Statistics of the number of equilibria in random social dilemma evolutionary games with mutation

In this paper, we study analytically the statistics of the number of equilibria in pairwise social dilemma evolutionary games with mutation where a game's payoff entries are random variables. Using the replicator-mutator equations, we provide explicit formulas for the probability distributions of the number of equilibria as well as other statistical quantities. This analysis is highly relevant assuming that one might know the nature of a social dilemma game at hand (e.g., cooperation vs coordination vs anti-coordination), but measuring the exact values of its payoff entries is difficult. Our delicate analysis shows clearly the influence of the mutation probability on these probability distributions, providing insights into how varying this important factor impacts the overall behavioural or biological diversity of the underlying evolutionary systems.


The replicator-mutator equation
The replicator-mutator equation is a set of differential equations describing the evolution of frequencies of different strategies in a population that takes into account both selection and mutation mechanisms. It has been employed in the study of, among other applications, population genetics [11], autocatalytic reaction networks [29], language evolution [21], the evolution of cooperation [14,20] and dynamics of behavior in social networks [22].
Suppose that in an infinite population there are n types/strategies S 1 , · · · , S n whose frequencies are, respectively, x 1 , · · · , x n . These types undergo selection; that is, the reproduction rate of each type, S i , is determined by its fitness or average payoff, f i , which is obtained from interacting with other individuals in the population. The interaction of the individuals in the population is carried out within randomly selected groups of d participants (for some integer d). That is, they play and obtain their payoffs from a d-player game, defined by a payoff matrix. We consider here symmetric games where the payoffs do not depend on the ordering of the players in a group. Mutation is included by adding the possibility that individuals spontaneously change from one strategy to another, which is modeled via a mutation matrix, Q = (q ji ), j, i ∈ {1, · · · , n}. The entry q ji denotes the probability that a player of type S j changes its type or strategy to S i . The mutation matrix Q is a row-stochastic matrix, i.e., n j=1 q ji = 1, 1 ≤ i ≤ n.
The replicator-mutator is then given by, see e.g. [17,15,16,24] x i = n j=1 where x = (x 1 , x 2 , . . . , x n ) andf (x) = n i=1 x i f i (x) denotes the average fitness of the whole population. The replicator dynamics is a special instance of (1) when the mutation matrix is the identity matrix.
In this paper, we are interested in random social dilemma games where T and S are uniform random variables in the corresponding intervals, namely • In PD games: • In SD games: • In SH games: T ∼ U (0, 1), S ∼ U (−1, 0), • In H games: T ∼ U (0, 1), S ∼ U (0, 1).
Random evolutionary games, in which the payoff entries are random variables, have been employed extensively to model social and biological systems in which very limited information is available, or where the environment changes so rapidly and frequently that one cannot describe the payoffs of their inhabitants' interactions [4,3,19,12,5,6,10,8]. Equilibrium points of such evolutionary system are the compositions of strategy frequencies where all the strategies have the same average fitness. Biologically, they predict the co-existence of different types in a population and the maintenance of polymorphism.
In this paper, we are interested in computing the probability distributions of the number of equilibria, which is a random variable, in the above random social dilemmas. Answering this question is of great importance in the context of social dilemmas since one might know the nature of the game, i.e. the payoff entries ranking in the game, but it might be difficult to predict or measure the exact values of these entries. When mutation is absent (q = 0), the answer is trivial because there is always a fixed number of equilibria depending on the nature of the social dilemmas [20,27,7]. As shown in our analysis below, this is however not the case any longer when mutation is non-negligible, and this number highly depends on the nature of the social dilemma too.
The following result [7] provides partial information about these probability distributions.
Theorem 1.1. (DuongHanDGA2020) Suppose that S and T are uniformly distributed in the corresponding intervals as above. Then According to the above theorem, SD games and H games are simple and complete. However, the probabilities of having 3 equilibria (or alternatively 1 equilibrium) in SH and PD games are left open in [7]. The key challenge is that the conditions for these games to have 3 equilibria (or alternatively 1 equilibrium) are much more complicated than those of 2 equilibria. The aim of this paper is to complete the above theorem, providing explicit formulas for these probabilities. As such, it will also allow us to derive other statistical quantities (such as average and variance), which are important to understand the overall distribution and complexity of equilibrium points in pairwise social dilemmas (with mutation). To this goal, we employ suitable changes of variables, which transform the problem of computing the probabilities to calculating areas, and perform delicate analysis.

Main results
The main result of this paper is the following.
Theorem 1.2. The probability that SH and PD games have 3 equilibria is given by, respectively The above theorem combines Theorem 2.3 (for SH-games) and Theorem 2.6 (for PD games), see Section 2. Theorems 1.1 and 1.2 provides explicitly the probability distributions of the number of equilibria for all the above-mentioned pairwise social dilemmas. In SH-games and PD-games, these distributions are much more complicated and significantly depend on the mutation strength. We summarize these results in the following summary box.
Box 1: Probability of having k equilibria in a pairwise social dilemma (p k ) • Harmony game (H) • Stag-Hunt game (SH) • Prisoner's Dilemma (PD) As a consequence, we can now derive other statistical quantities such as the mean value, ENoE, and the variance, VarNoE of the number of equilibria using the following formulas We depict these quantities in Figure 1 below. Our delicate analysis clearly shows the influence of the mutation on the probability distributions, thus on the complexity and bio-diversity of the underlying evolutionary systems. We believe that our analysis may be used as exemplary material for teaching foundational courses in evolutionary game theory, computational/quantitative biology and applied probability. The rest of the paper is organized as follows. In Section 2, after recalling some preliminary details, we present the proof of the main theorem 1.2, which we split into Theorem 2.3 for SH games in subsection 2.3 and Theorem 2.6 for PD games in subsection 2.4. Finally, in Section 3 we provide further discussion.
2 Probability of having three equilibria in SH and PD games

Joint probability density via change of variable
The following lemma is a well-known result to compute the probability density of random variables using change of variables. We state here for two random variables that are directly It is very small for PD and always equals 0 for SD and H games. The probability of having two equilibria, p 2 , is highest for SD and H. As a result, SH has the highest average number of equilibria across all games, for all 0 < q < 0.5 (it is minimum when q = 0.285). For most q, PD has the lowest number of equilibria (it is maximum when q = 0.438). All the figures are generated using analytical formulas derived in Box 1 (for p k ) and (4) for the average and variance. These analytical results are also in accordance with numerical simulation results provided in [7], obtained through samplings of the random payoff entries T and S. applicable to our analysis, but the result is true in higher dimensional spaces.
Then the joint probability distribution of Y 1 and Y 2 is where the Jacobian J is given by

Equilibria in Social dilemmas
By simplifying the right hand side of (3), equilibria of a social dilemma game are roots in the interval [0, 1] of the following cubic equation It follows that x = 0 is always an equilibrium. If q = 0, (5) reduces to Note that for SH-games and SH-games x * ∈ (0, 1), thus it is always an equilibrium. On the other hand, for PD-games and H-games, x * ∈ (0, 1), thus it is not an equilibrium If q = 1 2 then the above equation has two solutions x 1 = 1 2 and x 2 = T +S T +S−1 . In PD, SD and H games, x 2 ∈ (0, 1), thus they have two equilibria x 0 = 0 and x 1 = 1 2 . In the SH game: if T + S < 0 then the game has three equilibria x 0 = 0, x 1 = 1 2 and 0 < x 2 < 1; if T + S ≥ 0 then the game has only two equilibria x 0 = 0, x 1 = 1 2 . Now we consider the case 0 < q < 1 2 . For non-zero equilibrium points we solve the following quadratic equation Therefore, a social dilemma has three equilibria iff h has two distinct roots in (0, 1), which is equivalent to g having two distinct positive roots, or the following conditions must hold ∆ = (q + a − c) 2 + 4qc > 0, q + a − c < 0, c < 0.
LetT := T − 1 and The inverse transformation (X, Y ) → (T , S) is given bŷ Condition (7) is given by The domain D is illustrated in Figure 2. We apply Lemma 2.1 to find the joint distribution of X and Y . We compute the Jacobian of the transform (X, Y ) → (T , S), which is given by Hence if (T , S) has a probability density f (t, s) then (X, Y ) has a probability density g(x, y) = |J|f (T (x, y), S(x, y)).
We now apply this approach to SH and PD games.

The Stag Hunt (SH)
Proposition 2.2. The probability that SH games have 3 equilibria is given by where D is the subset of R 2 determined by According to (10), the joint probability distribution of (X, Y ) is We can characterise D 1 further by explicitly ordering the lower and upper bounds in the above formula. We have It follows that: Hence Thus, D 1 is the quadrilateral ABOC with vertices Theorem 2.3. The probability that SH games have 3 equilibria is given by Proof. According to Proposition 2.2, in order to compute p SH By comparing 2(q + √ 1 − 2q − 1) with −q, it follows that the point F is inside the edge OC if 0 ≤ q ≤ 4/9 and is outside OC whenever 4/9 < q < 0.5. On the other hand, the parabola By comparing x 1 and x 2 with −q we have G 1 is always in the edge AC, while G 2 is outside AC if 0 < q ≤ 4/9 and is inside AC if 4/9 < q < 0.5. In conclusion 1. for 0 < q ≤ 4/9: the intersection D ∩ D 1 is the domain formed by vertices A, B, E, F , and See Figure 4 for illustrations of the two cases above, with q = 0.3 and q = 0.45 respectively. We are now ready to compute the probability that the SH game has three equilibria.
For 0 < q ≤ 4/9: The probability that the SH game has three equilibria is We proceed by computing the areas in the expression above. Area of D 1 is Area of BOE is where H = (0, −q). Area of CF G is Substituting (13), (14), and (15) back to (12) we obtain, for 0 < q ≤ 4/9 Now we consider the remaining case 4/9 < q < 0.5. In this case where ♦G 1 G 2 is the domain with vertices G 1 and G 2 formed by the parabola y = − x 2 4q − q and the line y = qx−(1−2q) Substituting (13), (14) and (17) back to (16) we obtain, for 4/9 < q < 0.5, This finishes the proof of this theorem

Prisoner's Dilemma (PD)
Proposition 2.4. The probability that PD games have 3 equilibria is given by where D is defined above (as in the case of the SH games) and D 2 is the triangle MNO with vertices See Figure 5 for illustration of D 2 and its intersection with D for several values of q.
According to (10), the joint probability distribution of (X, Y ) is g(x, y) = |J|f (T (x, y), S(x, y)) = 1 (1 − 2q) 1 (x,y)∈D2 , where We now characterise D 2 further. We have Thus D 2 is the triangle MNO with vertices The probability that the SH game has three equilibria is thus This finishes the proof of this proposition.
The following elementary lemma provides an upper bound for p P D 3 , particularly implying that it tends to 0 as q goes to 0.
As a consequence, p P D 3 is always smaller or equal to 0.5 and tends to 0 as q tends to 0.
Proof. Area of D .
Theorem 2.6. The probability that PD games has three equilibria is given by The point F is inside the edge DC if 0 ≤ q ≤ 4/9 and is outside DC whenever 4/9 < q < 0.5.
On the other hand, the parabola y = − x 2 4q − q meets the line y = qx−(1−2q) 1−q at two points < q ≤ 4/9 and outside it otherwise. Therefore, we have three cases (see Figure 6 for illustration).
This completes the proof of this theorem.

Summary and outlook
It has been shown that in human behaviours and other biological settings, mutation is nonnegligible [31,25,33]. How mutation affects the complexity and bio-diversity of the evolutionary systems is a fundamental question in evolutionary dynamics [20,27]. In this paper, we have addressed this question for random social dilemmas by computing explicitly the probability distributions of the number of equilibria in term of the mutation probability. Our analysis based on random games is highly relevant and practical, because it is often the case that one might know the nature of a game at hand (e.g., a coordination or cooperation dilemma), but it is very difficult and/or costly to measure the exact values of the game's payoff matrix. Our results have clearly shown the influence of the mutation on the number of equilibria in SH-games and PD-games. The probability distributions in these games are much more complicated than in SD-games and H-games and significantly depend on the mutation strength. For a summary of our results, see again Box 1 and Figure 1. Our analysis has made use of suitable changes of variables, which expressed the probability densities in terms of area of certain domains. For future work, we plan to generalise our method to other approaches to studying random social dilemmas such as finite population dynamics and payoff disturbances [13,30,1,2], as well as to multi-player social dilemma games [23,28,18].