In lowest unique bid auctions, $N$ players bid for an item. The winner is
whoever places the \emph{lowest} bid, provided that it is also unique. We use a
grand canonical approach to derive an analytical expression for the equilibrium
distribution of strategies. We then study the properties of the solution as a
function of the mean number of players, and compare them with a large dataset
of internet auctions. The theory agrees with the data with striking accuracy
for small population size $N$, while for larger $N$ a qualitatively different
distribution is observed. We interpret this result as the emergence of two
different regimes, one in which adaptation is feasible and one in which it is
not. Our results question the actual possibility of a large population to adapt
and find the optimal strategy when participating in a collective game.Comment: 6 pag. - 7 figs - added Supplementary Material. Changed affiliations.
  Published versio

Bernhardsson, Sebastian

Galster, Gorm

Juul, Jeppe

Pigolotti, Simone

Vivo, Pierpaolo

Physical Review Letters

English

arXiv

In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid,
provided that it is also unique. We use a grand canonical approach to derive an analytical expression for
the equilibrium distribution of strategies. We then study the properties of the solution as a function of the
mean number of players, and compare them with a large data set of internet auctions. The theory agrees
with the data with striking accuracy for small population-size N, while for larger N a qualitatively
different distribution is observed.We interpret this result as the emergence of two different regimes, one in
which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large
population to adapt and find the optimal strategy when participating in a collective game.Postprint (published version

Juul, Jeep

UPCommons (Universitat Politècnica de Catalunya)

Equilibrium Strategy and Population-Size Effects in Lowest Unique Bid AuctionsSimone Pigolotti,1,2 Sebastian Bernhardsson,1,3 Jeppe Juul,1 Gorm Galster,1 and Pierpaolo Vivo41Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen, Denmark2Dept. de Fisica i Eng. Nuclear, Universitat Politecnica de Catalunya Edif. GAIA,Rambla Sant Nebridi s/n, 08222 Terrassa, Barcelona, Spain3Swedish Defence Research Agency, SE-147 25 Tumba, Sweden4Univ. Paris-Sud, CNRS, LPTMS, UMR8626, Orsay F-01405, France(Received 30 April 2011; published 22 February 2012)In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid,provided that it is also unique. We use a grand canonical approach to derive an analytical expression forthe equilibrium distribution of strategies. We then study the properties of the solution as a function of themean number of players, and compare them with a large data set of internet auctions. The theory agreeswith the data with striking accuracy for small population-size N, while for larger N a qualitativelydifferent distribution is observed. We interpret this result as the emergence of two different regimes, one inwhich adaptation is feasible and one in which it is not. Our results question the actual possibility of a largepopulation to adapt and find the optimal strategy when participating in a collective game.DOI: 10.1103/PhysRevLett.108.088701 PACS numbers: 02.50.Le, 89.65.Gh, 89.75.DaIn recent years, statistical physics has provided power-ful tools to study both equilibria [1,2] and dynamicalproperties [3–6] of games where the number of playersis large. So far, most of the efforts in this field have beenfocused on games in which interactions among players arepairwise, the most notable and studied example being theprisoner’s dilemma [3]. However, there are many ex-amples of collective games where a unique winner issingled out from a large population. In such cases, com-paring theory with empirical data is particularly challeng-ing: As the number of players increases, the statisticaldescription becomes more appropriate. However, the com-plexity of the equilibrium strategy may also increase, thusmaking it more difficult for real agents to infer it fromavailable information [7].Online auctions provide a fertile, yet scarcely investi-gated ground for exploring this problem [8–11]: The avail-ability of large data sets provides a unique opportunity tostudy whether strategies of real players actually maximizetheir winning chances. Here, we focus on lowest unique bidauctions. This game is interesting for two reasons. First, itis sufficiently simple to allow for a comprehensive mathe-matical analysis [12,13]. Second, many detailed auctionrecords are freely available online. The game is formulatedas follows: N players can bid for an item of value V. Thebid must be a multiple of a minimum amount, so that onecan effectively consider bid amounts as natural numbers.The winner (if any) is the player who places the lowest bid,which is unique, i.e., no other player bid on that amount.All players must pay a fee c to take part in the auction;additionally, the winner has to pay the bid amount.In this Letter, we derive an explicit analytic expressionfor the symmetric Nash equilibrium of the lowest uniquebid auction and explore its dependence on the total numberof players N. To achieve an explicit solution we assumethat N is not fixed, but fluctuates according to a Poissoniandistribution, in analogy with the choice of the grandcanonical ensemble in equilibrium statistical mechanics.We then compare the expression with a large data set [14]where players are informed in advance of the total numberof allowed bidsN. We find a remarkable change in the dataas N increases: In the low N regime (fewer than  200players), the theory predicts very well the bid distribution.In this regime, the game is effectively a lottery, sincewinning chances are evenly spread on any number.Conversely, in the large N regime, the data deviate fromthe theoretical solution and rather follow an exponentialdistribution. Here, players fail to adapt to the optimalstrategy and the winning chances are highly dependenton the chosen number.In a symmetric Nash equilibrium, all players adopt thesame strategy, and no player can benefit from changingstrategy unilaterally, i.e., should any player change strat-egy, his expected payoff would be equal or worse. ConsiderN individuals playing according to the same strategy p ¼ðp1; p2; p3; . . .Þ, where pk is the probability of bidding thenumber k. Then, the distribution of bids is multinomial:P ðnÞ ¼ N!n1!n2! . . .pn11 pn22 . . . ; (1)where n ¼ ðn1; n2; n3; . . .Þ are the number of bids placedon each number k. For convenience, we introduce thegenerating function:G NðxÞ ¼XfngP ðnÞxn11 xn22 ; . . . ;¼ ðx  pÞN: (2)PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 20120031-9007=12=108(8)=088701(5) 088701-1  2012 American Physical SocietyWe now assume that N is not fixed, but fluctuates accord-ing to a Poissonian distribution of mean , leading to agrand canonical generating function:~G ðxÞ ¼XNNeN!GNðxÞ ¼ exp½ðp  x 1Þ; (3)i.e., the number of bids on each number is also Poissondistributed with mean fk ¼ pk. By differentiating ~G withrespect to x, it is straightforward to compute the probabil-ities wk of k being the winning number, i.e., that there is aunique bid on k and no unique bids on lower numbers:wk ¼ fkefkYk1j¼1ð1 fjefjÞ: (4)It is also useful to compute the probability ck of k being apotential winning number, i.e., that there are no bids on kand no winners on lower numbers:ck ¼ efkYk1j¼1ð1 fjefjÞ: (5)We note that the probability of each bid to be a winner ona given number wk=fk is equal to the probability ck ofnumbers to be potential winning numbers, which is apeculiar property of Poisson games [15].The expected payoff for a player bidding k will beðV  kÞwk=fk  c. At equilibrium, the expected payoffshould be independent of k. Otherwise, players couldbenefit from bidding on numbers with high payoffs morefrequently. Imposing this condition leads to a recurrencerelation for the average bidding frequencies,fkþ1 ¼ lnðefk  fkÞ þ ln1 1V  k: (6)To avoid a negative number of bids, the support of thedistribution must be limited to the region where all fk’s arepositive. In this region k < V holds, so that the fk’s arestrictly decreasing. The initial condition f1 has tobe determined iteratively from the conditionPjfj ¼ .If the frequencies fk tend to zero for values of k muchsmaller than the value of the item V (a condition that isalways fulfilled in our data set, and will be consistentlyassumed in the following), the last term in (6) can bedisregarded. In this limit (formally corresponding toV ! 1) the recurrence relation has been derived in [13]and it is well defined for all values of k 2 N . We extendthis solution by noting that the normalization conditionPkfk ¼  implies an explicit expression of f1: Eq. (6) canbe written as fk ¼ efk  efkþ1, which summed over kyields the initial condition f1 ¼ lnðþ 1Þ. This allowsfor an explicit expression for any specific fk’s by iteration.Substituting the solution into (4) also shows that theaverage chance of winning of each bid is equal toðþ 1Þ1, which is also equal to the chance of having nowinner in the auction.Before discussing the comparison with the data, webriefly study some properties of the Nash equilibriumdistribution. When fk  1, a continuous approximation of(6) shows that fk decreases logarithmically with k. Forsmall fk  1, one can approximate fkþ1  f2k=2, showingthat the distribution has a superexponential cutoff. Thescaling of the value kðÞ, around which the cutoff occurs,can be estimated in the following way. By removing all theplayers that bid on k ¼ 1we change the average number ofplayers by the amount f1 giving new¼old lnðoldþ1Þ.This is equivalent to shifting the whole distribution by onealong the k axis, resulting in knew ¼ kold  1. This scalingtransformation in the continuum limit becomes d=dk ¼lnðþ 1Þ, with the solution kðÞ ¼ lið1þ Þ þ C, whereliðzÞ ¼ Rz0 dt= lnt is the logarithmic integral and C is aconstant of Oð1Þ.We now move to the comparison of the equilibriumsolution with empirical data from Ref. [14]. The data setincludes 724 online auctions from April 2007 to January2011 with a variable number of bids ranging from N ¼ 26to N ¼ 4748. The number of allowed bids in a particularauction is announced before bidding starts, and the auctioncloses when this number is reached. Each player is alloweda limited number of bids. The average number of bids perplayer in the full data set is only 2.47 and very weaklydependent on N. In the Supplemental Material [16], wedemonstrate with agent based simulations that allowing asmall number of multiple bids per player does not altersignificantly the equilibrium strategy. For this reason, inthe subsequent analysis, we will neglect the effect ofmultiple bids by the same player and treat the bids asstatistically independent.In Fig. 1 we compare the theoretical and empiricalbidding frequencies in different auctions having differentnumbers of players and different item values. In order tomake the histograms smoother, we averaged each panelover L different auctions having similar numbers of playersand same item value (shown in the figure).On the fine scale of single numbers, the data show astructure dictated by known psychological effects. Playerstend to favor odd numbers over even numbers. Somespecific numbers (like 17 and other primes) are perceivedto be ‘‘original’’ by some players, and are chosen withsignificantly larger probability than neighboring ones.On a coarser scale, the agreement between theory anddata is striking for smaller auctions [i.e., fewer than 200players, panels (A) and (B) of Fig. 1]. It is particularlyremarkable that the empirical histograms reproduce thesharp cutoff, considering the nontrivial dependence of kon the number of players.Theoretically, players should adjust their bidding strat-egies according to the number of players rather than thePRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-2item value, which can be assumed to be infinite. In allauctions in the data set, the cutoffs of the correspondingtheoretical distributions occur at bid values much smallerthan the item values (shown in panels). To test whether theempirical bidding distributions are independent of V, panel(E) compares two sets of auctions with the same number ofplayers, but with item values that differ by a factor of 3.The bidding distribution for the pricey item have a slightlyheavier tail, but overall the distributions are very similar.The agreement between theory and data progressivelydeteriorates as the number of players increases [see panels(C) and (D)]. For a large number of players (more than2000) an exponential distribution fits the data better, asshown in the lower panels of Fig. 1. This can be understoodby arguing (see, e.g., [17–19]) that players having incom-plete information about the game play according toa strategy defined as pk ¼ expðEkÞ=Pj expðEjÞwhere Ek is the expected payoff when placing a bid kand  quantifies the degree of uncertainty about thegame. If players have poor knowledge about the optimalstrategy, they could assume a uniform prior probability towin on each k, resulting in Ek decreasing linearly with kdue to the cost of the bid when winning. Substituting thisprior into the aforementioned strategy yields a bid distri-bution exponentially decreasing with bid size.A quantitative measure of how the data deviate fromtheory is shown in Fig. 2, where, for each auction, we plotthe l2 distance d between the theoretical probability distri-bution and the empirical one, d ¼ N2Pkðfk kÞ2,where k is the number of bids on k in a given auction.If bids were randomly drawn from the theoretical distribu-tion, the expected distance would decrease with the numberof players as hdi ¼ N1. Empirically, the distances have alarge spread around the expected value for small auctionsand are consistently larger than expected forN > 200. Thisoutcome cannot be simply explained by the larger numberof auctions for smaller N: As a consequence of players’turnover, the distributions at fixed N do not evolve withtime in a significant way, as we checked by comparingolder to more recent auctions.Another interesting quantity is the distribution of actualwinning numbers. At equilibrium, it should be equal to thedistribution of bids. As shown in Fig. 3, the empiricaldistribution of winning numbers supports this feature.The vast majority of winning numbers fall within thetheoretical cutoff (pink shaded area of Fig. 3). In thefigure, we also show the analytical estimate k ð1þ 1= lnNÞN= lnN based on the asymptotic expansionof the logarithmic integral. The black line is the averagewinning number, which in the relevant range, scales ap-proximately in the same way as the cutoff.Binning the data yields average winning numbers inexcellent agreement with the theory, even for large Nwhere the empirical distribution of bids departs from thetheory. However, the variance of winning numbers be-comes much smaller than the theoretical prediction forN > 103. To further explore this phenomenon, we computethe actual probabilities to win on a certain number, givenFIG. 2. l2 distance between the theoretical solution and theempirical histogram for each auction. The continuous line is thetheoretical expectation hdi ¼ N1 if all bids for each histogramwere randomly drawn for the theoretical distribution.(A) (B)(C) (D)(E) (F)FIG. 1 (color online). Histograms of bidding frequenciescompared with the theoretical equilibrium distribution. Panels(A)–(D) show average histograms of different auctions with asimilar number of players and same item value. Panel (E)compares histograms of auctions with a similar number ofplayers, but significantly different item values. Panel (F) issame as panel (D) but in linear-log scale, to illustrate theexponential shape of the empirical distribution at large N.PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-3the empirical distributions of bids for auctions of varioussizes. This probability is given bywk=fk which, since ck ¼wk=fk, is the same as the probability to win on a certainnumber for an additional player entering the game. Theresults are shown in Fig. 4 for the same examples consid-ered in Fig. 1. For auctions with few players, the largestchance of winning is not more than 4–5 times higher thanðN þ 1Þ1, the winning probability at the Nash equilib-rium. In this region, the game is not very different from alottery, as the winning chances do not depend much on thechosen number k  V. In contrast, for large auctions, thewinning chances on low and high numbers are very small,while intermediate bids may have a winning chance morethan 60 times higher than the average bid in the Nashequilibrium, providing opportunities for exploitation andthus potential room for adaptation.Summarizing, in this Letter we derived the analyticalequilibrium bidding distribution for the lowest unique bidauction and compared it with empirical data. The emergingpicture is that players are able to infer the optimal strategywith striking accuracy when the number of playersN is nottoo large. In the large N regime, the population-levelstrategy is highly nonoptimal and it seems to be determinedby the simple principle of not assuming a particular pref-erence for any number while avoiding the cost of largebids.This result raises nontrivial questions about the effec-tiveness of adaptive dynamics in collective games. While ithas been studied whether adaptation eventually drives thesystem to the optimal solution or generates more complexdynamical behaviors [20], it is also crucial to assess howfast the equilibrium is reached. A thorough study of adap-tive dynamics in such a system will be the subject of afuture study; in the Supplemental Material [16], we showthat indeed, in the simple case of replicator dynamics,adaptation becomes slower at larger N. Such slowdowncould prevent the inference of the equilibrium strategy forlarge populations in realistic time scales, and thus explainthe increasing lack of adaptation in our data as N grows.We acknowledge useful discussions with M. Marsili, K.Sneppen, C. Strandkvist, and P. Kempson.[1] D. Challet, M. Marsili, and R. Zecchina, Phys. Rev. Lett.84, 1824 (2000).[2] J. Berg and A. Engel, Phys. Rev. Lett. 81, 4999 (1998).[3] C. Hauert and G. Szabó, Am. J. Phys. 73, 405 (2005).[4] A. Traulsen, J. C. Claussen, and C. Hauert, Phys. Rev.Lett. 95, 238701 (2005).(A)(B)FIG. 3 (color online). Panel (A): Winning numbers as a func-tion of auction size. The continuous line is the theoreticalaverage and the shaded area denotes the region where bids arebelow the theoretical cutoff k. Panel (B): Average and standarddeviation of the winning numbers, compared with the respectivetheoretical lines.FIG. 4 (color online). Winning chances when bidding k. Blacklines indicate the same empirical data as panels (A)–(D) ofFig. 1. Gray (red) lines are the relative probabilities to winwhen betting on k given that all other players bid according tothe empirical distribution. To help the comparison, we haverenormalized the winning probabilities ck with their value forthe Nash equilibrium ðN þ 1Þ1. The shaded points mark theactual winning bids for each auction (arbitrary y position).PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-4[5] A. Melbinger, J. Cremer, and E. Frey, Phys. Rev. Lett. 105,178101 (2010).[6] N. Hanaki, A. Kirman, and M. Marsili, J. Econ. Behav.Organ. 77, 382 (2011).[7] S. K. Baek, P. Minnhagen, S. Bernhardsson, K. Choi, andB. J. Kim, Phys. Rev. E 80, 016111 (2009).[8] A. Namazi and A. Schadschneider, Int. J. Mod. Phys. C17, 1485 (2006).[9] I. Yang, H. Jeong, B. Kahng, and A.-L. Barabási, Phys.Rev. E 68, 016102 (2003).[10] J. Reichardt and S. Bornholdt, J. Stat. Mech. Theor. Exp.2007, P06016 (2007).[11] T. Galla, M. Leone, M. Marsili, M. Sellitto, M. Weigt, andR. Zecchina, Phys. Rev. Lett. 97, 128701 (2006).[12] S. K. Baek and S. Bernhardsson, Fluctuation and NoiseLetters 9, 61 (2010).[13] R. Östling, J. T.-y. Wang, E. Y. Chou, and C. F. Camerer,American Economic Journal: Microeconomics, 3, 1(2011).[14] Available at http://www.auctionair.co.uk.[15] R. B. Myerson, Int. J. Game Theory 27, 375 (1998).[16] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.108.088701 for addi-tional data analysis and simulation results.[17] R. D. McKelvey and T. R. Palfrey, Games Econ. Behav.10, 6 (1995).[18] H.-C. Chen, J.W. Friedman, and J.-F. Thisse, GamesEcon. Behav. 18, 32 (1997).[19] J. K. Goeree and C.A. Holt, Proc. Natl. Acad. Sci. U.S.A.96, 10564 (1999).[20] M.A. Nowak and K. Sigmund, Science 303, 793(2004).PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-5

Equilibrium strategy and population-size effects in lowest unique bid auctions

Bernhardsson, Per Johan Sebastian

Juul, Jeppe Søgaard

Copenhagen University Research Information System

Equilibrium Strategy and Population-Size Effects in Lowest Unique Bid Auctions

In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid,

provided that it is also unique. We use a grand canonical approach to derive an analytical expression for

the equilibrium distribution of strategies. We then study the properties of the solution as a function of the

mean number of players, and compare them with a large data set of internet auctions. The theory agrees

with the data with striking accuracy for small population-size N, while for larger N a qualitatively

different distribution is observed.We interpret this result as the emergence of two different regimes, one in

which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large

population to adapt and find the optimal strategy when participating in a collective game

UPCommons

6 pag. - 7 figs - added Supplementary Material. Changed affiliations. Published versionInternational audienceIn lowest unique bid auctions, $N$ players bid for an item. The winner is whoever places the \emph{lowest} bid, provided that it is also unique. We use a grand canonical approach to derive an analytical expression for the equilibrium distribution of strategies. We then study the properties of the solution as a function of the mean number of players, and compare them with a large dataset of internet auctions. The theory agrees with the data with striking accuracy for small population size $N$, while for larger $N$ a qualitatively different distribution is observed. We interpret this result as the emergence of two different regimes, one in which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game

HAL: Hyper Article en Ligne

Simone Pigolotti

Sebastian Bernhardsson

Jeppe Juul

Gorm Galster

Pierpaolo Vivo

Crossref

UPCommons. Portal del coneixement obert de la UPC

Equilibrium Strategy and Population-Size Effects in Lowest Unique Bid AuctionsSimone Pigolotti,1,2 Sebastian Bernhardsson,1,3 Jeppe Juul,1 Gorm Galster,1 and Pierpaolo Vivo41Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen, Denmark2Dept. de Fisica i Eng. Nuclear, Universitat Politecnica de Catalunya Edif. GAIA,Rambla Sant Nebridi s/n, 08222 Terrassa, Barcelona, Spain3Swedish Defence Research Agency, SE-147 25 Tumba, Sweden4Univ. Paris-Sud, CNRS, LPTMS, UMR8626, Orsay F-01405, France(Received 30 April 2011; published 22 February 2012)In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid,provided that it is also unique. We use a grand canonical approach to derive an analytical expression forthe equilibrium distribution of strategies. We then study the properties of the solution as a function of themean number of players, and compare them with a large data set of internet auctions. The theory agreeswith the data with striking accuracy for small population-size N, while for larger N a qualitativelydifferent distribution is observed. We interpret this result as the emergence of two different regimes, one inwhich adaptation is feasible and one in which it is not. Our results question the actual possibility of a largepopulation to adapt and find the optimal strategy when participating in a collective game.DOI: 10.1103/PhysRevLett.108.088701 PACS numbers: 02.50.Le, 89.65.Gh, 89.75.DaIn recent years, statistical physics has provided power-ful tools to study both equilibria [1,2] and dynamicalproperties [3–6] of games where the number of playersis large. So far, most of the efforts in this field have beenfocused on games in which interactions among players arepairwise, the most notable and studied example being theprisoner’s dilemma [3]. However, there are many ex-amples of collective games where a unique winner issingled out from a large population. In such cases, com-paring theory with empirical data is particularly challeng-ing: As the number of players increases, the statisticaldescription becomes more appropriate. However, the com-plexity of the equilibrium strategy may also increase, thusmaking it more difficult for real agents to infer it fromavailable information [7].Online auctions provide a fertile, yet scarcely investi-gated ground for exploring this problem [8–11]: The avail-ability of large data sets provides a unique opportunity tostudy whether strategies of real players actually maximizetheir winning chances. Here, we focus on lowest unique bidauctions. This game is interesting for two reasons. First, itis sufficiently simple to allow for a comprehensive mathe-matical analysis [12,13]. Second, many detailed auctionrecords are freely available online. The game is formulatedas follows: N players can bid for an item of value V. Thebid must be a multiple of a minimum amount, so that onecan effectively consider bid amounts as natural numbers.The winner (if any) is the player who places the lowest bid,which is unique, i.e., no other player bid on that amount.All players must pay a fee c to take part in the auction;additionally, the winner has to pay the bid amount.In this Letter, we derive an explicit analytic expressionfor the symmetric Nash equilibrium of the lowest uniquebid auction and explore its dependence on the total numberof players N. To achieve an explicit solution we assumethat N is not fixed, but fluctuates according to a Poissoniandistribution, in analogy with the choice of the grandcanonical ensemble in equilibrium statistical mechanics.We then compare the expression with a large data set [14]where players are informed in advance of the total numberof allowed bidsN. We find a remarkable change in the dataas N increases: In the low N regime (fewer than  200players), the theory predicts very well the bid distribution.In this regime, the game is effectively a lottery, sincewinning chances are evenly spread on any number.Conversely, in the large N regime, the data deviate fromthe theoretical solution and rather follow an exponentialdistribution. Here, players fail to adapt to the optimalstrategy and the winning chances are highly dependenton the chosen number.In a symmetric Nash equilibrium, all players adopt thesame strategy, and no player can benefit from changingstrategy unilaterally, i.e., should any player change strat-egy, his expected payoff would be equal or worse. ConsiderN individuals playing according to the same strategy p ¼ðp1; p2; p3; . . .Þ, where pk is the probability of bidding thenumber k. Then, the distribution of bids is multinomial:P ðnÞ ¼ N!n1!n2! . . .pn11 pn22 . . . ; (1)where n ¼ ðn1; n2; n3; . . .Þ are the number of bids placedon each number k. For convenience, we introduce thegenerating function:G NðxÞ ¼XfngP ðnÞxn11 xn22 ; . . . ;¼ ðx  pÞN: (2)PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 20120031-9007=12=108(8)=088701(5) 088701-1  2012 American Physical SocietyWe now assume that N is not fixed, but fluctuates accord-ing to a Poissonian distribution of mean , leading to agrand canonical generating function:~G ðxÞ ¼XNNeN!GNðxÞ ¼ exp½ðp  x 1Þ; (3)i.e., the number of bids on each number is also Poissondistributed with mean fk ¼ pk. By differentiating ~G withrespect to x, it is straightforward to compute the probabil-ities wk of k being the winning number, i.e., that there is aunique bid on k and no unique bids on lower numbers:wk ¼ fkefkYk1j¼1ð1 fjefjÞ: (4)It is also useful to compute the probability ck of k being apotential winning number, i.e., that there are no bids on kand no winners on lower numbers:ck ¼ efkYk1j¼1ð1 fjefjÞ: (5)We note that the probability of each bid to be a winner ona given number wk=fk is equal to the probability ck ofnumbers to be potential winning numbers, which is apeculiar property of Poisson games [15].The expected payoff for a player bidding k will beðV  kÞwk=fk  c. At equilibrium, the expected payoffshould be independent of k. Otherwise, players couldbenefit from bidding on numbers with high payoffs morefrequently. Imposing this condition leads to a recurrencerelation for the average bidding frequencies,fkþ1 ¼ lnðefk  fkÞ þ ln1 1V  k: (6)To avoid a negative number of bids, the support of thedistribution must be limited to the region where all fk’s arepositive. In this region k < V holds, so that the fk’s arestrictly decreasing. The initial condition f1 has tobe determined iteratively from the conditionPjfj ¼ .If the frequencies fk tend to zero for values of k muchsmaller than the value of the item V (a condition that isalways fulfilled in our data set, and will be consistentlyassumed in the following), the last term in (6) can bedisregarded. In this limit (formally corresponding toV ! 1) the recurrence relation has been derived in [13]and it is well defined for all values of k 2N . We extendthis solution by noting that the normalization conditionPkfk ¼  implies an explicit expression of f1: Eq. (6) canbe written as fk ¼ efk  efkþ1, which summed over kyields the initial condition f1 ¼ lnðþ 1Þ. This allowsfor an explicit expression for any specific fk’s by iteration.Substituting the solution into (4) also shows that theaverage chance of winning of each bid is equal toðþ 1Þ1, which is also equal to the chance of having nowinner in the auction.Before discussing the comparison with the data, webriefly study some properties of the Nash equilibriumdistribution. When fk  1, a continuous approximation of(6) shows that fk decreases logarithmically with k. Forsmall fk  1, one can approximate fkþ1  f2k=2, showingthat the distribution has a superexponential cutoff. Thescaling of the value kðÞ, around which the cutoff occurs,can be estimated in the following way. By removing all theplayers that bid on k ¼ 1we change the average number ofplayers by the amount f1 giving new¼old lnðoldþ1Þ.This is equivalent to shifting the whole distribution by onealong the k axis, resulting in knew ¼ kold  1. This scalingtransformation in the continuum limit becomes d=dk ¼lnðþ 1Þ, with the solution kðÞ ¼ lið1þ Þ þ C, whereliðzÞ ¼ Rz0 dt= lnt is the logarithmic integral and C is aconstant of Oð1Þ.We now move to the comparison of the equilibriumsolution with empirical data from Ref. [14]. The data setincludes 724 online auctions from April 2007 to January2011 with a variable number of bids ranging from N ¼ 26to N ¼ 4748. The number of allowed bids in a particularauction is announced before bidding starts, and the auctioncloses when this number is reached. Each player is alloweda limited number of bids. The average number of bids perplayer in the full data set is only 2.47 and very weaklydependent on N. In the Supplemental Material [16], wedemonstrate with agent based simulations that allowing asmall number of multiple bids per player does not altersignificantly the equilibrium strategy. For this reason, inthe subsequent analysis, we will neglect the effect ofmultiple bids by the same player and treat the bids asstatistically independent.In Fig. 1 we compare the theoretical and empiricalbidding frequencies in different auctions having differentnumbers of players and different item values. In order tomake the histograms smoother, we averaged each panelover L different auctions having similar numbers of playersand same item value (shown in the figure).On the fine scale of single numbers, the data show astructure dictated by known psychological effects. Playerstend to favor odd numbers over even numbers. Somespecific numbers (like 17 and other primes) are perceivedto be ‘‘original’’ by some players, and are chosen withsignificantly larger probability than neighboring ones.On a coarser scale, the agreement between theory anddata is striking for smaller auctions [i.e., fewer than 200players, panels (A) and (B) of Fig. 1]. It is particularlyremarkable that the empirical histograms reproduce thesharp cutoff, considering the nontrivial dependence of kon the number of players.Theoretically, players should adjust their bidding strat-egies according to the number of players rather than thePRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-2item value, which can be assumed to be infinite. In allauctions in the data set, the cutoffs of the correspondingtheoretical distributions occur at bid values much smallerthan the item values (shown in panels). To test whether theempirical bidding distributions are independent of V, panel(E) compares two sets of auctions with the same number ofplayers, but with item values that differ by a factor of 3.The bidding distribution for the pricey item have a slightlyheavier tail, but overall the distributions are very similar.The agreement between theory and data progressivelydeteriorates as the number of players increases [see panels(C) and (D)]. For a large number of players (more than2000) an exponential distribution fits the data better, asshown in the lower panels of Fig. 1. This can be understoodby arguing (see, e.g., [17–19]) that players having incom-plete information about the game play according toa strategy defined as pk ¼ expðEkÞ=Pj expðEjÞwhere Ek is the expected payoff when placing a bid kand  quantifies the degree of uncertainty about thegame. If players have poor knowledge about the optimalstrategy, they could assume a uniform prior probability towin on each k, resulting in Ek decreasing linearly with kdue to the cost of the bid when winning. Substituting thisprior into the aforementioned strategy yields a bid distri-bution exponentially decreasing with bid size.A quantitative measure of how the data deviate fromtheory is shown in Fig. 2, where, for each auction, we plotthe l2 distance d between the theoretical probability distri-bution and the empirical one, d ¼ N2Pkðfk kÞ2,where k is the number of bids on k in a given auction.If bids were randomly drawn from the theoretical distribu-tion, the expected distance would decrease with the numberof players as hdi ¼ N1. Empirically, the distances have alarge spread around the expected value for small auctionsand are consistently larger than expected forN > 200. Thisoutcome cannot be simply explained by the larger numberof auctions for smaller N: As a consequence of players’turnover, the distributions at fixed N do not evolve withtime in a significant way, as we checked by comparingolder to more recent auctions.Another interesting quantity is the distribution of actualwinning numbers. At equilibrium, it should be equal to thedistribution of bids. As shown in Fig. 3, the empiricaldistribution of winning numbers supports this feature.The vast majority of winning numbers fall within thetheoretical cutoff (pink shaded area of Fig. 3). In thefigure, we also show the analytical estimate k ð1þ 1= lnNÞN= lnN based on the asymptotic expansionof the logarithmic integral. The black line is the averagewinning number, which in the relevant range, scales ap-proximately in the same way as the cutoff.Binning the data yields average winning numbers inexcellent agreement with the theory, even for large Nwhere the empirical distribution of bids departs from thetheory. However, the variance of winning numbers be-comes much smaller than the theoretical prediction forN > 103. To further explore this phenomenon, we computethe actual probabilities to win on a certain number, givenFIG. 2. l2 distance between the theoretical solution and theempirical histogram for each auction. The continuous line is thetheoretical expectation hdi ¼ N1 if all bids for each histogramwere randomly drawn for the theoretical distribution.(A) (B)(C) (D)(E) (F)FIG. 1 (color online). Histograms of bidding frequenciescompared with the theoretical equilibrium distribution. Panels(A)–(D) show average histograms of different auctions with asimilar number of players and same item value. Panel (E)compares histograms of auctions with a similar number ofplayers, but significantly different item values. Panel (F) issame as panel (D) but in linear-log scale, to illustrate theexponential shape of the empirical distribution at large N.PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-3the empirical distributions of bids for auctions of varioussizes. This probability is given bywk=fk which, since ck ¼wk=fk, is the same as the probability to win on a certainnumber for an additional player entering the game. Theresults are shown in Fig. 4 for the same examples consid-ered in Fig. 1. For auctions with few players, the largestchance of winning is not more than 4–5 times higher thanðN þ 1Þ1, the winning probability at the Nash equilib-rium. In this region, the game is not very different from alottery, as the winning chances do not depend much on thechosen number k V. In contrast, for large auctions, thewinning chances on low and high numbers are very small,while intermediate bids may have a winning chance morethan 60 times higher than the average bid in the Nashequilibrium, providing opportunities for exploitation andthus potential room for adaptation.Summarizing, in this Letter we derived the analyticalequilibrium bidding distribution for the lowest unique bidauction and compared it with empirical data. The emergingpicture is that players are able to infer the optimal strategywith striking accuracy when the number of playersN is nottoo large. In the large N regime, the population-levelstrategy is highly nonoptimal and it seems to be determinedby the simple principle of not assuming a particular pref-erence for any number while avoiding the cost of largebids.This result raises nontrivial questions about the effec-tiveness of adaptive dynamics in collective games. While ithas been studied whether adaptation eventually drives thesystem to the optimal solution or generates more complexdynamical behaviors [20], it is also crucial to assess howfast the equilibrium is reached. A thorough study of adap-tive dynamics in such a system will be the subject of afuture study; in the Supplemental Material [16], we showthat indeed, in the simple case of replicator dynamics,adaptation becomes slower at larger N. Such slowdowncould prevent the inference of the equilibrium strategy forlarge populations in realistic time scales, and thus explainthe increasing lack of adaptation in our data as N grows.We acknowledge useful discussions with M. Marsili, K.Sneppen, C. Strandkvist, and P. Kempson.[1] D. Challet, M. Marsili, and R. Zecchina, Phys. Rev. Lett.84, 1824 (2000).[2] J. Berg and A. Engel, Phys. Rev. Lett. 81, 4999 (1998).[3] C. Hauert and G. Szabo´, Am. J. Phys. 73, 405 (2005).[4] A. Traulsen, J. C. Claussen, and C. Hauert, Phys. Rev.Lett. 95, 238701 (2005).(A)(B)FIG. 3 (color online). Panel (A): Winning numbers as a func-tion of auction size. The continuous line is the theoreticalaverage and the shaded area denotes the region where bids arebelow the theoretical cutoff k. Panel (B): Average and standarddeviation of the winning numbers, compared with the respectivetheoretical lines.FIG. 4 (color online). Winning chances when bidding k. Blacklines indicate the same empirical data as panels (A)–(D) ofFig. 1. Gray (red) lines are the relative probabilities to winwhen betting on k given that all other players bid according tothe empirical distribution. To help the comparison, we haverenormalized the winning probabilities ck with their value forthe Nash equilibrium ðN þ 1Þ1. The shaded points mark theactual winning bids for each auction (arbitrary y position).PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-4[5] A. Melbinger, J. Cremer, and E. Frey, Phys. Rev. Lett. 105,178101 (2010).[6] N. Hanaki, A. Kirman, and M. Marsili, J. Econ. Behav.Organ. 77, 382 (2011).[7] S. K. Baek, P. Minnhagen, S. Bernhardsson, K. Choi, andB. J. Kim, Phys. Rev. E 80, 016111 (2009).[8] A. Namazi and A. Schadschneider, Int. J. Mod. Phys. C17, 1485 (2006).[9] I. Yang, H. Jeong, B. Kahng, and A.-L. Baraba´si, Phys.Rev. E 68, 016102 (2003).[10] J. Reichardt and S. Bornholdt, J. Stat. Mech. Theor. Exp.2007, P06016 (2007).[11] T. Galla, M. Leone, M. Marsili, M. Sellitto, M. Weigt, andR. Zecchina, Phys. Rev. Lett. 97, 128701 (2006).[12] S. K. Baek and S. Bernhardsson, Fluctuation and NoiseLetters 9, 61 (2010).[13] R. O¨stling, J. T.-y. Wang, E. Y. Chou, and C. F. Camerer,American Economic Journal: Microeconomics, 3, 1(2011).[14] Available at http://www.auctionair.co.uk.[15] R. B. Myerson, Int. J. Game Theory 27, 375 (1998).[16] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.108.088701 for addi-tional data analysis and simulation results.[17] R. D. McKelvey and T. R. Palfrey, Games Econ. Behav.10, 6 (1995).[18] H.-C. Chen, J.W. Friedman, and J.-F. Thisse, GamesEcon. Behav. 18, 32 (1997).[19] J. K. Goeree and C.A. Holt, Proc. Natl. Acad. Sci. U.S.A.96, 10564 (1999).[20] M.A. Nowak and K. Sigmund, Science 303, 793(2004).PRL 108, 088701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending24 FEBRUARY 2012088701-5

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Equilibrium strategy and population-size effects in lowest unique bid auctions

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