Sequetial parimutuel games Rob Feeey ad Stephe P. Kig Departmet of Ecoomics The Uiversity of Melboure Parkville, Vic. 3052. Australia Abstract: I a parimutuel bettig system, a successful player s retur depeds o the umber of other players who choose the same actio. This paper examies a geeral solutio for two-actio sequetial parimutuel games, ad shows how the (uique) equilibrium of such games leads to simple patter of behavior. I particular, we show that there is a advatage to beig a early mover, that early players might choose actios with a ex ate low probability of success, ad that player actio choices ca flip with small chages i the parameters of the game. JEL umbers: C72, D84 Key Words: Parimutuel Bettig, Iterdepedecies, Ivestmet Clusters Correspodig author, Stephe P. Kig. We would like to thak Jeff Borlad for his useful commets. We would also like to thak the Faculty of Ecoomics ad Commerce at the Uiversity of Melboure for a grat to sposor this research.
1. Itroductio May ivestmets ivolve returs that are depedet o the actios of others. A simple case is a parimutuel gamblig system where bettors make a fiacial ivestmet o the outcome of a sportig evet. A example is the totalizer system at horse races where idividuals place bets o horses. If their chose horse wis, the the retur to a idividual is a fractioal proportio of the etire amout wagered o the race. The retur depeds o both the total bets of all gamblers ad the proportio who bet o the same horse. Parimutuel gamblig systems have log bee of iterest to ecoomists because they capture importat elemets of more geeral ivestmet decisios. They are aalogous to simplified fiacial markets where the scale of the pricig problem has bee reduced, with gamblers bettig o horses rather tha stocks ad comparig odds rather tha prices. Chadha ad Quadt (1996) argue that bettig markets ca provide a excellet test-bed for examiig market efficiecy. Parimutuel gamblig systems have also bee used to test for risk attitudes ad utility prefereces, for example Asch, Malkiel ad Quadt (1982) ad more recetly Hamid, Prakash ad Smyser (1996). Much of the theoretical work aalysig parimutuel systems, however, makes a critical small player assumptio that oe idividual caot ifluece the actios of others. I other words, there are always eough players i the system so that the effect of oe player s actios o the iformatio ad returs of other players ca be igored. For example, Potter ad Wit (1995) examie a parimutuel system where players idepedetly choose actios after receivig a idividual sigal that is draw from a commo distributio. But each player igores the cosequeces of their actio o the odds of the gamble. Similarly, Wataabe (1997) aalyses a parimutuel system with a cotiuum of players. 1 1 Plott, Wit, ad Yag (1997) cosider various models of parimutuel bettig markets, ad compare some experimetal results agaist models. They develop a model without the small player assumptio but are uable to solve for equilibrium i this model. Wataabe, ooyama, Mori (1994) cosider a fiite player game but where players have mutually icosistet beliefs. 1
The small player assumptio greatly simplifies the modelig of parimutuel systems, but it is very strog. For example, if there are few players or if some players wager relatively large sums of moey the the iterdepedecy of returs i a parimutuel system meas that the small player assumptio is likely to be violated. Further, as has bee observed i other cotexts, whe there is asymmetric iformatio, the actios of oe player ca result i sigificat iformatio trasmissio that affects the actios of other players eve whe returs are ot directly liked. The literature o herdig ad iformatio cascades aalyses this pheomeo (Baerjee (1992), Bikhchadai, Hirscheifer, ad Welch (1992, 1998), Welch (1992)). 2 While much of the work i related areas has focussed o iterdepedecy ad the trasmissio of iformatio, this paper focuses o iterdepedecy of player actios as govered by the parimutuel form of retur. We cosider a simple model of a parimutuel system where all players have idetical iformatio. This meas that each idividual s actio will affect the returs of all other players by (a) icreasig the size of the prize pool ad (b) raisig the expected retur associated with other actios. We cosider a sequece of players who must choose betwee two actios. Expected returs deped o the umber of players, the actios chose by all other players, ad a exogeous parameter such as the probability that oe actio is correct. Players must determie their optimal actio give their kowledge of the actios take by all precedig players ad their expectatio of the choices of all succeedig players. Our results highlight two importat features of sequetial decisio makig. First, players may cluster o choices, i that they will ofte make the same choice as the player who immediately precedes them. Such clusterig does ot reflect ay iformatio asymmetry or iferece process as i the literature o iformatio cascades. Rather, it reflects the iability of margial returs over actios to be perfectly equated i a fiite player game. Secodly, players who must make relatively early actio choices may choose a actio that ex ate appears to have a low expected payoff. Their decisio is drive by the ratioal belief that later players will cluster o the outcome that ex ate appears more 2 As the classic paper by Grossma ad Stiglitz (1980) poits out, eve if all players are price takers, i a ratioal expectatios equilibrium, aggregate actios by otherwise small agets ca covey 2
favorable. The iterdepedecy of returs meas that such clusterig ca result i a ex ate more favorable actio havig a lower expected retur ex post. To see a example of this secod pheomeo, cosider a situatio where three players sequetially choose the outcome of a football game betwee the Bears ad the Bulls. There is a total prize pool of $100, ad the ex ate probability of the Bears wiig is 60%. But returs are parimutuel, so that those players who correctly choose the wier of the game share the prize pool equally amog themselves. The i equilibrium, we would expect the first player to maximize his retur by choosig to back the Bulls while the secod ad third players support the favored Bears. Give the first player s choice, the secod ad third players idividually prefer to both support the Bears ad gai a expected retur of (at least) $30 rather tha to back the Bulls ad receive a expected retur of o more tha $20. The first player, who ca accurately ifer the behaviour of the other players, will back the Bulls ad receive a expected retur of $40. If, i cotrast, the first player had chose to support the favored Bears, the his expected retur would oly be $30 at best, as at least oe of the other players would also support the Bears. The first player maximizes his expected retur by isolatig himself through his choice of the ex ate worse alterative. Our results ca be applied to a variety of ivestmet situatios. For example, suppose that ivestors sequetially choose betwee two tows i which to locate similar retail stores. We should ot be surprised if early ivestors all choose the same tow ad later ivestors all choose the alterative locatio. Further, if oe tow has a larger populatio ad, as such, would appear ex ate to provide a preferred retail locatio, we should ot be surprised if early ivestors all choose to locate i the smaller tow. This behavior does ot reflect ay iformatio asymmetry or market-drive herdig. Rather, it simply reflects the parimutuel form of the payoffs associated with the ivestmets. iformatio. 3
2. The Model idividuals sequetially choose oe of two possible actios, a { A, B}. Each idividual receives a payoff that depeds o both the specific actio that they choose ad the umber of other players that have chose the idetical actio. Deote the umber of players choosig A as. The payoff to player i from choosig A is give by pm while the payoff from choosig B is ( 1 M where p ad M are exogeous parameters. All players ca observe the history of the game. We wish to examie the subgame perfect equilibria of this game. This game ca be iterpreted i (at least) two ways: Dividig a fixed moetary pool: each player kows ex ate the total payoff associated with ay actio. This is give by pm for actio A ad by (1-M for actio B. The payoff to a player i from actio a is equal to the total payoff associated with actio a divided by the umber of players who choose that actio. I other words, a idividual s retur from choosig A or B is pm ad ( 1 M respectively. A simple parimutuel game: There exists a true state of the world Ω where { α, β} where p [ 0,1] Ω. The probability that Ω= α is p ad the probability that Ω= β is 1 p. Idividuals are uaware of the true state ex ate but they do kow the value of p. Actio A is correct whe the true state is α, while actio B is correct whe the true state is β. Each idividual pays oe dollar to play the game. If a idividual i chooses actio A ad this is the correct actio the they receive a gross retur equal to M. If they choose B ad this is correct the they receive a gross retur of is pm ad M. So a idividual s expected retur from choosig A or B give ( 1 M respectively. 4
3. Geeral Results The followig two theorems apply to the model preseted above. The proofs are give i the appedix. Theorem 1 Give ad p, cosider such that + 1 p. 3 The: + 1 + 1 Theorem 2 a) If p ( 1 the there is a subgame perfect ash equilibrium where players 1,2,, choose A ad the remaider choose B. i.e. a equilibrium of the form AAA AB BBB. b) If p ( 1 the there is a subgame perfect ash equilibrium where players 1,2,,- choose B ad the remaider choose A. i.e. a equilibrium of the form BBB BA AAA. + 1 Give ad p, cosider such that < p < ad + 1 + 1 p ( 1. The the equilibrium defied i Theorem 1 is uique. These two theorems show that the equilibrium outcomes of the sequetial parimutuel ivestmet game will follow a simple patter. The players break ito two simple groups accordig to their order i the sequece ad their actio choice. Either the first players will all choose actio A with the remaider choosig actio B, or the first players will all choose B with the remaider choosig A. Further the subgame perfect equilibrium is geerally uique i the sese that give a fiite value of, multiple equilibria oly exist for a fiite umber of values of p. 3 ote that such a will always exist but eed ot be uique. 5
The equilibrium ivolves a early mover advatage. While the first group (either or - players) all receive the same payoff, this payoff is greater tha that received by the remaider of the players. But this advatage does ot mea that early movers choose the ex ate more likely outcome or the actio associated with the ex ate larger 1 pool of fuds. With p > 2, the first players may still all choose actio B i equilibrium ad receive a higher payoff tha those player who choose actio A. If we iterpret p as a probability, the the uique equilibrium ca ivolve early movers backig a extreme log-shot. For example, suppose that p = 0.99 ad = 101. The the two theorems imply that the uique equilibrium ivolves the first player choosig B ad the remaiig 100 choosig A. The theorems show how the equilibrium behavior of the players ca be very sesitive to the parameters of the game. The additio of a extra player, for example, ca cause the equilibrium to flip so that almost all players alter their choice. To see this, 11 suppose that p = 20. If there are iitially eight players the the uique equilibrium outcome is for the first four players to choose actio A ad the remaiig players to choose actio B. But if we add a extra player to the game, the the uique equilibrium ivolves the first four players choosig actio B while the remaiig players choose A. If the extra player moves last, the the additio of this player has led each of the origial players to chage their choice. Similarly, as p chages, the equilibrium actios will alter, followig a predictable patter. For example, as p falls from oe to oe-half, successively more players will choose actio B rather tha A. Further, the choice of early movers will flip betwee A 3 ad B as p falls. To see this, suppose that = 3. It is easy to cofirm that if p (,1] the the uique equilibrium outcome ivolves all three players choosig A. If p falls 2 3 so that p (, ) 3 4 the the uique outcome ivolves the first two players choosig A ad the last choosig B. But if p falls further, so that p ( 1, 2 ) will choose B ad the latter two players will choose A. 2 3 the the first player 4 6
Multiple equilibria oly exist for specific values of ad p. Whe p = ( 1 there are C equilibrium outcomes. I each, players choose actio A ad ( ) players choose actio B but all possible combiatios of orderigs ca 2 arise as equilibria. For example, with = 3 ad p = ay outcomes where oe player chooses actio B ad the other two players choose actio A ca occur i equilibrium. Whe p = ( 1 there are C +1 + C equilibrium +1 outcomes. I equilibrium either + 1 players choose A ad 1 players choose B or players choose A ad ( ) choose B. I either case, all possible combiatios of orderigs ca arise. 3 4. Coclusio I this paper we have aalyzed a sequetial parimutuel gamblig game ad have characterized the equilibria of this game. While our model is relatively simple, our results provide isight ito the behavior of other systems that ivolve iterdepedet ivestmet decisios. For example, our results ca explai why a early retailer of a ew product might prefer to locate i a relatively small tow rather tha close to a larger market, or why sequetial ivestmet decisios might appear to be characterized by clusterig subject to sudde switches i choice. Our results do ot deped o iformatio asymmetries or icosistet beliefs, but simply reflect the iterdepedet ature of ivestmet returs. I this sese, our results provide a simple explaatio for observed pheomeo, like clusterig. This said, the model could obviously be exteded to allow for iformatio asymmetry ad the aggregatio of iformatio as the game progresses. As oted i the literature o iformatio cascades, eve with idepedet returs, sequetial iformatio aggregatio ca ivolve sigificat imperfectios. This will be complicated by parimutuel returs as ay cascade dilutes the retur to early players. However, this remais the topic of future research. 7
Bibliography Asch, P., Malkiel, B., Quadt, R.(1982) "Racetrack Bettig ad Iformed Behavior", Joural of Fiacial Ecoomics, 10, 187-194 Baerjee, A. (1992) "A simple model of herd behavior", The Quarterly Joural of Ecoomics, 107, 797-817 Bikhchadai, S., Hirshleifer, D., Welch, I. (1992) "A Theory of Fads, Fashio, Custom, ad Cultural Chage as Iformatioal Cascades", The Joural of Political Ecoomy, 100, 992-1026 Bikhchadai, S., Hirshleifer, D., Welch, I. (1998) "Learig from the behavior of others: coformity, fads ad iformatioal cascades", Joural of Ecoomic Perspectives, 12, 151-170. Chadha, S., Quadt, R. (1996) "Bettig bias ad market equilibrium i racetrack bettig", Applied Fiacial Ecoomics, 6, 287 292. Hamid, S., Prakash, A., Smyser, M. (1996) "Margial risk aversio ad prefereces i a bettig market", Applied Ecoomics, 28, 371-376. Grossma, S., ad Stiglitz, J. (1980) "O the impossibility of iformatioally efficiet markets", America Ecoomic Review, 70, 393-408. Plott, C., Wit, J., Yag, W. (1997) "Parimutuel Bettig Markets as Iformatio Aggregatio Devices: Experimetal Results", Califoria Istitute of Techology Social Sciece Workig Paper 986, April. Potters, J., Wit, J. (1995) "Bets ad bids: favorite-logshot bias ad wier s curse", Katholieke Uiversiteit Brabat, etherlads: CetER, Discussio Paper, Jue. Wataabe, T. (1997) "A parimutuel system with two horses ad a cotiuum of bettors", Joural of Mathematical Ecoomics, 28, 85-100. 8
Wataabe, T., H. ooyama, M. Mori (1994) "A Model of a Geeral Parimutuel System: Characterizatios ad Equilibrium Selectio", Iteratioal Joural of Game Theory, 23, 237-260. Welch, I. (1992) "Sequetial sales, learig ad cascades", The Joural of Fiace, 47, 695-732. 9
Appedix. For the proofs of theorems 1 ad 2 it is coveiet to itroduce the followig otatio. Let P aη be the retur to a player choosig actio a whe η players i total choose actio a. 1. Proof of Theorem 1 Cosider the putative equilibrium where p ( 1 ad the first players choose A, ad the remaider choose B, i.e. a equilibrium of the form AAA AB BBB. (The proof of the symmetric case where p ( 1 is aalogous.) From our parameter restrictios it is easy to show that the followig iequalities hold: P A P B P A + 1 P B 1 P A 1 P B + 1 P A + x < P B x P A x > P B + x We ow show that o player will fid it desirable to uilaterally deviate from the putative equilibrium. First, cosider the players who choose A i the putative equilibrium. Suppose a subset of Z of these players deviate ad choose B. From the above iequalities, these players are oly strictly better off if at least Z + 1 players who would have chose B i the putative equilibrium ow choose A. ow, cosider the players who choose B i the putative equilibrium. Suppose a subset of Z of these players deviate ad choose A. From the above iequalities, these players are oly strictly better off if at least Z + 1 players who would have chose A i the putative equilibrium ow choose B. ote that this implies that ay deviatios must make some players worse off tha i equilibrium as it caot be the case that Z Z + 1 ad Z Z + 1 simultaeously. It remais to show that the first player who uilaterally deviates caot be made strictly better off. Let the first player deviate by choosig actio a rather tha actio a. The first player will oly strictly gai if all the players who choose actio a are strictly worse off. But this caot occur. To see this, suppose the coverse ad cosider the last player that chooses a. otig that P aη is decreasig i η ad from the above iequalities, this player will always gai by choosig a rather tha a regardless of the behavior of subsequet players. So this player will ot choose actio a ad we have a cotradictio. 10
As o player i the putative equilibrium ca ever strictly gai by deviatig, the putative equilibrium is a subgame perfect ash equilibrium. Q.E.D. 2. Proof of Theorem 2 The followig lemmas are useful i the proof of theorem 2. Lemma 1 + 1 Cosider, p, ad such that < p <. The, i ay subgame perfect + 1 + 1 ash equilibrium, players will choose A ad - players will choose B. Proof: From theorem 1 we kow that there exists a equilibrium where players choose A ad players choose B. Cosider ay other putative equilibrium. Suppose that i this putative equilibrium + x players choose A, ad x players choose B. Cosider the last player that chose A ad call this player i. If i deviates ad chooses B, the i is strictly better off regardless of the actios of ay subsequet players i+ 1,, sice P B x+ 1 > P A + x. Therefore i will deviate ad the putative equilibrium where more tha players choose A caot be a actual equilibrium. Similarly if -x players choose A. Therefore, i equilibrium, players choose A ad - players choose B. Q.E.D. Lemma 2. Cosider ay subgame of the whole game, begiig with the i th player ad ay subgame perfect ash equilibria of this subgame give the choices of players 1,, i 1. This equilibrium oly depeds o the umber of players before i who choose A ad B, ot their specific order. Proof: This trivially follows as the payoffs for each player oly deped o the umber of other players associated with each actio ad ot their order. Q.E.D. Proof of Theorem 2. From lemma 1 we ca restrict attetio to equilibria where players choose actio A ad the remaider choose actio B. We begi with the case where p > ( 1. From theorem 1 we kow that the relevat equilibrium exists. To show that it is uique, cosider ay other putative equilibrium. I this putative equilibrium, there must exist two players, i ad i+1, where i plays B ad i+1 plays A. 11
Cosider the last such pair of players. We show that such a player i will always fid it optimal to deviate so that the putative equilibrium is ot a actual equilibrium. To see this, if i deviates ad chooses A, the player i+1 chooses either B or A. Suppose i+1 chooses B, the by lemma 2 o player i+ 2,, will fid it desirable to deviate. But the player i receives p rather tha ( 1 ad is better off ad will therefore deviate. Alteratively, suppose player i+1 chooses A after player i deviates. Player i+1 will oly fid this optimal if p > (1, where x is the umber of players x + 1 i+ 2,, that deviate from B to A i the subgame perfect ash equilibrium after i+1 chooses A. This is because player i+1 could choose B ad get ( 1. But player i also gets p > (1 ad will therefore deviate. x + 1 As player i always deviates, there is o equilibrium with player i that plays B where i+1 plays A. The aalogous proof holds for the symmetric case where p > ( 1, ad a ( 1 p ) > p. Therefore, the equilibrium characterized i theorem 1 is uique. Q.E.D. 12