Expected Dynamics of an Imitation Model in 2x2 Symmetric Games

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The figure shows the actual (in blue) and expected (in black) proportion of -strategists in a population of individuals who, at each iteration (of time length ), are randomly matched in pairs to play a symmetric 2×2 game. The two possible actions (or pure strategies) in the game are labeled and . Thus, each individual in the population is either an -strategist or a -strategist. The payoffs of the game are , , , and (parameters), where, for instance, denotes the payoff obtained by an -strategist when he plays with a -strategist.


At the end of each iteration, after all individuals have played the game, one randomly selected player revises her strategy— or —according to the following rule: "I look at another (randomly selected) individual; if and only if she got a payoff higher than mine, I adopt her strategy".


Contributed by: Luis R. Izquierdo and Segismundo S. Izquierdo (May 2010)
Open content licensed under CC BY-NC-SA



The first three Snapshots show the Hawk-Dove game in a population of individuals ( denotes the Hawk strategy and denotes the Dove strategy).

Snapshot 1:

Snapshot 2:

Snapshot 3:

The last three Snapshots show the Prisoner's Dilemma game in a population of individuals ( denotes the cooperative strategy and denotes the defective strategy).

Snapshot 4:

Snapshot 5:

Snapshot 6:


[1] S. S. Izquierdo and L. R. Izquierdo, "Stochastic Approximation to Understand Simple Simulation Models," Journal of Statistical Physics, Dec 2012. dx.doi.org/10.1007/s10955-012-0654-z. Download preprint.

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