We introduce a new family of Parrondo's games of alternating losing
strategies in order to get a winning result. In our version of the games we
consider an ensemble of players and use "social" rules in which the
probabilities of the games are defined in terms of the actual state of the
neighbors of a given player.Comment: 4 pages (including 2 figures

Toral, R.

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arXiv

 We introduce a new family of Parrondo's games of alternating losing strategies in order to get a winning result. In our version of the games we consider an ensemble of players and use "social" rules in which the probabilities of the games are defined in terms of the actual state of the neighbors of a given player. </jats:p

RAÚL TORAL

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Fluctuation and Noise Letters

COOPERATIVE PARRONDO'S GAMES

Texto completo arXiv:cond-mat/0101435v2We introduce a new family of Parrondo’s games of alternating losing strategies in order to get a winning
result. In our version of the games we consider an ensemble of players and use “social” rules in which the
probabilities of the games are defined in terms of the actual state of the neighbors of a given playerThis work was supported by MCYT (Spain), grants numbers PB97-0141-C01-01 and BFM2000-1108.Peer reviewe

Toral, Raúl

Digital.CSIC

arXiv:cond-mat/0101435v2  19 Mar 2001COOPERATIVE PARRONDO’S GAMESRau´l ToralInstituto Mediterra´neo de Estudios Avanzados, IMEDEA (CSIC-UIB)Campus UIB, 07071-Palma de Mallorca, Spain,email: raul@imedea.uib.es, http://www.imedea.uib.es/PhysDeptWe introduce a new family of Parrondo’s games of alternating losing strategies in order to get a winningresult. In our version of the games we consider an ensemble of players and use “social” rules in which theprobabilities of the games are defined in terms of the actual state of the neighbors of a given player.Parrondo’s paradox [1–4] shows that the combination of two losing strategies can lead to a winning result. The paradox canbe phrased in terms of very simple gambling games in which some unit (say, 1 euro) is won or lost with a given probability.We can imagine that the games consist on tossing different biased coins and that a “capital” C(t) is built. Every time a game isplayed (a coin is tossed) time increases by one unit, t → t + 1, and the capital increases or decreases by 1, C(t) → C(t) ± 1.Games are classified as winning, losing or fair if the average capital 〈C(t)〉 increases, decreases or remains constant with time,respectively.The original version of the paradox is based on the two following basic games that can be played at any time t:• Game A: there is a probability p of winning.• Game B: If the capital C(t) is a multiple of 3, the probability of winning is p1, otherwise, the probability of winning is p2.Game A is fair if p = 1/2 and it is easy to prove that game B is fair if the condition (1−p1)(1−p2)2 = p1p22 holds. Choosing, forexample, p = 0.5−ǫ, p1 = 0.1−ǫ, p2 = 0.75−ǫ, with ǫ a small positive number it turns out that both game A and game B, whenplayed by themselves, are losing games. The surprise arises when game A and B are played alternatively, either in successionsuch as AABBAABBAABB. . . or randomly by choosing (with probability 1/2) the next game to be played. We will denote,for short, game “A+B” the case in which games A or B are chosen randomly at each time step and game “[2, 2]” corresponds tothe sequence AABBAABBAABB. . . In these cases of alternation in the game played, the paradoxical result is that the outcometurns out to be a winning game, although both games A and B are losing games. The paradox can be related to a discrete versionof the flashing ratchet [5] in which a Brownian particle is subjected, alternatively, to the effect of two potentials. The potentialsare such that each one, acting alone, induces a net movement for the particle in the direction of decreasing potential (say, to theleft). However, in the case of alternating potentials, the net movement is in the opposite direction (to the right). The possiblerelevance of the paradox to other situations of interest in fields such as Biology, Economy and Physics has been recently pointedout [6,7].A new version of the paradox has been introduced recently [6]. In this new version, game A is the same as defined before, butthe probabilities of game B do not depend directly on the actual value of the capital, but rather on the past history of winningand losing games. Let us classify the game played at any time t as a “loser” (“winner”) if it results in loss (win) of capital. Theprobabilities of game B depend then on the outcome of the last two games. More precisely, game B is modified as follows: theprobability of winning at time t is:• p1, if game at t− 2 was loser and game at t− 1 was loser,• p2, if game at t− 2 was loser and game at t− 1 was winner,• p3, if game at t− 2 was winner and game at t− 1 was loser,• p4, if game at t− 2 was winner and game at t− 1 was winner.The condition for the new game B to be a fair one is p1p2 = (1−p3)(1−p4). For instance, the choice p = 1/2−ǫ, p1 = 0.9−ǫ,p2 = p3 = 0.25− ǫ, p4 = 0.7− ǫ (for positive ǫ) results in A and B being both losing games. However, it has been shown that(for sufficiently small ǫ) the paradox appears again: the alternation games A+B or [2, 2] are both winning games.Inspired by this result, we introduce a new version of the games that we call “cooperative Parrondo’s games”. In this version,we consider an ensemble of N players, each one owning a capital Ci(t), i = 1, . . . , N . This capital evolves by combination oftwo basic games, A and B. First a player, i, is randomly chosen between the N players. Player i can then play either game A orgame B according to some rules. When a player wins in any particular game, it is labeled as “winner”, otherwise it is labeledas “loser”. The state of “winner” or “loser” for a particular player remains unchanged in time until that player has the chance toplay again. Game A is the same as before: there is a probability p (alternatively 1 − p) for player i to win (lose) one unit andincrease (decrease) his capital Ci(t) by 1. The new ingredient is that the probabilities of game B are defined according to thestate of the two neighbor players, i+ 1 and i− 1 (we assume periodic boundary conditions). The simplest version appears whenthe probability of winning at time t depends on the present state of the neighbors and it is given by:• p1, if player at site i− 1 is a loser and player at site i+ 1 is a loser,• p2, if player at site i− 1 is a loser and player at site i+ 1 is a winner,1• p3, if player at site i− 1 is a winner and player at site i+ 1 is a loser,• p4, if player at site i− 1 is a winner and player at site i+ 1 is a winner.We classify the games according to the behavior of the total capital C(t) =∑iCi(t). Therefore, a winning game is one forwhich the average value of the total capital 〈C(t)〉 increases with time, and similarly for losing and fair games.We will show that Parrondo’s paradox applies also to this kind of games: there are situations, i.e. values of the probabilitiesp, p1, p2, p3, p4, in which the alternation of game A and game B can perform better that games A and B separately. In particular,it is possible for games A and B to be losing or fair games, and that an alternation of games A and B leads to a winning result.Before presenting numerical evidence of this paradoxical result, we perform a mean-field type analysis that can capture someaspects of this new game.Let PAi (t), PBi (t) and PA+Bi (t) be the probability that the i player is labeled as winner at time t when playing only game A,only game B or game A+B, respectively. For game A it is PAi(t) = p for all times t. For games B and A+B the evolution ofthe probabilities is governed by master equations which can not be solved explicitly. In our mean-field approach we assume thatthe probabilities of being winner or loser at time t for the different players are equal for all of them, i.e. PBi(t) ≡ PB(t) andPABi(t) ≡ PAB(t). The mean-field equation for the evolution of the common probability in game B is the map:PB(t + 1) = (1 − PB(t))2p1 + PB(t)(1 − PB(t))(p2 + p3) + PB(t)2p4 ≡ F (PB(t)), (1)where the different terms of this equation reflect each of the 4 possibilities that have been spelt out in the definition of game B.In the random sequence defining game A+B, there is a probability 1/2 of choosing game A and a probability 1/2 of choos-ing game B at each time step. Therefore the corresponding mean-field equation for game A+B is the map PA+B(t + 1) =12[p + F (PA+B(t))]. The stationary probability of winning in game B is obtained by setting PB(t + 1) = PB(t) ≡ PB andsatisfies the equation PB = F (PB). Similarly, the mean field stationary solution PA+B for game A+B is obtained by solvingPA+B = 12[p+ F (PA+B)]. As for any quadratic map, there is at most one stable stationary solution of the maps of game B orA +B. It can be shown that for game B there is a stable solution PB ∈ [0, 1] for p4 ≥ 1/4, whereas for p4 < 1/4, there are nostable solutions for some values of the set of probabilities {p1, p2, p3} defining the game. On the other hand, there is always astable solution for the map A +B such that PA+B ∈ [0, 1].We will find a “weak” paradoxical result of the Parrondo type whenever the random alternation of game A+B performs betterthan the average of games A and B. This is to say, whenever the condition PA+B > 12(p + PB) is satisfied. This depends onthe set of numbers p, p1, p2, p3, p4. For instance, choosing p = 0.5, p2 = p3 = 0.16, p4 = 0.7 the inequality is satisfiedboth for p1 < 0.5395 and for p1 > 0.981.FIG. 1. Average capital per player, 〈C(t)〉/N , versus time, t, measured in units of games per player. The probabilities defining the gamesare: p = 0.5, p1 = 1, p2 = p3 = 0.16, p4 = 0.7. These results show that game A is fair, game B is a losing game, but that when games Aare B are played in random succession (game A+B) or in the [2, 2] alternation AABBAABB. . ., the result is a winning game. We show in thisgraph results for N = 50, 100, 200 players. Averages are performed with respect to M = 1000 realizations of the games using differentrandom numbers.We now describe the results obtained after computer simulations of the model. The numerical results, in accordance withthe mean field analysis, do show the existence of sets of probabilities p, p1, p2, p3, p4 for which the game A+B performsbetter than the average of games A and B played separately (“weak paradoxical result”). This happens, for instance, for p =0.5, p1 = 0.5, p2 = p3 = 0.16, p4 = 0.7 in which both games B and game A+B are losing games. More interestingly, there are1These numbers by no means constitute an exceptional example. Many other sets of probabilities lead to this paradoxical result.2situations in which the paradoxical result is enhanced by the fact that game B is a losing game whereas game A+B turns out tobe a winning game. This “strong” paradoxical result, which is observed in the numerical simulation, is not correctly predictedby the mean-field analysis2. As a way of example, we plot in figure 1 the average capital when playing the games using thevalues p = 0.5, p1 = 1, p2 = p3 = 0.16, p4 = 0.7. As shown in the figure, and for these values of the probabilities, it turns outthat game A is fair, game B is losing, but both A+B and [2, 2] are winning games. Mean field in this case, yields PB = 0.50386,PA+B = 0.50218, and it is able to predict only a weak paradoxical result: PA+B > (p+ PB)/2, but can not predict the strongparadoxical result PA+B > PB , as observed in the simulation. By choosing p = 0.5 − ǫ, for small enough ǫ, it is possible toturn game A into a losing game while still keeping game A + B as a winning game. The existence of this paradoxical resultwhen playing games whose probabilities depend on the present state of the neighbors is the main result of this paper.We next characterize the capital distribution between the different players. We expect, due to the law or large numbers,that the probability distribution P (Ci, t) of the capital Ci of player number i at time t follows in all cases, and in the limitof large N , a Gaussian distribution. Due to the symmetry between players, the average capital of a single player is equalfor all of them, 〈Ci(t)〉 = N−1〈C(t)〉. We have computed the variance of the single-player capital distribution σ2(t) =〈N−1∑iCi(t)2 − [〈C(t)〉/N ]2〉. For game A we have the exact result: N−1〈C(t)〉 = (2p− 1)t and σ2(t) = 4p(1− p)t. Wecan see in figure 2 that the variance of game B is, for the particular values of the probabilities taken here, larger than that of gameA. However, the random alternation of game A+B or the sequence [2, 2] both yield a similar variance which is smaller than thatof game B. We conclude that the distribution of capital between the different players for the combined games A+B or [2, 2] ismore uniform than that of game B. This interesting result excludes that a winning overall game is produced as a consequence ofa large fluctuation implying only a small number of players.FIG. 2. Time evolution of the variance σ2 of the single player capital distribution using games with the same probabilities than in figure 1and N = 50 players. For game A the exact result is σ2 = t. A linear fit to the law σ2 = αt gives α = 1.91 for game B and α = 1.15 for bothgames A+B and [2, 2]. This indicates that the spread of capital is less in the combined games than in the single game B.In conclusion, we have introduced a new family of games based upon the Parrondo’s strategy of alternating non-winninggames to generate a winning result. The novelty of our treatment lies on that a winning game results from the interactionbetween the different players. In that sense, we can consider that our results can serve as a model for the application of thegames to situations in which “social” rules for the definition of the success probabilities need to be considered. Extensions ofthe games in which the players are connected by more realistic small-world type networks can also be considered.Acknowledgements: This work was supported by MCYT (Spain), grants numbers PB97-0141-C01-01 and BFM2000-1108.[1] G.P. Harmer and D. Abbott, Nature 402, 864 (1999).[2] G.P. Harmer and D. Abbott, Stat. Sci. 14, 206 (1999).2This statement is based upon a numerical comparison of the mean field solutions. This analysis does not exclude completely that a strongparadoxical result can be obtained within the mean field theory in a tiny region of parameter space {p, p1, p2, p3, p4}, but we believe this is avery unlikely possibility.[3] G.P. Harmer, D. Abbott, P.G. Taylor and J.M.R. Parrondo in Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations, AmericanInstitute of Physics, Melville (New York) 2000, eds. D. Abbott and L.B. Kiss.[4] G.P. Harmer, D. Abbott, P.G. Taylor, C.E.M. Pearce and J.M.R. Parrondo in Proc. Stochastic and Chaotic Dynamics in the Lakes, AmericanInstitute of Physics (in the press), ed. P.V.E. McClintock.[5] R.D. Astumian and M. Bier, Phys. Rev. Lett. 72, 1766 (1994).[6] J.M.R. Parrondo, G. Harmer and D.Abbott, Phys. Rev. Lett. 85, 5226 (2000).[7] P. Davies, Physics and Life. Lecture in honor of Abdus Salam. In The First Steps of Life in the Universe, Proceedings of the Sixth TriesteConference on Chemical Evolution, eds. J. Chela-Flores, T. Tobias and F. Raulin, Kluwer Academic Publishers (in press).4

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