The Unification of Evolutionary Dynamics through the Bayesian Decay Factor in a Game on a Graph

Abstract

We unify evolutionary dynamics on graphs in strategic uncertainty through a decaying Bayesian update. Our analysis focuses on the Price theorem of selection, which governs replicator(-mutator) dynamics, based on a stratified interaction mechanism and a composite strategy update rule. Our findings suggest that the replication of a certain mutation in a strategy, leading to a shift from competition to cooperation in a well-mixed population, is equivalent to the replication of a strategy in a Bayesian-structured population without any mutation. Likewise, the replication of a strategy in a Bayesian-structured population with a certain mutation, resulting in a move from competition to cooperation, is equivalent to the replication of a strategy in a well-mixed population without any mutation. This equivalence holds when the transition rate from competition to cooperation is equal to the relative strength of selection acting on either competition or cooperation in relation to the selection differential between cooperators and competitors. Our research allows us to identify situations where cooperation is more likely, irrespective of the specific payoff levels. This approach provides new perspectives into the intended purpose of Price’s equation, which was initially not designed for this type of analysis.

Appendices

Appendix A

Proof of Proposition 1

We observe that

$$\begin{aligned}{} & {} \pi _i \gtreqless \pi _j \\{} & {} \Leftrightarrow p\left( w(1+\epsilon )-c\right) + (1-p)\left( w-c\right) \gtreqless pw+(1-p)w \\{} & {} \Leftrightarrow pw(1+\epsilon )-pc+w-c-pw+pc \gtreqless pw + w -pw \\{} & {} \Leftrightarrow pw\epsilon -c \gtreqless 0 \\{} & {} \Leftrightarrow p \epsilon w \gtreqless c. \end{aligned}$$

$\square$

Appendix B

Proof of Proposition 2

We observe that

$$\begin{aligned}{} & {} \pi _i \gtreqless \pi _j \\{} & {} \Leftrightarrow p\left( w(1+\epsilon )-c\right) x_{i}^{N-1} + (1-p)\left( w-c\right) \gtreqless pw+(1-p)w\left( 1-x_{i}^{N-1}\right) \\{} & {} \Leftrightarrow p(w(1+\epsilon )-c)x_{i}^{N-1} + (1-p)wx_{i}^{N-1} \gtreqless pw +(1-p)w \\{} & {} -(1-p)w +(1-p)c \\{} & {} \Leftrightarrow x_{i}^{N-1} \left( p(w(1+\epsilon )-c)+(1-p)w \right) \gtreqless pw +(1-p)c \\{} & {} \Leftrightarrow x_{i}^{N-1} \gtreqless \frac{c+p(w-c)}{w+p(\epsilon w-c}) \\{} & {} \Leftrightarrow x_{i} \gtreqless \left( \frac{c+p(w-c)}{w+p(\epsilon w-c}) \right) ^{\frac{1}{N-1}} \end{aligned}$$

$\square$

Appendix C

Proof of Proposition 3

We analyzed the stability conditions of the system by examining its fixed points,¹⁹ assuming $\delta =0.5$.²⁰ To accomplish this, we utilized the Jacobian matrix. We solved the system of differential equations $\dot{E}(p)=\{x_{i}, x_{j}\}$, which yielded the fixed point values $x_{i}^{\star }=\{0, \frac{-2+15p-15{p^2}-10px_{j}+10{p^2}x_{j}+30{p^2}x_{j}^{99}-20{p^2}x_{j}^{100}}{20p(1-x_{j}^{99}+px_{j}^{99})}\}$ and $x_{j}^{\star }=\{0, \frac{-2+15p-15{p^2}-10px_{i}+10{p^2}x_{i}+30{p^2}x_{i}^{99}-20{p^2}x_{i}^{100}}{20p(1-x_{i}^{99}+px_{i}^{99})}\}$. Our analysis revealed that the system possesses both a corner equilibrium and an interior equilibrium. However, we found that all fixed points are classified as saddle points due to the eigenvalues of the Jacobian matrix exhibiting opposite signs. At a saddle point, the system displays both stable and unstable behavior in different directions, causing trajectories near the point to either converge or diverge. It is worth noting that saddle points can lead to complex and unpredictable system behavior.

After analyzing the stability of fixed points in the system, we now turn our attention to the presence of attractors. In a system of differential equations, an attractor is a set of states or trajectories that the system tends to converge towards over time, regardless of its initial conditions. Fixed points can also act as attractors in a system. At $p=0.25$, we observe an attractor with coordinates $\{x_{i}, x_{j}\}=(0.38,0.38)$. Similarly, at $p=0.50$ and $p=0.75$, we have attractors with coordinates $\{x_{i}, x_{j}\}=(0.3,0.3)$ and $\{x_{i}, x_{j}\}=(0.13,0.13)$, respectively. At $p=1$, we observe $\{x_{i}, x_{j}\}=(0,0)$.

A noteworthy finding is that as the probability of encountering a cooperator increases, the population dynamics system tends to converge towards the interior equilibrium, which approaches zero as p approaches one. The system exhibits a bifurcation around the system attractor point. However, when players are guaranteed to encounter a cooperator, incentivizing free-riding behavior, only the corner equilibrium attracts the system. This results in both populations of players competing against each other, ultimately leading to the extinction of one or both populations. $\square$

Appendix D

Proof of Proposition 4

The stability conditions of the system were investigated by analyzing its fixed points, while assuming a value of $\delta =0.5$. The Jacobian matrix was utilized for this purpose. We solved the differential equations $\dot{E}(p)=\{x_{i}, x_{j}\}$ and obtained the values of the fixed points as $x_{i}^{\star }=\{0, \frac{-2x_{j}+15px_{j}-15{p^2}x_{j}-10px_{j}^{3}+10{p^2}x_{j}^{3}+30{p^2}x_{j}^{100}-20{p^2}x_{j}^{102}}{20px_{j}^{2}(1-x_{j}^{99}+px_{j}^{99})}\}$ and $x_{j}^{\star }=\{0, \frac{-2x_{i}+15px_{i}-15{p^2}x_{i}-10px_{i}^{3}+10{p^2}x_{i}^{3}+30{p^2}x_{i}^{100}-20{p^2}x_{i}^{102}}{20px_{i}^{2}(1-x_{i}^{99}+px_{i}^{99})}\}$. Our analysis indicated that the system possesses both a corner equilibrium and an interior equilibrium. Our analysis indicates that, apart from the coordinates of the attractor, all other fixed points were found to be saddle points, as evidenced by the opposite signs of the eigenvalues of the Jacobian matrix. At a saddle point, the system exhibits both stable and unstable behavior in different directions, leading to trajectories that either converge or diverge near the point. It should be noted that saddle points can result in complex and unpredictable system behavior.

Following our analysis of the stability of fixed points in the system, we now shift our focus to the presence of attractors. In a system of differential equations, an attractor is defined as a set of states or trajectories that the system tends to converge towards over time, regardless of its initial conditions. Fixed points can also serve as attractors in a system. At $p=0.25$, we observe an attractor with coordinates $\{x_{i}, x_{j}\}=(0.62,0.62)$. Similarly, at $p=0.50$ and $p=0.75$, we have attractors with coordinates $\{x_{i}, x_{j}\}=(0.55,0.55)$ and $\{x_{i}, x_{j}\}=(0.37,0.37)$, respectively. At $p=1$, we observe $\{x_{i}, x_{j}\}=(0.03,0.03)$.

A key observation is that the population dynamics system typically approaches the interior equilibrium in situations where there is uncertainty about encountering a cooperator. The interior equilibrium approaches zero as p approaches one. The system as a whole exhibits attraction towards a stable attractor point. Nevertheless, when players are guaranteed to interact with cooperators, thereby incentivizing free-riding behavior, only the corner equilibrium attracts the system. This leads to both player populations competing against each other, ultimately resulting in the extinction of one or both populations. $\square$

Appendix E

Proof of Proposition 5

The stability conditions of the system were investigated by analyzing its fixed points, while assuming a value of $\delta =0.5$. We recall that the mutation is considered to occur for sure. The Jacobian matrix was utilized for this purpose. We solved the differential equations $\dot{E}(p)=\{x_{i}, x_{j}\}$ and obtained the values of the fixed points as $x_{i}^{\star }=\{0, \frac{-x_{j}+5px_{j}-5p^{2}x_{j}-5px_{j}^{2}+5p^{2}x_{j}^{2}+10p^{2}x_{j}^{100}-10{p^2}x_{j}^{101}}{5p(2x_{j}-3)(1-x_{j}^{99}+px_{j}^{99})}\}$ and $x_{j}^{\star }=\{0, \frac{-x_{i}+5px_{i}-5p^{2}x_{i}-5px_{i}^{2}+5p^{2}x_{i}^{2}+10p^{2}x_{i}^{100}-10{p^2}x_{i}^{101}}{5p(2x_{i}-3)(1-x_{i}^{99}+px_{i}^{99})}\}$. Our analysis indicated that the system possesses an unstable corner equilibrium, for the eigenvalues of the Jacobian matrix were positive. When a fixed point is unstable, it means that small perturbations from the fixed point will cause the system to move away from the fixed point rather than converge to it. In out case, the unstable fixed point is a source, such that the system exhibits a behavior similar to that of a repeller: trajectories diverge away from the fixed point in all directions.

Following our analysis of the stability of fixed points in the system, we now shift our focus to the presence of attractors. In a system of differential equations, an attractor is defined as a set of states or trajectories that the system tends to converge towards over time, regardless of its initial conditions. In this system, the fixed points do not serve as attractors. At $p=0.25$, we observe an attractor with coordinates $\{x_{i}, x_{j}\}=(1,1)$. Similarly, at $p=0.50$ and $p=0.75$, we have attractors with coordinates $\{x_{i}, x_{j}\}=(1,1)$ and $\{x_{i}, x_{j}\}=(1,1)$, respectively. At $p=1$, we observe $\{x_{i}, x_{j}\}=(1,1)$.

The finding of this study indicates that the population dynamics system tends to converge towards an unstable attractor, regardless of the probability of encountering a cooperator. An unstable attractor is a point in a dynamical system that attracts nearby trajectories, but any slight perturbation to the system can cause the trajectories to diverge and move away from it. Assuming a certain probability of mutation towards cooperation during the game, the system continuously eliminates competitors who keep returning as free-riders because they are certain to encounter cooperators, thus driving the system towards a transient full population density. However, when players are guaranteed to interact with cooperators, incentivizing free-riding behavior, only mutation can bring them back to cooperation. $\square$

Appendix F

Proof of Proposition 6

The stability of the system was investigated by examining its fixed points, while assuming a fixed value of $\delta =0.5$ and assuming that mutation towards cooperation always occurs. The Jacobian matrix was utilized to determine the stability conditions. The differential equations $\dot{E}(p)=\{x_{i}, x_{j}\}$ were solved, and the fixed points were obtained as $x_{i}^{\star }=\{0, \frac{-x_{j}+5px_{j}-5p^{2}x_{j}-5px_{j}^{3}+5p^{2}x_{j}^{3}+10p^{2}x_{j}^{100}-10{p^2}x_{j}^{102}}{5p(2x_{j}^{2}-3)(1-x_{j}^{99}+px_{j}^{99})}\}$ and $x_{j}^{\star }=\{0, \frac{-x_{i}+5px_{i}-5p^{2}x_{i}-5px_{i}^{3}+5p^{2}x_{i}^{3}+10p^{2}x_{i}^{100}-10{p^2}x_{i}^{102}}{5p(2x_{i}^{2}-3)(1-x_{i}^{99}+px_{i}^{99})}\}$. Our analysis revealed the presence of an unstable corner equilibrium, as the eigenvalues of the Jacobian matrix were found to be positive. An unstable fixed point implies that small perturbations from the fixed point will cause the system to move away from it instead of converging towards it. In our case, the unstable fixed point is a source, resulting in a behavior similar to that of a repeller, whereby trajectories diverge away from the fixed point in all directions.

After analyzing the stability of the fixed points in the system, we proceed to investigate the presence of attractors. An attractor is a set of states or trajectories to which a system of differential equations converges over time, regardless of its initial conditions. Fixed points can also serve as attractors in a system, but such is not the case in this particular instance. At $p=0.25$, we observe an attractor located at the coordinates $\{x_{i}, x_{j}\}=(1,1)$. Similarly, at $p=0.50$ and $p=0.75$, we find attractors at $\{x_{i}, x_{j}\}=(1,1)$ and $\{x_{i}, x_{j}\}=(1,1)$, respectively. When $p=1$, the attractor is located at $\{x_{i}, x_{j}\}=(1,1)$.

The research finding suggests that the population dynamics system tends to converge towards an unstable attractor, regardless of the probability of encountering a cooperator. An unstable attractor is a point in a dynamical system that attracts nearby trajectories, but any slight perturbation to the system may cause the trajectories to diverge and move away from it. Assuming a certain probability of mutation toward cooperation during the game, the system continuously eliminates competitors who persist as free-riders due to their certain encounters with cooperators, driving the system towards a transient full population density. When players are guaranteed to interact with cooperators, which incentivizes free-riding behavior, only mutation can prompt them to revert to cooperation. Interestingly, Bayesian updating has no effect on population dynamics in the case of certain mutation. $\square$

Appendix G

Proof of Proposition 7

Through the combination of Eqs. (9) and (10), a general equality can be obtained. It is known that $\Sigma _{i} px_{i}\pi _{i}=cov(\pi _{i},p) + \bar{\pi }\bar{p}$, where covariance is given by the expression $cov(\pi _{i},p)=\Sigma _{i} px_{i}\pi _{i}-\bar{\pi }\bar{p}$. The identification of all the covariances that are implicit in the equation allows for the derivation of $\delta$, which is obtained through the solution of $cov(\pi _{i},p)\delta = cov(\pi _{j},p)(1+\delta ) \Leftrightarrow \delta = \frac{cov(\pi _{j},p)}{cov(\pi _{i}-\pi _{j},p)}$. In the context of the Price equation, the covariance term represents the impact of natural selection on the alteration in the trait values. Selection is interpreted as the difference between the expected payoff of a model-player and the average payoff within the population. It can be concluded that the equivalence holds when the rate of transition from competition to cooperation corresponds to the relative strength of selection exerted on competition in proportion to the selection differential between the cooperators and competitors. $\square$

Appendix H

Proof of Proposition 8

The combination of Eqs. (6) and (11) leads to a general equality. Specifically, it is established that $\Sigma _{i} px_{i}\pi _{i}=cov(\pi _{i},p) + \bar{\pi }\bar{p}$, where covariance is expressed as $cov(\pi _{i},p)=\Sigma _{i} px_{i}\pi _{i}-\bar{\pi }\bar{p}$. By identifying the implicit covariances in the equation, we can solve for $\delta$ using $cov(\pi _{i},p)(1+\delta ) = cov(\pi _{j},p)\delta \Leftrightarrow \delta = \frac{cov(\pi _{i},p)}{cov(\pi _{j}-\pi _{i},p)}$. In the context of the Price equation, the covariance term indicates the effect of natural selection on changes in trait values. Selection is represented as the difference between the expected payoff of a model-player and the average payoff within the population. The equivalence is valid when the transition rate from competition to cooperation corresponds to the relative strength of selection acting on cooperation relative to the selection differential between competitors and cooperators.$\square$

Appendix I

Proof of Corollary 1

In order to determine the range of values for $\pi _i$ and $\pi _j$ that satisfy $x_i^{\star } = \frac{\pi _j}{\pi _j-\pi _i} \in [0,1]$, we must solve the inequalities $0 \le x_i^{\star } \le 1$. Substituting the expression for $x_i^{\star }$ into the inequalities, we obtain $0 \le \frac{\pi _j}{\pi _j-\pi _i} \le 1$. If $\pi _i > \pi _j$, the denominator is negative, and $x_i^{\star }$ is undefined. Similarly, if $\pi _i = \pi _j$, the denominator is zero, and $x_i^{\star }$ is undefined. Therefore, we require $\pi _i < \pi _j$ to ensure that $x_i^{\star } \in [0,1]$. By combining these conditions, we can conclude that $x_i^{\star } \in [0,1]$ if and only if $\pi _i=0$. In this case, $x_i^{\star } =1$, which implies full density of cooperators. While a zero payoff for cooperation might initially discourage cooperative behavior, the full density of cooperators can still emerge under certain conditions in an evolutionary context. This typically requires a sufficiently large population, the ability to adopt a mixed strategy that includes cooperation, and the benefits of mutual cooperation outweighing the costs of being exploited by defectors. This implies that $w(1+\epsilon )-c=w-c$. This equation leads to the conclusion that the reward for cooperation is zero. Another way to come to this result is by posing $w(1+\epsilon )-c=0 \Leftrightarrow \epsilon = \frac{c-w}{w}$. Since $\epsilon$ cannot be negative, it follows that $w=c$ or $\epsilon =0$, which suggests that the benefits of mutual cooperation outweigh the costs of being exploited by competitors. Given the conditions described, cooperators are expected to form clusters within the network where the need for rewards is eliminated, as they will interact primarily with other cooperators, thus avoiding exploitation. This phenomenon can only occur when the game is played on a graph with a structured population. In light of the fact that $x_i^{\star }=1$, we can infer that the entire network can be viewed as a singular, vast cluster of cooperators. $\square$