In game theory, fictitious play is a learning rule first introduced by George W. Brown. In it, each player presumes that the opponents are playing stationary (possibly mixed) strategies. At each round, each player thus best responds to the empirical frequency of play of their opponent. Such a method is of course adequate if the opponent indeed uses a stationary strategy, while it is flawed if the opponent's strategy is non-stationary. The opponent's strategy may for example be conditioned on the fictitious player's last move.
Brown first introduced fictitious play as an explanation for Nash equilibrium play. He imagined that a player would "simulate" play of the game in their mind and update their future play based on this simulation; hence the name fictitious play. In terms of current use, the name is a bit of a misnomer, since each play of the game actually occurs. The play is not exactly fictitious.
Convergence propertiesedit
In fictitious play, strict 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)
Generalized Rock Paper Scissors
A
B
C
a
0, 0
2, 1
1, 2
b
1, 2
0, 0
2, 1
c
2, 1
1, 2
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 a potential game (Monderer and Shapley 1996-a,1996-b)
The game has generic payoffs and is 2 × N (Berger 2005)
Fictitious play does not always converge, however. Shapley (1964) proved that in the game pictured here (a nonzero-sum version of Rock, Paper, Scissors), if the players start by choosing (a, B), the play will cycle indefinitely.
Terminologyedit
Berger (2007) states that "what modern game theorists describe as 'fictitious play' is not the learning process that George W. Brown defined in his 1951 paper": Brown's "original version differs in a subtle detail..." in that modern usage involves the players updating their beliefs simultaneously, whereas Brown described the players updating alternatingly. Berger then uses Brown's original form to present a simple and intuitive proof of convergence in the case of two-player nondegenerate ordinal potential games.
The term "fictitious" had earlier been given another meaning in game theory. Von Neumann and Morgenstern [1944] defined a "fictitious player" as a player with only one strategy, added to an n-player game to turn it into a (n + 1)-player zero-sum game.
Referencesedit
Berger, U. (2005) "Fictitious Play in 2xN Games", Journal of Economic Theory 120, 139–154.
Berger, U. (2007) "Brown's original fictitious play", Journal of Economic Theory 135:572–578
Brown, G.W. (1951) "Iterative Solutions of Games by Fictitious Play" In Activity Analysis of Production and Allocation, T. C. Koopmans (Ed.), New York: Wiley.
Fudenberg, D. and D.K. Levine (1998) The Theory of Learning in Games Cambridge: MIT Press.
Monderer, D., and Shapley, L.S. (1996-a) "", Games and Economic Behavior 14, 124-143.
Monderer, D., and Shapley, L.S. (1996-b) "Fictitious Play Property for Games with Identical Interests 2021-05-13 at the Wayback Machine", Journal of Economic Theory 68, 258–265.
Nachbar, J. (1990) "Evolutionary Selection Dynamics in Games: Convergence and Limit Properties", International Journal of Game Theory 19, 59–89.
von Neumann and Morgenstern (1944), , Princeton and Woodstock: Princeton University Press.
Robinson, J. (1951) "An Iterative Method of Solving a Game", Annals of Mathematics 54, 296–301.
Shapley L. (1964) "" In Advances in Game Theory M. Dresher, L.S. Shapley, and A.W. Tucker (Eds.), Princeton: Princeton University Press.
External linksedit
Game-Theoretic Solution to Poker Using Fictitious Play
January 01, 1970
fictitious, play, this, article, includes, list, references, related, reading, external, links, sources, remain, unclear, because, lacks, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, july, 2023, learn, when, r. This article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help improve this article by introducing more precise citations July 2023 Learn how and when to remove this message In game theory fictitious play is a learning rule first introduced by George W Brown In it each player presumes that the opponents are playing stationary possibly mixed strategies At each round each player thus best responds to the empirical frequency of play of their opponent Such a method is of course adequate if the opponent indeed uses a stationary strategy while it is flawed if the opponent s strategy is non stationary The opponent s strategy may for example be conditioned on the fictitious player s last move Contents 1 History 2 Convergence properties 3 Terminology 4 References 5 External linksHistory editBrown first introduced fictitious play as an explanation for Nash equilibrium play He imagined that a player would simulate play of the game in their mind and update their future play based on this simulation hence the name fictitious play In terms of current use the name is a bit of a misnomer since each play of the game actually occurs The play is not exactly fictitious Convergence properties editIn fictitious play strict 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 Generalized Rock Paper Scissors A B C a 0 0 2 1 1 2 b 1 2 0 0 2 1 c 2 1 1 2 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 2 N Berger 2005 Fictitious play does not always converge however Shapley 1964 proved that in the game pictured here a nonzero sum version of Rock Paper Scissors if the players start by choosing a B the play will cycle indefinitely Terminology editBerger 2007 states that what modern game theorists describe as fictitious play is not the learning process that George W Brown defined in his 1951 paper Brown s original version differs in a subtle detail in that modern usage involves the players updating their beliefs simultaneously whereas Brown described the players updating alternatingly Berger then uses Brown s original form to present a simple and intuitive proof of convergence in the case of two player nondegenerate ordinal potential games The term fictitious had earlier been given another meaning in game theory Von Neumann and Morgenstern 1944 defined a fictitious player as a player with only one strategy added to an n player game to turn it into a n 1 player zero sum game References editBerger U 2005 Fictitious Play in 2xN Games Journal of Economic Theory 120 139 154 Berger U 2007 Brown s original fictitious play Journal of Economic Theory 135 572 578 Brown G W 1951 Iterative Solutions of Games by Fictitious Play In Activity Analysis of Production and Allocation T C Koopmans Ed New York Wiley Fudenberg D and D K Levine 1998 The Theory of Learning in Games Cambridge MIT Press Monderer D and Shapley L S 1996 a Potential Games Games and Economic Behavior 14 124 143 Monderer D and Shapley L S 1996 b Fictitious Play Property for Games with Identical Interests Archived 2021 05 13 at the Wayback Machine Journal of Economic Theory 68 258 265 Nachbar J 1990 Evolutionary Selection Dynamics in Games Convergence and Limit Properties International Journal of Game Theory 19 59 89 von Neumann and Morgenstern 1944 Theory of Games and Economic Behavior Princeton and Woodstock Princeton University Press Robinson J 1951 An Iterative Method of Solving a Game Annals of Mathematics 54 296 301 Shapley L 1964 Some Topics in Two Person Games In Advances in Game Theory M Dresher L S Shapley and A W Tucker Eds Princeton Princeton University Press External links editGame Theoretic Solution to Poker Using Fictitious Play Retrieved from https en wikipedia org w index php title Fictitious play amp oldid 1194705537, wikipedia, wiki, book, books, library,
