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Fictitious play:
two viewpoints and two versions
Erik Quaeghebeur ∗
October 9, 2003
I will be focusing on non-cooperative two-player games in strategic form.
An important type of player behavior in iterated games of this type is de-scribed by so-called fictitious play. In this context, the players act myopically,in the sense that they are only interested in maximizing their immediate ex-pected pay-off.
There are two viewpoints, which differ in the interpretation given to mixed Using a randomization device
In the first viewpoint, a player faces the same opponent iteratively, and bothplayers can play mixed strategies using a randomization device.
In the classical version of fictitious play, the players use a precise Dirichlet model (PDM). This means that a Dirichlet distribution over the strategy sim-plex of their opponent is used to represent their own beliefs about the mixedstrategy their opponent is going to play. Maximizing their expected payoffthen corresponds to playing a best reply to the expected strategy of their oppo-nent under this PDM. After observing the pure strategy played by his oppo-nent, the player adapts his beliefs by modifying the (parameters of the) PDMpreviously used.
The parameters of a PDM correspond to the expected strategy and to the strength of their current belief. The choice for initial values of these parametersis problematic, for example, the initial choice of expected strategy will alwaysbe arbitrary.
To remove the problem of choosing an initial strategy, fictitious play can be modified by letting the players use an imprecise Dirichlet model (IDM) ;[4]. This means that they use a set of Dirichlet distributions instead of one.
Initially the set can be chosen such that all strategies are expected under oneof the distributions in the set. To maximize his expected pay-off, the playerchooses a strategy from the set of best replies to the expected strategies. Eachtime a game is played, the player updates all the distributions forming hisIDM.
SYSTeMS research group, Ghent University.
Using a population of players
In the population dynamics viewpoint, players can only play pure strategiesand are faced with a whole population of possible opponents; [4]. All the play-ers in the population, whose strategies are fixed, play among themselves insome sort of competition. A new player enters the competition each iteration;[3].
Classically, a PDM is again used by the new player to choose his strategy upon entering the competition. The PDM represents the beliefs of the outsideworld about the relative frequencies of players in the population playing a cer-tain pure strategy. The new entrant will play a best reply to the mixed strategythat corresponds to the expectation of these frequencies. After observing thepure strategy chosen by the last new entrant, the outside world updates itsPDM for later entrants.
Unless the outside world has observations starting from the first entrant to the competition, the use of an IDM can also remove the arbitrariness of thechoice of initial expected frequencies in this case.
Convergence results
The interesting convergence results of Fudenberg and Kreps [1] for the PDM-version remain valid in a slightly modified form for the IDM-version; [4].
These results guarantee that if convergence occurs, it is to an equilibrium. Theyhold for both viewpoints, which are mathematically equivalent.
Acknowledgements
The research of the author is financed by a Ph.D. grant of the Institute for thePromotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).
References
[1] DREW FUDENBERG AND DAVID M. KREPS Learning mixed equilibria.
Games and Economic Behaviour, 5:320–367, 1993.
[2] JOSEF HOFBAUER AND KARL SIGMUND Evolutionary game dynamics.
Bulletin of the American Mathematical Society, 40(4):479–519, 2003.
[3] JOSEF HOFBAUER Stability for the Best Response Dynamics. Preprint, [4] ERIK QUAEGHEBEUR AND GERT DE COOMAN Game-theoretic Learn- ing Using the Imprecise Dirichlet Model. In ISIPTA’03 – Proceedings ofthe Third International Symposium on Imprecise Probabilities and TheirApplications, Jean-Marc Bernard, Teddy Seidenfeld, and Marco Zaffalon(Eds.), 452–466. July 2003. Available at http://allserv.UGent.be/~equaeghe/.

Source: http://ac.erikquaeghebeur.name/content/EQ-2003-DynComp-abstract.pdf