is one-dimensional. Hence, under standard evolutionary dynamics like the replicator dynamic or the logit dynamic, we can write the basin of attraction of \(e_1\) as \((y_1,1]\) while that of \(e_2\) as \([0,y_1)\). For the logit dynamic, these basins of attraction would hold up to an approximation. Thus, the basin of attraction of \(e_1\) is larger if \(1-y_1>y_1\) or \(y_1<\frac{1}{2}\). This is the condition for \(e_1\) to be risk dominant.
 
24
But it is still possible for the equilibrium that is not risk dominant to be selected. It is just that its basin of attraction is smaller.
 
25
Thus, in (23), if c is close to V, then the hare payoff is close to zero.
 
26
In a two-strategy game, \(m^2=1-m^1\).
 
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Metadata
Title
Group Selection Under the Replicator Dynamic
Author
Ratul Lahkar
Publication date
20-02-2024
Publisher
Springer US
Published in
Dynamic Games and Applications
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-024-00556-9

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