Game theory and non-linear dynamics: the Parrondo Paradox case study P. Arena, S. Fazzino, L. Fortuna * , P. Maniscalco Dipartimento Elettrico, Elettronico e Sistemistico, System and Control Group, Universit  a degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy Abstract In this paper a new research topic is explored on the role of chaos in a particular game problem: the Parrondo Paradox. In the original formulation of this paradox, it has been proved that two separate losing games can be combined following a random or periodic strategy in order to have a resulting winning game. In this paper, three key points will be dealt with. The first one regards the introduction of a strategy based on various chaotic time series: this couldimprovethegainintheclassicaltwogamesParrondoproblem.Thesecondoneconcernswiththeintroductionof a third loosing game based on the history of the game and not on the capital as in the classical Parrondo two games Problem.Finally,theParrondoParadoxhasbeengeneralizedfor N gamesandanalgorithmhasbeenproposedinorder toinvestigatethroughanoptimizationapproachtheregionofprobabilityparameterspaceinwhichParrondoParadox can occur. Ó 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction ParrondoÕsParadox[1–4]isanewresearchtopicinGameTheorydevisedbyParrondoasapedagogicalillustration oftheBrownianratchet.Itoccurswhentwostatisticallylosinggamesofchance,saygameAandgameB,arecombined following a random or periodic strategy in order to have a resulting winning game. This is best demonstrated in the original version of ParrondoÕs paradox by tossing coins where the coins are biased towards winning or losing. In particular: • game A consists of a biased coin, say coin 0, which has probability p of winning; • game B can be described by the following statement. If the present capital is a multiple of M then the chance of winning is p 1 , if it is not a multiple of M the chance of winning is p 2 . It has been proved [2] that game A results a loosing game when the following condition is met: 1  p p > 1 ð1Þ while game B is loosing when: ð1  p 1 Þð1  p 2 Þ M1 p 1 p M1 2 > 1 ð2Þ * Corresponding author. Tel.: +39-095-339535; fax: +39-095-330793. E-mail address: lfortuna@dees.unict.it (L. Fortuna). 0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII:S0960-0779(02)00397-1 Chaos, Solitons and Fractals 17 (2003) 545–555 www.elsevier.com/locate/chaos