Dynamics of Three-Strategy Symmetric Games
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration shows the phase portrait in the 2D simplex of the single-sampling "imitate if better" dynamics in three-strategy symmetric games for large populations.[more]
Consider a population of agents (players) who, in every time step, are randomly matched in pairs for an interaction that can be modeled as a symmetric game. The ordinal preferences over outcomes for the row agent are summarized in the payoff matrix , where a higher number indicates a higher preference. At the end of every time step, after all individuals have played the game, one randomly selected player revises their strategy according to the following rule: I look at another (randomly selected) individual and adopt their strategy if and only if they got a higher payoff than I did. (This is the so-called "single-sampling imitate if better" rule.) With probability , the revising agent adopts a random strategy rather than the one prescribed by the previous rule.
This Demonstration shows the mean dynamic for the agent-based model described. One time unit in the mean dynamic corresponds to one revision for each agent in the population on average. The program also provides a numerical approximation to the critical points of the system and to their corresponding eigenvalues, which are helpful in assessing the dynamic stability of the critical point.[less]
Contributed by: Luis R. Izquierdo, Segismundo S. Izquierdo and William H. Sandholm (January 2020)
Open content licensed under CC BY-NC-SA
This model is analyzed in detail in .
Snapshot 1: a game with seven critical points
Snapshot 2: rock paper scissors game without noise
Snapshot 3: rock paper scissors game with small noise
Snapshot 4: a game with one critical point and critical regions
Snapshot 5: a game with an internal critical region
Snapshot 6: a game with critical regions only
 S. S. Izquierdo and L. R. Izquierdo, "Stochastic Approximation to Understand Simple Simulation Models," Journal of Statistical Physics, 151(1–2), 2013 pp. 254–276. doi:10.1007/s10955-012-0654-z.