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 law enforcement officials. If the punters have naive beliefs uncorrelated with the Nature’s draw of the teams’ winning odds, the underdog will be bribed, provided the impact of 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 the Nature’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 unconventional monitoring 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, AS2 Level 6, 1 Arts Link, Singapore 117570; E-mail: [email protected]† Durham University Business School, Durham, U.K. DH1 3LB; E-mail: [email protected]
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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.
∗Department of Economics, National University of Singapore, Faculty of Arts and Social Sciences, AS2Level 6, 1 Arts Link, Singapore 117570; E-mail: [email protected]†Durham University Business School, Durham, U.K. DH1 3LB; E-mail: [email protected]
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/-
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
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
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:
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.
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