Dynamics of Three-Strategy Symmetric Games

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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.


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.


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 [1].

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


[1] 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.


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