1. Introduction
The gambling industry is a significantly large industry and has been growing its online presence consistently over the past years. The estimated amount of gross gambling revenue, being amount wagered less payouts, for online gambling businesses in UK and EU was around twenty-six billion euro during 2020 [
1]. Around 40% of this is is generated from sports betting. The growth in online gambling is bound to continue as more markets are opening up since gambling licences are being generally viewed as a means to increase revenue for states [
2]. It is therefore fairly straightforward to note that online sports betting is a multi-billion dollar industry.
Sports betting has therefore been a focus of interest in different academic circles with significant emphasis on problem gambling (for example [
3,
4]); profiling gamblers (for example [
5,
6]); the inefficient functioning of betting markets (for example [
7,
8,
9]); or strategies to beat markets (for example [
10,
11]). In that respect, the focus has been on the clients of these services or the actual numbers/predictions generated by the services.
It would be expected that the solvency of a multi-billion dollar industry would also be of great interest to academics and regulators. Yet there has been scarce research on the solvency of the providers of this service, the sports betting companies themselves, with notable exceptions being [
12,
13]. Ref. [
12] introduces a theoretical quantitative framework that can be used to measure the potential risk of a sports betting company from its trading operations by subdividing its portfolio of wagers into bundles according to their likelihood size. Ref. [
13] is an in-depth overview and comparison of the perceived risks faced by banks and gambling companies ranging from compliance to operational risks.
The rationale for this vacuum in bookmaker solvency could be due to bookmakers being considered as the ‘bad’ guys that do not contribute to the welfare of society and the lack of systemic effects should a bookmaker fail. However, bookmakers are large corporations whose ruin, or inefficiencies, are bound to affect financial markets. The current state of affairs with respect to reserving is leading to an inefficient use of capital. Moreover, it would be ideal to have fair, stable, strong bookmakers that are able to enforce responsible gaming and limit access to money-laundering rather than bookmakers that are inefficient in their processes.
In this paper, we introduce the use of perturbed betting processes to determine bookmaker’s capital more accurately and the likelihood of ruin by taking example of a series of accumulators (also known as multiples) on soccer matches. The results can be extended to other sports or events—we used soccer as it is the most popular sport and one of the few sports in which three outcomes are possible due to the prevalence of draws.
In addition to traditional representations and calculations, the paper focuses on Dirac notations and Feynman’s path calculations in betting market scenarios. Although there is a vast literature on the use of these techniques applied in finance (see [
14,
15,
16,
17]), Dirac notations and Feyman’s path calculations are not commonly applied in insurance and betting industries. Utev and Tamturk applied them into actuarial risk and capital modeling [
18,
19]. In follow-up joint papers, the approach was used for insurance claim analysis [
20], catastrophic modelling by combining epidemic and actuarial models [
21] and stock market data analysis [
22].
2. Bookmaker’s Expected Profit via Matrix Representation
A soccer match can end in one of three outcomes: home team win, draw and away team win. Let us assume that the probabilities for each one of these three outcomes are
,
and
. However, bookmakers would use inflated probabilities when determining odds in order to guarantee a profit. We use standard notation as described in [
11,
23] to define implied probabilities a
for
and
.
European odds are defined by
. Let
be the total value in units of wagers placed on outcome
i. By all definitions, in terms of betting company, the revenue is
and expected payout is
. Hence, the profit of the company can be defined simply for a single match by
When
n soccer matches are considered, we have
different scenarios (tickets) because there are 3 options in a match as home team win, draw and away team win as seen in
Figure 1.
The betting of one stake based on multiple outcomes is called a multiple or an accumulator. Accumulators allow bettors to place one small stake to potentially large returns but exponentially low likelihoods. Bettors can put money on
different options for
n matches under assumption that they bet with
n matches. By default, the expected profit for
n matches is defined by
where
is probability of being
n independent events, so
for
Notice that
Similarly, representing multiply of n European odds on each match, and for
Bettors do not have to select to bet on every match. If the total number of matches is
n, bettors can create tickets by selecting
or 1 matches as desired, so the total number of options is
In this circumstance, the formula of the expected capital of betting company can be modified by
However, for simplicity, we will continue with the formula in (
1) instead of the formula in (
3).
From (
1) and (
2), the expected profit is
The company is expected to profit if
while negative values of
k can cause ruin of the company [
12]. However, the optimisation of
plays a vital role in terms of profitability of the company. In the case of a variable
, the formula in (
4) can be modified as
We assume that the bookmaker has data to compute of and can control by k. Even though is computable based on historical data, error term in the computation can produce surprising results.
As mentioned before, let
be the total value in units of wagers for outcome
i. Then,
matrix of wagers values
Let
and
be a probability matrix and European odds matrix respectively for
ith match. Then, probability matrix
P for
n matches is defined via tensor product
One can note that
P and
E are
dimensional matrices. Expected profit of the betting company is shown via a trace function representing sum of the diagonal elements of
.
3. Taking More Uncertainty into Account
In betting modeling, analysis of historical data does not reflect current situation perfectly. For example; referee mistakes [
24], weather on the day of the match [
25], distance travelled to match [
26] and injuries of players can directly affect the outcome of the match. Therefore, even if a precise prediction is made, we need to modify the stochastic process with random uncertainty to make more realistic. This issue will be discussed more comprehensively in
Section 5.
As mentioned before, for the
ith match, probability of home win is
, probability of draw is
and probability of away win is
. Let
be probabilities of unpredictable things, and
are modified probabilities by
as
where we assume that
and
for
. Under these conditions, the corresponding matrix for the
ith match can be written as
Modified probabilities for
n matches can be shown in the following matrix via tensor operator
According to
and
random variables, we can obtain different results, and get the average value by using simulation methods like Monte Carlo.
where
M is a big enough number.
Notice that even if we modified P, E was not exposed to change because E is only computed by historical data while based on historical data analysis and unpredictable things in recent time.
4. Bookmaker’s Profit via Path Calculation
The path calculation approach has been applied to many different disciplines. To apply the path calculation approach in gaming, we have to think of the outcome of each match as a location or value at a different time. For example, let us consider five betting tickets for four matches as per the
Table 1 below.
In this table,
represent home win, draw and away win, respectively. Now, we can consider the betting process via path calculation as per
Figure 2. For example, the first ticket assumed the home teams will win the first and third matches, the away team will win the second match, and the last match will end in a draw. To make the our approach more clear, the betting process can be imagined as a time series by considering the matches as time, and results as locations or values.
Between two spacetime points
at time
and
at time
, the propagator of the quantum system can be defined as the probability transition amplitude via the wavefunction [
27]
where
K is called as Feynman kernel. By using Hamiltonian operator with bra-ket notations, the propagator can be written as
In other words, a propagator makes the state function at spacetime point move to another spacetime point .
Dirac defined notations about vectors in Hilbert space that corresponds to a physical system [
28]. In Dirac formalism,
and
represent column and row vectors, which are used to show quantum states. A bra is the Hermitian conjugate transpose of the corresponding ket. Elements of
or
consist of 1 in position
i and 0 elsewhere on computational basis. Surely, a ket can be defined in different ways (e.g., Hadamard and Circular basis [
29]). A quantum operator like
represents a matrix
In the propagator in (
6),
is an operator in semigroup form with Hamiltonian operator
H. In physics, the Hamiltonian of a system is defined as the sum of kinetics and potential energies [
28],
where
K and
V are representing kinetic and potential energy of the system respectively.
Furthermore, Hamitonian operators provide a way to go to quantum formalism from a classical Markovian approach, which is characterized by the continuous semigroup
with generator
Q. By taking minus generator operator equal to the corresponding Hamiltonian
, we have
. In general, the Hamiltonian operator
H does not need to be a generator, specifically for the cases of non-embeddable Markov chains [
18].
Throughout the quantum parts of this paper, we work on a Hilbert Space that is a complete vector space. Regardless of finite or infinite dimensional, a Hilbert space is separable if its basis is countable [
28].
The scalar product of the basis states
and
is known as Dirac-delta function, and with the Fourier representation, it is written by [
14]
where
is momentum space basis with the completeness equation
. Inner products of basis state and momentum basis state are
where
is the eigenstate of the momentum operator (e.g., let
P be a momontum operator, so
).
For actuarial random walks, the following propagator has been computed by Tamturk and Utev in the previous papers [
18,
19] and follow-up joint papers [
20,
21] with respect to several Hamiltonian operators with the help of the completeness equation and corresponding eigenvectors of different Hamiltonian operators,
where
and
are eigenvalue and eigenvector of the Hamiltonian operator,
i is complex unit.
4.1. Transition Probability via Path Calculation
Regarding the path calculations, the main difference between gambling and the other disciplines is that order of the games does not change the results. Let us consider the following propagator with a matrix operator by assuming that
is location (result) of
i-th match as
Notice that all rows are taken as the same since we assume that the location of the
th match is independent from the
ith match. Sum of probabilities of the all the possible paths can be showed by
For the sake of simplicity, let us assume
denoted by
because
where
and
. Note that this statement is not true for diagonal operators
.
4.2. Hamiltonian Operator for Expected Profit
If an operator is self-adjoint (adjoint-operator is equal to itself), it can be called as Hermitian operator [
28]. A Hermitian Hamiltonian matrix can be diagonalised with help of the following transformation
where
U is a unitary matrix. If
U is a square matrix,
and
. Elements of
D are eigenvalues representing energy levels.
where
U is a
unitary matrix,
is its adjoint. Similarly,
and
can be written in the following forms:
and
Then,
where
is
unitary matrix.
Hamiltonian operator can be defined for the
ith match by
However,
cannot be generalized for multimatches case by
because
does not give the total wage amount for each possible result; so instead of that, the
dimensional
W matrix mentioned in
Section 2 is taken into account.
where
V is
unitary matrix, and the diagonal elements can be computed as
Regarding finding the expected profit and capital of the betting company, we will use Hamiltonian operators
and
for transition probabilities and corresponding odds. Now let us find probability of a single match via the transformation in Equation (
7).
where the main complication is finding the eigenvalues of the Hamiltonian operators.
For a single match, we defined
as
where it can be noticed that we added
inside of
to set up an independence between
and
.
When
is substituted into Equation (
9), we will obtain
For
n matches, probability of
n events via path approach,
can be computed by multiplying
n integrals which will cause the long computation time, so for the sake of simplicity, we consider
for the
n events, and define new
for
n matches as
where
and
.
Similarly, the odds for
n matches can be computed by following path approach with Hamiltonian
operator
In this circumstance, the expected profit is
where
is
diagonal element of
W.
Example 1. Let us assume the probabilities of three matches are as in Table 2, and let us calculate probability of being draw for the first match, away win for the second match and home win for the third match. In this circumstance, for draw, away win and home win, respectively,
With
P and
, the probability of
respectively for the three matches is found by
The profit of the betting company can be computed as
4.3. Ruin Probability of the Betting Company
Let
,
be numbers of the matches in the
i-th week. Then, ruin probability of the company at the end of the m weeks with assumption of 0 initial reserve can be computed by
where
T is the ruin time defined by
As seen from the ruin formula in (
14), we consider weekly capital of the company as integer values—this can be done by truncated numerical approach. However, finding the probability of the capital is based on complex computations for large time scale and high number of matches. Another challenge in terms of the computation is dynamic odds. We refer this to future researches.
5. Perturbation of Betting Process
Classical Markov processes may be perturbed to produce new Markov chain with similar statistical properties [
30,
31]. Perturbation of Markov process is quite an important approach in order to take unpredictable things into account. Therefore, the logic is simply based on modification of the random process without changing the general properties.
In continuous time, the transition matrix can be found via the generator matrix.
where
Q is called the generator of the continuous time Markov process
The sum of the elements in each row of
Q is zero with
where
As mentioned in
Section 4,
, so the stochastic process can be perturbed by modifying the generator matrix. For example, let us consider
generator matrix on
Then, as mentioned in Feigin and Rubinstein’s article [
30], perturbed generator
can be found by
where
R is a replacement matrix. With different versions of the replacement function, we can modify the stochastic process in various ways. However, we follow a different way. As mentioned in
Section 4, Hamiltonian operator
H does not need to be
, which us gives more flexibility. The perturbed Hamiltonian operator can be defined by
where
V is a Hermitian operator. For the
ith match, the probabilities of the changes caused unpredictable situations is displayed by
where
and
represent the unpredictable part for
i-th game. By default,
+
. Furthermore, more restrictions can be added like
for
and
. However, this is not a research interest for us at this stage.
The total change for
n matches via tensor product is
In this circumstance, the profit of the company after the
n matches is
The equation above can be computed with respect to different Hamiltonian operators. This gives us a flexibility to play on. Of course, the perturbed process can be defined in more complex ways by taking seasonal weather situation, pandemic and macro economic variables, etc. into account in further research.
6. Conclusions
The betting sector is growing in a phenomenal rate, especially in the US [
2], and notwithstanding the size of the industry, quantitative developments have focused on getting accurate odds or profiling customers. There is little to no work on the potential ruin of a bookmaker. Moreover there is a huge gap between the few theoretical approaches and applications in the betting sector.
In this paper, the betting process has been considered as a time series to apply the path approach with Dirac notations. We think that this kind of new approach will attract the interest of researchers in other disciplines to this sector, and more collaborative work will be produced in interdisciplinary work environments.
Furthermore, the application of such techniques can help in comparing the performance and comparative risk of different bookmakers, which could be used to better manage the financial performance of a company. Currently, the gambling market is in a phase of mergers and acquisitions at a valuation of more than one billion dollars per activity (example [
32,
33]). The application of more complex techniques such as those presented here could provide a relative edge when considering larger trades.