Inefficiency of voting in Parrondo games

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1 Physica A 343 (24) Inefficiency of voing in Parrondo games Luis Dinı s, Juan M.R. Parrondo Dep. de Física Aómica, Molecular y Nuclear, Grupo Inerdisciplinar de Sisemas Complejos (GISC), Universidad Compluense de Madrid, 284-Madrid, Spain Received 9 Augus 23; received in revised form 3 April 24 Available online 14 July 24 Absrac We sudy a modificaion of he so-called Parrondo s paradox where a large number of individuals choose he game hey wan o play by voing. We show ha i can be beer for he players o voe randomly han o voe according o heir own benefi in one urn. The former yields a winning endency while he laer resuls in seady losses. An explanaion of his behaviour is given by noing ha selfish voing prevens he swiching beween games ha is essenial for he oal capial o grow. Resuls for boh finie and infinie number of players are presened. I is shown ha he exension of he model o he hisory-dependen Parrondo s paradox also displays he same effec. r 24 Elsevier B.V. All righs reserved. PACS: 2.5. r; 5.4. a; Ge Keywords: Parrondo s paradox; Majoriy rule; Brownian raches 1. Inroducion The dynamics of a flashing rache [1 4] can be ranslaed ino a counerinuiive phenomenon in gambling games which has recenly araced considerable aenion. I is he so-called Parrondo s paradox [5 8] consising of wo losing games, A and B, ha yield, when alernaed, a winning game. Corresponding auhor. address: parr@seneca.fis.ucm.es (J.M.R. Parrondo) /$ - see fron maer r 24 Elsevier B.V. All righs reserved. doi:1.116/j.physa

2 72 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) In game A, a player osses a coin and makes a be on he hrow. He wins or loses 1 euro depending on wheher he coin falls heads or ails. The probabiliy p 1 of winning is p 1 ¼ 1 2 wih p 1; so game A is fair when ¼ and losing when 4. By losing, winning, and fair games here we mean ha he average capial is a decreasing, increasing, and a consan funcion of he number of urns, respecively. The second game or game B consiss of wo coins. The player mus hrow coin 2 if his capial is no a muliple of hree, and coin 3 oherwise. The probabiliy of winning wih coin 2 is p 2 ¼ 3 4 and wih coin 3 is p 3 ¼ 1 1. They are called good and bad coins respecively. I can be shown ha game B is also a losing game if 4 and ha ¼ makes B a fair game [7,8]. The rules of boh games A and B are depiced in Fig. 1. Surprisingly, swiching beween games A and B in a random fashion or following some periodic sequences produces a winning game, for 4 sufficienly small, i.e., he average of player earnings grows wih he number of urns. Therefore, from wo losing games we acually ge a winning game [5 7]. This indicaes ha he alernaion of sochasic dynamics can resul in a new dynamics, which differs qualiaively from he original ones. Alernaion is eiher periodic or random in he flashing rache [1 4] and in he paradoxical games [5 7]. On he oher hand, we have recenly sudied he case of a conrolled alernaion of games, where informaion abou he sae of he sysem can be used o selec he game o be played wih he goal of maximising he capial [9]. This problem is rivial for a single player: he bes sraegy is o selec game A when his capial is a muliple of hree and B oherwise. This yields higher reurns han any periodic or random alernaion. Therefore, choosing he game as a funcion of he curren capial presens a considerable advanage wih respec o blind sraegies, i.e., sraegies ha do no make use of any informaion abou he sae of he sysem, as i is he case of he periodic and random alernaion. Analogously, in a flashing rache, swiching on and off he rache poenial depending on he locaion of he Brownian paricle allows one o exrac energy from a single hermal bah, in apparen conradicion wih he second law of hermodynamics [1]. This is nohing Game A Game B Is X() a muliple of hree? No Yes 1/2-1/2+ win lose 3/4-1/4+ 1/1-9/1+ win lose win lose Fig. 1. Rules of he wo Parrondo games.

3 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) bu a Maxwell demon, who operaes having a his disposal informaion abou he posiion of he paricle; and i is he acquisiion or he subsequen erasure of his informaion wha has an unavoidable enropy cos [11], prevening any violaion of he second law. Whereas a conrolled alernaion of games is rivial for a single player, ineresing and couner-inuiive phenomena can be found in collecive games. In Ref. [9], we considered a collecive version of he original Parrondo s paradox. In his model, he game A or B ha a large number N of individuals play can be seleced a every urn. I urns ou ha blind sraegies are winning whereas a sraegy which chooses he game wih he highes average reurn is losing [9]. In he presen paper, we exend our invesigaion of conrolled collecive games considering a new sraegy based on a majoriy rule, i.e., on voing. This ype of rule differs from he one considered in Ref. [9] and has been proved o be relevan in several siuaions, such as he modelling of public opinion [12,13] or he design of muli-layer neural neworks by means of commiee machines [14,15]. We will show ha, in conrolled games, he rule is very inefficien: if every player voes for he game ha gives him he highes reurn, hen he oal capial decreases, whereas blind sraegies generae a seady gain. As menioned above, for a single player, he majoriy rule does defeas he blind sraegies. The inefficiency of voing is consequenly a purely collecive effec. In conras wih he shor range opimisaion sraegy in Ref. [9] where he average reurns are maximised, he players now choose he game according o heir own benefi in he nex coin oss and never care abou how bad is ha decision for he res of he communiy. Hence, he inefficiency of he conrol is now sronger: he shor-range opimisaion is only worse han he blind sraegies if jus a random seleced se of he players are allowed o play he games a each urn whereas he majoriy rule is losing no maer how many of he players are allowed o voe and play. Finally, he same effec can be found for he capial-independen games inroduced in Ref. [8], showing ha he mechanism underlying he inefficiency of voing is general and can be exended o oher sysems while reaining is main feaures. The paper is organised as follows. In Secion 2, we presen he model and he couner-inuiive performance of he differen sraegies. In Secion 3, we discuss and provide an inuiive explanaion of his behaviour. In Secion 4, we analyse how he effec depends on he number of players. In Secion 5, we exend hese ideas o he capial-independen games inroduced in Ref. [8]. Finally, in Secion 6 we presen our main conclusions. 2. The model The model consiss of a large number N of players. In every urn, hey have o choose one of he wo original Parrondo games, described in he Inroducion and in Fig. 1. Then every individual plays he seleced game agains he casino.

4 74 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) Capial per player Majoriy Rule ABBABB... Random Fig. 2. Evoluion of he capial per player in an infinie ensemble for ¼ :5 and he hree sraegies discussed in he ex. We will consider hree sraegies o achieve he collecive decision. (a) The random sraegy, where he game is chosen randomly wih equal probabiliy. (b) The periodic sraegy, where he game is chosen following a given periodic sequence. The sequence ha we will use hroughou he paper is ABBABB::: since i is he one giving he highes reurns. (c) The majoriy rule (MR) sraegy, where every player voes for he game giving her he highes probabiliy of winning, wih he game obaining he mos voes being seleced. The model is relaed o oher exensions of he original Parrondo games played by an ensemble of players, such as hose considered by Toral [16,17]. However, in our model he only ineracion among players can occur when he collecive decision is made. Once he game has been seleced, each individual plays, in a compleely independen way, agains he casino. Moreover, in he periodic and random sraegies here is no ineracion a all among he players, he model being equivalen o he original Parrondo s paradox wih a single player. The MR makes use of he informaion abou he sae of he sysem, whereas he periodic and random sraegies are blind, in he sense defined above. One should hen expec a beer performance of he MR sraegy. However, i urns ou ha, for large N, hese blind sraegies produce a sysemaic winning whereas he MR sraegy is losing. This is shown in Fig. 2, where he capial per player as a funcion of he number of urns is depiced for he hree sraegies and an infinie number of players (see Appendix A for deails on how o obain Fig. 2). 3. Analysis How many players voe for each game? The key magniude o answer his quesion and o explain he sysem s behaviour is p ðþ, he fracion of players whose money is a muliple of hree a urn. This fracion p ðþ of players voe for game A in order o

5 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) avoid he bad coin in game B. On he oher hand, he remaining fracion 1 p ðþ voe for game B o play wih he good coin. Therefore, if p ðþx1=2, here are more voes for game A and, if p ðþo1=2, hen game B is preferred by he majoriy of he players. Le us focus now on he behaviour of p ðþ for ¼ when playing boh games separaely. If game A is played a large number of imes, p ðþ ends o 1/3 because he capial is a symmeric and homogenous random walk under he rules of game A. On he oher hand, if B is played repeaedly, p ðþ ends o 5/13. This can be proved by analyzing game B as a Markov chain [7,8]. I is also remarkable ha, for p ðþ ¼5=13, he average reurn when game B is played is zero. Fig. 3 represens schemaically he evoluion of p ðþ under he acion of each game, as well as he prescripion of he MR sraegy explained above. Now we are ready o explain why he MR sraegy yields worse resuls han he periodic and random sequences. We see ha, as long as p ðþ does no exceed 1=2, he MR sraegy chooses game B. However, playing B akes p closer o 5=13, well below 1=2, and hus more han half of he players voe for game B again. Afer a number of runs, he MR sraegy ges rapped playing game B forever. Then p asympoically approaches 5/13, and as his happens, game B urns ino a fair game when ¼. As a consequence, he MR will no produce earnings any more, as can be seen in Fig. 4. The inroducion of 4 urns game B ino a losing game if played repeaedly. Consequenly, he MR sraegy becomes a losing one as in Fig. 2. To overcome his losing endency, he players mus sacrifice heir shor-range profis, no only for he benefi of he whole communiy bu also for heir own reurns in he fuure. Hence, some kind of cooperaion among he players is needed o preven hem from losing heir capial. A similar effec has been found by Toral in anoher version of collecive Parrondo s games. There, sharing he capial among players induces a seady gain [16]. In our case, he sriking resul is ha no complex cooperaion is necessary. I is enough ha he players agree o voe a random. We would like o sress he differences wih he shor-range opimisaion considered in Ref. [9]. In ha paper, he criical value of p ðþ was 5/13 insead of 1/2. As a consequence, he voing sraegy is even less efficien han he shor-range opimisaion. For insance, he shor-range opimisaion is winning if he whole Majoriy Rule: Play B Play A 1/2 A B 1/3 5/13 Fig. 3. Schemaic represenaion of he evoluion of p ðþ under he acion of games A and B. The prescripion of he MR is also represened.

6 76 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) π () Majoriy Rule ABBABB... Random Capial per player Majoriy Rule ABBABB... Random Fig. 4. Evoluion of p ðþ (lef) and he capial per player (righ) for N ¼1, ¼ for he MR and random sraegies. The MR chooses game B when p is below he sraigh line depiced a 1/2 and game A oherwise. ensemble plays in each urn. To achieve he counerinuiive phenomena ha blind sraegies perform beer han he shor-range opimisaion, in Ref. [9] we were forced o include he following rule: in each urn only a fracion g of he individuals in he ensemble play he game. For he voing sraegy, he inclusion of his rule is no necessary. 4. Finie number of players In he previous analysis an infinie number of players has been considered. Remarkably, for jus one player he MR sraegy rivially performs beer han any periodic or random sequence, since i compleely avoids he use of he bad coin. In his secion, we analyse he crossover beween he winning behaviour for a small number of players and he losing behaviour when his number is large. Fig. 5 shows numerical resuls of he average capial per player for an increasing number of players ranging from 1 o 1. One can observe ha, he larger he number of players, he worse he resuls for he MR sraegy, becoming losing for a number of players beween 5 and 1. The above discussion for an infinie ensemble allows us o give a qualiaive explanaion. The difference beween large and small N is he magniude of he flucuaions of p ðþ around is expeced value. If game B is chosen a large number of imes in a row, hen he expeced value of p ðþ is 5/13. On he oher hand, he MR selecs B unless p ðþ is above 1/2. Therefore, for he MR o selec A, flucuaions mus be of order 1=2 5=13 ¼ 3=26 :115. For N players, he fracion of players wih capial muliple of hree, p ðþ, will be a random variable following a binomial disribuion, a leas if B has been played a large number of imes in a row. If he expeced pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value of p ðþ is 5/13, flucuaions of p ðþ around his value are of order 5=13 8=13 1=N. Then, flucuaions will allow he MR sraegy o choose A if N 2. Far above his value, flucuaions ha drive p ðþ above 1/2 are very rare, and MR chooses B a almos every urn, as can be seen in Fig. 6. On he oher hand,

7 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) N=1 Capial per player 1 5 Majoriy Rule ABBABB... Random N=1 N=5 N=1 N= Fig. 5. Simulaion resuls for he average capial per player for N ¼ 1, 5, 1, and 1 players, ¼ :5, and he hree differen sraegies. The simulaions have been made over a variable number of realizaions, ranging from 1 realizaions for N ¼ 1 o 1 realizaions for N ¼ 1. Simulaions for he random and periodic sraegies have been made wih N ¼ 1 players and averaging over 1 realizaions. For hese blind sraegies, he resul does no depend on he number of players N. 1.8 π () Fig. 6. Simulaion showing he evoluion of p for N ¼ 7. Flucuaions ha drive p above.5 are very rare, i.e., game A is seldom chosen. for N around or below 2, here is an alernaion of he games ha can even bea he opimal periodic sraegy. We see ha he MR sraegy can ake profi of flucuaions much beer han blind sraegies, bu i loses all is efficiency when hese flucuaions are small. We believe

8 78 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) ha his is closely relaed o he second law of hermodynamics. The law prohibis any decrease of enropy only in he hermodynamic limi or for average values. On he oher hand, when flucuaions are presen, enropy can indeed decrease momenarily and his decrease can be exploied by a Maxwell demon. 5. Hisory dependen games A similar phenomenon is exhibied by he games inroduced in Ref. [8], whose rules depend on he hisory raher han on he capial of each player. Game A is sill he same as above, whereas game B is played wih hree coins according o he following Table 1. Inroducing a large number of players bu allowing jus a randomly seleced fracion g of hem o voe and play, he same voing paradox is recovered for sufficienly small g. Again, blind sraegies achieve a consan growh of he average capial wih he number of urns while he MR sraegy reurns a decreasing average capial, as is shown in Fig. 7. The explanaion of he phenomenon goes quie along Table 1 Before las Las Prob. of win Prob. of loss 2 1 a a Loss Loss p 1 1 p 1 Loss Win p 2 1 p 2 Win Loss p 2 1 p 2 Win Win p 3 1 p 3 p 1 ¼ 9 1, p 2 ¼ 1 4, and p 3 ¼ Capial per player Majoriy Rule ABBABB... Random 5 1 Fig. 7. Evoluion of he average money of he players in he hisory-dependen games for g ¼ :5, ¼ :5 and hree differen sraegies.

9 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) he same lines as for he original games. The selfish voing prevens he necessary alernaion of games, alhough now is game A which is seleced oo many imes in a row. 6. Conclusions We have shown ha he paradoxical games based on he flashing rache exhibi a counerinuiive phenomenon when a large number of players are considered. A majoriy rule based on selfish voing urns o be very inefficien for large ensembles of players. We have also discussed how he rule only works for a small number of players, since in ha case i is able o exploi capial flucuaions. The inefficiency of voing encounered in our model could be raher general. The mechanism behind his inefficiency is he following. Those individuals wih a capial non-muliple of hree voes for game B, ensuring heir own benefi in he presen urn. However, even if hese players conform he majoriy, heir selfish voe is harmful for he whole ensemble. They should sacrifice heir shor-erm profis o help he ohers o avoid he use of he bad coin of game B (coin 3). The blind sraegy (eiher random or periodic) forces his sacrifice, yielding a seady gain for he ensemble. The same mechanism operaes in he capial-independen games, as shown in Secion 5. A similar behaviour is found in he so-called Braess paradox, where he maximizaion of individual profi yields overall losses for he communiy [18,19]. As in he Braess paradox, he model presened here shows ha cooperaion among individuals via a collecive decision can be beneficial for everybody. In his sense, he model is also relaed o ha presened by Toral in Ref. [17]. Since John Maynard Smih firs applied game heory o biological problems [2], games have been used in ecology and social sciences as models o explain social behaviour of individuals inside a group. Some generalizaions of he voing model migh be useful for his purpose. For insance, i could be ineresing o analyse he effec of mixing selfish and cooperaive players or he inroducion of players who could change heir behaviour depending on he fracion of selfish voers in previous urns. The model can also be relevan in random decision heory or he heory of sochasic conrol [21] since i shows how periodic or random sraegies can be beer han some kind of opimisaion. In his sense, here has been some work on general adapive sraegies in games relaed wih Parrondo s paradox [9,22,23]. Finally, his model and, in paricular, he analysis for N finie, promps he problem of how informaion can be used o improve he performance of a sysem. In he models presened here, informaion abou he flucuaions of he capial is useful only for a small number of players, ha is, when hese flucuaions are significan. I will be ineresing o analyse his crossover in furher deail, no only in he case of he games bu also for Brownian raches. Work in his direcion is in progress.

10 71 L. Dinís, J.M.R. Parrondo / Physica A 343 (24) Acknowledgemens The auhors graefully acknowledge fruiful discussions wih Chrisian Van den Broeck, who suggesed us he inroducion of he MR sraegy. We also hank H. Leonardo Marı nez for valuable commens on he manuscrip. This work has been financially suppored by Gran BFM C2-2 from Miniserio de Ciencia y Tecnologı a (Spain) and by a Gran from he New del Amo Program (Universidad Compluense). Appendix A. Evoluion equaions In his secion, we describe he semi-analyical soluion of he model for an infinie number of players, used o depic Fig. 2. Le p i ðþ, be he fracion of players whose capial a urn is of he form 3n þ i wih i ¼ ; 1; 2 and n an ineger number. If game A is played in urn, hese fracions change following he expression [8]: p ð þ 1Þ 1=2 þ 1=2 p ðþ B C B CB p 1 ð þ 1Þ A 1=2 1=2 þ A@ p 1 ðþ A ða:1þ p 2 ð þ 1Þ 1=2 þ 1=2 p 2 ðþ which can be wrien in a vecor noaion as! p ð þ 1Þ! ¼PA p ðþ : ða:2þ Similarly, when B is played, he evoluion is given by! p ð þ 1Þ! ¼PB p ðþ ða:3þ wih 1 1=4 þ 3=4 B C P B 1=1 1=4 þ A : ða:4þ 9=1 þ 3=4 Now we can wrie he evoluion equaion for each sraegy. For he random sraegy! p ð þ 1Þ ¼ 1 2 ½P A þ P B Š! p ðþ : ða:5þ For he periodic sraegy (ABBABB..)! p ð3ð þ 1ÞÞ ¼ P 2 B P! A p ð3þ : ða:6þ Finally, wih he MR sraegy he ensemble plays game A if p ðþx1=2 and B oherwise. Therefore, (! P! A p ðþ if p ðþx1=2 ; p ð þ 1Þ ¼ P! ða:7þ B p ðþ if p ðþo1=2 : Noice ha he MR sraegy is he only one inducing a nonlinear evoluion in he populaion fracions. To calculae he evoluion of he capial, we compue he

11 winning probabiliy in each game p A win ðþ ¼ 1 2 ARTICLE IN PRESS L. Dinís, J.M.R. Parrondo / Physica A 343 (24) p B win ðþ ¼ 1 1 p ðþþ 3 4 ð1 p ðþþ : ða:8þ Finally, he average capial hx ðþi per player evolves as hxð þ 1Þi ¼ hxðþi þ 2p win ðþ 1 ða:9þ and p win ðþ is replaced by p A win ðþ or pb winðþ, depending on he game played a urn in each sraegy. References [1] R.D. Asumian, P. Hanggi, Phys. Today 55 (11) (22) 33. [2] P. Reimann, P. Hanggi, Appl. Phys. A 75 (22) 169. [3] P. Reimann, Phys. Rep. 361 (22) 57. [4] A. Ajdari, J. Pros, C.R. Acad. Sci. Paris II 315 (1993) [5] G.P. Harmer, D. Abbo, Sa. Sci. 14 (1999) 26. [6] G.P. Harmer, D. Abbo, Naure 42 (1999) 846. [7] G.P. Harmer, D. Abbo, Flucuaion and Noise Le. 2 (22) R71. [8] J.M.R. Parrondo, G.P. Harmer, D. Abbo, Phys. Rev. Le. 85 (2) 24. [9] L. Dinis, J.M.R. Parrondo, Europhys. Le. 63 (23) 319. [1] J.M.R. Parrondo, B.J. de Cisneros, Appl. Phys. A 75 (22) 179. [11] H.S. Leff, A.F. Rex, Maxwell s Demon. Enropy, Informaion, Compuing, Adam Hilger, Brisols, 199. [12] S. Galam, Physica A 32 (23) 571. [13] P.L. Krapivsky, S. Redner, Phys. Rev. Le. 9 (23) [14] E. Barkai, D. Hansel, H. Sompolinsky, Phys. Rev. A 45 (1992) [15] C. Van den Broeck, J.M.R. Parrondo, Phys. Rev. Le. 71 (1993) [16] R. Toral, Flucuaion and Noise Le. 1 (21) L7. [17] R. Toral, Flucuaion and Noise Le. 2 (22) L35. [18] D. Braess, Unernehmensforschung 12 (1969) 258. [19] J.E. Cohen, Proc. Nal. Acad. Sci. USA 95 (1998) [2] J. Maynard Smih, Evoluion and he Theory of Games, Cambridge Universiy Press, Cambridge, [21] D.J. Whie, Markov Decision Processes, Wiley, New York, [22] E. Behrends, Parrondo s paradox: a priori and adapive sraegies, preprin. [23] S. Rahmann, Opimal adapive sraegies for games of he parrondo ype, preprin.