Fictitious Play
A learning process in which players best-respond to the empirical distribution of past opponent play.
Definition
At each round , player forms beliefs as the empirical frequency of opponents' past actions and best-responds: .
Convergence in beliefs need not imply convergence in actions: the path can cycle.
Intuition
Players track what opponents have done historically and choose the action that would have been best against that history.
It captures a simple form of experiential learning without Bayesian updating.
Worked example
Fictitious play converges to Nash in zero-sum games (Robinson 1951), potential games (Monderer-Shapley), and games with strategic complements.
In Rock-Paper-Scissors, fictitious play cycles without converging.
The math
Brown (1951): originally proposed as a method to compute the value of zero-sum games.
Fictitious play is a best-response dynamic; smooth fictitious play replaces argmax with stochastic choice (QRE-like).
Where it is used
Algorithmic game theory (computing equilibria), multi-agent learning, and economic modeling of boundedly rational learning.
Connects to reinforcement learning and regret minimization.
More in Game theory
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