Match-fixing under competitive odds

Transcription

1 Match-fixing under competitive odds Parimal Kanti Bag a,, Bibhas Saha b a Department of Economics, National Universit of Singapore, Facult of Arts and Social Sciences, AS Level 6, Arts Link, Singapore 7570; [email protected] b School of Economics, Universit of East Anglia, Norwich, NR4 7TJ, U.K.; [email protected] Abstract Two bookmakers compete in Bertrand fashion while setting odds on the outcomes of a sporting contest where an influential punter (or betting sndicate) ma bribe some plaer(s) to fix the contest. Zero profit and bribe prevention ma not alwas hold together. When the influential punter is quite powerful, the bookies ma coordinate on prices and earn positive profits for fear of letting the lemons (i.e., the influential punter) in. On the other hand, sometimes the bookies make zero profits but also admit match-fixing. When matchfixing occurs, it often involves briber of onl the strong team. The theoretical analsis is intended to address the problem of growing incidence of betting related corruption in world sports including cricket, horse races, tennis, soccer, basketball, wrestling, snooker, etc. JEL Classification: D4, K4. Ke Words: Sports betting, bookie, punters, corruption, match-fixing, lemons problem.. Introduction Match-fixing and gambling related corruption often grab news headlines. Almost an sport horse races, tennis, soccer, cricket, to name a few is susceptible to negative external influences. Someone involved in betting on a specific sporting event ma have access to Corresponding author; tel.: , fax: See Race-fixing probed in Fallon trial and similar reports at com/news/articles/sports/racefixing+probed+in+fallon+trial+/ See also a BBC panorama on this subject ( For tennis, see reports such as Tennis chiefs battle match-fixers and ITF working with ATP, WTA and Grand Slam Committee to halt match-fixing in tennis ( ; In March 009, Uefa president Michel Platini publicl issued the following warning: There is a grave danger in the world of football and that is match-fixing. Uefa general secretar sas, We are setting up this betting fraud detection sstem across Europe to include 7,000 matches in the first and second division in each national association. See Latest, police carried out 50 raids in German, the UK, Switzerland and Austria: Prosecutors believe a 00-strong criminal gang has bribed plaers, coaches, referees and officials to fix games and then made mone b betting Preprint submitted to Elsevier March 5, 0

2 plaer(s) and induce underperformance through briber. In high visibilit sports man unsuspecting punters, bookmakers and the general viewing public ma therefore be defrauded in the process. When there is a real threat of match-fixing, how do bookmakers respond when their onl instruments are the betting odds the set on the competing teams wins? Would the set odds so that a match-fixer is discouraged from fixing the match? Ordinaril bookmakers can be expected to price aggressivel, but with potential match-fixing in the background such aggressive price competition can be trick: it makes one of the teams vulnerable to significant betting b the match-fixer. Fearing this the bookmakers ma abstain from price cutting, or competitive forces could still prevail lowering odds to trigger match-fixing. Which outcome is more likel to happen and when? Also, how damaging is match-fixing to the bookmakers in terms of profits? Competition usuall means less profits. Is it an different with matchfixing? In this paper, we develop a model of briber and corruption in sports to address the above questions. We adapt the horse race betting models due to Shin (99; 99) to analze match-fixing in team or individual sporting contests. In Shin (99) a monopolist bookmaker sets odds on each one of two horses winning a race, whereas in Shin (99) two bookmakers simultaneousl set odds, as in Bertrand competition, in an n-horse race game (n > ). 3 In both models, there is an insider who knows precisel which horse would win the race, while the remaining are noise punters with their different exogenous beliefs about the horses winning probabilities that are uncorrelated with the true probabilities. The bookmaker(s) know onl the true winning probabilities. 4 Rather than assuming an insider who knows before betting the identit of the winner (as in Shin s models), we consider the prospect of a gambler influencing the contestants winning on the results. See the report (dated 0 November, 009), For match-fixing in cricket, see For basketball, Wolfers (006) estimated that nearl percent of all games in NCAA Division one basketball (about 500 games between 989 and 005) involved gambling related corruption. A striking account of match-rigging in Sumo wrestling in Japan appears in Duggan and Levitt (00). And at the time of revising this draft, snooker got tainted with the revelation of betting related briber (see news/ stm). Corruption in sports has been onl occasionall highlighted b economists without the consideration of its causal relationship to betting see Duggan and Levitt (00), and Preston and Szmanski (003). Strumpf (003), Winter and Kukuk (008), and Wolfers (006) are few exceptions. 3 To be precise, Shin s (99) price-setting game is slightl different from one-shot Bertrand game: the bookmakers first submit bids specifing a maximum combined price for bets on all the horses, the low bidder wins and then sets prices for individual bets so that the total for all bets combined does not exceed the winning bid. 4 In an empirical framework, Shin (993) provides estimates for the incidence of insider trading in UK betting markets.

3 odds through briber and match-fixing. Ex ante (before bribing), this gambler, to be called the influential punter, is no better informed than the bookmakers and is less privileged than Shin s insider. However, different from Shin s framework, the influential punter ma become better informed than the bookmakers through his secret dealings with one of the contestants, if chance presents it and bookmakers odds make it worthwhile. Thus, we shift the focus from the use of insider information (i.e. pure adverse selection) to manipulative action that generates inside information for the influential gambler. As actions are choices, our bookmakers can control these b appropriatel setting their odds a possibilit absent in models of pure adverse selection such as Shin s. Moreover, there is an issue of legalit. While betting on the basis of inside information ma not be illegal, match-fixing through briber clearl is. However, the dominant focus of the theoretical literature on betting so far (including Shin s works) has been to explain the empirical regularit of the favorite-longshot bias in race-track betting. 5 We raise concern about the unfairness of contests due to match-fixing not onl for the sake of unsuspecting bettors but also for the viewing public and the media that rel on the public s interest in sports. 6 Our model involves two bookmakers (or bookies) and two tpes of punters (ordinar/naive and influential). The bookies set fixed odds and the punters place bets on the outcome of a sporting contest between two teams (or contestants). 7 The influential punter (or equivalentl a large betting sndicate), who shares the same beliefs as the bookies (and the plaers) about the teams winning chances, ma be able to gain access to some members in one of the teams and bribe them to sabotage the team. 8 When bribing a team, the influential punter would place a bet on the other team. The anti-corruption authorit ma investigate the losing team and punish the match-fixing punter and the corrupt plaer(s) whenever it catches them. 9 5 Ottaviani and Sorensen (008) is a detailed surve of alternative explanations. 6 In a parimutuel market setting, Winter and Kukuk (008) allow a participant jocke to underperform, but the do not consider enforcement: the cheating joke does not face the prospect of being found out for the deliberate underperformance. Winter-Kukuk model is not thus adequatel rich to analze cheating incentives formall in the stle of standard cheating/punishment models. 7 Fixed-odds betting, as opposed to parimutuel betting, is a more relevant format for analsis of matchfixing where bookies pla a significant role without direct involvement in the act of briber and/or placement of surrogate bets. For a contrast between how odds are set (or determined) in these two betting markets (but without the issues of match-fixing), see Ottaviani and Sorensen (005). 8 In contests involving rival firms or lobbies, sabotage is a well-studied theme; see, for instance, Konrad (000). Our sports contest model is much simpler than the effort contest games (such as the one analzed b Konrad) in that we assume exogenous winning probabilities of the contestants due to their inherent skills (or characteristics), and sabotage is a deliberate underperformance relative to one s own skills. 9 The law enforcement is one of investigation rather than monitoring (Mookherjee and Png, 99). 3

4 With the threat of match-fixing looming, in selecting odds the bookies take into account both the benefit and the danger of undercutting each other. When the influential punter cannot place too large a bet, the following results occur. If, ex ante, teams are relativel more even, competition ields zero expected profits for the bookies without attracting the risk of briber and match-fixing: prices of both the tickets corresponding to fair odds tend to be too high for the influential punter to bribe and bet. 0 But if (ex ante) teams are more uneven, Bertrand competition cannot guarantee elimination of briber; the bookies make zero profits and the influential punter earns rent. Match-fixing will occur with positive probabilit, and it can be attributed to opportunism. If undercutting triggers match-fixing, its adverse impact (loss) is shared b both bookies, but if match-fixing is not triggered then the gain is exclusive (positive profit). Whenever match-fixing occurs, it involves briber of onl the strong team. On the other hand, when the influential punter can place a significantl large bet, the adverse impact of match-fixing could be so severe that undercutting becomes ver risk. In particular, in contests that are ex ante nearl even, the bookies will coordinate on prices strictl above fair odds and sustain, non-cooperativel, positive profits and prevent briber. Positive profits seem to go against common wisdoms of competition. Here, the fear of triggering (the lemons of) match-fixing forces the bookies to coordinate on prices. Ironicall, without the corrupting influential punter the bookies would compete awa profits. There is also another possibilit that the bookies set prices inducing briber of either team and make zero expected profits. With this latter equilibrium, the chance of match-fixing remains rather high (as the influential punter will bribe whenever he has an access to a team) and ticket prices are set above the respective teams uncorrupted winning odds to make up for the potential loss to the match-fixing punter. It is difficult to cleanl predict, though, which of the two equilibria positive profit bribe prevention or zero profit match-fixing is likel to happen. A natural question to ask is what happens if there is onl one bookmaker. In those instances where competition increases the risk of match-fixing, avoidance of competition should reduce this risk. However, in some of these situations the monopolist ma have a perverse incentive to encourage match-fixing and thus gain from the defrauded naive punters. Analsis of the monopol case involves different complexities and we comment on some likel results towards the end of section 4. Before we proceed to detailed analsis, we would like to note that in practice bookmakers are well aware of the potential risks of the influential punter s involvement and as a precaution 0 In Shin (99; 99), prices exaggerate the odds. 4

5 the ma limit the size of trades at posted prices. Even more, the bookmakers ma set new odds seeing the increasing volume of bets being placed on a particular outcome so that the influential punter ma face a quantit-price trade-off. Further, odds revisions ma generate and disseminate new information even among the ordinar punters leading to an erosion of the value of insider information, similar to the market micro structure literature (Glosten and Milgrom, 985; Kle, 985). While our model does not incorporate these features emploed in models of financial economics, we do not see the basic insights of our analsis changing qualitativel even if a more sophisticated and much more complex model were formulated. We also assume exogenous investigation probabilities and fines b the prosecution authorities, to keep the analsis tractable. Nor do we model the role of sports bodies that ma regulate the betting market in large to prevent cheating. These considerations are important no doubt, but beond the scope of the present work. In section we present the model, followed b an analsis of the betting and bribing decisions in section 3. In section 4 we analze the Bertrand duopol competition. Section 5 concludes. The formal proofs appear in the Appendix.. The Model There are two bookmakers, called the bookies, who set the odds on each of two teams winning a competitive sports match (equivalentl, set the prices of two tickets); the match being drawn is not a possibilit. Ticket i with price π i ields a dollar whenever team i wins the contest and ields nothing if team i loses, with 0 π, π. To keep the notations simple, bookie indices will be omitted from the prices. There are a continuum of naive punters, to be described as punters or sometimes ordinar punters, parameterized b individual belief (i.e., the probabilit) q that team will win ( q is the probabilit that team will win); q is distributed uniforml over (0, ). Ordinar punters stubbornl stick to their beliefs. There is also a knowledgeable and potentiall corrupt/influential punter, to be referred as punter I, who ma influence a team s winning chances b bribing its corruptible plaers to underperform. Punter I gains access to team i with probabilit 0 µ i ; with probabilit µ µ, he fails to gain an access. At best, punter I can access onl one team. The bookies and the prosecution authorit know onl (µ, µ ). The distribution of ordinar punters wealth is uniform over [0,, with a collective wealth of dollars; the wealth of punter I is z = dollars. This ma be difficult to implement, however, as an corrupt betting sndicate ma have multiple punters on its team. 5

6 In the absence of an external influence, the probabilit that team will win is 0 < p < and the corresponding probabilit for team is p = p. The bookies, punter I, and the plaers all initiall observe the draw p. The prosecution need not observe p. Even when the prosecution observes p, it does not emplo sophisticated game-theoretic reasoning to infer if match-fixing has occurred given the betting odds and p. The prosecution authorit (or ACU, anti-corruption unit) investigates team i onl when team i loses the contest. Assume that the probabilit of investigation of team i, 0 < α i <, is known to all, and the investigation detects briber, if an, with probabilit one. investigation probabilit ma differ across teams. 3 Given our focus on the bookies pricing strategies, we take the prosecution to be a non-strategic rule-book follower. On conviction, the corrupt plaer (or plaers) will be imposed a total fine 0 < f f and 4 punter I is fined 0 < f I f. When punter I gets access to team i, b making a bribe promise of b i conditional on team i losing he can lower the probabilit of team i winning from the true probabilit p i to λ i p i, where 0 λ i <, provided the corrupt plaers of team i cooperate with punter I in undermining the team performance. λ i depends on the susceptibilit to corruption and briber of team i s members, i.e., whether a small or a significant section of the team takes part in undermining the team cause. Also, the particular plaer (or plaers) to whom punter I is likel to have an access ma be of varied importance to the team s overall performance. We take λ i to be exogenous and common knowledge. The bookies, two tpes of punters and the corruptible team members all are assumed to be risk-neutral and maximize their respective expected profits/paoffs. Define the betting and briber game, Γ, as follows: The Levitt (004) recognizes that bookmakers are usuall more skilled at predicting match outcomes than ordinar punters. In an case, without such confidence in abilities the bookies won t be in the business. 3 This difference could be due to the teams different susceptibilit to corruption. 4 The finding of briber is assumed to reveal the identit of punter I. Such an assumption ma not be unrealistic, as the enforcement authorit is unlikel to let go the trail of briber that the come to unearth. Alternativel, the bookies can alert enforcement authorities to unusuall high bets placed b particular punters who then ma be investigated to see an potential link to the plaers. There are instances of bookmaking firms voiding bets in tennis suspecting wrongdoing; see several reports in The Independent, a UK newspaper, including Wimbledon on high alert over suspected match-fixing rings (8 June, 009). This particular report mentions three Betfair customers placing bets of the order of $540, 94, $368, 036 and $53, 833 on a 007 tennis match between Nikola Davidenko (No. 5 in world ranking) and Martin Vassallo Arguello (ranked 87th) in favor of Arguello, who then went on to win the match; see also for a similar instance of voiding of bets on another tennis match b the bookmaking firm, William Hill. For instances in snooker and football, refer 6

7 Stage. Nature draws p and reveals it to the bookies, punter I and the plaers; the ordinar punters draw their respective private signals q. Then the bookies simultaneousl announce the prices (π, π ). Stage. Punter I learns about his access to team or team or neither, 5 subsequentl decides whether to bribe the team or not (in the event of gaining access). Stage 3. The ordinar punters as well as punter I place bets according to their eventual beliefs. When the bookies charge the same price for a given ticket, market is evenl shared, and when the charge unequal prices, lower price captures the whole market. The match is plaed out according to winning probabilities (p, p ) or (λ p, λ p ) (where team is bribed), or ( λ p, λ p ) (where team is bribed) and the outcome of the match is determined. Stage 4. Finall, the ACU follows its investigation polic, (α, α ). On successful investigation, fines are imposed on the corrupt plaer(s) and punter I. See also Fig.. and p drawn prices I s access briber? match plaed ACU exams. posted to teams bets placed bets settled penalties appl Figure : Time line Thus the bookies move simultaneousl in stage, then punter I decides on briber in stage followed b betting in stage 3 and finall the prosecution moves, defining the extensive form. Simultaneous moves (in stage ) and punter I s betting based on privatel held beliefs about the teams winning odds make the game an imperfect information game between the two bookies and punter I. So we will solve for the subgame perfect equilibrium (SPE) strategies in prices, briber and betting. 3. Betting and Bribing Decisions 3. Ordinar Punters Betting Decision The ordinar punters adopt the following betting rule: 6 5 The timing of the influential punter s access to teams (after or before the odds are posted) is not going to matter. What is important is that the bookies do not know whether, or to which team, the influential punter will have an access. 6 This is same as the betting rule b the Outsiders in Shin (99). 7

8 If q π but q < π, bet on team ; If q π but q < π, bet on team ; If q π and q π, then bet on team if If q < π and q < π, do not bet on either team. q π q π and bet on team if q π q π ; 3. Plaer Incentives for Bribe-taking and Sabotage Given the prosecution s investigation strateg, let us consider the incentives of plaers to accept bribes. In a team context, the incentives concern the corruptible member(s) of a team. We assume a single corruptible member in each team; the analsis applies equall to a consortium of corruptible members. Suppose the corruptible plaer of team i (with whom punter I establishes contact) gets the reward w in the event team i wins, and receives nothing if team i loses. 7 Given an belief p i, a bribe b i is accepted and honored b the corruptible plaer b underperforming if and onl if (λ i p i )w + ( λ i p i )(b i α i f) p i w + ( p i )(b i α i f) i.e., if and onl if b i w + α i f. () A plaer can renege on his promise to underperform even after entering into an agreement with punter I. The right-hand side of () recognizes this possibilit. A plaer will be penalized for taking bribes, even if he might not have deliberatel underperformed. 8 The minimum bribe required to induce the corruptible plaer to accept the bait is b i = w + α i f. That is, the reservation bribe covers the loss of the prize w and the expected penalt. We assume that punter I holds all the bargaining power so that b i = b i Influential Punter s Betting and Bribing Incentives First consider the betting incentives. Having learnt the true probabilities p i and observed the prices π i (i =, ), if punter I fails to contact either team or decides not to bribe, he will bet z on team i if and will bet on neither if p i max{, p j }, i j, 0 π i π j p i <, i =,. π i 7 The prize w includes both direct and indirect rewards, with the latter in the form of lucrative endorsement opportunities for commercials. The plaer ma additionall receive unconditional retainer wage/appearance fee that does not affect the plaer s bribe-taking incentives. In the case of a two-plaer contest such as tennis, the index i will refer to the plaer, with w i reflecting the particular plaer s reputation/stake. 8 To prove that a plaer has deliberatel underperformed is ver difficult. On the other hand, briber can be established based on hard evidence. 9 Our analsis can be easil extended to bargaining over bribe. 8

9 The expected profit to punter I from betting exclusivel on team i is EΠ I z 0i = p i π i z, i =,, and zero when he bets on neither. His expected profit from not bribing is max{eπ I 0i, EΠ I 0j, 0}. If, however, punter I contacts a corruptible member of team i, offers him a bribe and places a bet on team j, his expected profit equals: EΠ I (b i ) = ( λ i p i ) [ z π j b i α i f I z. Substituting b i = w + α i f, EΠ I (b i ) = ( λ i p i )z [ π j Ω i z = ( λi p i )z [ π j φ i, where Ω i = w + α i(f + f I ), and φ i = z λ i p i + ( λ i p i )Ω i. Clearl, if EΠ I (b i ) > 0 and greater than the profit from not bribing, he will bribe (upon access). But there are several situations of indifference, for which we impose two tie-breaking rules: Assumption. (Tie-breaking rule I) If bribing and betting and betting without bribing ield identical and positive expected profits for the influential punter, then he will choose bribing and betting. (Tie-breaking rule II) If bribing and betting and betting without bribing ield zero expected profits for the influential punter, then he will not bet at all. The first rule would bring to bear the full impact of the (negative) influence. An adverse consequence of bribing for punter I, such as getting caught leading to jail and banning from sports betting etc., is captured b the penalt term f I. So leaning towards betting and bribing to break the indifference should be reasonable. Tie-breaker-II is to ensure that the bribe prevention prices are well-defined. To analze various plaers decisions we impose the following assumption, which, along with Assumption, will be maintained throughout the paper. The assumption is based on sound economic principles. Assumption. (Dutch-book restriction) The bookies must alwas choose prices 0 π, π, both on- and off-the-equilibrium path, such that π + π. The Dutch-book restriction (or rather the absence of the Dutch book) can be defended as follows. If instead π + π <, it gives rise to the mone pump scenario impling someone 0 When pi π i = pj π j, the punter is indifferent between two teams. Betting on both teams ields the same profit as exclusive betting, given the betting rule specified above. 9

10 who otherwise might not have bet on the sporting event (for reasons of risk aversion and the likes) can make free mone b spending less than a dollar to earn a dollar for sure. It is also conceivable that if one of the bookies violates the Dutch-book restriction, the other bookie can bet large sums of mone and drive his competitor out of business. It is reasonable to assume that the bookies will have reserves of funds to engage in this predator behavior. We now consider three scenarios relevant for punter I s briber decisions. Bribe prevention: Suppose π i p i, i =, (such that max{eπ I 0i, EΠ I 0j, 0} = 0). Then punter I does not bribe team i, if and onl if EΠ I (b i ) 0, i.e., π j Tie-breaker-II applies when EΠ I (b i ) = 0. λ i p i + ( λ i p i )Ω i φ i. () Bet reversal: Alternativel, suppose π i < p i (and π j > p j, b Assumption ) such that max{eπ I 0i, EΠ I 0j, 0} = z( p i π i ) > 0. Then punter I bribes team i and bets on team j (as opposed to betting on team i), if and onl if EΠ I (b i ) z( p i π i ), i.e., π j Tie-breaker-I applies when EΠ I (b i ) = z( p i π i ). ( λ i p i )π i p i + ( λ i p i )Ω i π i ψ i (π i ). (3) Bet accentuation: Continuing with the assumption that π i < p i (and π j > p j ) such that max{eπ I 0i, EΠ I 0j, 0} = z( p i π i ) > 0, punter I bribes team j and bets on team i if and onl if EΠ I (b j ) z( p i π i ), i.e., π i ( λ j)p j h j. (4) ( λ j p j )Ω j Tie-breaker-II applies when EΠ I (b j ) = z( p i π i ). Interpretations: Condition (), the bribe prevention condition, sas that b setting the price of ticket j high enough, team i can be protected from match-fixing, and b doing so for both tickets punter I can be altogether kept out of the market. Indeed, that will be the outcome in an equilibrium featuring bribe prevention. If punter I does not bribe but bets on team i, he must earn strictl positive profit. This is possible if and onl if π i < p i, which is clearl loss-making for the bookies. Thus, if briber is prevented, punter I will not participate at all. Condition (3) is the condition for bribe inducement of team i, when team i is otherwise attractive to bet on. Essentiall b reducing the price of ticket j below a threshold level, the 0

11 betting incentive of punter I is reversed (hence the term, bet reversal). The threshold level will evidentl depend on the price of ticket i. In particular, ψ i < φ i for π i < p i ; if π i = p i then φ i = ψ i. How the threshold price for bet reversal compares with the bribe prevention price threshold can be seen in Fig.. Here, given π < p and π > p, team is bribed for π < ψ (π ). p p p h f (p ) p p ~ p / Figure : Bet reversal Finall, (4) is the condition for bribe inducement of team j, when prima facie team i is attractive to bet on. Here b reducing the price ticket of i below a threshold level (so that briber can b financed from the potential gains), punter I s incentive to bet on team i is further strengthened b prompting him to bribe team j (bet accentuation). An implication of the tie-breaking rule II, which is also evident in the briber conditions (3) and (4), is that punter I will bet onl if his expected profit is strictl positive. This also means: Fact. When punter I places a bet, the bookies expected profits from an potential trade with punter I must be negative. 4. Bertrand Competition in Bookmaking In this section, our principal observations will be on two important issues concerning the effects of competition. First, a basic fact of (Bertrand) competition is that firms earn zero

12 profits. But here the profitabilit of trades b bookmakers depends on the tpe of trades. The prices ma endogenousl lead to match-fixing and informed trading b select punters. So whether price competition will lead to zero profits or not cannot be answered independentl of the related match-fixing/corruption implications: does competition in the betting market impl a corruption-free pla of the sports contest? While under certain conditions competition ensures zero profits (to the bookies) and prevention of briber and match-fixing (Proposition ), either of these two results ma fail to obtain in isolation (Propositions 3 and 4) under complementar conditions, that is, briber/match-fixing ma be triggered with positive probabilit or firms ma make positive expected profits. Moreover, for the scenarios that we stud positive expected profits and briber/match-fixing do not occur at the same time. In the remainder of this section, we analze these possibilities. Before we sa what might happen in equilibrium, we can sa the following. Lemma. There cannot be an equilibrium (featuring bribe inducement or bribe prevention) in which π + π <, where (π, π ) are the minimal of two sets of prices charged b the two bookies. It can be readil seen that if π + π < were to hold in equilibrium, then it must be the case that either each ticket or at least one ticket is underpriced relative to its corrupted or uncorrupted probabilit of winning. An underpriced ticket must be loss-making, and one of the bookies can alwas profitabl deviate b raising the price. Hence, π + π < cannot arise in equilibrium. This result will be useful for our analsis later on, and is not directl implied b, nor does it rel on, the Dutch-book restriction; even if the ticket prices set b each bookie individuall satisf Assumption, the lower price of each ticket ma add up to less than. 4. Bribe Prevention with Zero Profit Figure 3: Ordinar punter s betting rule

13 Let us first determine the prices at which expected profit is zero and briber is prevented. Focusing on identical prices (and therefore suppressing bookie indices), consider the posting of (π, π ) in the region of π + π. An ordinar punter s optimal betting rule is indicated in Fig. 3. So the bookie s objective function can be written as: EΠ BP = [ π ( p π ) dq + [ π 0 ( p π ) dq = [ 3 π π p π p π. The bribe prevention constraints are: π max{p, φ (p )}, π max{p, φ (p )}. From the objective function one might expect that competition in each market should induce π = p and π = p (i.e., prices equal the true probabilities of winning) leading to EΠ BP = 0. But to ensure such an outcome, the prices must also prevent briber. To analze the possibilit of such an equilibrium, let us introduce two critical probabilities (refer Fig. 4): Definition. Let p be the unique p such that φ (p ) = p, and ˆp be the unique p such that φ (p ) = p. Further, p and ˆp can be calculated as follows : p = [( λ ) + Ω [ + λ 4Ω λ Ω λ ( + Ω ), ˆp = [( λ ) + Ω [ + λ 4Ω λ Ω λ ( + Ω ). It can be readil checked that φ (p ) > 0, φ (p ) < 0 with φ (0) > 0 and φ () <, as shown in Fig. 4. Therefore, a unique p must exist and is in (0, ). It then follows that at all p < p, φ (p ) > p, and at all p > p, φ (p ) < p. Similarl, φ (p ) < 0, φ (p ) < 0 with φ (0) > 0, φ () <. Therefore, ˆp also exists and it is unique. Further, at all p > ˆp, φ (p ) > p, and at all p < ˆp, φ (p ) < p. If Ω i is large enough, which requires z to be small relative to w + α i (f + f I ), p will be smaller than ˆp. In other words, the influential punter should not be too powerful (in terms of wealth); in Fig. 4, as z becomes large, both φ and φ curves shift upwards, pushing p to the right and ˆp to the left, thus shrinking and even flipping the ( p, ˆp ) interval. Until specified otherwise, we will assume: Assumption 3. (Not too powerful punter I) Let Ω > ( λ ) λ and Ω > ( λ ) λ so that p < < ˆp. 3

14 , p p ~ ^ p p Figure 4: Region of bribe prevention, zero profit equilibrium Fig. 4 is drawn on the basis of Assumption 3. Clearl, over the interval [ p, ˆp, representing close contests, the zero-profit prices π i = p i (i =, ) prevent briber, and these can be sustained as Bertrand equilibrium b appling the usual logic: unilateral price increase(s) b a bookie do not improve profits, and an price reduction(s) inflict losses (in addition to violating the Dutch-book constraint). The closeness of the contest means if prices are set according to fair odds, both the ticket prices tend to be rather high for the influential punter to bribe one team, bet on the other team and make a profit; hence match-fixing is prevented. But outside [ p, ˆp we cannot have bribe prevention, along with zero profits, in equilibrium a direct implication of the constructed interval [ p, ˆp. So now the question is, outside the interval [ p, ˆp can briber be prevented with certaint while profit remains positive? The answer is no. Below we provide detailed reasons; a formal short proof appears in the Appendix. If bribe prevention with positive profits were to be an equilibrium for sufficientl unbalanced contests, then we must have one of the following three possibilities: (i) both tickets are generating profits; (ii) onl one ticket ields positive profit, while the other ticket is generating a loss, but the overall profit is positive; and (iii) onl one ticket ields positive profit, while the other ticket ields zero profit. In all three cases, given Assumption 3 and also as is evident from Fig. 4, onl one team, i.e. the weak team, needs to be protected from the influential punter s betting b raising its price well above its uncorrupted probabilit of win, and thus ielding positive profits for the bookmakers. For instance in the region [0, p ), ticket needs to be protected. For the other ticket (namel ticket when p < p ), competition will wither awa profits. Therefore, possibilit (i) is ruled out. Possibilit (ii) is also ruled out, because one can raise the price of the loss-making ticket and lose the market 4

15 altogether. So, we are left with onl possibilit (iii). For the sake of concreteness consider p < p. Here as argued above, the profit-generating ticket must be ticket, due to the bribe prevention constraint π φ (recall ()). But it can be easil seen that competition will force the constraint to bind. Thus, we will have π = φ > p. For ticket we have π = p. Now, from this proposed equilibrium, we argue that both tickets can be undercut without increasing the prospect of briber and profit will improve. To see that, suppose one of the bookies reduces π slightl below p and takes a small loss on ticket. But simultaneousl he makes the bribe prevention constraint π = φ irrelevant. As punter I now can gainfull bet on ticket without committing briber, the new bribe prevention constraint should be π > ψ (recall (3) and tie-breaking rule I). As can be checked from () and (3), ψ < φ as long as π < p. Therefore π can be suitabl reduced to π (in accordance with the reduction in π below p ) such that ψ < π < φ. Thus, briber is still prevented and the bookie full captures both markets. As long as price reductions are of small order, loss in ticket will be approximatel zero, and gains from ticket will be bounded awa from zero, because the new profit from ticket is almost twice as large. Hence, possibilit (iii) is also ruled out. Proposition. (Bribe prevention) Suppose Assumption 3 holds. (i) For p [ p, ˆp, the unique and smmetric equilibrium under Bertrand competition is π = p and π = p, such that briber is prevented surel and each bookie earns zero expected profit. (ii) For p outside the interval [ p, ˆp, there is no pure strateg equilibrium under Bertrand competition in which briber is prevented with probabilit one. Thus, it is possible that the influential punter will not bribe so that match-fixing is not a threat and Bertrand competition leads to zero profits with prices of bets equalling fair odds. In contrast, in Shin (99; Proposition ) the presence of an insider meant distortion in the prices of bets awa from fair odds, in particular, prices reflected the favorite-longshot bias. The main reason for the difference in our result is that, different from Shin (99) and similar other analsis of competitive bookmaking (e.g., Ottaviani and Sorensen, 005), our bookmakers are as well informed as the influential punter (i.e., the insider) when match-fixing is deterred. The basis for price distortion is thus removed. 4. Bribe Inducement with Zero Profit Outside [ p, ˆp, we will look for a (pure strateg pricing) equilibrium in which briber occurs with positive probabilit. Also, we will be interested in the qualitative equilibrium 5

16 properties characterizing briber. Studing when briber actuall occurs, and not just talk about the implications of its potential threat, should be highl relevant in the context of growing incidence of betting related match-fixing in sports as we have cited in the Introduction. Let us start b asking: Which team will be bribed? The following result will be useful in analzing the briber prospect of the longshot. Lemma. Suppose Assumption 3 holds (i.e., the influential punter is not too powerful). Then at all p p it must be that h < p, where h ( λ )p ( λ p )Ω as in (4). If the longshot (i.e. team when p < p ) is to be bribed, ticket price, π, must be lowered below h b the bet accentuation condition (4). B Lemma h < p and therefore h < p. This suggests that a large rent has to be transferred to punter I to induce him to bribe the longshot. This ma not be optimal. Proposition. (Briber of the strong team) Suppose Assumption 3 holds. Then in an equilibrium (of the Bertrand game) there is either briber of onl the strong team or none at all. To understand wh onl the strong team is bribed, consider first briber of the longshot (bet accentuation). If team (the longshot) were to be bribed, ideall the bookies would have liked to lure the naive punters to bet on team b setting π appropriatel low and/or π high. But to encourage punter I to bribe team and bet on team, the bookies must do the opposite set π low and π high. This conflict makes briber of the underdog lossmaking and unsustainable for the bookies. Next, consider the prospect of either team being bribed. That would require setting not just π low, but also π low. There are no pair of prices feasible for this to happen, as the Dutch-book restriction will be violated. Thus, if briber is to occur, it must involve onl the favorite. Studing data on German horse race betting, Winter and Kukuk (008) found some evidence of cheating b the favorites when races are ver uneven. Suspecting cheating b one of the favorite jockes, the bettors tend to bet disproportionatel more on the longshots. Winter and Kukuk s empirical observation that the favorite(s) cheat bears some resemblance to our theoretical observation above. However, their analsis is mainl for parimutuel betting whereas ours is for fixed-odds betting. In parimutuel betting, market odds are generated endogenousl based on actual bets placed (rather than bookie-determined odds). Therefore, which team is bribed should depend on the distribution of naive punters beliefs about the teams chances, their beliefs about others beliefs and their betting strategies, etc. Intuitivel, if a majorit of naive punters is expected to take a bet on a particular team, be it favorite or not, then an influential punter would be tempted to bribe this team, bet on the other team 6

17 and profit from the endogenousl determined low odds on the eventuall winning team. That is, bribing decision in parimutuel betting should depend on the general direction of bettor strategies in the market due to externalities generated through endogenous odds, which is quite distinct from the considerations relevant for fixed-odds markets. To the best of our knowledge there is no formal theoretical analsis of match-fixing for parimutuel betting, so a clear differentiation between the two market forms must await further research. To continue with our analsis, when the favorite (team ) is to be bribed, π must be above some threshold level (h ) and π must be below a similar threshold level (ψ ). Can the bookies then make positive expected profits, and can such profit-generating prices be sustained as a Nash equilibrium? The answer is no. Even though the Dutch-book restriction limits the scale of undercutting, it turns out that there will alwas be some incentive for slight undercutting on one or both tickets. If ticket were to generate positive profit alone or along with ticket (remember, team s winning prospect is enhanced with the briber of team ), then a bookie can easil steal this market without altering punter I s incentive to bribe team (simpl lower π slightl). On the other hand, if onl ticket generates profit then a coordinated undercutting on both tickets (if necessar) becomes profitable, despite a loss on ticket (a formal argument appears in the proof). In either scenario, positive profit cannot be sustained. It is also the case that in a bribe inducement equilibrium, the favorite must be underpriced (π < p ) and the longshot then must be over-priced (π > p ) relative to the uncorrupted probabilities of winning. The reason is, the bookies need to attract (ordinar) punters to bet on a losing cause, i.e. the favorite, and at the same time discourage them from betting on the longshot whose probabilit of winning has secretl gone up. These also impl that punter I must be earning a strictl positive expected profit in a bribe inducement equilibrium, regardless of whether the bookies make zero or positive expected profits. Given Proposition and the first observation above, we are left with onl the possibilit of a zero profit, bribe inducement equilibrium. Below we first define this equilibrium, followed b formal statements of the above observations in lemmas (as we will need part of the equilibrium definition to prove the lemmas), and then we develop the equilibrium argument. Equilibrium E. Suppose Assumption 3 holds and p [0, p ). Smmetric equilibrium prices (π 0, π 0 ) with π 0 < p and π 0 > p are such that on access onl team (the strong B smmetr we mean smmetr across bookies. We do not analze whether there might be an asmmetric equilibrium in which briber is induced (with positive probabilit), at least one bookie is the sole server in one market and each bookie makes zero expected profit. Such an equilibrium, if it exists, will be similar in spirit, as far as briber is concerned, to the equilibrium E. Further, smmetric prices must generate zero profit in each market, otherwise deviation would occur. 7

18 team) will be bribed, whereupon punter I will bet on team (the weak team) and otherwise (i.e., team is not accessed) he will bet on team. Each bookie makes zero expected profit in each market, that is, EΠ b = { ( π0 ) [ pb + µ z [ ( λ p )} = 0, (5) π 0 π 0 EΠ b = { ( π0 ) [ pb + ( µ )z [ p } = 0, (6) π 0 π 0 where p b = [µ ( λ p ) + ( µ )p and p b = p b = [µ λ p + ( µ )p are the ex-ante probabilities of team and team winning. Lemma 3. Suppose Assumption 3 holds. Then there is no equilibrium with match-fixing in which the bookies earn positive (expected) profits. Lemma 4. Suppose Assumption 3 holds, and p [0, p ). Then there cannot be a bribe inducement equilibrium such that π p and π p. In other words, an bribe inducement equilibrium must involve π < p and π > p, with punter I earning a positive (expected) profit. Equilibrium E construction. The two prices defined b (5) and (6) must satisf condition (3) for briber of team : π 0 ψ (π 0 ), and violate condition (4) so that team is not bribed: π 0 > h. From (5) and (6) we can identif the bounds for π 0 and π 0. In the event of gaining access to team, punter I will bribe team and bet on team if the price of ticket is smaller than the corrupted probabilit of team winning, i.e. π 0 < ( λ p ). So the second term in (5), which refers to profit from punter I, must be negative, and therefore, the first term in (5) (i.e. profit from the naive punters) must be positive, ) which implies π 0 > p b. Therefore, π 0 must lie within the interval (p 3 b, ( λ p ). Similarl, in the event of not gaining access to team, punter I will bet on team if π 0 < p. This makes the second term (representing profit from punter I) in (6) negative, and therefore the first term in (6), which represents profit from the naive punters, must be positive impling π 0 > p b. That is 3 It is evident that p < p b < λ p. 8

19 to sa, π 0 must belong to the interval are in conformit with the Lemma 4 observation. (p b, p ). The constructed ranges for π 0 and π 0 The above bounds ensure that π 0 + π 0 > p b + p b =. Further, since EΠ b and EΠ b are continuous functions of π and π respectivel, there must exist at least one π 0 and one π 0 within the above specified intervals solving (5) and (6). In fact, the solution (π 0, π 0 ) is unique. 4 Before we start to analze various deviation incentives, we impose a mild assumption on (µ, µ ) and (λ, λ ). Assumption 4. (A threshold on strong team s ex-ante winning odds, with corruption) For all p p < /, the following holds: p [µ λ + ( µ µ ) + µ ( λ p ) > p, or equivalentl p [ ( µ ) + µ λ < p [ ( µ ) + µ λ. That is, despite the prospect of match-fixing the strong team does not become weaker than where the weak team was in the absence of corruption. 5 Under smmetr of µ and λ, the assumption is automaticall satisfied. Unlike the textbook Bertrand model, analzing deviations is much more complex in our setting due to the interdependence between the two markets: gains from undercutting in one market must be evaluated in light of the possible briber implication in the other market, and moreover a bookie will have the option of altering the two prices in various combinations (lower/increase, lower/sta-put, lower/lower, etc.). In addition, if a deviation alters the status-quo briber or no-briber situation, the deviation profit of the bookie can rise or fall discontinuousl. Starting at zero-profit prices (π 0, π 0 ), neither bookie would gain b deviating unless it alters the briber incentive of punter I. We must therefore protect our posited equilibrium against (onl) the following three deviations altering the briber incentive: 4 Eqs. (5) and (6) are quadratic in π 0 and π 0 respectivel, solving which we obtain: π 0 = { ( + k ) ± ( k ) + 4µ λ p z } π 0 = { ( + k ) ± ( k ) + 4( µ )p z }, where k = p b + µ z and k = p b + ( µ )z. For each price, one of the two roots exceeds. 5 Note, however, that with briber possible, the ex-ante probabilit of team winning ma well be lower than that of team because µ ma be high, µ low while λ could be low and λ high. 9

20 Starting from h < π 0 < p so that team is not bribed, and p < π 0 ψ (π 0 ) so that team is bribed, (i) Deviation leading to briber of team instead of team : π is reduced to π and π is chosen to be some appropriate π such that π h and min{π, π 0 } > ψ (π ); 6 (ii) Deviation leading to briber of either team: π is reduced to π and π is appropriatel chosen to be some π such that π h and p < min{π, π 0 } ψ (π ); (iii) Deviation leading to briber of neither team: π is reduced to π and π is reduced to π (or unchanged so that π = π 0 ) such that h < π and p < ψ (π ) < π. 7 It turns out that deviations (i) and (ii) are either infeasible (i.e. violate the Dutch-book restriction) or clearl unprofitable (Assumption 4 and Lemma will be used to establish this in the proof of Proposition 3 in the Appendix). It is deviation (iii) that requires additional attention. It is quite possible that if π 0 is sufficientl high, then one of the bookies can undercut in such a manner that punter I will no longer find it optimal to bribe, and et the deviating bookie will make a positive expected profit. We therefore identif an upper bound for π 0, below which deviation (iii) will not be profitable. The following two definitions will be used to determine the relevant upper bound. Definition. (A bound for π 0 ) Fix an 0 < p < p. Let ( π, π ) be such that π + π = and π = ψ ( π ). B construction the prices ( π, π ) are unique, and π > p, π < p. Definition 3. (An alternative bound for π 0 ) Fix an 0 < p < p, and assume that π. There exists a unique pair of prices, (π +, π ) = (π M0, π M0 ), at which the dual objectives of EΠ M = 0 and π = ψ (π ) will be met, where EΠ M represents no-briber, monopol profit with punter I betting on team and given b EΠ M = [ 3 π π p π p π + z [ p π, (7) where p < π M0 < φ and 0 < π M0 < p. 6 Note that as π is lowered to π, ψ (π ) < ψ (π 0 ); it is conceivable that ψ (π ) < π 0 < ψ (π 0 ) in which case the deviating bookie ma even set π π 0. Setting π in excess of π 0 would impl though losing the sale of ticket, which is likel to be an unwise move. 7 To deviate and set π > π 0 will alwas be dominated b π = π 0 for the no-briber deviation considered. 0

21 In Fig. 5a displaed, the relevant upper bound on π 0 will be π (see Definition ). Uniqueness of ( π, π ) is evident from the construction. Next in Fig. 5b, the relevant upper bound on π 0 is π M0 (see Definition 3). Recall, b engaging in deviation (iii) a bookie expects to receive the monopol profit, EΠ M, with briber eliminated. It can be checked that the iso-profit curve, EΠ M = 0, intersects the π + π = line at two points (π = p, π = p ) and (π =, π + = ). Depending on the parameter values we will have p + < π < + (as drawn in Fig. 5a), or p < < π + (as drawn in Fig. 5b). Since EΠ M is defined over the region π + π, we discard the segment of the iso-profit curve that falls below the π + π = line. It can also be verified that the iso-profit curve will be concave at π (assuming p < + representing EΠ M + ). Further, an price pair that lies outside (or to the left of) the curve = 0 will ield negative profits (under monopol and no-briber), and an price pair ling inside (or to the right) will ield positive profits under monopol and no-briber. 8 When π <, + πm0 does not exist, as in Fig. 5a. Point w represents a (zero profit) bribe inducement equilibrium, where π 0 < π. To the south-west of point w there is no price pair at which the incentive for bribing the favorite team (team ) is violated and at the same time the Dutch-book restriction is met. Put another wa, b moving south-west there is no wa one can cross over to the other side of the ψ curve and still be on the right-hand side of the π + π = line. But if the equilibrium was at point w instead of w, in which case π 0 > π, then a deviation to a point like d would have been possible. Point d satisfies the Dutch-book restriction and it also ields a positive profit, no-briber monopol outcome. Thus, when π <, the restriction π + 0 π is both necessar and sufficient to rule out deviation (iii). The set of sustainable bribe inducement equilibrium prices is given b the shaded area (not including π = p b, π = p b ). The possibilit of π + is drawn in Fig. 5b. The equilibrium price is again denoted b point w, which shows that π 0 < π M0. Starting from point w if one moves south-west (i.e. undercutting on both tickets), one cannot violate the incentive for bribing the favorite team (b crossing over the ψ curve) without crossing over the EΠ M = 0 curve. That is to sa, deviation to no-briber prices will onl fetch negative profits. Similarl, if π 0 was greater than π M0, as is the case with point w, then deviation to point d, where briber does not occur, is perfectl possible and it will be profitable as well. As before, π 0 π M0 is thus the necessar and sufficient condition for ruling out deviation (iii). The set of sustainable 8 For ease of exposition we ignored the briber indifference condition while drawing the iso-profit curve. But the underling probabilit of a team winning that determines the no-briber monopol profit will depend on whether the prices remain above the briber indifference curve ψ. We consider this aspect in our formal argument.

22 p -l p b p EP M =0 /(+) d w f ~p b w b p p l p ~p p p Figure 5a: Zero profit, bribe inducement equilibrium p b p -l p EP M =0 M 0 p ~p /(+) b d w w b p f p M 0 ~p p l p p p Figure 5b: Zero profit, bribe inducement equilibrium

23 equilibrium prices is given b the shaded area (not including π = p b, π = p b ). As a numerical illustration of the equilibrium for this case as well as the first case, we provide an example in the Appendix (see Example ). Proposition 3. (Match-fixing Equilibrium) Suppose Assumptions 3 and 4 hold and consider an p < p. Then (π 0 > p, π 0 < p ) satisfing (5) and (6), and thus meeting the Dutch-book restriction, constitute a unique competitive equilibrium denoted b E if and onl if In equilibrium, π 0 ψ (π 0 ), π 0 > h, and { π0 π if π < + ; π 0 π M0 if π +. (i) punter I will bribe the strong team, team, whenever he gets an access to team ; (ii) each bookie will earn zero expected profit; and (iii) punter I will earn strictl positive expected profit. Thus, under plausible economic situations price competition among bookmakers in the sports betting market ma endogenousl lead to match-fixing initiated b a corrupt punter. And in conformit with common perceptions, with contests uneven the strong team is bribed. Finall, with the possibilit of profitable match-fixing open, the influential punter can also profitabl bet even when he actuall fails to get an access to the strong team and thus fails to bribe. That is, ticket prices are such that the influential punter profits either wa. This is in contrast with the bribe prevention equilibrium of Proposition in which the influential punter is full kept out. One ma be tempted to conclude that the above result is an evidence against the conventional wisdom that more competition means less corruption (Rose-Ackerman, 978). But there is little to relate our model to this usual corruption/competition stor where incentives for bribe-giving ma be created because of an exclusive valuable asset (sa a monopol service provision right), access to which can be restricted b regulation and where some ke government official decides arbitraril, without transparenc, who should get access. 4.3 Positive Profit, Bribe Prevention Equilibrium In this section we show that if Assumption 3 is violated and if in particular ˆp < < p, there are some interesting implications, especiall for contests that are close. Violation of Assumption 3 occurs if Ω i is relativel small, i.e., z is large relative to (w + α i (f + f I )) so 3

24 that the influential punter is quite powerful in terms of wealth. Fig. 6 presents this case. 9 We will restrict attention to the case of close contests: p (ˆp, p )., p p p m d e p m c d e a b n g ˆp m ~ p p Figure 6: Positive profit, bribe prevention prices Here we identif the possibilit of two tpes of equilibrium. In one, either team will be bribed b punter I upon access but the bookies earn zero expected profit. In the other, briber is prevented with certaint and the bookies earn positive expected profit each. The first possibilit is similar to our bribe inducement equilibrium in highl uneven contests (Proposition 3) except that now even the weak team is bribed and then punter I will bet on the strong team; in the event of not getting access to an team he will bet on neither. Punter I will also earn a strictl positive (ex-ante) expected profit. This equilibrium can be sustained if the bookies, unilaterall, cannot profitabl undercut in one or both prices to eliminate punter I s briber incentive for at least one team. The proof of this claim is similar to the proof of Proposition 3, so we include the formal result and its proof in an on-line supplementar material file. Instead, our focus here is going to be on the other equilibrium involving positive profit and bribe prevention that contrasts with the finding of Proposition (under the opposite assumption of not too powerful influential punter, i.e., Assumption 3) Bribe prevention then requires ticket prices to be set high, with φ, φ shifting upwards; contrast Fig. 6 with Fig. 4, especiall the reversal of positions of p and ˆp. 30 It is difficult to ascertain in the general case whether these two equilibria will hold in isolation or simultaneousl. In our on-line supplementar file we provide an example of multiple equilibria a highprice, bribe prevention equilibrium and a low-price, bribe inducement equilibrium either of which ma 4

25 Our proposed bribe prevention equilibrium is (π = φ, π = φ ); this generates strictl positive profit since φ > p, φ > p. More precisel, each bookie s equilibrium profit is: EΠ BP = k can be large around p = /. 3 [ 3 φ φ p φ p φ k. We need to show that this equilibrium is immune to all possible undercutting, i.e., undercutting on both tickets and undercutting on each ticket separatel. In what follows we provide an informal argument b suggesting conditions that will ensure immunit against all tpes of undercutting. The formal proof and the precise conditions are provided in the Appendix. Slight undercutting on both tickets: First consider undercutting on both tickets. In Fig. 6, let us select p to be m, at which φ = b and φ = a. Suppose one bookie deviates from the equilibrium b undercutting π slightl below a, and π slightl below b. As long as the reduced π i is strictl greater than p i, b the violation of bribe prevention constraint () punter I will be strictl better off b bribing. Therefore, team will be bribed with probabilit µ and team with probabilit µ. Having monopolized both markets the undercutting bookie will face significant losses to punter I with probabilit µ + µ, against significant gains from ordinar punters with probabilit ( µ µ ). Intuitivel, it then seems that his expected overall profit is likel to be smaller than the non-deviation duopol profit if µ + µ is sufficientl high. Let µ be such that the deviation profit EΠ BI is just equal to the duopol profit k, and for µ + µ > µ, EΠ BI < k. Thus, a lower bound on the total probabilit of access seems in order. In the Appendix we formalize this logic. occur depending on how the bookies coordinate. Given that bribe prevention ields positive profits for the bookies while bribe inducement ields zero profits, the bookies ma be able to coordinate on the bribe prevention equilibrium. However, one might wonder if the standard perturbation technique of global games will help achieve a unique equilibrium; see an extensive treatment on global games b Morris and Shin (003). In our context one ma plausibl consider that the bookies observe a nois signal of the uncorrupted real odds but the influential punter observes the odds without noise, much like the government in the currenc attack model of Morris and Shin (998). But there are several complications. First, in our case the bookies have multiple and interdependent action sets (i.e. two prices). This interdependence, it can be checked, results in the bookies paoffs under briber not being supermodular in the price, π, and the probabilit, p. Thus, the bookies overall expected paoffs will fail the supermodularit test, an important requirement in global games. Second, the bookies actions, i.e. the prices, are not alwas (strictl) strategic complements: if one of the bookies lowers one price, the other bookie ma not follow suit because the price reduction might invite informed betting b the influential punter. Third, in global games the state determines optimal actions; in our case, the state (i.e., the probabilit of a team winning) ma change due to the influential punter s action. Because of these complications it is difficult to speculate whether the perturbation technique would achieve a unique equilibrium in our setting. The question was raised b one of the referees and we hope to pursue it in future. 3 Note that even if the volume of bets on the two tickets ma be fairl even if the ticket prices, for p = /, are smmetric (or fairl close), large prices of bets means the bookies profits ma be substantial. 5

26 Slight undercutting on a single ticket: Suppose the deviating bookie lowers π slightl from b while maintaining π = φ = a. As long as π > p (which is indeed possible at p = m in Fig. 6), an slight reduction in π from φ will trigger briber of team with probabilit µ (but team l will not be bribed). In the event of briber capturing market becomes a curse, and therefore if µ is sufficientl high the bookie will be deterred from such undercutting. Let the critical value of µ be denoted as µ, such that at all µ µ, EΠ BI k. Smmetricall, let µ be the critical value of µ such that at all µ µ slight undercutting on ticket is deterred. Thus, we need to have lower bounds on individual µ i s as well as their sum (µ + µ ) to support the proposed equilibrium. Large-scale undercutting: In addition to slight undercutting, we need to consider largescale undercutting as well. In Fig. 6 π can be reduced from b to some π and π can be reduced from a to some π such that π +π. For example, π = n and π = c can be one such deviation, π = e and π = d is another deviation. 3 With such deviation the deviating bookie will earn monopol profit (from both markets) with the prospect of match-fixing for either team. Identifing conditions under which this deviation (monopol) profit falls short of the duopol profit k proves to be difficult under the general case. But we can sa that if the monopol profit from deviation (admitting briber of either team) is increasing at π = φ and π = φ, then the deviating bookie would prefer to undercut slightl rather than substantiall. The same argument applies when we consider undercutting on a single ticket (admitting briber of onl one team). Based on the monotonicit argument of the deviation profit (as above), we can restrict attention to small-scale undercutting. In the following proposition we provide a sufficient condition to this effect. Then supporting of the bribe prevention, positive profit equilibrium requires ruling out slight undercutting on one or both tickets which will be ensured b the lower bound restrictions on µ and µ discussed earlier. In Example in the Appendix we demonstrate this tpe of equilibrium numericall. Proposition 4. (Bribe prevention and positive profit) Suppose ˆp < < p, and p (ˆp, p ). That is, the influential punter has significant wealth and the contests are close. If (i) µ and µ exceed some threshold levels (to be precisel determined in the Appendix), and 3 At π = d < p and π = e > p, the Dutch-book restriction is satisfied. To see this we can map π = e onto the horizontal axis and arrive at point g which is above the 45 line and has coordinates (e, d). Since point g is outside the unit simplex, e + d must be greater than. 6

27 (ii) ρ + zµ ( λ p ) φ, and ( ρ) + zµ ( λ p ) φ, (8) where ρ = µ λ p + µ ( λ p ) + ( µ µ )p is the ex-ante probabilit of team winning when either team can be bribed, then π = φ and π = φ is a bribe prevention equilibrium in which each bookie makes a positive expected profit. The above is a possibilit result which, to our knowledge, is new. One would normall expect competition to drive down smmetricall informed bookmakers profits to zero (as was the result in Shin s (99) exogenous insider information model, for instance). Our intuition is that high chances of corruption make undercutting a risk proposition as it ma create a lemon (Akerlof, 970) and give rise to an adverse selection problem similar to the credit rationing stor of Stiglitz and Weiss (98). Potential entr b the influential punter who has significant wealth and who can fix the match works as a disciplining influence deterring deviation b the bookies from the implicit collusive equilibrium. Bookmakers tendencies to resist excessive price reduction is similar to banks resisting lowering of interest rates that ma invite high-risk borrowers. It ma also be noted that positive profits for the bookies are generated not due to an asmmetr of information between the bookies about the teams winning odds, although such a situation can be easil visualized when sometimes bookies ma have differing private information due to their expertise (or the lack of it) on underworld/illegal betting sndicates. Our result thus differs from that of Dell Ariccia et al. (999) who studied Bertrand competition between two incumbent banks (who are somewhat better but asmmetricall informed about the potential customers riskiness due their differing existing market shares) and an entrant (with no information) and showed that the dominant incumbent bank would earn positive profits due to its superior information. We can now assess the reasons for possible multiple equilibria. In the not-so-powerful influential punter case, for close contests, competition had driven prices all the wa down to zero-profit levels and all along briber of both teams were prevented. Then, having reached the zero-profit prices, further unilateral price reduction(s) that could potentiall lead to bribe inducement are not feasible as that would violate the Dutch-book restriction. In contrast, in the powerful influential punter case (and close contests), the bookies could be restrained from competitive price cutting (as deviation profit would be lower, even when strictl positive) well before individual prices reach the uncorrupted win probabilities due to the adverse selection problem noted above; the significance of the influential punter s wealth 7

28 is of relevance here. But it is also conceivable that if the bookies start from smmetric prices at which briber is alread induced and profits are positive, the competitive pressure will lead prices all the wa down to zero-profit levels as there is no further fear of inducing briber. So far we have not commented on the favorite-longshot bias. In the bribe prevention equilibrium of Proposition, the bias clearl disappears. But in Propositions 3 and 4, the favorite-longshot bias reappears. 4.4 Uneven Contest and the Possibilit of Bribing the Favorite Now we consider uneven contests continuing with the assumption that ˆp < < p. In particular, we restrict attention to p < ˆp (i.e., highl uneven contests). What can we sa about match-fixing or bribe prevention? We discuss the following possibilities informall. Bribe prevention: As argued in section 4., here too bribe prevention with zero profit is not possible, because at all p < ˆp, we have p < φ and φ < p. Prevention of briber of team requires setting π φ if π = p. But then ticket will ield positive profit. Similarl, bribe prevention with positive profit is also not possible. If the prices are set such that π = p and π = φ > p (ielding positive profit on ticket ), one of the bookies can slightl undercut on ticket (from π to π ) and simultaneousl undercut on ticket in such a manner that π > ψ (π ). This undercutting is clearl profitable and et briber is avoided, as shown earlier in section 4.. Thus, here too bribe prevention with positive profit is ruled out. Bribe inducement: We can confirm that Lemma 4 will continue to hold, so an bribe inducement equilibrium must involve π < p and π > p. But then we can rule out equilibrium involving briber of the underdog: briber of the underdog (team ) would impl punter I betting on the favorite (team ), and since π < p, this would impl, as shown in the proof of Proposition, ticket will be loss-making. Also it can be checked that this observation does not rel on Assumption 3. How about equilibrium involving briber of either team or onl the favorite? Note that when punter I is quite powerful (i.e., z is large relative to w + α(f + f I )), Ω i falls below and our Lemma ma not necessaril hold; h can be larger than p or even larger than p. In section 4.3, a low value of h was particularl helpful in eliminating the incentive to bribe the underdog (i.e., briber of the bet accentuation variet), which in turn helped us to rule out the possibilit of briber of either team (Proposition ) and protect the equilibrium involving briber of the favorite alone (Proposition 3). Here too, we can protect this result, as long as h p. From (4) we can determine that if Ω [ λ λ p,, then h p. Then 8

29 our result in Proposition eliminating briber of either team and bribe inducement positive result of Proposition 3 both go through. Basicall, if Ω falls below but does not fall too far, our qualitative results of section 4.3 remain unaffected. What happens if Ω falls further? Clearl, h will rise above p, and at some point one of the sufficient conditions of our match-fixing equilibrium, π 0 > h, ma fail. In effect, ticket price does not have to be reduced too far to trigger briber of team. This is a deviation that could be profitable and thus the match-fixing equilibrium ma be destroed, and along with it the existence of a pure-strateg equilibrium ma be ruled out. 4.5 Comparing Bertrand Competition with Monopol So far we have not commented on an important case of the match-fixing problem: monopol bookmaking. In a related work (Bag and Saha, 00), we stud the monopol problem. Based on the progress of this work we can sa the following. A monopolist bookmaker can control the influential punter s briber incentive without having to worr about losing sales to a rival. But even then it is conceivable that sometimes the monopolist ma want to engineer match-fixing. Intuitivel, b inducing match-fixing a monopolist would lose to the influential punter but gain from naive punters. But whether the net gain will be large enough to justif match-fixing will depend on, among other things, how close the contest is. Suppose the influential punter is not-so-powerful (low z) and the contest is close. In the absence of briber the bookie does not expect to make much profit as naive punters bets nearl cancel out on average (due to their uniform beliefs). But b inducing briber and lowering the price of the bribed team majorit of naive punters can be directed to bet on the losing team, and the expected profit from naive punters will rise significantl, albeit at a cost the paout to the influential punter. For small z, the paout will be small, and bribe inducement is likel to be optimal. This will be opposite to the competitive outcome of no briber (Proposition ). For highl uneven contests, in the absence of briber the monopolist tends to gain significantl b inducing most of naive punters to bet on the weak team. Here, the influential punter poses a threat to upset this calculation b bribing the strong team and betting on the weak team. But as long as z is low, the monopolist can withstand the threat b marginall raising the price of the weak team (up to φ i ). So bribe prevention is likel to be optimal here, again an outcome different from the competitive case (that of match-fixing, as shown in Proposition 3). Of course, the precise nature of the results will depend on several factors, such as µ i, λ i etc. But broadl speaking, we expect to see different results for monopol when the influential punter is not so powerful. But for the case of high z (i.e. quite powerful influential punter), the monopol outcome 9

30 ma be of the following tpes. In close contests match-fixing is likel to be an unattractive option because of the heav loss to the influential punter; this would be similar to the bribe prevention equilibrium under competition as shown in Proposition 4. However, under competition bribing can also occur (due to multiple equilibrium) unlike in monopol. In highl uneven contests as well, the monopolist ma not prefer match-fixing for the same reason (high z resulting in high paout), but here the size of the access probabilit and other details can be crucial. Overall, competition in bookmaking raises distinct strategic considerations not present in monopol and ma thus give rise to equilibrium outcomes that are ver different compared to monopol. 5. Conclusion Table : Summar of match-fixing results Close contests Highl uneven contests Punter I is not Bribe prevention Bribe inducement of the too powerful with zero profit favorite and zero profit Punter I is Bribe prevention with Bribe inducement of the powerful positive profit, and/or favorite and zero profit bribe inducement of either team and zero profit Match-fixing in a number of sports and its implications for betting have attracted a great deal of media attention in recent times. Building on Shin s (99; 99) horse race betting model with fixed odds, we analze the match-fixing and bribing incentives of a potentiall corrupt gambler and show how competition in bookmaking affects match-fixing, taking the anti-corruption authorit s investigation strateg as exogenous. At the set prices, the bookies are obliged to honor the bets using deep pockets. The bookies pricing decisions determine whether the corrupt influence comes into pla or kept out. We show that competition ma not alwas ensure zero profit, nor does it alwas prevent briber. And when match-fixing is induced, often the strong team will be bribed, though in some close contests either team ma be bribed. In Table we present a broad summar of our results. Acknowledgements The work carried out in this paper was partl supported b an NUS Isaac Manasseh Meer Fellowship to Saha. Earlier versions were presented at the 009 RES conference 30

31 (U.K.), Applied Theor conference (CSSS, Kolkata), Growth and Development conference (ISI, New Delhi), 00 AEI-Four Joint Workshop on Economic Theor (KIER, Koto), SAET conference (Singapore), and in seminars at IGIDR (Mumbai) and UEA. The manuscript is extensivel revised based on the critical reviews of two anonmous referees and an Advisor Editor. We also received thoughtful comments from Adita Goenka. We thank them all. An remaining mistakes are entirel ours. Appendix A. Proof of Proposition. Part (i): B construction, for an p [ p, ˆp, the ticket prices (π p, π p ) impl that π φ and π φ (see Fig. 4). So b condition () briber cannot occur. Then Bertrand competition would lead to the zero-profit equilibrium prices (π = p, π = p ) for both the bookies; unilateral price reduction(s) from (π = p, π = p ) is not feasible as it violates the Dutch-book restriction. The equilibrium is unique just like in the standard Bertrand competition game. Part (ii): Consider p < p, where the focus is restricted to briber of team onl; smmetric argument will appl to p > ˆp. If there is a pure strateg bribe prevention equilibrium, it must be from one of the following two price configurations: [ π = φ and π = p. (We can rule out π > φ and/or π > p due to Bertrand competition in each of ticket and respectivel.) [ π (ψ, φ ) and π < p, such that punter I finds bribing team (when accessed) less profitable than betting on team ; ψ is defined in (3). Of these configurations, [ cannot be equilibrium: starting from π < p, a bookie can raise onl π and avoid the loss on ticket. For configuration [, starting from (π = φ (p ), π = p ) we show that a deviation in the form of slight undercutting on both tickets b one of the bookies will be profitable. In the posited equilibrium each bookie earns an overall profit (ticket index used as subscript) EΠ = ( φ ) [ p φ > 0, since p < p. Now consider the following deviation: lower π slightl to π = p ɛ and choose δ > 0 appropriatel so that ψ (p ɛ) < π = φ δ < φ ; this is feasible since ψ is continuous and increasing in π. Further, as ɛ becomes small, the permissible δ will also become small. With this deviation briber of team is not induced; but it results in monopolization of both the markets b the deviating bookie. From ticket the loss is ( p + ɛ)[ ɛ p ɛ, which is approximatel equal to zero for ɛ is ver small. But from ticket the profit will be EΠ = ( φ + δ)[ p φ δ, which is approximatel EΠ (i.e. twice the posited 3

32 equilibrium profit), for δ sufficientl small. This is a contradiction. Thus, at no prices briber is prevented with certaint. Q.E.D. Proof of Lemma. Consider p p < /. That h ( λ )p λ ( λ p )Ω < p, i.e., λ p < Ω follows from the fact that at p = / we have λ < Ω (b Assumption 3) and λ p = ( λ ) ( λ ) λ λ p is increasing in p. Q.E.D. Proof of Proposition. First we show that briber involving onl the underdog cannot be an equilibrium. Consider p [0, p ) where p < /, and suppose there is an equilibrium in which onl the underdog is bribed. Then we must have, b (3) and (4), π h, π < p, and π > ψ (π ) p, where (π, π ) corresponds to the smmetric price, if the markets are shared, and corresponds to the minimum price of each ticket, if prices are asmmetric leading to monopol of at least one market. Then punter I will alwas bet on team, either b bribing team or without bribing, and will never bet on team. Thus, if market is shared, the expected profit of each bookie from ticket is EΠ = [ ( π ) + z [ µ ( λ p ) + ( µ )p, π and if market is not shared, one of the bookies gets EΠ from market. In either case, since π < p < ( λ p ), it must be that EΠ < 0 at all π that permits briber of onl team. This, we are going to argue, cannot be part of an equilibrium. To see wh, first consider the case where ticket market is shared b the bookies. Then one of the bookies can deviate and set a higher price for ticket dumping all of his ticket sale onto the other bookie and avoid the loss from ticket while not affecting his profit from ticket ; this is not consistent with equilibrium. Next consider monopol sale of ticket b one of the bookies (call him bookie ), who must also have a positive share in ticket sale ielding him a strictl positive profit because otherwise the bookie can alwas do better to quit both markets b setting high enough prices and avoid losses. In fact, given that bookie makes a strictl positive profit from ticket sale, it must be that ticket market is shared with bookie (b charging the same price for ticket as bookie, bookie can make a strictl positive profit which is better than his default option of no sale of ticket ). Now bookie can lower π slightl to steal ticket market from bookie, ielding him an improved profit; but then, again, this deviation possibilit cannot be consistent with the hpothetical equilibrium. 3

33 A smmetric argument as above will appl for p (ˆp, which is the same thing as p [0, p ), ruling out briber of onl team which is the weak team. Finall, as shown in Proposition, for p [ p, ˆp there cannot be an briber in equilibrium. Next we show that briber involving either team cannot be an equilibrium either. Suppose not, and consider p p < /. Then we must have, b (), (3) and (4), and Lemma, one of the following three possibilities: π h, π < p, and p < π ψ (π ), (A.) π h, π < p, and p < π ψ (π ), (A.) π > p, π > p, π < φ (p ), and π < φ (p ), (A.3) where (π, π ) refers to the smmetric price, if the markets are shared, and refers to the minimum price of each ticket, if prices are asmmetric leading to monopol in at least one market. First, consider (A.). Lemma together with p < / impl that h < p < p, and b construction ψ (π ) < φ < p (Fig. ). Therefore, equilibrium prices π +π ψ (π )+h < p + p = (the strict inequalit follows from Lemma ), contradicting Lemma. Consider (A.). We can write π ψ (π ) < φ < p (since π < p and the last inequalit follows from Fig. 4). This contradicts the fact that π > p. Finall, consider (A.3). For p p, observe that φ (p ) < p < π (the first inequalit follows from Fig. 4), but this contradicts π < φ (p ). For p (ˆp, (i.e., equivalentl p [0, p )) appl a smmetric argument as above to rule out briber of either team, and for p [ p, ˆp again appl Proposition to rule out briber in equilibrium. This completes the proof. Q.E.D. Proof of Lemma 3. Suppose not, and assume there is a positive profit equilibrium. Suppose the prices are smmetric and the are (π, π ). 33 Given Proposition, we onl need to consider the case where onl the favorite (i.e., team ) is bribed. There are three possibilities for a positive profit equilibrium: positive profit (i) from both tickets, (ii) from ticket onl (zero profit from ticket ), and (iii) from ticket onl (zero profit from ticket ). 33 The treatment of the bookies charging different prices will be similar. 33

34 In all scenarios we must have, b (3) and (4), p < π ψ (π ), h < π < p, π π 0, π π 0, with at least one strict inequalit and π + π >, where (π 0, π 0 ) ield zero expected profit on each ticket (for both duopol and monopol). Formall, (π 0, π 0 ) is obtained from (5) and (6). (Later in the text it will be asserted that the solution (π 0, π 0 ) is unique and it can be verified that the profit expressions (5) and (6) are increasing in prices at the zero-profit solution. Hence, the lower bounds π 0 and π 0.) If scenario (i) or (ii) occurs in equilibrium so that π > π 0, then a bookie can slightl undercut ticket price, leave the briber incentives unchanged and capture market ; this will be a profitable deviation. So neither (i) nor (ii) can be part of an equilibrium. So consider scenario (iii). Suppose (π 0, π ) is an equilibrium price vector such that profit from ticket is zero and profit from ticket is strictl positive (π > π 0 ). We will then have, b (3) and (4), h < π < p, and either (iii.a) p < π 0 < ψ (π ), or (iii.b) p < π 0 = ψ (π ). In the case of (iii.a), a bookie can profitabl deviate b slightl reducing π to π ɛ while leaving π unchanged and maintaining π < ψ (π ɛ) (for ɛ small enough). Ticket (which ields positive profit) will be monopolized, while ticket still ields zero profit; clearl this will be a profitable deviation. Thus, (iii.a) is ruled out. Finall, consider (iii.b), and the following deviation: π is reduced to π ɛ and π is reduced to π 0 δ such that π 0 δ = ψ (π ɛ), where ɛ is small. Both tickets and will be monopolized b the deviating bookie, and briber incentives will be unaffected: δ(ɛ) = π 0 ψ (π ɛ) is a continuous and increasing function of ɛ (as ψ (.) is continuous and increasing), and in particular as ɛ 0, δ(ɛ) 0. Now compare the gain in market with the loss in market. The gain in market is EG(ɛ) = ( π + ɛ) [ pb + ( µ )z [ p π ɛ ( π ) [ pb π = ( π ) [ pb π ɛ + ( µ ) z ( µ ) z [ p π ɛ π ɛ [ p π where p b = [µ λ + ( µ )p, and A = ɛ [ { ( )} pb (π ɛ) + π ( µ ) z + A, (A.4) p π (π ɛ). In the market for ticket, the equilibrium price (π 0 ) ields zero profit. B deviating 34

35 from this price one earns an expected profit from ticket equal to EΠ M (ɛ) = ( π 0 + δ) [ pb + µ z [ ( λ p ) π 0 δ π 0 δ = ( π 0 ) [ pb + µ z [ ( λ p ) [ p b + δ, (A.5) π 0 δ π 0 δ π 0 δ where p b = p b. As ɛ 0, δ(ɛ) 0, and from (A.5) it is clear that EΠ M approaches twice the level of expected profit per bookie from ticket market before the deviation, which is equal to zero. On the other hand, from (A.4) we see that lim ɛ 0 EG(ɛ) = ( π ) [ p b + ( µ [ π ) z p π > 0 (b the premise of the equilibrium). The overall gain from deviation is lim ɛ 0 EG(ɛ) > 0, hence (iii.b) cannot arise in equilibrium. Q.E.D. Proof of Lemma 4. Suppose not. Suppose there is an equilibrium (π, π ) with π p and π p, and it induces briber of team. Suppose prices are asmmetric giving rise to monopol in market. Then the expected profit from ticket is EΠ b = ( π ) [ µ ( λ p ) + ( µ )p π + µ z [ ( λ p ) π. Alternativel, if prices are smmetric, expected (duopol) profit from ticket is EΠ bd = EΠ b /. In either case, since π p < ( λ p ), it follows that EΠ b < 0. Then a bookie can raise π and avoid making losses, a contradiction. Q.E.D. Based on Definition 3, next we establish a technical result on (π M0, π M0 ) to be used below in the proof of Proposition 3. Lemma 5. (Existence and Uniqueness of (π M0 is unique, if and onl if π +., π M0 )) The bound π M0 exists and it Proof. Recall from Definition 3, (π M0, π M0 ) simultaneousl solve π = ψ (π ) and EΠ M = 0, where EΠ M is given b (7). Substituting π = π in (7) write EΠ M = [ ( + )π π ( π ) + ( + p ( + ))π p. Below we first observe certain properties of the EΠ M (.,.) function with direct reference to the (π, π )-plane so that the proof of this lemma can be understood with the help of Figs. 5a and 5b; these Figures are not exhaustive though. 35

36 Properties: [ There are onl two solutions to EΠ M (π, π ) = 0 and π + π =, viz., (π = p, π = p ), (π = +, π = + ). Given these and since EΠ M π +π = < 0 at π = 0 and π =, we conclude that: If p < + then EΠ M > 0 at all π (p, + ) and π = π ; EΠ M < 0 at all π < p as well as π > + and π = π. Alternativel, if p > + then EΠ M > 0 at all π ( +, p ) and π = π ; EΠ M < 0 at all π < as well as π > p and π = π. + [ Graphicall, on the (π, π )-plane the iso-profit curve EΠ M = 0 intersects the π +π = at two points identified in propert [. The segment of the iso-profit curve ling strictl below the π +π = line violates the Dutch-book restriction and therefore is discarded in our search for (π M0, π M0 ), as indicated b the dotted part (of the curve) in Figs. 5a,b. [3 EΠ M is continuous at all {(π, π ) π + π, π, π }. Further, from EΠM + p π we can conclude that EΠ M is increasing (decreasing) in π at all π < (>) p /. Similarl, from EΠM π = [ + p π we can conclude that EΠ M is increasing (decreasing) in π at all π < (>) p. π = [4 Consider a subset of the feasible prices: π p and π p in the region π + π. Then properties [ and [3 together impl that all points to the right of the iso-profit curve EΠ M = 0 ield EΠ M > 0 and all points to the left of the iso-profit curve EΠ M = 0 ield EΠ M < 0 (since EΠ M is increasing in π ). [5 We also specificall note the following ((b) follows from propert [4): (a) At π = p and π =, EΠ M = 0. 36

37 (b) Given π = p, at all π (p, ) we have EΠ M = (π p ) π ( π ) > 0; in particular, at π = p and π = φ, EΠ M > 0 (since p < φ < at p < p ). Next, we search for (π, π ) such that EΠ M = 0 where We claim the following: π p, π p, and π + π. For an π max{p, }, if + EΠM (π, π = π ) 0 then there exists a unique π p such that EΠ M (π, π ) = 0. B propert [, at an π max{p, } we have + EΠM (π, π = π ) 0, and b propert [5.b we have EΠ M (π, p ) 0. Now, since EΠ M (., π ) is continuous in π over [ π, p for an π max{p, conclude the following: that [A For ever π [max{p, }, b appling the intermediate value theorem we can + + },, there must exist some ˆπ (π ) [ π, p such EΠ M (ˆπ (π ), π ) = 0. Implicitl ˆπ (π ) solves EΠ M (π, π ) = 0. Note in particular that ˆπ (π = p ) = p, ˆπ (π = ) = p, ˆπ ( + ) = + and also ˆπ (φ ) < p (the last inequalit being implied b properties [5.b and [4). The solution ˆπ (π ) is also unique because, b propert [3, EΠ M is increasing in π < p /. This establishes our above claim. Further, since EΠM π 0 b appling the implicit function theorem we conclude: [A ˆπ (.) is a continuous function. Now we are going to show that (refer Definitions and 3): If π (or equivalentl π + ), there must exist a unique value of π + π, namel π M0, such that π = ψ (ˆπ (π )). First consider the case p >. Since π + > p, it follows that π >. It is also clear + that ψ (ˆπ (p )) = ψ (p ) = φ > p, and ψ (ˆπ (φ )) < φ because ˆπ (φ ) < p (as observed above) and ψ (.) is increasing in π. Now define the composite function η(π ) = ψ (ˆπ (π )) π, which will be continuous in π (using [A), and using the fact that η(p ) > 0 and η(φ ) < 0, we can appeal to the intermediate value theorem to conclude that there must exist at least 37

38 one π M0 (p, φ ) such that η(π M0 ) = 0. Denote π M0 = ˆπ (π M0 ); it can be verified that π M0 ( π, p ). (Note that this case is not covered b Figs. 5a,b.) Next, consider the case p, and continue to assume that π + (or equivalentl + π ). We note that ψ + (ˆπ ( )) = ψ + ( ) ψ + ( π ) = π. That is, η( ) 0, + + where η(π ) is the same composite function as defined in the earlier case. On the other hand, ψ (ˆπ (φ )) < φ (alread noted) impling η(φ ) < 0. Given that π < φ (b definition) and π, we have < φ + +. B the intermediate value theorem, once again there exists at least one π M0 [, φ + ) such that η(π M0 ) = 0. Again, denote π M0 = ˆπ (π M0 ). (This case is shown in Fig. 5b.) At the minimal (as well as maximal) π M0 such that η(π M0 ) = 0, it is eas to see that we must have η (π ) < 0. This implies ψ (.) dˆπ dπ <. Since = ψ ψ (π ) (π ) (because of continuit and monotonicit of ψ (.)), at π M0 the following must hold when dˆπ dπ > 0 : dˆπ dπ < ψ (.), or equivalentl dπ dπ > ψ (.). (A.6) EΠ M =0 Note that this condition is met at π M0 and the inequalit is maintained thereafter. That is to sa, π M0 is unique. This is confirmed b checking the curvatures of the zero iso-profit curve and the ψ (.) curve and the fact that ˆπ (φ ) < p and ψ (φ ) = p (see Fig. 5b). On the (π, π )-plane the iso-profit curve is concave when rising and convex when declining. This can be verified (which we leave out) b differentiating the following slope expression: dπ dπ EΠ M =0 = ( π ) [ p π π π p As shown in Fig. 5b these two curves can intersect onl once and the iso-profit curve will cut the briber indifference curve from below, which confirms the slope condition (A.6). If the were to intersect twice, the iso-profit curve must fall below the briber indifference curve at π = p ; but we know that is impossible, because at π = p and π (p, φ ) the no-briber monopol profit is strictl positive b propert [5.b. On the other hand, if π <, which is shown in Fig. 5a, at all π + [ π, EΠ M (π, π ) 0 (b propert [) and therefore there cannot be an ˆπ (π ) such that. + ), ψ (ˆπ (π )) = π, because ˆπ (π ) does not exist (in our feasible region). That is to sa, π M0 does not exist. Now consider an π [, φ +, if this range is non-empt. Here too, π M0 does not exist because η(π ) < 0. Q.E.D. Proof of Proposition 3. The uniqueness of (π 0, π 0 ) is guaranteed b the uniqueness of solution to eqs. (5) and (6) (see footnote 4). The three conditions together are both 38

39 necessar and sufficient. The necessit of the first two conditions is obvious. If π 0 > ψ (π 0 ) and/or π 0 h, team will not be bribed and/or team will be bribed due to violations of conditions (3) and/or (4). For the third condition, the necessit part will be established along with sufficienc. For the first two deviations, onl sufficienc needs to be established. In what follows we address each deviation.. Ruling out of deviation (i). Suppose bookie engages in deviation (i). As a result, he captures all of ticket sale. As for ticket market, there are three subcases: (a) π > π 0 ; (b) π = π 0 ; (c) π < π 0. In subcase (a), bookie off-loads all of ticket sale to bookie. Ticket will be bought b punter I (whether he manages to bribe team or not) and a section of the ordinar punters. The deviation profit of bookie from the sale of ticket is [ ( π ) + z [ µ ( λ p ) + ( µ )p. π This expression is negative because π < π 0 < p < µ ( λ p ) + ( µ )p. Thus deviation subcase (a) is ruled out. In the subcases (b) and (c), bookie at least shares the market for ticket. That is, he will be selling both tickets. We know, under this deviation π h and b Lemma h < p, so together π < p. As for ticket price, π 0 ψ (π 0 ) < φ for π 0 < p (from () and (3), it follows that ψ (π 0 ) < φ ). Also, φ < p for p < / (see Fig. 4), and so π 0 < p. Thus π + π π 0 + π < p + p =, which is a violation of the Dutch-book restriction, hence the deviations (b) and (c) cannot be feasible.. Ruling out of deviation (ii). Next, consider deviation (ii) (again b bookie ), so that p < min{π, π 0 } ψ (π ) and π h < p (the last inequalit follows from Lemma ). Punter I will bribe whichever team he gets access to, and bet on the other team; and if I gets access to neither team, he will bet on team. As before bookie will capture all of market, and will face one of three following scenarios in market depending on π : (d) π > π 0 ; (e) π = π 0 ; (f) π < π 0. First consider the subcases (e) and (f), where π π 0. Here too bookie sells both tickets, and once again we will have the violation of the Dutch-book restriction: π + π π 0 + π < p + p = (note that π 0 < p follows precisel the same wa as derived above when ruling out subcases (b) and (c) for deviation (i)). So the deviations (e) and (f) will not be feasible. 39

40 Finall, consider subcase (d) where bookie does not sell ticket. Bookie s paoff from deviation is calculated as follows: ( π ) [ µ λ p + µ ( λ p ) + ( µ µ )p π +( µ µ )z [ p + µ z [ λ p. (A.7) π π Profit from punter I (i.e., the last two bracketed terms of (A.7)) is negative, given that π < p < ( λ p ). The profit from the naive punters (which is given b the first term) will also be negative if π < µ λ p + µ ( λ p ) + ( µ µ )p. (A.8) We claim that the RHS of (A.8) is strictl greater than p, i.e., p [ ( µ ) + µ λ + p [ µ µ λ > p, or, p [ ( µ ) + µ λ > p [ ( µ ) + µ λ, which is true b Assumption 4. On the other hand, using Lemma, π h < p. So (A.8) is established and hence profit from the naive punters is negative, making the proposed deviation unprofitable. 3. Ruling out of deviation (iii). Finall, consider deviation (iii) (again b bookie ), following which there will be no attempt at briber. There are two cases to consider. Case ( π < ). Deviation to no-briber prices is ruled out if and onl if π + 0 π. First consider sufficienc. We begin b noting that since π 0 + π 0 >, π 0 π implies π 0 > π (= π ). Now let the deviation prices (π, π ) be such that π π 0 π, π < π 0, and contrar to the claim suppose π > ψ (π ), which can be written as π < ψ (π ). But ψ (π ) ψ ( π ) = π, and hence π < π. Therefore, π + π < which violates the Dutch-book restriction, and thus this deviation is not possible. Now consider the necessit part. Suppose π 0 > π (as in w in Fig. 5a); then there is a pair of prices (π, π ) such that π π 0, π < π 0, satisfing π > ψ (π ), the Dutch-book restriction, and ielding, as shown in the proof of Lemma 5, EΠ M > 0 (see point d in Fig. 5a). Thus, the proposed deviation is profitable. Case ( π ). Deviation to no-briber prices is ruled out if and onl if π + 0 π M0. We begin with sufficienc. Suppose there is a pair of prices (π, π ) such that π π 0, π < π 0, satisfing π > ψ (π ), π + π and ielding, contrar to 40

41 our claim, EΠ M > 0. As EΠ M (π, π ) > 0, we must have ˆπ (π ) < π (because b reducing π, profit can be reduced to zero; see Fig. 5b). Then we should also have ψ (ˆπ (π )) < π leading to η(π ) = ψ (ˆπ (π )) π < 0. But this is a contradiction to the fact that η(π ) 0 at π π M0 (as π M0 was the minimal π at which η = 0, as argued in the proof of Lemma 5). Now consider the necessit part. Suppose π 0 > π M0 (as in w in Fig. 5b); then there is a pair of prices (π, π ) such that π π 0, π < π 0, satisfing π > ψ (π ), the Dutch-book restriction, and ielding EΠ M > 0 (see point d in Fig. 5b). Thus, the proposed deviation is profitable. This completes the proof of Proposition 3. Q.E.D. Proof of Proposition 4. Below we derive conditions that would guarantee the particular tpe of positive profit, price coordination equilibrium in which briber is prevented. Example towards the end of this Appendix shows that the conditions are not vacuous. Slight undercutting on both tickets: B undercutting on both tickets, π (p, φ ), π (p, φ ), bookie earns the following profit: p π EΠ BI = µ [ π ( λ p ) dq + π π 0 ( ( λ p ) dq } {{ } +µ [ π k ( ( λ p ) ) dq + π π 0 π ( λ p ) dq } {{ } k +( µ µ )[3 π π p +z [ µ { ( λ p ) π π p π } + µ { ( λ p ) π π }. (A.9) Let π = φ ɛ and π = φ ɛ, ɛ and ɛ both arbitraril small. Since [3 π π p k, we rewrite (A.9) as: π EΠ BI µ k +µ k +( µ µ )k z [ µ ( λ p ) φ } {{ } >; φ < λ p ( λ p ) +µ (µ +µ ). (A.0) φ } {{ } >; φ < λ p The first (and second) term(s) in (A.0) indicate expected profit from naive punters when team (team ) is bribed. The third term captures the no-briber profit; this is twice the bribe prevention duopol profit due to monopolization of both markets. The fourth term 4

42 is the expected net paout to punter I which is positive-valued. If µ + µ is sufficientl large (sa, µ + µ ), the magnitude of EΠ BI will cruciall depend on the magnitude of µ k + µ k. If max{k, k } is not too large relative to k (or is smaller than k), then clearl EΠ BI < k = EΠ BP (note that the third term approaches zero while the fourth term is negative-valued). 34 On the other hand, b letting µ + µ 0 we will get EΠ BI = k > k. Therefore, b the intermediate value theorem: There exists µ + µ = µ such that EΠ BI ( µ) = k. If there are multiple µ then we take the largest µ to be our threshold µ + µ. 35 Thus, slight undercutting on both tickets is unprofitable, if µ + µ µ. Slight undercutting on ticket alone: Now consider the possibilit that the price of ticket is reduced slightl below φ, while the price of ticket is held at φ. The market for ticket is captured, but then team will be bribed with probabilit µ in which case punter I will bet on ticket. Let us set in eq. (A.9), π = φ, µ = 0, and adjust for sharing of market. Then we write the first bookie s deviation paoff as: EΠ BI µ k + ( µ )k µ z[ ( λ p ) φ µ {( φ )( λ p φ )} + ( µ ) {( φ )( p φ )}. The first, third and fourth terms together capture the profit in the event of briber; of these the third term indicates the net loss to punter I. Here since market is not captured, the profit is less than k. The second and fifth terms together give the profit in the event of no-briber. The no-briber profit is greater than k because of the capturing of market. Therefore, EΠ BI < k if µ satisfies the following condition: { ( φ )( p φ µ ) } { ( φ )( p φ ) } { + ( φ )( λ p φ ) } + z { ( λ p ) φ } + (k k ) µ. 34 For example, consider µ = µ = /, λ = λ = 0, and Ω = Ω = Ω (= F/z where F = w+α(f +f I )). Then we have φ = φ = /( + Ω). This gives k = Ω( Ω) +Ω and EΠ BI = k zω. Now EΠ BI < EΠ BP requires 3z ( F )z + F > 0, which will be definitel true for z near both zero and one. 35 Under plausible assumptions µ can be unique. For example, when µ = µ ( µ), EΠ BI µ = k + k 4k z [ λ p φ + λp φ < 0 would guarantee a unique threshold µ. More generall, µ and µ can be varied in the same proportion to check how EΠ BI behaves with respect to µ + µ. 4

43 Upon simplification, we obtain µ = { ( φ )( p φ ) } p ( λ ) [ φ +φ φ + zω ( λ p ). (A.) Slight undercutting on ticket alone: The analsis is similar to the previous case. Now ticket price is lowered slightl below φ, while π = φ. Bookie s deviation profit can be calculated (b setting µ = 0 in (A.9)) as EΠ BI µ k + ( µ )k µ z [ ( λ p ) φ { µ ( φ )( λ p ) } + ( µ ) { ( φ )( p ) }. φ φ The deviation can be ruled out if { ( φ )( p φ µ ) } { ( φ )( p φ ) } { + ( φ )( λ p φ ) } + z { ( λ p ) φ } + (k k ) µ. Upon simplification, we obtain µ = { ( φ )( p φ ) } p ( λ ) [ φ +φ φ + zω ( λ p ). (A.) Large-scale undercutting on both tickets: However, the above conditions do not appl to large-scale deviations. What if the prices are significantl reduced and profit rises? Let ρ denote the ex-ante probabilit of team winning (from the bookie s point of view) when either team ma be bribed, where ρ = µ λ p + µ ( λ p ) + ( µ µ )p. The deviating bookie s bribe inducement problem is to maximize: EΠ BI = [ 3 π π ρ π ( ρ) [ ( λ p ) ( λ p ) z µ + µ π π (µ π + µ ), subject to p π < φ and p π < φ. The unconstrained solutions (ignoring the two constraints) are: π = ρ + zµ ( λ p ), π = ( ρ) + zµ ( λ p ). If π φ and π φ as in (8), then EΠ BI must be non-decreasing at π φ and π φ. Therefore, the deviating bookie would like to capture both markets onl b undercutting slightl. 43

44 There are two other possible deviations: large-scale undercutting of ticket onl, and of ticket onl. We next show that b ruling out large-scale undercutting of both tickets, these last two deviations are also ruled out. Large-scale undercutting on ticket alone: As before, we should take into account undercutting of onl one ticket. Suppose onl ticket is undercut, and ticket s price is held at φ. In this case, punter I will bribe team (on access) and bet on team. Market is shared, but market is full captured. Let the probabilit of team winning in this case be denoted as ρ. We can easil calculate Now maximize EΠ BI = ρ = µ λ p + ( µ )p. [ ( φ )( ρ φ ) + [ ρ π ( ρ ) π µ z [ ( λ p ) π with respect to π subject to the constraints: π < φ, π = φ and π + φ. The unconstrained solution for π is π = ( ρ ) + zµ ( λ p ). Large-scale undercutting on ticket alone: Similarl consider the case where onl ticket is undercut, and ticket s price is held at φ. Here, team will be bribed (on access) and bets will be placed on team (b punter I). Market is shared, but market is captured. Let the new probabilit of team winning be denoted as ρ, where ρ = µ ( λ p ) + ( µ )p. The bookie should then choose π to maximize EΠ BI = [ + ρ π ρ [ + ( φ )( ( ρ ) ) µ z [ ( λ p ) π φ π subject to the constraints: π < φ, π = φ and π + φ. The unconstrained solution for π is π = ρ + zµ ( λ p ). It can be readil seen that since λ p < p < λ p, we have ρ < ρ < ρ and 44

45 ( ρ ) < ( ρ) < ( ρ ). This implies ρ + zµ ( λ p ) ( ρ) + zµ ( λ p ) < < ρ + zµ ( λ p ), ( ρ ) + zµ ( λ p ). Therefore, if large-scale undercutting on both tickets is ruled out b ensuring ρ + zµ ( λ p ) φ and ( ρ) + zµ ( λ p ) φ, then large-scale single price undercutting is also ruled out. Q.E.D. Example. To illustrate the existence of the equilibrium in Proposition 3, we consider some numerical parameter configurations in Table that satisf the equilibrium conditions. Suppose conditions are smmetric for both teams. First assume Ω = Ω =, µ = µ = 0.5, λ = λ = 0.5, and z = 0.3 (and = 0.7); see the first column. In the second column λ, λ are reduced to 0.3. In the third column λ and λ are both further reduced to zero, but at the same time z is raised to 0.5 (and lowered to 0.5) leading to a fall in Ω and Ω to.. For the parameter specification in column, p = 0.8 and ˆp = 0.7. This column represents the case of π < = 0.4. Now consider p + = 0.05 < p. At this probabilit the zero-profit prices of ticket and ticket are 0.5 and respectivel, obtained b solving equations (5) and (6). π 0 is strictl less than p = 0.95 and it gives rise to the highest bribe inducement price of ticket, ψ = 0.55; ψ is defined in (3). That ψ is strictl greater than π 0 = 0.5 implies that if team is accessed it will be bribed. Further, that team will not be bribed is evident from the fact that π 0 > h = Further, it is ensured that undercutting on both tickets and inducing briber of either team, are not possible. Minimum prices to do so (π = h = 0.03 and π = ψ (h ) = 0.007) do not satisf the Dutch-book restriction. It is also not possible to deviate to no-briber scenario, because π 0 < π = 0.3. Also note that the zero-profit prices lie within the intervals specified earlier: π 0 > µ λ p + ( µ )p = ( p b ) = 0.87 and µ ( λ p ) + ( µ )p = p b = 0.3 < π 0 < ( λ p ) = In the two successive rows (in column ) we consider two higher values of p (p = 0.0 and p = 0.6 respectivel). With higher p the gap between the two (zero profit) prices gets narrower; π 0 increases and π 0 decreases. But in all cases π 0 < p, and π 0 remains 45

46 Table : Zero profit, bribe inducement (of team ) prices Parameters Parameters Parameters z=0.3, = =0.5, z=0.3, = =0.5, z=0.5, =0.5, =0.5 =0.3 =0 ~ ^ p =0.8, p =0.7 /(+)=0.4 Bribe inducement range of p : p < 0.8 Prob. Zero profit prices p = =0.5 p = =0.935 p b =0.3 p b =0.87 p =0.0 p =0.90 p b =0.7 p b =0.83 p =0.6 p =0.84 p b =0. p b =0.78 Constraints: ~ =0.3 ( 0 )=0.55 h =0.03 (h )= = =0.885 Constraints: ~ =0.4 ( 0 )=0.64 h =0.06 (h )= =0.5 0 =0.8 Constraints: ~ =0.54 ( 0 )=0.64 h =0.043 (h )=0.08 Comments: (i) At or above p =0.7 up to p = 0.8 no (pure strateg) equilibrium exists. (ii) The bribe inducement equilibrium exists at all p <0.7. ~ ^ p =0.3, p =0.7 p =0.45, p =0.55 /(+)=0.4 /(+)=0.33 Bribe inducement range of p : p < Bribe inducement range of p : 0. p < 0.45 Prob. Zero profit prices Prob. Zero profit prices p =0.05 p =0.95 p b =0.5 p b =0.85 p =0.0 p =0.90 p b =0.0 p b =0.80 p =0.3 p =0.87 p b =0. p b = =0.9, 0 =0.935 Constraints: ~ =0.63 ( 0 )=0.9 h =0.08 (h )= =0.36, 0 =0.875 Constraints: ~ =0.7 ( 0 )=0.93 h =0.036 (h )= =0.6, 0 =0.875 Constraints: ~ =0.75 ( 0 )=0.99 h =0.047 (h )=0.037 Comments: (i) At or above p =0.5 up to p = 0.30 no (pure strateg) equilibrium exists. (ii) The bribe inducement equilibrium exists at all p <0.5. ~ p =0.05 p =0.95 p b =0.9 p b =0.8 p =0.0 p =0.90 p b =0.4 p b =0.76 p =0. p =0.89 p b =0.4 p b =0.76 ^ 0 =0.35, 0 =0.94 Constraints: ~ =0.37 M0 =0.385 ( 0 )=0.45 h =0.04 (h )= =0.385, 0 =0.9 Constraints: ~ =0.378 M0 =0.396 ( 0 )=0.457 h =0.083 (h )= =0.393, 0 =0.9 Constraints: ~ =0.379 M0 =0.397 ( 0 )=0.459 h =0.09 (h )=0.09 Comments: (i) At or above p =0. up to p = 0.45 no (pure strateg) equilibrium exists. (ii) The bribe inducement equilibrium exists at all p <0.. 46

47 strictl less than ψ (π 0 ), confirming the inducement of briber of team. Briber of team is also ruled out for π 0 being greater than h. Deviation to no-briber is also ruled out because π 0 < π. However, at higher p beond 0.6 one of the constraints will be violated and our proposed equilibrium does not exist. Column shows the effect of a decline in λ i, while still representing the case π < Here λ and λ fall to 0.3 and corruption becomes more costl. If punter I bribes team and bets on ticket, the bookies expected loss to punter I will rise; to offset that the must get +. a higher expected profit from the naive punters. Therefore, π 0 must rise. At p = 0.05, π 0 rises to 0.9. Here too we see that all the relevant constraints are satisfied to ensure that no possible deviation will occur to undermine the zero profit, bribe inducement equilibrium. In column 3, we consider a special case, where λ = λ = 0; that bribing will lead to certain defeat of the team. We also increase the size of the influential punter s wealth z from 0.3 to 0.5; this implicitl reduces Ω and Ω to. and + to Here we get π > + so that the relevant upper bound (for preventing no-briber deviation) is π M0 instead of π. Larger values of z and smaller (λ, λ ) increase the potential damage from briber; for the same reasoning applied to column, price of ticket increases further. In fact, it rises sharpl to 0.35 (at p = 0.05). Here, π M0 > π in all three values of p (0.05, 0.0, and 0.). π 0 is strictl less than π M0 in all three situations. All other relevant constraints are also satisfied. In the last row, we note the ranges of p (restricting attention to p p ) at which the bribe inducement equilibrium does not exist. We also know that there cannot be an other equilibrium. Comparing column with column we see that as λ falls from 0.5 to 0.3, the range for the non-existence of equilibrium expands from [0.7, 0.8 to [0.5, Further, in column 3 where λ drops to zero (and also z increases) the range for no equilibrium significantl expands to [0., It is also worth emphasizing that if the bribe inducement equilibrium exists at some p, then at all lower values of p (assuming p < p ) the bribe inducement equilibrium should exist. This is an insight we gain from our numerical exercise. Example. Suppose λ = λ = 0, Ω = Ω = Ω. Then φ = φ =, and p +Ω = and ˆp = Ω. If Ω <, then ˆp +Ω < p. As in Proposition 4, we consider p (ˆp, p ). Next, it can be shown that at the proposed bribe prevention equilibrium π = π =, +Ω [ each bookie s expected profit is EΠ BP = Ω( Ω) +Ω. Now consider a unilateral deviation from the proposed equilibrium b slight undercutting on both tickets. This gives an expected profit approximatel, EΠ BI = [ Ω( Ω) +Ω z(µ + µ )Ω. Such deviation is unprofitable, if +Ω 47

48 Table 3: Feasible (, ) for positive profit, bribe prevention Parameter specification Case : = =0, z=0.5, =0.8 ^ ~ p 0. 44, p Parameter specification Case : = =0, z=0.4, =0.5 ^ ~ p 0. 33, p , and 0. 5, and p = , p = , , p = , p = , p = , p = , p = , p = , p = , p = , = = = Feasible Set = Case : p = = =0.5 Figure 7: Feasible access probabilities 48

49 EΠ BP EΠ BI or equivalentl, [ Ω (µ + µ ). z + Ω Further, slight undercutting on ticket and ticket each is unprofitable if µ > µ and µ > µ respectivel, where µ and µ are given in (A.) and (A.) respectivel. For this example, these critical values of µ and µ reduce to µ = [ p ( + Ω) ( + Ω)[p + z, µ = [ p ( + Ω) ( + Ω)[p + z. In addition, to prevent large-scale undercutting condition (8) must be satisfied, which in this case gives rise to the following two restrictions: [ µ p + z ( + Ω) p p + [p + z µ, µ [ [p + z + µ p ( + Ω). p Now we construct two sets of numerical examples in Table 3 and verif that the set of (µ, µ ), which satisfies each of the above-mentioned conditions, is indeed non-empt. Two columns present two cases or examples each with a series of probabilit values considered. In column, we set w + α(f + f I ) = 0. and z = 0.5. This specification gives rise to Ω = 0.8, and ˆp = Ω = 0.44, p +Ω = = Moreover, φ +Ω = φ = = 0.56, which +Ω is equal to our proposed equilibrium prices. We then consider several values of p from the interval (0.44, 0.56) and show that at each of these p values there is a non-empt set of µ and µ values such that no deviation from π = π = 0.56 is profitable. Since these prices exceed p and p, expected profit for each bookie under bribe prevention is strictl positive. Under this parameter specification one constraint on (µ, µ ) that would commonl occur at all p (0.44, 0.56) is µ + µ 0.67; this ensures that slight undercutting on both tickets is not profitable. Then there are four additional constraints to examine, which will var depending on p. Consider p = There are individual restrictions on µ and µ to rule out undercutting on a single ticket, as given in (A.) and (A.). The other two constraints are given b (8), which rules out large-scale undercutting. The same constraints are then reproduced at higher values of p in the interval (ˆp, p ). As can be seen, in each case, the feasible set of (µ, µ ) is non-empt. Next, in column we set z = 0.4 leaving everthing else unchanged. As punter I s wealth increases, Ω falls (to 0.5) leading to an expansion of the interval (ˆp, p ) to (0.33, 0.67). 49

50 Condition µ + µ 0.5 prevents slight undercutting on both tickets. Other constraints on µ and µ are derived considering five different values of p from 0.36 and At each of these values of p the feasible set of (µ, µ ) that would support the proposed bribe prevention equilibrium is shown to be non-empt. Fig. 7 illustrates the case of p = 0.63 on the (µ, µ ) plane for the scenario considered in column, i.e. (z = 0.4, Ω = 0.5). Supplementar material The online version of this article contains additional supplementar material. Please visit doi:... References Akerlof, G.A., 970. The market for lemons, qualitative uncertaint and the market mechanism. Quart. J. Econ. 84, Bag, P.K., Saha, B., 00. Should a bookie induce match-fixing? The monopol case. mimeo. Dell Ariccia, G., Friedman, E. and Marquez, R., 999. Adverse selection as a barrier to entr in the banking industr. RAND J. Econ. 30, Duggan, M., and Levitt, S.D., 00. Winning isn t everthing: Corruption in Sumo wrestling. Amer. Econ. Rev. 9, Glosten, L., Milgrom, P., 985. Bid, ask and transaction prices in a specialist market with heterogeneousl informed traders. J. Financial Econ. 4, Konrad, K., 000. Sabotage in Rent-seeking contests. J. Law, Econ. Organization 6, Kle, A., 985. Continuous auctions and insider trading. Econometrica 53, Levitt, S.D., 004. Wh are gambling markets organised so differentl from financial markets? Econ. J. 4, Mookherjee, D., Png, I., 99. Monitoring versus investigation in law enforcement. Amer. Econ. Rev. 8, Morris, S., Shin, H., 998. Unique equilibrium in a model of self-fulfilling currenc attacks. Amer. Econ. Rev. 88, Morris, S., Shin, H., 003. Global games: Theor and applications. In: Advances in Economics and Econometrics, the Eighth World Congress, eds. M. Dewtripont, L. Hansen and S. Turnovsk, Cambridge Universit Press. 50

51 Ottaviani, M., Sorensen, P.N., 005. Parimutuel versus fixed-odds markets. Working paper, Marco.aspx. Ottaviani, M., Sorensen, P.N., 008. The favorite-longshot bias: An overview of the main explanations. In: Handbook of Sports and Lotter Markets, ed. Donald B. Hausch and William T. Ziemba, Elsevier North Holland. Preston, I., Szmanski, S., 003. Cheating in contests. Oxford Rev. Econ. Polic 9, Rose-Ackerman, S. Corruption: A Stud in Political Econom. Academic Press, New York (978). Shin, H., 99. Optimal betting odds against insider traders. Econ. J. 0, Shin, H., 99. Prices of state contingent claims with insider traders, and the favouritelongshot bias. Econ. J. 0, Shin, H., 993. Measuring the incidence of insider trading in a market for state-contingent claims. Econ. J. 03, Stiglitz, J.E., Weiss, A., 98. Amer. Econ. Rev. 7, Credit rationing in markets with imperfect information. Strumpf, K.S., 003. Illegal sports bookmakers. Working paper, cigar/. Winter, S., Kukuk, M., 008. Do horses like vodka and sponging? On market manipulation and the favourite-longshot bias. Applied Econ. 40, Wolfers, J., 006. Point shaving: Corruption in NCAA basketball. Amer. Econ. Rev. Papers and Proceedings 96,