Convergence Properties
In fictitious play Nash equilibria are absorbing states. That is, if at any time period all the players play a Nash equilibrium, then they will do so for all subsequent rounds. (Fudenberg and Levine 1998, Proposition 2.1) In addition, if fictitious play converges to any distribution, those probabilities correspond to a Nash equilibrium of the underlying game. (Proposition 2.2)
| A | B | C | |
|---|---|---|---|
| a | 0, 0 | 1, 0 | 0, 1 |
| b | 0, 1 | 0, 0 | 1, 0 |
| c | 1, 0 | 0, 1 | 0, 0 |
Therefore, the interesting question is, under what circumstances does fictitious play converge? The process will converge for a 2-person game if:
- Both players have only a finite number of strategies and the game is zero sum (Robinson 1951)
- The game is solvable by iterated elimination of strictly dominated strategies (Nachbar 1990)
- The game is a potential game (Monderer and Shapley 1996-a,1996-b)
- The game has generic payoffs and is 2xN (Berger 2005)
Fictitious play does not always converge, however. Shapley (1964) proved that in the game pictured here (a limit case of generalized Rock, Paper, Scissors games), if the players start by choosing (a, B), the play will cycle indefinitely.
Read more about this topic: Fictitious Play
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