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Topic: Fictitious play

	Multi-Agent Decision Making
		Fictitious play is a substantial improvement on Cournot dynamics that allows each agent to remember everything that went on before.
		In essence, fictitious play changes the Cournot model so that, rather than placing all weight on the previously played action, weight is shifted between actions in such a way that a probabilistic model is produced.
		The behavior of the fictitious play algorithm is typical; agents jump around a lot until eventually the probabilties converge to the Nash equilibrium values.
	students.cs.byu.edu /~cs670ta/FictitiousPlay.html (2609 words)

	Fictitious play - Wikipedia, the free encyclopedia
		In it, each player presumes that her opponents are playing stable (possibly mixed) strategies.
		Brown first introduced fictitious play as an explanation for Nash equilibrium play.
		In fictitious play Nash equilibria are absorbing states.
	en.wikipedia.org /wiki/Fictitious_play (519 words)

	Learning and Evolution in Games
		Fictitious play is a kind of quasi-Bayesian model: Agents behave as if they think they are facing an exogenous, stationary, unknown, distribution of opponents' strategies.
		Another response is to think that fictitious play does describe the long-run outcome, and players will not notice that their fictitious play model is badly off, provided that their payoffs are not worse than the model predicts.
		This is "exponential fictitious play": each strategy is played in proportion to an exponential function of the utility it has historically yielded; it corresponds to the logit decision model that has been extensively used in empirical work.
	www-cs-students.stanford.edu /~erikr/ESSAY7.html (4655 words)

	U of M News Service
		The learning model of fictitious play is related to the work of John Nash, the subject of the movie, "A Beautiful Mind." Fictitious play models how people might interact to discover a Nash equilibrium, which is the best joint strategy for games with two or more players that reach an outcome to mutual advantage.
		Using computer modeling, U-M engineers used fictitious play to simulate the traffic conditions through 75 traffic lights in the city of Troy, Mich. Smith said the result was at least a 20 percent savings in travel time, just by adjusting the traffic signals.
		But by using fictitious play, decisions about which direction to turn are anticipated in parallel at every intersection, and the previous decisions fuel the next until all but the best options are weeded out.
	www.umich.edu /news?Releases/2005/May05/r052405a (643 words)

	Theory of Learning Errata (Site not responding. Last check: 2007-10-20)
		The bottom of the page asserts that fictitious play converges to a single point, the correlated equilibrium that puts equal weight of 1/6 on 6 of the pure strategies.
		If in fact fictitious play converges to a point, it must converge to a Nash equilibrium; it could not converge to a correlated equilibrium that is not Nash, such as the one putting weight 1/6 on 6 of the pure strategies.
		Play cycles near this manifold, albeit at a slower rate as time passes.
	www.dklevine.com /papers/errata.htm (153 words)

	Aner Sela - Abstracts (Site not responding. Last check: 2007-10-20)
		The definition of fictitious play may depend on first move rules, initial beliefs, weights assigned to initial beliefs, and tie-breaking rules determining the particular best replies chosen at each stage.
		Using the original definition of Brown (1951) in which the first moves are chosen arbitrarily and no tie-breaking rules are assumed, we give an example of a fictitious play process in a 2x2 game that does not converge to equilibrium.
		It is proved that if every player is using either a belief-based learning scheme with bounded recall or a generalized fictitious play learning scheme, then after sufficiently long time, the players' bids are in equilibrium in the one-shot auction in which the types are commonly known.
	econ.bgu.ac.il /facultym/anersela/abstracts.htm (790 words)

	XML Topic Maps (XTM) 1.0 (Site not responding. Last check: 2007-10-20)
		The role that a topic plays as a member of an association; the nature of its involvement in that association.
		Superclass-subclass is a class of association that expresses superclass-subclass relationships between topics that play the roles of superclass and subclass respectively.
		The core concept of superclass; the role of superclass as played by one of the members of a superclass-subclass association.
	www.topicmaps.org /xtm/1.0 (7677 words)

	Consistency and Cautious Fictitious Play (Site not responding. Last check: 2007-10-20)
		Abstract: We study a variation of fictitious play, in which the probability of each action is an exponential function of that action's utility against the historical frequency of opponents' play.
		Regardless of the opponents' strategies, the utility received by an agent using this rule is nearly the best that could be achieved against the historical frequency.
		Fictitious play shares these properties provided it switches "infrequently" between actions.
	levine.sscnet.ucla.edu /papers/consist.htm (111 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Optimization using Weighted Fictitious Play Bradley A. Belsky and Theodore J. Lambert III Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI bbelsky@umich.edu, lambertt@umich.edu Abstract Fictitious play is a model traditionally used to describe learning.
		A recent development of fictitious play is its use as an optimization heuristic.
		The belief distributions from the fictitious play heuristic converge to the set of Nash Equilibrium, which we view as a local optimum.
	www.gtcenter.org /Archive/Conf03/belsk-abs.txt (269 words)

	Come out and play—fictitiously (Site not responding. Last check: 2007-10-20)
		The learning model of fictitious play is related to the work of John Nash, the subject of the movie, "A Beautiful Mind." Fictitious play models how people might interact to discover a Nash equilibrium, which seeks the best joint strategies for games with two or more players that reach an outcome to mutual advantage.
		Using computer modeling, U-M engineers used fictitious play to simulate the traffic conditions through 75 traffic lights in Troy, Mich. Smith says the result was at least a 20 percent savings in travel time, just by adjusting the traffic signals.
		By applying the concept of fictitious play, decisions about which direction to turn are anticipated in parallel at every intersection, and the previous outcomes fuel the next until all but the best options are weeded out.
	www.umich.edu /~urecord/0405/Jun13_05/13.shtml (607 words)

	National Theatre : Platform Papers : Michael Frayn on Democracy (Site not responding. Last check: 2007-10-20)
		In this play the events are presented as being actual events, happening in front of your eyes, so it’s a different structure in that sense.
		The play is about how difficult it is for everyone to come to any common policy on anything, not just in political life but how you arrange the furniture in the living room or where you go on holiday.
		MF There is a plan to open the play in Germany on, I think, 6 May 2004, which is the 30th anniversary of Brandt’s resignation.
	www.nationaltheatre.org.uk /?lid=7664 (4932 words)

	Abstracts of 1999 Conference Papers (Site not responding. Last check: 2007-10-20)
		In weighted fictitious play, players take an arithmetic average of what others have done in the past, perhaps weighting more recent observations more heavily, to make a guess about what other players will do in the future; then they choose a strategy which has a high payoff given that guess.
		Thus, reinforcement and fictitious play learning are closely linked; they vary (almost) only in the degree of weight given to foregone payoffs (expressed by delta).
		Using data from human subjects playing simple strategic games repeatedly (from 10 to 100 periods), we estimate the value of the parameters, and forecast "out of sample" the last 30% of the periods in each experiment, for 8 different games (a total of more than 10,000 observations).
	sqab.psychology.org /abstracts-1999.html (6624 words)

	Department of Economics - Faculty - H. Peyton Young (Site not responding. Last check: 2007-10-20)
		Consider a population of agents who play a game through repeated interactions, and adapt their behavior based on information about other agents' previous behavior.
		It is shown by example that learning rules of the fictitious play type fail to converge in certain kinds of coordination games.
		Variants of fictitious play in which past actions are eventually forgotten and which incorporate small stochastic perturbations are better behaved for this class of games: over the long run, players manage to coordinate with probability one.
	www.econ.jhu.edu /people/Young/Abstracts.html (589 words)

	Robert L. Smith
		Fictitious Play (FP), which is an iterative process originating in game theory, executes a non-cooperative game repeatedly among players represented by a partition of the decision variables of the underlying system optimization problem.
		The basic paradigm that will be investigated derives from Fictitious Play which is an adaptive procedure wherein each player assumes that other players will play according to the empirical distribution of their previous plays.
		The Fictitious Play method is a novel paradigm for optimization that draws from several distinct disciplines and application areas, including classical optimization, game theory, transportation science, and queueing network protocols.
	www-personal.engin.umich.edu /~rlsmith (4978 words)

	Learning in Games
		The random matching in a large population provides a rationale for the assumption that players play myopically when considering the effect that their current actions may have on the future play of opponents.
		Ficitious play plays a best-response to the historical frequency of opponents' play.
		This is a classic example of the "tragedy of the commons" that plays a central role in much of economics.
	levine.sscnet.ucla.edu /Papers/SantaCruz/santacruz.htm (976 words)

	SFB 303 - One against All in the Fictitious Play Process (Site not responding. Last check: 2007-10-20)
		Abstract: There are only few "positive" results concerning multi-person games with the fictitious play property, that is, games in which every fictitious play process approaches the set of equilibria.
		We show that if all sub-games are either zero-sum games, weighted potential games, or games with identical payoff functions, then the fictitious play property holds for the associated game.
		Keywords: fictitious play, zero-sum games, games with identical payoff functions, weighted potential games, weak weighted potential games.
	www.wiwi.uni-bonn.de /sfb/papers/abstracts/bonnsfb380.html (159 words)

	Fictitious Play (Site not responding. Last check: 2007-10-20)
		EconPapers: The Rate of Convergence of Continuous Fictitious Play...
		Fictitious Play Property for Games with Identical Interests...
		AMCA: Loss Aversion Fictitious Play by Rimawan Pradiptyo...
	www.scienceoxygen.com /math/577.html (147 words)

	Game Theory
		The main purpose of game theory is to consider situations where instead of agents making decisions as reactions to exogenous prices ("dead variables"), their decisions are strategic reactions to other agents actions ("live variables").
		An agent is faced with a set of moves he can play and will form a strategy, a best response to his environment, which he will play by.
		Introduced "fictitious play" or the "best response dynamic" as a device that converges to Nash equilibrium.
	cepa.newschool.edu /het/schools/game.htm (1057 words)

	RePEc (Site not responding. Last check: 2007-10-20)
		In particular, exponential fictitious play and a suitably perturbed reinforcement model have the same expected motion and therefore will have the same asymptotic behaviour.
		It is also shown that more general models of stochastic fictitious play and perturbed reinforcement learning have identical local stability properties.
		The main identifiable difference between the two models is speed: stochastic fictitious play gives rise to faster learning.
	www.inomics.com /cgi/repec?handle=RePEc:edn:esedps:42 (154 words)

	Research (Site not responding. Last check: 2007-10-20)
		As an example, we study the convergence and stability properties of Sj\"ostr\"om's (1994) mechanism, on the assumption that boundedly rational players find their way to equilibrium using monotonic learning dynamics and also with fictitious play.
		This mechanism implements most social choice functions in economic environments using as a solution concept the iterated elimination of weakly dominated strategies (only one round of deletion of weakly dominated strategies is needed).
		With fictitious play, if the agents have completely mixed prior beliefs, beliefs and play converge to the outcome the planner wants to implement.
	www.econ.upf.es /eng/research/onepaper.php?id=221 (227 words)

	AMCA: Loss Aversion Fictitious Play by Rimawan Pradiptyo (Site not responding. Last check: 2007-10-20)
		Fictitious play has been derived from the implicit assumption that players follow von Neumann-Morgenstern expected utility (EU) function.
		We extended fictitious play in which the players' characteristic of loss aversion has been taken into account in the learning process.
		Loss Aversion Ficitious Play (LAFP) did not only depend on the structure of the game and the players' beliefs, but also on the loss aversion coefficient and the reference point (aspiration level).
	at.yorku.ca /c/a/f/c/40.htm (192 words)

	Performing Playwrights
		The children study the writing process, elements of fiction, read plays throughout the year, and attend performances with an awareness that they will eventually write their own class play.
		The class gathers to discuss the structure necessary to write a play and to review the impor tant components of a fictitious play.
		The next step is to write the opening scene together as the teacher models the development of characters and plot and the collaborative play writing process.
	www2.lhric.org /impact/perform.htm (503 words)

	St John: The Craft: To Play the Villain (Site not responding. Last check: 2007-10-20)
		And there you are dressed and face painted, or hidden, and voice prepared take the stage like the best whore in town demand it satisfies you first and foremost for in obeying that demand the public will adore you.
		Throughout the play your presence must be felt, your trademark pithy lines delivered exquisitely ("I know not what they call these in Franconia, I never visited the King of Burgers") your crossbows cocked to 90 degrees from the horizontal and then the public will remember you to their graves.
		And when, as he mustm the villain dies- ensure that you keep your lines bold to the end, raise your hand in the time honoured claw of death as the life ebbs from you and ensure that the script has allowed you a suitable final line, laughter or shriek.
	www.anycities.com /user/thorkwind/craft-02.htm (772 words)

	Fictitious play in 2xn games
		It is known that every continuous time fictitious play process approaches equilibrium in every nondegenerate 2x2 and 2x3 game, and it has been conjectured that convergence to equilibrium holds generally for 2xn games.
		"Fictitious play in 2 x n games," Journal of Economic Theory, Elsevier, vol.
		"Fictitious play and- no-cycling conditions," Sonderforschungsbereich 504 Publications 97-12, Sonderforschungsbereich 504, University of MannheimSonderforschungsbereich 504, Universität Mannheim.
	ideas.repec.org /p/wpa/wuwpga/0303009.html (292 words)

	Fictitious Play in Coordination Games
		We study the Fictitious Play process with bounded and unbounded recall in pure coordination games for which failing to coordinate yields a payoff of zero for both players.
		It is shown that every Fictitious Play player with bounded recall may fail to coordinate against his own type.
		On the other hand, players with unbounded recall are shown to coordinae (almost surely) against their own type as well as against players with bounded recall.
	ideas.repec.org /p/bon/bonsfb/423.html (372 words)

	Laura's Play (Site not responding. Last check: 2007-10-20)
		This play is about a recently qualified teacher who accepts a drama teacher’s job at an ostensibly prestigious grammar school.
		Unfortunately, she soon discovers that the school is rife with staff politics and that the exterior appearance of this fine school is underpinned with, racism, mistreatment of staff, sexual harassment, in-fighting and back biting and a turgid slurry of disdain against an incompetent headmaster.
		The story, all names, characters and incidents portrayed in this play are fictitious.
	www.now-im-here.com /laurasplay.htm (393 words)

	Iterative Algorithms for Two-Person Zero-Sum Games - Storming Media
		Although it is an effective method, the fictitious play algorithm converges slowly to the value of the game.
		Gass, Zafra, and Qiu reported that their modified algorithm converges faster than the original fictitious play on a collection of randomly generated games.
		When game values are far away from zero, this thesis empirically shows that the original fictitious play algorithm can outperform the modified one.
	www.stormingmedia.us /01/0192/A019263.html (215 words)

	Learning and Evolution In Games (Site not responding. Last check: 2007-10-20)
		payoffs in fictititous play and the "infrequent-switching" condition
		appendix on Dirichlet priors and the Bayesian interpretation of fictitious play
		extensions of the fictitious play model to partial sampling
	fudenberg.fas.harvard.edu /learn.htm (289 words)

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