Corrupt Bookmaking in a Fixed Odds Illegal Betting Market

Parimal Kanti Bag∗ Bibhas Saha†

March 16, 2016

Abstract

Where sports gambling is illegal, underground betting may involve corrupt book-makers secretly fixing popular contests. In such scenarios which team is likely to bribed– the favorite or the underdog? The answer depends on the nature of the punters’ be-liefs. The analysis makes two main observations assuming exogenous monitoring by lawenforcement officials. If the punters have naive beliefs uncorrelated with the Nature’sdraw of the teams’ winning odds, the underdog will be bribed, provided the impactof sabotage is moderate. If the impact of sabotage is severe (alternatively marginal),almost all (no) contests will be fixed. If the punters’ beliefs are correlated with theNature’s draw, the favorite will be bribed. Such pattern of fixing, particularly the sur-prising possibility of bribing the underdog, highlights the need for some unconventionalmonitoring ex post.

JEL Classification: D42, K42. Key Words: Illegal (sports) betting, bookie,punters, corruption, match-fixing, sabotage, whistleblowing, naive/correlated beliefs,big upsets.

∗Department of Economics, National University of Singapore, Faculty of Arts and Social Sciences, AS2Level 6, 1 Arts Link, Singapore 117570; E-mail: ecsbpk@nus.edu.sg†Durham University Business School, Durham, U.K. DH1 3LB; E-mail: b.c.saha@durham.ac.uk

1 Introduction

Deliberate underperformance can occur in any contest for a variety of reasons. In many

professional sports, such as football, cricket, tennis, snooker and horse racing, alleged un-

derperformance under the influence of unscrupulous bettors is a common occurrence.1 In

legal betting markets, corrupt influence may be spotted by bookmakers or the gambling

regulatory authority. But in countries (some in Asia) where gambling is not legal but people

still gamble, the bookmakers themselves can try to manipulate the outcome of a contest.

Some of the spot-fixing controversies in India’s high profile cricket league, IPL, suggest

that underground bookmakers themselves were involved in fixing (see Hawkins, 2013). Head-

lines such as “Football’s authorities fighting $1 trillion crime wave powered by illegal bet-

ting markets in Asia” (http://www.telegraph.co.uk/sport/football/international/9848868/-

Footballs-authorities-fighting-1trillion-crime-wave-powered-by-illegal-betting-markets-in-Asia.html),

or the BBC report that “According to the Doha-based International Centre for Sports Secu-

rity, India’s illegal sports betting market is worth some $150bn a year” (http://www.bbc.com/-

news/business-35493058) suggest that illegal gambling-related corruption is too big an issue

to be ignored.

For horse races, Shin (1991, 1992) had modeled the problem of insider betting in fixed

odds markets under monopoly and competitive bookmaking.2 But insider betting has more

to do with using privileged information rather than exerting influence to alter the outcome of

a contest, which match-fixing is all about. Extending Shin’s framework Bag and Saha (2011,

2015) modeled match-fixing under competition and monopoly respectively. In both papers,

the bookmakers are honest, and their pricing strategy recognizes the threat of match-fixing

coming from an anonymous punter.3 Nature of the market equilibrium and admissibility of

match-fixing were the key focus in those two papers.

In this paper, we consider an environment where betting is organized through a secretive

network due to legal prohibition on gambling. Secret network allows the bookmaker to not

only enjoy sufficient market power, but also exert corrupt influence on a contest if he so

1BBC online news have many such reports: http://news.bbc.co.uk/1/hi/programmes/panorama/2290356.stm(horse races); http://news.bbc.co.uk/sport1/hi/tennis/7035003.stm or “Tennis match-fixing ‘a secret on thetour everybody knows’ ” at http://www.bbc.com/sport/tennis/35356550 (corruption in tennis);

http://news.bbc.co.uk/sport1/hi/football/europe/4989484.stm (fixing in football);http://news.bbc.co.uk/sport1/low/cricket/719743.stm (fixing in cricket);http://news.bbc.co.uk/2/hi/uk news/8656637.stm (snooker).Formal studies of sports corruption are by Wolfers (2006), and Duggan and Levitt (2002). Strumpf (2003)

and Winter and Kukuk (2008) study the betting markets and discuss how betting odds may be affected ifillegal activities have taken place.

2See also Glosten and Milgrom (1985) for insider trading in financial markets, and Ottaviani and Sorensen(2005, 2008) for analysis of parimutuel betting markets.

3Konrad (2000) is a theoretical analysis of sabotage in general contests with no reference to betting.

1

wishes. The contest in question is a two-team sports match, with the bookmaker offering

fixed odds bets on either team’s win. We ask: Would the monopolist stay honest or resort

to corrupt bookmaking, and if he turns corrupt what type of contest is he likely to fix?

To answer this question we consider a setup where the bookmaker has the precise knowl-

edge of the probability of each team’s win (i.e. Nature’s draw), while the bettors do not. In

addition, he has links with corrupt players of either teams which he can exploit to enhance

his informational advantage. To study his incentive to do so we consider two belief structures

of the punters. In one, their beliefs are exogenous. The average belief of the bettors in this

case is in general different from the Nature’s draw. In the other belief structure, bettors’

beliefs are correlated with the Nature’s draw, and the correlation is such that the average

belief of the bettors is exactly equal to the true probability. In either case, the bettors

are unaware of the bookmaker’s corrupt link. The exogenous or naive belief structure has

been utilized in Shin (1991, 1992) and Bag and Saha (2011, 2015). Bag and Saha (2015)

shows, in particular, that if bettors are partially rational and conscious of the possibility

of match-fixing (i.e. non-strategic), the betting market becomes very thin. Naivety and

non-strategic bettors are therefore two important reasons both for bettor participation and

bookmaker’s profitability in sports betting. We would therefore stay with the non-strategic

bettors assumption even for the correlated beliefs formulation.

� Results, intuitions and policy implications. In either of the two belief structures,

bookie’s profit from honest pricing is maximum when the contest is most lopsided, so that

almost all bettors can be induced to bet on the losing team. So when the contest is not

lopsided, it can be made one, via match-fixing, as far as possible, given the impact of

sabotage. In the exogenous beliefs case, where the average belief can be different from the

Nature’s draw, bribing the underdog makes the contest most lopsided ensuring a higher

chance of its defeat and in turn guiding the majority of bettors to bet on it. This leads to

a surprising result that the bookie would prefer to bribe the underdog. One explanation for

this puzzling result is that when bribery does not ensure defeat with certainty, a bookmaker

will be apprehensive about bribing the favorite because in case it goes on to winning the

contest and many have bet due to its attractive pricing the bookie will incur a huge loss.

With the longshot this risk is lower. This way, the bookie’s belief (after fixing) moves away

as far as possible from the belief of the average bettor.

In the case of correlated beliefs, the bettors’ beliefs are concentrated around the Nature’s

draw within a small band. Bettors here are partially informed, and their average belief

coincides with the true probability. Honest pricing leads to splitting the bets equally with

the prospect of profit being greatly dampened. Match-fixing enables the bookie to make

the punters en masse bet on the losing team, without having to reduce the price of the bet

2

proportionately. That is to say, while the winning probability of the bribed team can be

knocked down by a significant proportion and well below the most pessimistic bettor’s belief,

to induce en masse betting price does not have to fall below the most pessimistic belief. This

also means the wedge between the ‘rigged probability’ and the ‘lowest belief’ will only get

larger when the bribed team is favorite. Even the most pessimistic bettor’s belief then is so

high that a higher price can be set to reduce the expected payout, while lowering its win

probability as far as the impact of the sabotage permits. Thus, in this case favorite is the

bribe target, which is consistent with conventional wisdom.

Of course, in either belief settings, if the impact of sabotage is very high or very low the

prediction is straightforward. In the severe impact case, any team can be bribed, barring

a small set of highly lopsided contests (which do not need to be fixed). In the low impact

case, the gains are so small that fixing is not worth it.

While our analysis are carried out under the two polar assumptions, exogenous vs. cor-

related beliefs, the reality must lie somewhere in between with a section of bettors partially

informed while others may be naive. Thus, match-fixing prediction from an outside ob-

server’s point of view is likely to be far from clear. Nevertheless, the insights to be derived

from the analysis should provide some guidance.

Another point to come out of our exercise is a lesson on enforcement. Investigation

of match-fixing is assumed exogenous. Should enforcement be outcome dependent, say the

enforcement authority choose to investigate only when there is an upset? One of our results

suggests that such biased enforcement could be quite misplaced as sometimes the bookie

prefers to bribe the weak team. In fact, as a recent work by Chassang and Miquel (2013) has

shown, it might be better not to make enforcement too sensitive.4 Otherwise the bookmaker

is likely to bribe the team whose loss is less likely to raise the enforcement authority’s

suspicion. The takeaway message is thus the intricate inconsistency between ex-ante and

ex-post monitoring: unless enforcement authorities can commit to a monitoring mechanism,

it is going to be plagued by instability.

In section 2 we present the model. The main analysis appears in sections 3 and 4, with

conclusions in section 5. Formal proofs appear in an Appendix.

4The authors’ analysis is about whistleblowing, and how making principal’s investigation too sensitiveto whistleblower’s reports undermines monitoring. Our adoption of the exogenous investigation strategyroughly incorporates the lessons from Chassang and Miquel.

3

2 The model

The most likely scenario of our model is the one where gambling is illegal and betting is or-

ganized through a secretive and personalized network giving rise to a captive (i.e. monopoly)

market. The sole bookmaker, called the bookie, sets the odds on each of two teams winning

a contest, a sports match for example. Odds setting is equivalent to setting the prices of

two tickets; ticket i with price πi yields a dollar whenever team i wins the contest and yields

nothing if team i loses. The match being drawn is not a possibility (by assumption).

In the absence of any external influence, the probability that team 1 will win, as drawn

by Nature, is p1; for team 2 it is p2 = 1−p1. These probabilities are known to the bookie and

the team players, but not the punters or bettors.5 Both teams have some corrupt players,

who are willing to underperform for secret financial gains.

The bookmaker has links with the corrupt players and may bribe a particular team of

his choice to shed their probability of winning from pi to λpi, (0 ≤ λ < 1), where λ is

exogenously given. After fixing the match the bookie posts the prices, following which the

punters bet.

There are a continuum of punters, having 1 dollar each to bet and y dollars collectively.

They are parameterized by individual belief (i.e., the probability) q that team 1 will win; q

is distributed over [0, 1] by a cumulative distribution function F(q) with probability density

f(q) > 0 for q ∈ [0, 1]. Importantly, q is a private signal of the punters and it is uncorrelated

to p1.

We are making two important assumptions about the punters. First, they believe that

their private signal is team 1’s true winning probability. This assumption is due to Shin

(1991), for which punters are called ‘naive’ or non-rational. As in Shin (1991) this assump-

tion helps to model profitable bookmaking in a simpler way. Later on we will relax this

assumption to some extent. Our second assumption is that punters are completely unaware

of the possibility of match-fixing. We will comment on the implications of modifying this

assumption.

One reason for making these two assumptions is to see whether the bookmaker would

always engage in match-fixing, considering that his punters are extremely naive and un-

suspecting. There is also anecdotal evidence from Asia suggesting a strong possibility that

despite several well publicized betting scandals bettors still believe the process to be by and

large fair.

The bookmaker, however, is not fully unchecked in its corrupt engagement. There is an

5Levitt (2004) recognizes that bookmakers are usually more skilled at predicting match outcomes thanordinary punters.

4

enforcement authority that is aware of illegal betting and match-fixing. We assume that

they follow a simple rule:

Exogenous enforcement: Check randomly if betting is taking place and investigate the

losing team, both with an exogenous probability α, and uncover any foul play.

For organizing betting the bookie will be fined sB; the bettors are, however, spared of any

fines or prosecution. So the bookie’s expected fine for organizing betting is αsB, which we

denote as s. Further, if match-fixing is uncovered there is an additional penalty on the bookie

and the participating players. Clearly, all these fines should be factored in the bribe amount

the bookie agrees to pay to the player(s). Let the aggregate expected cost of match-fixing,

contingent on losing, be denoted as c. Following Bag and Saha (2011, 2016) we assume that

though the match-fixing agreement is made ex ante the promised bribe is paid ex post only

if the contacted team loses.6

Finally, we impose a restriction on the distribution function of the punters’ beliefs, which

ensures certain monotonicity properties of the bet prices.7

Assumption 1. 1−F(x)f(x)

and F(1−x)f(1−x)

are decreasing in x, where x is a generic variable.

Punters’ Betting Decision. The punters adopt the following rule: Bet on team 1 if and

only if qπ1≥ max{ 1−q

π2, 1}; bet on team 2 if and only if 1−q

π2≥ max{ q

π1, 1}.

Throughout we impose the following restriction known as the Dutch-book restriction:

Assumption 2. The bookie must choose prices 0 ≤ π1, π2 ≤ 1 such that π1 + π2 ≥ 1.

This assumption avoids the problem of free money. With this (Dutch-book) restriction,

the punters with belief q ≥ π1 will buy ticket 1 (betting 1 dollar each), and punters with

belief 1− q ≥ π2 (or q ≤ 1− π2) will buy ticket 2. Punters with beliefs 1− π2 < q < π1 do

not bet at all.

3 To bribe or not to bribe

Before we analyze the bookie’s decision problem, let us first describe the game.

6This arrangement, as shown in Bag and Saha (2011), is incentive compatible for the corrupt players tounderperform. To provide some details on the expected cost c, assume that the player forgoes a prize moneyw by losing, and if detected (with probability α) he will have to pay sP as a fine. So the bribe amount mustbe at least w + αsP. Suppose the bookie pays only the minimum bribe. Then in addition to this amounthe faces a penalty αsM for match-fixing. Thus c = w + α(sM + sP). This is bookie’s expected cost ofmatch-fixing conditional on the bribed team losing.

7Uniform distribution satisfies this assumption.

5

Game Γ

Stage 1. Nature draws p1 and reveals it to the bookie, and the players; the punters draw

their respective uncorrelated private signals q.

Stage 2. The bookie decides whether to engage in match-fixing or not.

Stage 3. Prices (π1, π2) are set for the tickets on respective teams’ win, where 0 ≤ π1, π2 ≤1.

Stage 4. The punters place bets according to their beliefs. The match is played out

according to the teams’ winning probabilities (p1, 1 − p1) or (λp1, 1 − λp1) (where team

1 is bribed), or (1 − λp2, λp2) (where team 2 is bribed) and the outcome of the match is

determined.

Stage 5. Finally, the prosecution follows its anti-corruption investigation policy. ||

� Honest bookmaking. First we consider the case where the bookie does not fix the

match. In accordance with the Nature’s draw of p1, he posts (π1, π2), and the punters bet

according to their decision rule specified earlier. This gives rise to the following expected

profit of the monopolist bookmaker:

EΠh = y

∫ 1π1

(1−p1

π1)f(q)dq+ y

∫ 1−π20

(1−p2

π2)f(q)dq− s

= y

[1− F(π1) −

p1

π1+ p1

F(π1)

π1+ F(1− π2) − p2

F(1− π2)

π2

]− s.

It is clear from the above that to get nonnegative profit from each ticket, the bookie

must set π1 ≥ p1 and π2 ≥ p2 or p1 ≥ 1− π2. None of the bettors whose beliefs lie between

1 − π2 and π1 will bet. Intuitively, such beliefs are sufficiently close to the bookie’s belief

(and the true probability), and therefore the bookie cannot make a profit out of them. Of

course, then by reducing the ticket prices, the gap between π1 and 1− π2 can be reduced to

attract more betting. This will increase the revenue, but will also increase the payout. So

there is an optimal gap between π1 and 1 − π2 (or equivalently an optimal volume of bet)

that maximizes profit.

The first-order conditions for profit maximization are

∂EΠh

∂π1= y · p1f(π1)

π21

[1− F(π1)

f(π1)−

(π1 − p1)

p1π1

]= 0, (1)

∂EΠh

∂π2= y · p2f(1− π2)

π22

[F(1− π2)

f(1− π2)−

(π2 − p2)

p2π2

]= 0. (2)

The following lemma shows that optimal prices πi (i = 1, 2) are strictly increasing in pi

6

and bounded between 0 and 1.

Lemma 1 (Monotonic prices). Optimal prices (πh1 , πh2 ) solving (1) and (2) satisfy the

following properties: ∂πh1/∂p1 > 0, ∂πh2/∂p2 > 0 (i.e. ∂πh2/∂p1 < 0). Also, πh1 (0) = 0 and

πh1 (1) = 1. Symmetrically, πh2 (p2 = 0) = 0 and πh2 (p2 = 1) = 1, or πh2 (p1 = 1) = 0 and

πh2 (p1 = 0) = 1.

The resultant expected profit of the bookie will be a U-shaped curve with the highest

profit being equal to y− s at either end of the probability spectrum.

Substituting πh1 (pi) in EΠ(.) we write EΠh(p1) ≡ EΠ(πh1 (p1), πh2 (p2)) and apply the

envelope theorem to derive:

∂EΠh(p1)

∂p1= y

[−1− F(πh1 (p1))

πh1 (p1)+F(1− πh2 (p2))

πh2 (p2)

]. (3)

Proposition 1 (Bookie’s preference for unbalanced contests). The bookie’s profit

EΠh(p1) is U-shaped and has a unique minimum at p∗1 where the following relation holds:

1− F(πh1 (p1))

πh1 (p1)=F(1− πh2 (p2))

πh2 (p2).

If f(.) is symmetric, then p∗1 = 1/2 and πh1 (1/2) = πh2 (1/2).

Proposition 1 forms the basis on which we build the rest of our results. What this key

result suggests is that the more “uneven” the contest is, the better it is for the (honest)

bookie. The reason is that in a lopsided contest most bettors can be steered to bet on the

team that the bookie believes highly likely to lose.

For the rest of the analysis, we allow (honest) bookmaking to be always profitable despite

its illegality, by assuming that the fine on betting is not large.

Assumption 3. The minimum profit from honest bookmaking is positive: EΠh(p∗1) > s.

� Match-fixing. Match-fixing is manipulation of the winning probability and deviating

from the Nature’s draw. If firm 1 is bribed deviation occurs (downwardly) from p1 to λp1

and if team 2 is bribed deviation occurs (upwardly) from p1 to (1 − λ(1 − p1)). These

deviations allow the bookie to arrive at different points on the EΠh curve. Though EΠh

will no longer be relevant under match-fixing, as we show shortly the expected profits under

match-fixing will be a simple translation of EΠh, where the translation factor is λ.

7

Taking leads from Proposition 1 we can imagine a symmetric U-shaped expected profit

curve with minimum occurring at p1 = 1/2. Now let us do a thought experiment. Suppose

Nature picks p1 = 4/10 (i.e. team 1 is an underdog) and assume λ = 1/2. If the bookie

bribes team 1, he reduces p1 to 1/5. Alternatively, if he bribes team 2, p1 will rise to

(1− 3/10) = 7/10. If we ignore the cost of bribery, clearly we have EΠh(1/5) = EΠh(4/5) >

EΠh(7/10). Therefore, as long as the expected bribery cost is not overriding, the bookie

should bribe team 1, not team 2.

That means, the bookie would prefer to bribe the team which is already an underdog.

Intuitively, bribery of underdog makes the game more lopsided.This is by and large our key

observation of this section.

Without loss of generality let us consider the case of bribing team 1. The analysis of

bribing team 2 is symmetric. Upon bribery, team 1’s probability of winning now falls to

λp1 and team 2’s probability rises to (1− λp1). The bookie will also have to account for an

additional match-fixing cost, c, which is incurred with probability (1 − λp1) (i.e. if team 1

loses). So his expected profit changes to

EΠb = y

[1− F(π1) −

λp1π1

+ λp1F(π1)

π1+ F(1− π2) − (1− λp1)

F(1− π2)

π2

]− (1− λp1)c− s.

The first-order conditions for profit maximization will be the same as Eqs. (1) and (2),

with the adjustment that p1 is replaced by λp1 and p2 by (1− λp1). Therefore, the optimal

prices, denoted (πb1 , πb2), will have the same properties as the honest prices (πh1 , π

h2 ) as stated

in Lemma 1.

In fact, πb1(p1) = πh1 (λp1) and πb2(p1) = πh2 (1 − λp1). Therefore, πb1(p1) ≤ πh1 (p1) and

πb2(p1) ≥ πh2 (p1). As λp1 ∈ [0, λ], we will have πb1(p1 = 1) = πh1 (p1 = λ) and πb2(p1 = 1) =

πh2 (p2 = 1 − λ). As p1 rises from 0 to 1, πb1 rises from zero to πh1 (λ) (which is less than 1),

and πb2 falls from 1 to πh2 (1− λ) (which is positive).

Under bribery (of team 1) the price of ticket 1 will be reduced to attract more bets on

team 1, as its probability of winning has fallen, and correspondingly the price of ticket 2 will

be raised to discourage betting on team 2, because its winning chance has risen.

Since πb is a translation of πh, profit from bribery at a given p1 can be calculated

by plugging λp1 at the profit function under honest bookmaking and then subtracting the

expected cost of bribery. The following proposition makes this claim precise.

Proposition 2 (Odds rigging). Bribery of a team is accompanied by lowering the price

of the bribed team’s ticket and raising the price of the rival team’s ticket. Specifically, the

8

optimal prices under bribery of team 1 are:

πb1(p1) = πh1 (λp1), πb2(p1) = π

h2 (1− λp1)

Thus, πb1 ≤ πh1 , πb2 ≥ πh2 , and ∂πb1/∂p1 > 0, ∂πb2/∂p1 < 0. Also, πb1(0) = 0 and πb1(1) =

πh1 (λ), and πb2(p2 = 0) = 0 and πb2(1) = πh2 (1− λ).

The expected profit from bribery (of team 1) can be written as

EΠb(p1) = EΠh(λp1) − (1− λp1)c. (4)

When is bribery profitable? To answer this question we are now going to compare

EΠb(p1) and EΠh(p1). Crucially, the answer will depend on the uncorrupted probability p1

and the impact of sabotage λ.

We relegate the mathematical details to Appendix and here we motivate our claim graph-

ically. Fig. 1a depicts the EΠb curve against p1 for different values of λ.

Sabotage with maximum impact. If λ = 0, which signifies maximum impact of sabotage,

then EΠb(p1) = EΠh(0) − c = y − s − c is constant at all p1. In other words, the expected

profit from bribery will be a flat line below the highest value of the EΠh curve. In Fig. 1a

the topmost line represents this profit curve.

Sabotage with moderate impact. With λ > 0 the expected penalty is actually lower, but

the profit from the punters is also lower. On balance profit is lower than the maximum

impact case. Three curves show how λ affects EΠb. With λ > 0, but still very small,

expected profit falls everywhere, as shown by the second curve from the top (Fig. 1a). After

a critical value of λ the profit curve becomes nearly U-shaped. With further increase in λ

the profit curve slides leftward by displacing the minimum point.

What is the critical value of λ? Since EΠb is a function of λp1, instead of p1 and λ

separately, if EΠb has a minimum at some λp1, the minimum would remain invariant to

individual changes in p1 or λ. Suppose the minimum occurs at λp1 = k1. Then for any

given λ, we can trace the minimum of EΠb at p1 = k1/λ. If k1 = λ, the minimum of EΠb

occurs at p1 = 1, and if k1 > λ the minimum occurs in the interior as shown by two curves

corresponding to λ ′ and λ ′′ (λ ′′ < λ ′) in Fig. 1a. Thus, k1 is the critical value of λ.

Shape of the bribery profit curve. We can easily verify the almost U-shape of the EΠb

curve, when it has an interior minimum. Its slope and curvature are linked to that of EΠh

9

p1

EPb

y-c-s

(a) Profit from match-fixing

1

l=0

l<k1

p1

EP0, EPb

y-c-s

(b) Shrinking range of bribery

1

EPb(l=0)

EPb(l<k1)

EPb( )

p1*

y-s EP0

''

1 / lk

''' ll

1

' kl

'

1 / lk 1p

l

1

'' kl

a b

Figure 1: Regions of bribery

in a particular way, as can be seen from Eq. (4):

EΠ ′b(p1) = λ[EΠ ′h(λp1) + c]; (5)

and EΠ ′′b(p1) = λ2EΠ ′′h(λp1) > 0. (6)

The slope of the EΠb curve at p1 would be the λ fraction of the slope of the EΠh curve at

λp1 plus c. Thus, when EΠb is at its minimum at p1 = k1λ

, EΠh(.) must have a negative

slope equal to c at p1 = k1. Likewise, when EΠh(.) achieves its minimum at p1 = p∗1, the

EΠb curve must be rising atp∗1λ

.

Further, as Eq. (6) shows EΠ ′′b(.) > 0 (i.e. convex) the zero slope of EΠb at p1 =k1λ

must

yield a unique minimum. Also as EΠh is convex ( and it is U-shaped) EΠb is also convex,

and so when it has an interior minimum, it must be nearly U-shaped.

Comparison of two profit curves. Fig. 1b compares the two profit curves for different

values of λ. As the EΠb curve will have less curvature (see Eq. (6)) than the EΠh curve, the

marginal profit under bribery will change at a slower rate, and thus it takes a longer range

of p1 to reach its minimum.

We compare profits for three values of λ, namely 0, a λ between 0 and k1 and a limiting

value of λ called λ, at which the two profit curves become tangent to each other.

Sabotage with maximum impact. If λ = 0 (and c is not large), then the two curves – EΠh

and EΠb – must cross each other twice at points such as a and b. Between a and b the

bookie will bribe (team 1). Contests outside this interval, which are also highly unbalanced

contests, will never be fixed.

Sabotage with moderate impact. Now consider λ > 0 but still less than k1. In Fig 1b from

the intersection of EΠ0 and EΠb(λ < k1) we see that the region of bribery, the interval [a, b],

10

gets smaller and the scope of bribing the favorite diminishes much more than the scope for

bribing the underdog.

As λ increases further, the interval [a, b] continues to shrink, and ultimately disappears

after a critical λ say λ; the region of bribery shrinks to a single point like p1. At λ > λ

nowhere is bribery optimal. We refer to λ as the smallest impact of sabotage that makes

match-fixing optimal, and we call the contest p1 as the most vulnerable contest, because it

will be fixed at all λ ∈ [0, λ].

It is clear that p1, which must correspond to the tangency point of the two curves as

shown in Fig. 1b, must be less than p∗1, and if we do not allow too much asymmetry in the

belief distribution, p1 must be less than 1/2. That means, generally one would expect that

the most vulnerable contest will feature the bribed team to be an underdog.

We now state our key observations in two propositions, and then we explain this seemingly

‘counter intuitive’ result of bribing an underdog to lose.

Proposition 3 (Optimality of fixing). Suppose λ is sufficiently small, so that the impact

of sabotage is high. Then there exists a set of contests, belonging to the interval [a, b]

(a > 0, b < 1), in which fixing is preferred by the bookie. Equally, there will be a set of

highly unbalanced contests that will never be fixed.

A positive message of the above proposition is that despite having the punters naive and

unsuspecting, it is never the case that bookie will defraud them in all contests. Moreover,

there is a range of (high) λ that renders no contest worth fixing.

The next proposition characterizes the set of contests optimal for fixing, under the as-

sumption of symmetric belief distribution.

Proposition 4 (Range of profitable bribery). Assume f(q) is symmetric so that p∗1 =

1/2. Then between the option of bribing team 1 and not bribing, bookie’s preferred option

is to bribe, provided team 1 is neither too strong nor too weak and the impact of sabotage is

significant.

Formally, for any given λ < λ the bookmaker will bribe team 1 to reduce its probability of

winning from p1 to λp1 (λ < 1) at all p1 ∈ [a1(λ), b1(λ)], where 0 < a1(.) and b1(.) < 1 and

λ is such that a1(λ) = b1(λ) < 1/2. Moreover, a ′1(λ) > 0 and b ′1(λ) < 0. For λ > λ, bribery

of team 1 is not optimal.

While in general the bookie might not discriminate between favorite and underdog for

bribery, our prediction is that the contest that is most vulnerable to fixing (i.e. a(λ1) or p1)

features the bribed team as a moderate underdog. But this prediction may not conform to

the usual perception of match-fixing where the favorite is generally bribed. This wisdom

11

holds when both the bettors and the bookie have a similar belief about which team is favorite.

In that case, by secretly bribing the favorite the bookie can defraud the bettors.

However, that is not the case with our bettors, who are naive and whose (exogenous)

beliefs may widely diverge from that of the better informed bookie. So the bookie tries to

make a weak team weaker and then entices everybody to bet on it. If the punters tried

to infer the true odds from the quoted price, such a strategy would not work, and in that

environment it is not clear how and when the bookie will be able to make money. So long

as we accept the premise that enough naive bettors must be present for the bookie to do his

business, our unconventional prediction holds.

From a different perspective, that is of enforcement, our prediction actually gains more

weight, if we replace our assumption of exogenous enforcement by the strategy of ‘investigate

only if there is a surprise outcome,’ which invariably implies going after the favorite (when

it loses). Argument for such a strategy is that if a weak team loses there could be no case

for spending public money to investigate. We can see that the corrupt bookie will then

have stronger incentives to bribe the underdog to keep off the radar. Thus perhaps the

enforcement is best kept inscrutable, i.e., exogenous.

That outcome dependent monitoring, instead of exogenous monitoring (as we have as-

sumed), can be problematic has been shown by Chassang and Miquel (2013) in a principal-

agent model with a whistleblower. They argue that the principal’s investigation strategy

must not be too sensitive to the whistleblower’s report. Otherwise anticipation of any retali-

ation from the reported agent might dissuade whistleblowing in the first place. In our context

a similar strategic interaction between investigation and corruption decision is present, al-

though as such there is no explicit whistleblower in our model.

� Which team to bribe? The above analysis can be easily extended to bribing either

team, and then the result of bribing the underdog gets even sharper. Consider bribing team

2 as an alternative option. Applying the symmetry we can say that the optimal prices will

then be

πb21 (p1) = π∗1(1− λp2), πb22 (p1) = π

∗2(λp2)

which result in the following profit and the resulting profit

EΠ2(p1) = EΠ0(1− λp2) − (1− λp2)c− s. (7)

As before, by comparing profit in (7) with that from honest bookmaking, and applying the

same reasoning we will arrive at a similar interval of bribery p2 ∈ [a2, b2], or p1 ∈ [b2, a2]

for λ < λ. If we assume symmetry, then a2 > 1/2 and a2 = 1− a1 and b2 = 1− b1.

12

p1

EP0, EPb

y-c-s

1

EP2

EP1

p1*

=1/2

y-s EP0

a1 a2 b2 b1

Bribe team 1 Bribe team 2

Figure 2: Targeting of bribery

When the interval [b2, a2] overlaps with the interval [a1, b1], and if a contest belongs to

that interval, then the bookie has a choice of which team to bribe. It is not hard to see that

depending the value of λ, (usually at small values) the two intervals will overlap. Without

being unnecessarily too formal, let us focus on a scenario where such an overlap exists and

assume symmetry.

That means, p∗1 = 1/2, a1 ≤ b2 and b1 ≤ a2. By symmetry, of course, a2 = 1 − a1 and

b2 = 1 − b1. Then, a1 ≤ 1 − b1 or a1 + b1 ≤ 1, which also implies b1 ≤ 1 − a1 = a2. In

Fig. 2 we present such a scenario. Honest profit is represented by the U-shaped curve EΠ0

and corrupt profits are represented by EΠ1 (when team 1 is bribed) and EΠ2 (when team 2

is bribed). Two curves intersect at p∗1. Hence, at all p1 ∈ [a1, p∗1] team 1 will be bribed, and

at all p1 ∈ (p∗1, a2] team 2 will be bribed. Outside this interval, prices will be set honestly.

What is interesting to note when bribery occurs, never a favorite is bribed. In particular,

within the interval [b2, b1] where the bookie has a choice of teams, he picks the underdog.8

Proposition 5 (Bribe the underdog). Assume symmetry so that p∗1 = 1/2 and suppose

λ is such that a1 < b2 < b1 < a2, so that the bookie has a choice of team to bribe at all

p1 ∈ [b2, b1]. Then he will bribe team 1 at p1 ∈ [b2,12] and team 2 at p1 ∈ ( 1

2, b1]. In either

case the bribed team is an underdog.

Proposition 5 confirms the bookie’s preference for underdog as a bribe target in the

context of choosing between two teams. We have already noted in Proposition 4 that when

only one team is considered for bribing, the frequency of bribery largely correlates to the

8At p1 = 1/2 he is indifferent between the two teams.

13

frequency of the bribed team being an underdog. This preference leads to a clear demarcation

in terms of never picking a favorite to bribe, when both teams are available for bribing.

We have already explained that this apparently puzzling phenomenon is a natural conse-

quence of the assumption of bettors being naive and the bookie having superior information.

However, we should not overlook that our result does not entirely contradict the con-

ventional wisdom on match-fixing (i.e. bribe the favorite) when bribery induces the bribed

team’s defeat with certainty. But if bribery does not guarantee a loss, the preferred bribery

may be in the unanticipated direction than commonly assumed.

� Example: Uniform distribution. We illustrate Proposition 4 with an example. Sup-

pose the bettors’ belief q is uniformly distributed over [0, 1], with f(q) = 1 and F(q) = q. Un-

der honest bookmaking the bookie maximizes EΠ = y[3−π1−π2−p1π1− p2π2]−s and sets prices

as π∗1 =√p1 and π∗2 =

√p2. This leads to honest profit equal to EΠ0 = y[3−2

√p1−2

√p2]−s.

This function has a minimum at p1 =12

and the minimum value is EΠ0 = y(3− 2√2) − s.

Now consider pricing under bribery of team 1. The bookie will then maximize EΠ =

y[3 − π1 − π2 −λp1π1

− (1−λp1)π2

] − (1 − λp1)c − s, resulting in optimal prices πb1 =√λp1 and

πb2 =√1− λp1. After substituting these values into the profit function we obtain his bribery

profit as EΠ1 = y[3− 2

√λp1 − 2

√1− λp1

]− (1− λp1)c− s.

However, for profit comparison between the two options we need to resort to simulation,

which is reported in Table 1. We present three sets of parameters: (y = 0.80, c = 0.20), (y =

0.80, c = 0.30), (y = 0.90, c = 0.20). We show for each set of parameters the region of bribery

(of team 1) for some select values of λ until λ is reached. When y = 0.80 and c = 0.20,

the largest region of bribery is [0.018, 0.982] which ultimately contracts to a singular point

p1 = 0.30 at λ = 0.36. Beyond λ = 0.36 bribery (of team 1) is not optimal.

If the cost of bribery rises (to c = 0.30), the limiting λ falls, as λ is drastically reduced

to 0.19; however, the limiting contest becomes a little less uneven, as p1 corresponding to

λ increase to 0.34. The reason is that with higher cost of bribery, the expected profit from

corruption falls, and hence the tangency between EΠb and EΠ0 occurs at a higher value of

p1.

A similar effect is observed with an increase in the punters’ collective wealth (y = 0.9).

This also helps to sustain bribery much longer, as λ increases to 0.405. As can be seen, in

all three cases the limiting probability is strictly less than 1/2. That is, the limiting contest

features team 1 as a ‘moderate underdog’.

14

Table 1: Bribe inducement range of p1y = 0.80 c = 0.20 λ = 0.36

λ 0 0.05 0.10 0.15 0.20 0.25 0.30 0.36a1 0.018 0.033 0.05 0.06 0.08 0.11 0.15 0.30b1 0.982 0.850 0.77 0.70 0.64 0.57 0.49 0.30

y = 0.80 c = 0.30 λ = 0.19

λ 0 0.05 0.08 0.10 0.12 0.15 0.18 0.19a1 0.045 0.09 0.11 0.13 0.15 0.20 0.28 0.34b1 0.955 0.77 0.70 0.65 0.61 0.50 0.42 0.34

y = 0.90 c = 0.20 λ = 0.405

λ 0 0.05 0.10 0.15 0.20 0.25 0.35 0.405a1 0.014 0.025 0.035 0.05 0.06 0.08 0.15 0.35b1 0.986 0.87 0.80 0.73 0.67 0.61 0.48 0.35

4 Partially informed bettors

In this section we modify the naive beliefs assumption for bettors. Their private signal q

is now correlated with the Nature’s draw of p1, though as before they believe their signal

to be the true probability and they do not suspect any foul play. Thus the bettors will bet

“informatively” in the sense that the beliefs are closely aligned with the true probability

(and the bookie’s belief). In the real world, such correlated beliefs may arise when bettors

are knowledgeable about the contestants’ strengths and weaknesses. We will see that the

bookie’s informational advantage substantially diminishes in this environment; however, due

to the unsuspecting nature of the bettors he still can engage in match-fixing; but his target

of bribery will be different.

� Correlated beliefs. Our game Γ is modified only in terms of bettors’ beliefs. The bettors

draw their private signal q from an interval [p1 − δ, p1 + δ], where 0 < δ < 1/2 and p1 is

drawn by Nature, which is precisely observed by the bookie and the players as before.

We assume for simplicity that q is uniformly distributed over the interval [p1− δ, p1+ δ]

with density 1/2δ. The average belief of the bettors is precisely p1. However, for consistency

this assumption is applicable to the interval δ ≤ p1 ≤ 1− δ. For p1 outside this interval we

modify our assumption slightly. For p1 < δ, q is uniformly distributed over [0, p1 + δ) with

density 1/(p1+ δ) and for p1 > 1− δ, q is uniformly distributed over (p1− δ, 1] with density

1/(p2 + δ).

Let the collective wealth of the punters be denoted as y and their betting decisions are

guided by the same principle as laid out in section 2. When prices (π1, π2) are such that

π1 ≤ p1 + δ and 1 − π2 ≥ p1 − δ, punters with q ∈ [π1, p1 + δ] bet on team 1, and punters

15

with q ∈ [p1 − δ, 1− π2] bet on team 2; punters whose beliefs fall in between 1− π2 and π1

do not bet. The assumption of π1 + π2 ≥ 1 applies, which ensures that 1− π2 ≤ π1.When p1 ∈ [δ, 1−δ] and π1 ≤ p1+δ the proportion of punters betting on team 1, n1(π1),

is p1+δ−π12δ

. Similarly, when p1 ≤ δ, the same proportion is calculated as p1+δ−π1p1+δ

, and finally

when p1 ≥ 1− δ the proportion is 1−π1p2+δ

.

Combining all three cases into one, and likewise doing the same calculation for betting

on team 2, we write the general expression for the mass of bettors betting on team 1 and

team 2, respectively, as

n1(π1) =min {1, p1 + δ}− π1

min {1, p1 + δ}− max {0, p1 − δ}, n2(π2) =

1− π2 − max {0, p1 − δ}

min {1, p1 + δ}− max {0, p1 − δ}. (8)

It is noteworthy that in contrast to the previous case, the proportion of bettors now

directly depends on p1; n1(π1) + n2(π2) ≤ 1. The bookie takes into account the above

betting pattern generated by (π1, π2) and the draw p1, and then posts his optimal prices.

Note that he will not set π1 below (p1 − δ) and above (p1 + δ), because that would not

increase the sale of ticket 1, but there will be loss of revenues.

First consider honest bookmaking. Assuming both tickets are on offer, the bookie maxi-

mizes the following:

maxπ1,π2

EΠ ≡ y[n1(π1)(1−

p1

π1) + n2(π2)(1−

p2

π2)

]− s (9)

subject to

max {0, p1 − δ} ≤ π1 ≤ min {p1 + δ, 1},

max {0, p1 − δ} ≤ 1− π2 ≤ min {p1 + δ, 1},

π1 + π2 ≥ 1.(10)

The solution to this problem is given in the following proposition. The intuition is that

for π1 the upper bound of the belief distribution, p1+ δ or 1, matters while for π2 (or rather

1−π2) it is the lower bound, whether 0 or p1− δ, matters. These considerations lead to two

different optimal price functions for each ticket.

Proposition 6. The optimal prices under honest bookmaking are

π01 =

{ √p1(p1 + δ), ∀p1 ∈ [0, 1− δ)√p1, ∀p1 ∈ [1− δ, 1],

π02 =

{ √p2, ∀p1 ∈ [0, δ)√p2(p2 + δ) ∀p1 ∈ [δ, 1],

(11)

16

resulting in the following profit function:

EΠ0 =

y

p1+δ

[2+ p1 + δ− 2

√p1(p1 + δ) − 2

√p2

]− s for p1 < δ

yδ

[1+ δ−

√p1(p1 + δ) −

√p2(p2 + δ)

]− s for δ ≤ p1 ≤ 1− δ

yp2+δ

[2+ p2 + δ− 2

√p1 − 2

√p2(p2 + δ)

]− s for 1− δ < p1.

(12)

EΠ0 is symmetric and U-shaped with its minimum occurring at p1 = 1/2.

It is straightforward to check that the highest profit occurs at p1 = 0 and p1 = 1, and it

is equal to y− s, just as was under the naive bettor case. However, elsewhere the expected

profit is likely to be lower now. In particular, the minimum profit (at p1 = 1/2) will be less

than the minimum profit from the previous section (i.e. EΠh) under uniform distribution.

That is because the bookie has now less informational advantage. We assume that the

minimum profit is strictly positive.

� Match-fixing. Now consider the pricing problem for bribing team 1. As before, p1 is

reduced to λp1. One concern in the present case is that the bookie may not always offer both

bets. Depending on the value of λ and p1, it may be more profitable to set π1 = (p1 − δ)

and induce everybody to bet on team 1, and not offer any option to bet on team 2. Mind

that to allow bets on both teams, he must set (generally) p1 − δ < 1− π2 ≤ π1 < p1 + δ. It

is also possible that he can set π2 = p2 − δ and sell no ticket for team 1. However, we shall

see that the allowing the latter is never optimal for him. He will either sell both tickets, or

only ticket 1.

The bookie maximizes the following subject to constraints (10):

EΠb = y

[n1(π1)(1−

λp1

π1) + n2(π2)(1−

1− λp1π2

)

]− (1− λp1)c− s. (13)

Because constraints (10) involve several bounds, we have to pay attention to the com-

plication of which constraint binds and when. It turns out that team 1 will always attract

bets, i.e. n1(π1) > 0, and even there are situations when π1 is restricted to p1 − δ rendering

n1(π1) = 1 and n2(π2) = 0. That is, all bettors are induced to bet on team 1. There is also

the possibility that 1− π2 may be restricted to p1 − δ rendering n2(π) = 0.

Proposition 7. The optimal match-fixing prices are

πb1 =

{max {p1 − δ,

√λp1(p1 + δ)}, ∀p1 ∈ [0, 1− δ)

max {p1 − δ,√λp1}, ∀p1 ∈ [1− δ, 1],

(14)

17

πb2 =

{ √1− λp1, ∀p1 ∈ [0, δ)

min {p2 + δ,√(1− λp1)(p2 + δ)}, ∀p1 ∈ [δ, 1].

(15)

Betting pattern. It might be instructive to discuss the betting pattern induced by the

pricing rule stated in Proposition 7, since in the present context offering bet only on one

team can be optimal. The following discussion should be read in reference to Fig. 3 where

we depict the (p1, λ) configurations for either one bet or both bets being active.

• Betting on team 1 will always be active. Ticket 1 will always be sold. This is

evident from the fact that at all p1 ≤ 1− δ, πb1 < p1+ δ, and at all p1 > 1− δ we have

πb1 < 1, where 1 is the upper bound of the belief distribution. In Fig. 3 in all regions,

A, B and C bets will be placed on team 1.

• Betting on team 2 will also be active at low values of p1. Ticket 2 is bought

and sold if the following configuration holds:

max[0, p1 − δ] < 1− πb2 ≤ πb1 < min[p1 + δ, 1].

From Eq. (15) we see that when p1 ≤ δ, πb2 =√1− λp1, which is clearly positive and

greater than p2 + δ, which implies 1 − πb2 > 0 ≥ p1 − δ (when p1 ≤ δ). So regardless

of the value of λ > 0 team 2 will be bet on, provided p1 ≤ δ.

• When p1 > δ, betting on team 2 will be active if λ exceeds a critical value.

In this case, betting on team 2 will be active if√(1− λp1)(p2 + δ) ≤ p2 + δ, (or

equivalently 1− πb2 ≥ p1 − δ) which yields the condition λ ≥ p1−δp2

. This is the concave

curve separating regions B and C. In region C bets on both teams are active, but in

regions B and A bets on team 1 only are offered and accepted.

Here we need to take note of another constraint 1−πb2 ≤ πb1 (the Dutch-book assump-

tion). It turns out that λ ≥ p1−δp2

will suffice for this as well.

• When p1 > δ, all bettors bet on team 1 if λ is sufficiently small. For this to

occur we must have πb1 = p1 − δ which would occur if√λp1(p1 + δ) ≤ p1 − δ. This

implies λ ≤ (p1−δ)2

p1(p1+δ)for p1 < 1− δ and λ ≤ (p1−δ)

2

p1for p1 ≥ 1− δ. This combined curve

separates region A and B in Fig. 3. In region A all bettors bet on team 1.

• When p1 > δ and λ is moderate, only team 1 is bet on, but some bettors

abstain. This is given by region B, where πb1 > p1 − δ so that the bettors with belief

between p1 − δ and πb1 do not bet on team 1. But in this region πb2 = p2 + δ (or

18

p1

l

1

1 1-d d

1-d

(1-d)2

d/(1-l0) p1

**

l0

(p1-d)/p1

(p1-d)2/p1

(p1-d)2 p1(p1+d)

A

B

C

Region C: Both tickets are sold Region B: Only ticket 1 is sold, but some bettors abstain Region A: All bettors bet on team 1

(1-2d)2 (1-d)

Figure 3: Betting patterns under match-fixing

1 − πb2 = p1 − δ) so that betting on team 2 is not profitable for any bettor. So those

bettors who cannot bet on team 1, must abstain.

The bookie’s profit will vary considerably depending on whether betting occurs in region

A, B or C, and also on the magnitude of λ. For example, if λ ≥ (1− δ) we have a clear-cut

case of both bets being sold at all p1; but that is not so elsewhere. For economy of space,

here we describe the profit function for some arbitrary λ < (1− 2δ)2/(1− δ), such as λ0 :

EΠb =

yp1+δ

[2+ p1 + δ− 2

√λp1(p1 + δ) − 2

√1− λp1

]− (1− λp1)c− s ∀ p1 < δ

yδ

[1+ δ−

√λp1(p1 + δ) −

√(1− λp1)(p2 + δ)

]− (1− λp1)c− s ∀ δ ≤ p1 ≤ δ

(1−λ)

y2δ

[p1(1+ λ) + δ− 2

√λp1(p1 + δ)

]− (1− λp1)c− s ∀ δ

1−λ < p1 < p∗∗1

y[1− λp1

p1−δ

]− (1− λp1)c− s ∀ p∗∗1 ≤ p1 ≤ 1.

(16)

In the above, both bets are positive (n1 > 0,n2 > 0) up to p1 = δ/(1−λ). Thereafter, only

n1 > 0; from p∗∗1 onward n1 = 1, meaning that all bettors bet on team 1. Here we assumed,

as shown in Fig. 3, p∗∗1 < 1 − δ. At other values of λ, such as λ ∈ ( (1−2δ)2

1−δ, (1 − δ)2), the

critical value p∗∗1 is greater than (1− δ). So there, the market capture by ticket 1 occurs at

a higher level of p1.

What is significant in the case of correlated beliefs is that at higher values of p1 the

probability of team 1’s win falls below p1 − δ, and therefore all bettors can be induced to

19

bet on team 1 with greater ease (at higher values of p1). Match-fixing can be profitable only

if a large number of punters can be lured into betting on the ‘losing’ team. This is not easy

to do at lower values of p1. Take for example, p1 = 0.25, δ = 0.2 and λ = 0.4. So if team 1

is bribed, its winning probability falls to 0.10, which falls within the support of the punters’

belief distribution, which is [0.05, 0.45].

To make profit, the bookie must set π1 > 0.10 and π2 > 0.90. So there is a clear limit

on the bookie’s ability to expand the sale of ticket 1. In contrast, suppose p1 = 0.50. After

bribery team 1’s probability of win falls to 0.20; the support of the punters’ belief distribution

is [0.30, 0.70]. So it is quite possible to set π1 = 0.30 and induce all punters to bet on team

1. However, whether that is optimal or not depends on whether p1 = 0.5 exceeds p∗∗i or

not. Thus, with correlated beliefs bribing the favorite and inducing most punters to bet on

it becomes easier and profitable.

� Match-fixing or honest bookmaking? Now we examine at what values of p1 profit

from match-fixing exceeds profit from honest bookmaking. We first note two extreme cases:

λ → 1 and λ = 0. When λ → 1 at all p1 profit from match-fixing is strictly less than the

profit from no corruption, i.e. EΠb < EΠ0 as πbi → π0i and c > 0. This is no different from

the naive punters case. Similarly, when λ = 0, the bookie earns EΠb = y−c−s regardless of

p1. As the bribed team loses with certainty, the bookie always sets π1 = p1 − δ and induces

all bettors to bet on team 1. So his revenue is y, and the only cost he faces is the bribe

and the expected penalty from investigation. As under the naive punters case, here too, the

range of optimal bribery (assuming c not prohibitively large) is the largest: [a1, b1]. Since

EΠ0 is nearly U-shaped, EΠb = y − c cuts through EΠ0 at two points 0 < a1 < 1/2 and

1/2 < b1 < 1 as shown in Fig. 4. Note that compared to the naive punters case, EΠ0 now is

much steeper at very low and very high values of p1 and flatter around 1/2.

As λ rises from zero, EΠ1 shifts down. We formally establish in the proof of Proposition 8

that EΠ1 is declining at p1 close to zero and EΠb is increasing at p1 ≥ δ/(1− λ) (see region

B in Fig. 3), where only ticket 1 is sold. But this implies a1 increases and b1 decreases;

that is, the region of bribery contracts as was the naive punters (see Fig. 1b). But what

is crucially different now is that the minimum of EΠb occurs at some p1 < 1/2. This has

an important implication for the limiting contest (or the most vulnerable contest) optimal

for bribery: it will be an uneven contest where team 1 is favorite. Fig. 4 shows that at the

limiting λ, denoted λ, we have the unique point of optimal bribery a1 = b1 = a(λ) belonging

to the second half of the probability spectrum.

Proposition 8. The bookmaker will bribe team 1 at all p1 ∈ [a1(λ), b1(λ)], where 0 < a1(.)

and b1(.) < 1, if λ < λ. The critical λ gives the limiting contest of bribery: a1(λ) = b1(λ) =

20

p1

EP

y-s

y - c - s

1 1/2 )ˆ(a

EP0

EPb(=0)

EPb(>0)

)ˆ(bEP

a1

b1

Figure 4: Regions of bribery for correlated beliefs

a > 1/2; for λ > λ, bribery of team 1 is not optimal. This implies the contest that is most

vulnerable to bribery features team 1 as favorite.

� Which team to bribe? As in previous section here also we ask: when both teams

are accessible for fixing, which team the bookie is going to pick? Without repeating the

argument we can see that if we consider a λ < λ and two bribery profit curves – one for

each team, then there will be an interval (a, 12) when bribing team 2 is most profitable and

another interval ( 12, b) when bribing team 1 would be profitable.

Proposition 9 (Bribe the favorite). Suppose the bettors’ beliefs are correlated with the

Nature’s draw of p1, and the bookie can select the team to bribe at will. Then there exists a

range of λ and a range of contests such that bribery is preferred to honest bookmaking and

the bribed team will be the favorite.

� Simulation. To illustrate the key points of bribery in the correlated beliefs case, we

present some numerical examples in Table 2. We start with y = 0.8, c = 0.2, and δ = 0.2.

Team 1 is being bribed. The maximum λ up to which bribery is optimal is λ = 0.59, and

the bribery region starts with [0.003, 0.997] at λ = 0 and eventually contracts to a singular

point at p1 = 0.80. Then we raised the punters’ wealth to 0.9 holding everything else

unchanged. The limiting λ increases to 0.61 and the limiting contest becomes a bit less

uneven (p1 = 0.776), as was seen in the naive punters.

Next, we study the effect of an increase in the cost of bribery. As c is raised to 0.3

resetting y = 0.8, λ falls to from 0.59 to 0.50; bribery is sustained less longer, predictably.

21

Table 2: Bribe inducement range of p1y = 0.80 c = 0.20 δ = 0.20 λ = 0.59

λ 0 0.10 0.20 0.30 0.40 0.50 0.58 0.59a1 0.003 0.01 0.020 0.060 0.01 0.458 0.706 0.80b1 0.997 0.99 0.986 0.997 0.96 0.922 0.826 0.80

y = 0.90 c = 0.20 δ = 0.20 λ = 0.61

λ 0 0.10 0.20 0.30 0.40 0.50 0.60 0.61a1 0.003 0.007 0.140 0.025 0.09 0.390 0.68 0.776b1 0.997 0.993 0.987 0.977 0.961 0.929 0.82 0.776

y = 0.80 c = 0.30 δ = 0.20 λ = 0.50

λ 0 0.10 0.20 0.30 0.40 0.49 0.50a1 0.009 0.030 0.140 0.33 0.510 0.75 0.80b1 0.991 0.985 0.975 0.96 0.935 0.86 0.80

y = 0.80 c = 0.30 δ = 0.40 λ = 0.27

λ 0 0.10 0.20 0.25 0.265a1 0.018 0.06 0.460 0.595 0.70b1 0.982 0.96 0.913 0.831 0.70

All of these results are similar to the naive punters case; but one key difference is that in

all cases here, the limiting contest is strictly greater than 1/2. That is, the limiting contest

features the bribed team as sufficiently favorite, as we have predicted from our model.

We also show what happens if the support of the punters’ beliefs expands (from 0.40 to

0.80). With a greater support, the profit curve under honest bookmaking achieves greater

curvature. Generally, if the support is shorter, the distribution gets very compressed, and

the effect of an increase in p1 on profit becomes more muted especially between δ and (1−δ).

On the other hand, profit under bribery behaves a bit differently. When δ is smaller, there

is a greater opportunity of selling only ticket 1 and also capturing the entire market. This

makes bribery profitable, and it is evident from the rising segment of the profit curve. With

an increase in δ, the scope of selling only ticket 1 reduces, and the profit does not rise as

sharply as before. Hence the tangency occurs at a lower value of p1 (0.70 in our example)

and also the critical λ falls to 0.27.

5 Conclusion

Illegal gambling gives rise to not only corrupt practices of the key market players, but also a

concentrated market structure. So both from the consumers’ point of view and the govern-

22

ment’s revenue perspective, legalization is essential.9 It will pave the way for competition,

transparency and regulation. It will also attract more bettors into the market. Above all, it

would also enable the law enforcement authority to observe the posted betting odds and try

to extract the information about the bookmaker’s beliefs. Thus, there is scope for making

enforcement intelligent, which in turn will make the fixer more careful.10

All in all, our analysis should be seen as clearing some of the myths in illegal sports

betting. First, even in a one-shot model examined in this paper, the bookmaker does not

necessarily defraud the punters. Second, when he does find defrauding profitable it could be

that he engages in a liaison with an unlikely partner, the weak team. Third, what strategy is

profitable for the bookie is quite sensitive to the gamblers’ knowledge-base. Fourth, although

type of monitoring should matter, it is unclear whether changing from exogenous enforcement

to outcome-contingent monitoring is any better. Finally, in the age of connected world,

ignoring problems of illegal sports betting corruption in Asia should come at a cost to legal

betting markets in Europe and elsewhere with some sports being vulnerable to match-fixers

in far away lands.

Appendix

Proof of Lemma 1. First, we show that the second-order conditions (s.o.c.) are satisfied

by Assumption 1 and the fact of πi > pi:

∂2EΠh

∂π21= −

[f(π1)]2 + (1− F(π1))f

′(π1)

[f(π1)]2−2π1 − p1p1

< 0,

∂2EΠh

∂π22= −

[f(1− π2)]2 − F(1− π2)f

′(1− π2)

[f(1− π2)]2−2π2 − p2p2

< 0.

Next, we show the existence of a unique interior solution. Consider Eq. (1). For any

given p1 ∈ (0, 1) neither π1 = 0 nor π1 = 1 can be a solution. At π1 = 0 the bracketed

expression is [ 1f(0)

− 0] > 0, and at π1 = 1 it is [0 − 1−p1p1

] < 0. Further, ∂EΠh

∂π1is continuous

and strictly decreasing in π1 by the s.o.c. Hence, by the intermediate value theorem, there

must be a unique π1 ∈ (0, 1) such that ∂EΠh

∂π1= 0. Similar reasoning applies Eq. (2) for π2.

To establish monotonicity of π1(p1) we derive from Eq. (1)

∂2EΠh

∂π21

∂πh1∂p1

+∂2EΠh

∂π1∂p1= 0, or

∂2EΠh

∂π21

∂πh1∂p1

+(πh1 )

2

p21= 0.

9For economics of illegal activities see Becker (1968), and for the arguments surrounding the legalizationof drugs see Miron and Zweibel (1995) and Becker et al. (2006).

10When enforcement is intelligent, a monopolist bookie’s pricing strategy will involve a sophisticated gamebetween him, the corrupt players and the enforcement authority, which is a subject of separate research.Our present model sidesteps this problem.

23

Since ∂2EΠ∂π21

< 0, we must have∂πh1∂p1

> 0.

To see πh1 (0) = 0 and πh1 (1) = 1 rewrite Eq. (1) as [p1(1− F(π1)) − (π1 − p1)π1f(π1)] = 0.

Let p1 → 0; then we must have πh1 → 0 since f(.) > 0. Similarly, let p1 = 1 and from the

above we get, [1−F(π1)−(π1−1)π1f(π1)] = 0. Now if πh1 6= 1, then πh1 < 1 (since πhi cannot

exceed 1), in which case [1 − F(π1) − (π1 − 1)π1f(π1)] > 0. Hence, π1 must be raised until

πh1 = 1.

Symmetric argument establishes∂πh2∂p2

> 0 from Eq. (2). Q.E.D.

Proof of Proposition 1. From the derivative of EΠh as given in (3) we derive

∂

∂p1

1− F(πh1 )

πh1= −

πh1f(πh1 ) + (1− F(πh1 ))

(πh1 )2

πh1′(p1) < 0,

∂

∂p1

F(1− πh2 )

πh2= −

πh2f(1− πh2 ) + (F(1− πh2 ))

(πh2 )2

πh2′(p1) > 0.

This implies that the EΠh(p1) curve is convex:

∂2EΠ(πh1 , πh2 )

∂p21= −

∂

∂p1

1− F(πh1 )

πh1+

∂

∂p1

F(1− πh2 )

πh2> 0.

Further, since π2(p2 = 0) = π1(p1 = 0) = 0 and π1(p1 = 1) = π2(p2 = 1) = 1 we get

as p1 → 01− F(πh1 )

πh1→∞ and

F(1− πh2 )

πh2→ 0

as p1 → 11− F(πh1 )

πh1→ 0 and

F(1− πh2 )

πh2→∞.

So the EΠh(p1) curve must be U-shaped. At its minimum1−F(πh1 )

πh1=

F(1−πh2 )

πh2, and they

are equal only at a unique value of p1, say p∗1. If f(q) is symmetric, p∗1 = 1/2 implying

f(1− πh2 ) = f(πh1 ) and F(1− πh2 ) = 1− F(π

h1 ). Therefore, πh1 = π

h2 at p1 = 1/2. Q.E.D.

Proof of Proposition 4. The proof proceeds in several steps, which we establish in the

form of lemmas. First we prove that EΠb curve will be either declining or (nearly) U-shaped

(with an interior minimum) (Lemma 2). Then we establish (in Lemmas 3–5) that EΠb > EΠh

only when the EΠb is declining, and they can cross each other at most twice, thus creating

a convex and compact set of contests that are optimal for fixing.

Lemma 2. The EΠb curve is convex, and it has a unique interior minimum if λ > k1, where

k1 solves EΠ ′b(p1 = 1) = 0 for λ. If λ ≤ k1, EΠb is declining at all p1 ∈ [0, 1].

Proof. First we show that as EΠh(.) is a convex function (by Proposition 1), EΠb(.) is also

24

convex. That is, for any θ ∈ [0, 1] and for any (p1, p′1), we must have θEΠh(p1) + (1 −

θ)EΠh(p′1) > EΠh(p

01), where p01 = θp1 + (1 − θ)p ′1. Subtracting (1 − p01)c from both sides

of it we write

θEΠh(p1) + (1− θ)EΠh(p′1) − (1− p01)c > EΠh(p

01) − (1− p01)c.

The right-hand side expression is nothing but the EΠb evaluated at p1 =p01λ

:

RHS : EΠh(p01) − (1− p01)c = EΠb(

p01λ).

The left-hand side expression of the above inequality can be rearranged as:

LHS : θEΠh(p1) + (1− θ)EΠh(p′1) − [θ+ (1− θ) − (θp01 + (1− θ)p ′1)]c

= θ[EΠh(p1) − (1− p1)c] + (1− θ)[EΠh(p′1) − (1− p ′1)c]

= θEΠb(p1

λ) + (1− θ)EΠb(

p ′1λ).

Combining the left-hand side and the right-hand side we get

θEΠb(p1

λ) + (1− θ)EΠb(

p ′1λ) > EΠb(

p01λ),

which says that EΠb(.) is convex.

Now define k1 to be the value of λ that solves EΠ ′b(p1 = 1) = 0. That is EΠ ′b(.) =

EΠh(λ) − (1 − λ)c = 0. Then when EΠb has an interior minimum, it must be at λp1 = k1,

or at p1 =k1λ< 1, if λ > k1. It has to be a minimum, because EΠb is convex. However, for

λ < k1, the minimum point k1λ> 1; hence EΠb must then be declining at all p1.

Lemma 3. Consider p1 and p ′1 with p1 < p′1. Suppose EΠb(p1) > EΠh(p1) and EΠb(p

′1) >

EΠh(p′1). Then EΠb(p

′1) < EΠb(p1).

Proof. Write using Eq. (4), EΠb(p1) = EΠh(λp1) − (1 − λp1)c, and EΠb(p′1) = EΠh(λp

′1) −

(1− λp ′1)c. Set EΠb(p1) > EΠh(p1) and using the above expression for EΠb(p1) obtain

EΠh(λp1) > EΠh(p1) + (1− λp1)c. (17)

Similarly, setting EΠb(p′1) > EΠh(p

′1) and rewriting EΠb(p

′1) we get

EΠh(λp′1) > EΠh(p

′1) + (1− λp ′1)c. (18)

25

Now subtracting Eq. (18) from Eq. (17) we arrive at

EΠh(λp1) − EΠh(λp′1) > λ(p

′1 − p1)c+ [EΠh(p1) − EΠh(p

′1)].

There are two possibilities with [EΠh(p1)−EΠh(p′1)]: either EΠh(p1) ≥ EΠh(p ′1) or EΠh(p1) <

EΠh(p′1). If EΠh(p1) ≥ EΠh(p ′1), the above inequality implies

EΠh(λp1) − EΠh(λp′1) > λ(p

′1 − p1)c.

Rearranging terms and subtracting c from both sides we write this inequality as

EΠh(λp1) − (1− λp1)c > EΠh(λp′1) − (1− λp ′1)c or, EΠb(p1) > EΠb(p

′1).

Next, consider EΠh(p1) < EΠh(p′1). Using this fact we rewrite Eq. (18) as

EΠh(λp′1) > EΠh(p

′1) + (1− λp ′1)c > EΠh(p1) + (1− λp ′1)c

or, EΠh(λp′1) > EΠh(p1) + (1− λp ′1)c. (19)

Subtract each side of (19) from the respective sides of (17) to obtain:

EΠh(λp1) − EΠh(λp′1) > λ(p

′1 − p1)c.

Then rearrange terms and subtract c from both sides to arrive at EΠb(p1) > EΠb(p′1).

Lemma 4. There cannot be any p1 at which EΠ ′b(p1) > 0 and EΠb(p1) ≥ EΠh(p1).

Proof. Consider p1 and p1λ

; clearly p1λ> p1. As EΠ ′b(p1) > 0, p1 must be greater than the

minimum point k1/λ (assuming λ > k1), and we have EΠb(p1) < EΠb(p1λ).

Now Suppose contrary to our claim, EΠb(p1) ≥ EΠh(p1). Then we must also have

EΠb(p1

λ) > EΠb(p1) ≥ EΠh(p1), or, EΠb(

p1

λ) > EΠh(p1).

Substitute EΠb(p1λ) = EΠh(p1)− (1−p1)c in the above and obtainEΠh(p1)− (1−p1)c >

EΠh(p1) which is a clear contradiction. Hence, EΠb(p1) < EΠh(p1) when EΠ ′b(p1) > 0. �

Lemma 5. Suppose for any p1 and p ′1, EΠb(p1) ≥ EΠh(p1) and EΠb(p′1) ≥ EΠh(p ′1). Then

for p01 = θp1 + (1− θ)p ′1, where θ ∈ [0, 1], we must have EΠb(p01) > EΠh(p

01).

Proof. Assume without loss of generality p1 < p ′1, and we know by the implication of

Lemma 4, if EΠb exceeds EΠh at p1, p′1 and all points in between, then EΠb must be

decreasing. However, EΠh can be rising or falling.

26

We also know that EΠb(p1) < EΠh(p1) when p1 → 0 as well as when p1 → 1. Then if at p1

and p ′1, EΠb(.) ≥ EΠh(.), then at some point, saym, 0 < m ≤ p1 we have EΠb(m) = EΠh(m)

and at another point, say m ′, p ′1 ≤ m ′ < 1 we also have EΠb(m′) = EΠh(m

′).

By Lemma 2 EΠb(.) is either declining all through or has an interior minimum, while

by Proposition 1 EΠh(.) is U-shaped. Therefore, these two curves can cross each other at

most twice. So, if EΠb(p1) ≥ EΠh(p1) and EΠb(p′1) ≥ EΠh(p ′1), then for every θ ∈ [0, 1] and

p01 = θp1+(1−θ)p ′1, we must also have EΠb(p1) ≥ EΠh(p01). If we did not, then EΠb would

cut EΠh four times; but that is impossible given the convexity of the two curves.�

With the help of these four lemmas we now prove Proposition 4. We first need to establish

that λ exists. For that we begin by noting from Eq. (4) that if λ = 0, EΠb = y−s−c ≥ EΠhover a range of p1. Let this interval be [a1(0), b1(0)]. It must be convex and compact.

Now consider a strictly positive λ ≤ k1, such that EΠb shifts down everywhere, but it

does not have an interior minimum; at p1 = 1, EΠb = EΠh(λ)−(1−λ)c < EΠh(1) = y−s. So

there must be a unique interval [a1(λ), b1(λ)] such that EΠb ≥ EΠh at all p1 ∈ [a1(λ), b1(λ)].

Clearly, a1(λ) > a1(0) and b1(λ) < b1(0). Also b1(λ) > p∗1, because EΠb falls below EΠh

when EΠh is rising, and at p∗1 EΠh is at its minimum.

Next, consider a λ, say λ ′ ∈ (k1, 1] where EΠb has a minimum at p1 = k1/λ′ < 1. But by

Lemma 4 we can rule out the upward sloping segment of EΠb to exceed EΠh. Hence, when

EΠb = EΠh, EΠb must be declining. That means, b1 < k1/λ′.

As EΠb continually shifts down with an increase in λ, the interval [a1, b1] must contract,

i.e. a ′1(λ) > 0 and b ′1(λ) < 0. Since at λ = 1 the interval [a1, b1] does not exist, there must

exist a critical λ between 0 and 1, namely λ such that a1(λ) = b1(λ). Clearly, λ > k1. Q.E.D

Proof of Proposition 6. Eq. (11) is the unconstrained solution to the bookie’s problem,

and it can be verified that they are valid within the relevant interval of p1.

Substituting (π01, π120) in EΠ0 derive Eq. (12). Differentiating EΠ0 w.r.t. p1 we get

EΠ ′0(p1) =

y

p1+δ

[1− 2p1+δ

π1− 1

π2

]−A z

(p1+δ)2< 0 for p1 < δ

yδ

[− 2p1+δ

π1+ 2p2+δ

π2

]for δ ≤ p1 ≤ 1− δ

yp2+δ

[−1+ 2p2+δ

π2+ 1

π1

]+ B y

(p2+δ)2> 0 for 1− δ < p1,

(20)

where

A =[2+ p1 + δ− 2

√p1(p1 + δ) − 2

√p2

], B =

[2+ p2 + δ− 2

√p1 − 2

√p2(p2 + δ)

].

Over the interval p1 ∈ [p1 − δ, p1 + δ], EΠ′0(p1) < 0 at p1 < 1/2 and EΠ ′0(p1) > 0 at

p1 > 1/2; but EΠ ′0(.) = 0 at p1 = 1/2. So the EΠ0 is minimum at p1 = 1/2. Q.E.D.

27

Proof of Proposition 8. The proof of the proposition will involve showing that the

expected profit falls in λ for any given p1 and c > 0, and it is a convex curve (against p1)

with an interior minimum. This minimum occurs at p1 < 1/2. The proof is established

through the following lemmas.

Lemma 6. There exists a critical value of λ, say λ such that EΠb(λ) is tangent to EΠ0.

Proof. First consider the case of λ = 0. From Eq. (16) that EΠb = y− c− s at all p1. Since

this cuts the U-shaped EΠ0 curve twice (see Fig. 4) bribery is optimal at all p1 = [a1, b1].

Now let λ > 0 and note that EΠb is decreasing in λ at any given p1. In the limit, as

λ → 1, EΠb → EΠ0 − (1 − p1)c − s, rendering match-fixing not optimal. Therefore, by the

intermediate value theorem, there must exist a critical λ at which EΠb = EΠ0 at only one

value of p1, i.e. the tangency point of the two curves.

Lemma 7. EΠb must have a minimum at some interior p1.

Proof. This can be seen from the facts that EΠb is decreasing at p1 close to zero and increasing

at p1 close to 1. Differentiate EΠb for the segment p1 < δ and let p1 → 0, EΠ ′b(p1)→ −∞.

Similarly, consider EΠb for p1 > 1 − δ when all bettors buy ticket 1 (Region A of Fig. 3):

EΠb = y[1 − λp1p1−δ

] − (1 − λp1)c − s, and EΠ ′b(p1) = y λδ(p1−δ)2

+ λc > 0. Hence, there must

be a minimum strictly inside the interval (0, 1). In fact, we just established the minimum

cannot be in the region A. A similar argument can be made for regions B and C with p1

close to 1. �

Lemma 8. The minimum of EΠb cannot occur in region B, i.e. at p1 ≥ δ1−λ

.

Proof. In region B of Fig. 3, where ticket 2 is not bought at all and some bettors don’t buy

ticket 1 either, EΠb is increasing in p1. To establish this we will argue that EΠb is strictly

increasing at the lower end of the relevant p1-range, and since it is a convex curve (which

we also show) it must be increasing over the entire range.

The relevant range of p1 here is δ1−λ≤ p1 ≤ p∗∗1 for any λ ≤ (1 − δ). For economy of

space, we will discuss the range δ1−λ≤ p1 ≤ 1− δ. Write from Eq. (16)

EΠb =y

2δ

[p1(1+ λ) + δ− 2

√λp1(p1 + δ)

]− (1− λp1)c− s,

and its slope is

EΠ ′b(p1) =y

2δ

[1+ λ−

λ(2p1 + δ)

π1

]+ λc

=y

2δπ1

[(1+ λ)

√λp1(p1 + δ) − λ(2p1 + δ)

]+ λc

28

where π1 =√λp1(p1 + δ). Now consider the bracketed term [

√λp1(p1 + δ)(1+λ)−λ(2p1+

δ)] and substiute the lowest value of p1, i.e. p1 =δ1−λ

. The bracketed term becomes

δ√λ

1− λ

[√(2− λ)(1+ λ) −

√λ(3− λ)

].

It is easy to verify that√

(2− λ)(1 + λ) >√λ(3 − λ) reduces to (1 − λ)2 > λ(1 − λ)2 or

λ < 1. Thus, EΠb is increasing at p1 =δ1−λ

.

Next, we show that EΠb is a convex curve, i.e. EΠ ′′b(p1) > 0 in region B. Differentiate

further EΠ ′b(p1) =y2δ

[1+ λ− λ(2p1+δ)

π1

]+ λc, and obtain

EΠ ′′b(p1) = −yλ

2δπ21

[2π1 −

λ(2p1 + δ)2

2π1

]= −

yλ

4δπ31

[4π21 − λ(2p1 + δ)

2]=yλ2δ

4π31> 0.

Since EΠb is convex, it must be increasing at all p1 > δ/(1− λ) in region B. �

Lemma 9. The minimum of EΠb cannot occur at p1 ≥ 1/2.

Proof. By the implication of Lemma 8 the minimum must occur at region C, where both

tickets are sold, which means the minimum should occur at some p1 <δ1−λ

.

Note δ1−λ

will be greater or smaller than 1/2 depending on λ. If λ ≤ 1 − 2δ, then

p1 = δ1−λ

< 1/2, and in that case we have proved in Lemma 8 that the minimum of EΠb

occurs at p1 < 1/2.

The other possible case is λ > 1− δ such that 12< δ

1−λ. Let us now consider p1 ∈ (δ, δ

1−λ)

where both tickets are sold (region C in Fig. 3). We show that EΠb is increasing at p1 = 1/2,

and also EΠb can be verified to be convex. Let us rewrite EΠb from Eq. (16) as

EΠb = EΠ0 +y

δ

[A ′√p1 + δ− B

′√p2 + δ

]− (1− λp1)c− s

where A ′ =√p1 −

√λp1 and B ′ =

√(1− λp1) −

√p2. Differentiating the above we derive

EΠb′(p1) = EΠ0

′(p1) + λc+y

2δ×[

A ′√p1 + δ

+B ′√p2 + δ

+

√p1 + δ√p1

(1−√λ) +

√p2 + δ

{λ√

1− λp1−

1√p2

}].

Let us evaluate EΠ ′b(p1) at p1 = p2 = 1/2 and using the fact EΠ0′(1/2) = 0 obtain

EΠb′(p1 = 1/2) = λc+

y

δ√2+ δ

[1+ λδ− (1+ δ)

√λ(2− λ)√

2− λ

].

29

Now ascertain the sign of the numerator of the bracketed term at λ = 1 (the highest value)

and λ = 1− 2δ (the lowest value):

at p1 = 1 [1+ λδ− (1+ δ)√λ(2− λ)] = 0

at p1 = 1− 2δ [1+ δ− 2δ2 − (1+ δ)√1− 4δ2] > 0.

That is, EΠb′(p1 = 1/2) > 0 at all λ close to 1 as well as to (1−2δ). Further, by monotonicity

(w.r.t. λ) also at all intermediate values of λ between (1− 2δ) and 1, EΠb′(p1 = 1/2) > 0.

Hence, for any given λ the minimum must occur at p1 < 1/2 in region C. �

The implications of Lemmas 6–9 are that with successive increase in λ the range of

contests, [a1(λ), b1(λ)] must shrink and at λ collapse to a single point a1 = b1 = a. Since

the minimum of EΠb occurs at p1 < 1/2 at all values of λ, the tangency point a must be

greater than 1/2. Q.E.D.

References

[1] Bag, P.K. and Saha, B. (2011). Match-fixing under competitive odds. Games and Eco-

nomic Behavior 73, 318-344.

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