An Evolutionary Game With the Game Transitions Based on the Markov Process

TL;DR

This article considers the game transitions of individuals in evolutionary processes to capture the changing psychology of individuals in reality and explores the impact of transition rates between different game states, payoff parameters, the reputation mechanism, and different time scales of strategy updates on cooperative behavior.

Abstract

The psychology of the individual is continuously changing in nature, which has a significant influence on the evolutionary dynamics of populations. To study the influence of the continuously changing psychology of individuals on the behavior of populations, in this paper, we consider the game transitions of individuals in evolutionary processes to capture the changing psychology of individuals in reality, where the game that individuals will play shifts as time progresses and is related to the transition rates between different games. Besides, the individual's reputation is taken into account and utilized to choose a suitable neighbor for the strategy updating of the individual. Within this model, we investigate the statistical number of individuals staying in different game states and the expected number fits well with our theoretical results. Furthermore, we explore the impact of transition rates between different game states, payoff parameters, the reputation mechanism, and different time scales of strategy updates on cooperative behavior, and our findings demonstrate that both the transition rates and reputation mechanism have a remarkable influence on the evolution of cooperation. Additionally, we examine the relationship between network size and cooperation frequency, providing valuable insights into the robustness of the model.

Figures (12)

• Figure 1: Game state transitions of individuals. Each individual in the network will have the game state transition, the arrows signify the transitions from one game state to another, where the letters $\lambda_i$ and $\mu_{i+1}$ above or below the arrows denote the rate of transition from game state $G_i$ to $G_{i+1}$ and the rate of transition from game state $G_{i+1}$ to $G_i$ ($i=0, 1, \cdots, n-1$), respectively.
• Figure 2: An illustration of the model. Grey, orange, and blue durations represent the prisoner's dilemma game (PDG), the snowdrift game (SDG), and the stag-hunt game (SHG) durations of individuals, respectively. The red or green border indicates that the strategy adopted by the individual is defection or cooperation, respectively. The game state of each individual will be transformed from one to another at a specific rate. We chose evolution times $t = 40$ and $t = 80$ to observe snapshots of the network. The individual circled by the red dashed line updates his/her strategy at the next moment, and he/she will choose the more reputable individual among his/her neighbors with a higher probability to compare his/her payoff and decide whether to adopt the neighbor's strategy.
• Figure 3: An example diagram of the evolutionary time of an individual's game state and strategy. This figure illustrates the game transition and strategy update process for individual F. The game transition of individual F is determined by his/her specific transition rate, while the timing of strategy updates can follow different rules.
• Figure 4: Evolutionary curves of the individual number staying in different game states over time with different parameters. (a) shows the evolution of the individual number obtained from the initial setting of the parameters with $\lambda_0$=0.02, $\lambda_1$=0.06, $\mu_1$=0.04, $\mu_2$=0.08, and the other subplots are acquired by adjusting a parameter by the control variable method on the initial parameter settings. Specifically, (b) illustrates the individual number with $\lambda_0$ changed to 0.04. (c) shows the individual number with $\lambda_1$ adjusted to 0.12. (d) is the individual number with $\mu_1$ changed to 0.08. And (e) demonstrates the individual number with $\mu_2$ changed to 0.16. As time progresses, the number of individuals in each of the three game states gradually becomes stable.
• Figure 5: Statistical results of the individual number staying in different game states with different $\lambda_0$s. In this figure, we present the statistical distribution of the individual number in the network in the three game states with different parameters $\lambda_0$s. The $x$-axis and $y$-axis are set as the number of individuals and probability, respectively. (a) is the statistical distribution of the individual number staying in the PDG. (b) displays the statistical distribution of the individual number located in the SDG. And (c) demonstrates the statistical distribution of the individual number located in the SHG.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• Theorem 1
• Theorem 2
• proof