On the effect of taxation in the online sports betting market
Abstract
We analyze the effect of taxation in the online sport betting market. A relevant characteristic of this market is its negligible marginal cost on bet volume. Taxation can be on gross profit (Gross Profit Tax) or on volume (General Betting Duty). We model the two most popular online sport betting bets: fixedodds and spread, as compared with another traditional sport betting: parimutuel. We characterize the odds and the bookmaker’s payoff in (strong) subgame perfect equilibrium for each of the three types of bets under both taxation schemes. The results show that taxation on gross profit maximizes the utilitarian social welfare. Moreover, the three types of bets are equivalent when the market is symmetric.
Keywords
Taxation Online betting market Sport betting BookmakerJEL Classification
C72 D42 L831 Introduction
A remarkable feature of online betting (which includes sports, casino and card games such as poker) is that their operators require little more than an internet web page to enter a new market. As opposed to offline operators, they do not need to open physical selling points. Under an unregulated market, the cost of offering a bet is inelastic with respect to the bet volume, i.e. the total sum of betting stakes. This is because online betting users can bet from any internet terminal, even at home. As opposed, offline betting users need to be physically at a selling point.
Things may be different in a regulated market. Over the last years, many European countries have been regulating their online betting and gaming sector. However, this regulation has not been done in a uniform way throughout the different countries.
In general, the basic taxation scheme is based on two types of taxes: the General Betting Duty (GBD) is levied as a proportion of betting stakes; whereas the Gross Profits Tax (GPT) is levied as a proportion of the net revenue of the operators.
Taxation schemes for online sports betting
Country  General Betting Duty (GBD)  Gross Profits Tax (GPT) 

UK  6.75% (until 2001)  15% (since 2001) 
Italy  2–5% (general)  20% (spread) 
France  8.5%  – 
Germany  5%  20–80% 
Spain  22% (parimutuel) \(+\) 0.1% (all)  25% (fixed odds and spread) 
In the Spanish case, GBD has been the taxation scheme in the most traditional offline sport betting (la quiniela), which takes a parimutuel structure.
In a parimutuel market, a winning bet pays off a proportional share of the total stake on all outcomes. However, the most popular online sport operators are specialized in another two markets: Fixedodds and spread. In a fixedodd market, the operator sets the odds for each possible outcome of the match, and the bettors decide whether they accept or not these odds. In a spread market, the operator acts as an intermediator among the users, who bargain the odds.
For sport matches, a bet of 1 monetary unit on a particular team yields a return of \(\frac{1}{\pi }\) monetary units in case the team wins the match, and 0 otherwise. In this context, an odd \(\pi \in \left( 0,1\right) \) is defined as the probability assigned by the market. Notice that any riskneutral bettor would find it profitable to bet at odd \(\pi \) when her private probability estimation is higher than \(\pi \).
In parimutuel and spread bets, the operator’s profit comes from a commission on either the amount at risk or the winning amount (typically around 5% in online spread operators). In fixedodds bets, the operator’s profit comes from the odds, which should sum up more than \(100\%\) ^{2} for all the possible outcomes of the sport match.^{3}
In this paper, we model the three types of market in a general setting. The regulator decides on the general taxing scheme (either GBD or GPT) and the operators decide on their commission (parimutuel and spread operators) or odds (fixedodds operators). We assume that the spread betting commission is applied to the winning bets (as it is typical in online spread operators), whereas commission in parimutuel betting applies to the total amount (as in the Spanish regulation).
We show that, from the online bettor’s point of view, it is preferable a GPT scheme, in the following sense: In equilibrium, the odds are not affected by the taxation under GPT; whereas a GBD scheme would reduce the odds and hence the bettors’ utility.
These results agree with the ones presented by Smith (2000) and Paton et al. (2001, 2002), whom analyse the effect of the different taxation schemes in Australian, UK and USA betting markets. These results, however, are more focused on offline betting operators and government revenue. Moreover, they take into account the marginal cost of each bet. As opposed, we assume that these marginal costs are negligible.
There are other works that focus on parimutuel markets: Ottaviani and Sørensen (2009) provide a model that explains the empirical evidence of underdogs overbet. These authors argue that this bias may be due to privately informed bettors. As opposed, we prove (Corollary 1) that the spread bet operator would get a higher profit if the underdog wins the game.
Other works concentrate on fixedodd markets. For example, Bag and Saha (2011, 2016) study the externalities due to bribery in sports; and Levitt (2004) argues that the operators may achieve higher profits by an accurate prediction of the match outcome.
As far as we know, no similar research has been addressed for spread markets.
The rest of the paper is organized as follows. In Sect. 2 we describe the model. In Sect. 3, we characterize the equilibrium payoffs in each of the markets. In Sect. 4 we provide the main results. In Sect. 5, we study the symmetric case. In Sect. 6, we present some concluding remarks. Technical proofs are deferred to the “Appendix”.
2 The model
Two teams (home and away) play a competitive sport match; the match being drawn is not a possibility.
There are three types of agents in the model: A continuum set \({\mathcal {B}}\) of bettors are interested in betting, but only if the odds are attractive; a finite set K of bookmakers that offer bets; and a regulator (Government) that decides on taxes.
We assume that bettors are risk neutral and try to maximize their expected profit. Each bettor \(i\in {\mathcal {B}}\) is characterized by her individual belief (i.e. the probability) \(x_{i}\) that the home team wins (\(1x_{i}\) is the probability that the away team wins); \(x_{i}\) is distributed following an absolutely continuous cdf with probability density function f and full support over (0, 1).
 Riskadverse case:
Bookmakers do not have any belief on the true probability for the home team to win. Hence, it is not possible to estimate an expected profit for them. Instead, we assume that each bookmaker tries to maximize her monetary profit under the worst possible outcome of the match.^{4}
 Riskneutral case:
Bookmakers have a precise common estimation of the true probability \(q\in (0,1)\) for the home team to win. This estimation may arise from their own expertise on the sport discipline plus a detailed study of the match, or by a previous sampling among users with the most accurate bet record, or both. In this case, we assume that each bookmaker is riskneutral and tries to maximize her expected final monetary profit.
The third type of agent is the regulator, or Government, that looks for the social welfare via taxes. We consider that the regulator has the same attitude towards risk as the bookmaker, i.e. riskadverse when the bookmakers are riskadverse, and riskneutral when the bookmakers are riskneutral. As a way to measure social welfare, we consider two criteria: the total tax income and the utilitarian social welfare function. Our aim is to estimate the optimal tax (GBD and/or GPT) in order to maximize each of these two criteria.
2.1 The noncooperative game
Assume the regulator announces a tax, that could be a percentage \(\upsilon \) on volume (GBD), a percentage \(\rho \) on gross profit (GPT), or both. The (noncooperative) game has two steps:
Step 1 Each bookmaker \(k\in K\) observes \(\upsilon \) and \(\rho \) and announces her odds (fixedodds) or commission (spread/parimutuel). Let \(s_k\) denote this choice and let \(s_K = \left( s_k\right) _{k\in K}\).
Step 2 Each bettor \(i\in {\mathcal {B}}\) observes \(s_K\), and chooses whether to participate or not, and, in the former case, with which bookmaker and in which team she bets on. Let \(s_i(s_K)\) denote this choice.
For each \(L\subseteq K\), let \(s_L = \left( s_k\right) _{k\in L}\) denote a strategy profile for bookmakers in L. Analogously, for each \({\mathcal {C}} \subseteq {\mathcal {B}}\), let \(s_{{\mathcal {C}}}(\cdot ) = \left( s_i(\cdot )\right) _{i\in {\mathcal {C}}}\) denote a strategy profile for bettors in \({\mathcal {C}}\).
Following Neyman (2002), we assume that, for any bettors’ strategy profile, the set \({\mathcal {C}}\) of bettors that give any particular signal is always Borelmeasurable,^{5} and we denote its volume as \(\left\ {\mathcal {C}}\right\ \).
For any set S, we denote as \({\mathbb {R}}^S\) the Euclidean space where the coordinates are indexed by the elements of S. Given an admissible strategy profile \(s=\left( s_K,s_{{\mathcal {B}}}(\cdot )\right) \), we denote as \(u(s) \in {\mathbb {R}}^{K\cup {\mathcal {B}}}\), or simply u, the final payoff allocation of the noncooperative game.
2.2 The equilibrium concept
We will work with the standard concept of subgame perfect equilibrium and a natural extension of it, named bettorstrong subgame perfect equilibrium. Notice that the only proper subgames arise in Step 2.
Definition 1
 1.For each \(i\in {\mathcal {B}}\), each bookmakers’ strategy profile \({\widetilde{s}}_K\) and all bettor i’s strategy \({\widetilde{s}}_i(\cdot )\),$$\begin{aligned} u_i\left( {\widetilde{s}}_K, {\widetilde{s}}_i\left( {\widetilde{s}}_K\right) , s_{{\mathcal {B}}\setminus \{i\}}\left( {\widetilde{s}}_K\right) \right) \le u_i({\widetilde{s}}_K, s_{{\mathcal {B}}}\left( {\widetilde{s}}_K\right) ). \end{aligned}$$
 2.For each \(k\in K\) and all bookmaker k’s strategy \({\widetilde{s}}_k\),$$\begin{aligned} u_k\left( {\widetilde{s}}_k, s_{K\setminus \{k\}}, s_{{\mathcal {B}}}\left( {\widetilde{s}}_k, s_{K\setminus \{k\}}\right) \right) \le u_k\left( s_K, s_{{\mathcal {B}}}\left( s_K\right) \right) . \end{aligned}$$
The first part of the definition states that no bettor has incentives to deviate in Step 2, even if the bookmakers did. The second part states that no bookmaker has incentives to deviate in Step 1.
Subgame perfect equilibria is a standard solution concept. However, we will also focus on a refinement of it. Notice that one of the assumptions is that each bettor has the same amount of money to bet. But this is obviously not a realistic assumption. Another interpretation is that the bettors are in fact minimal bet stakes, or coins, willing to be spend by the actual users, each of them owning many coins. Hence, it is obvious that different coins can coordinate their strategies, being held by the same user. Our next definition of equilibrium allows to capture this coordination. It also covers situations where the bettors increase their stakes when the bookmaker improves the odds (or decreases the commission).
Definition 2
 1.For each \({\mathcal {C}}\subseteq {\mathcal {B}}\), all bookmakers’ strategy \({\widetilde{s}}_K\) and all strategy profile \({\widetilde{s}}_{{\mathcal {C}}}(\cdot )\),for all \(i\in {\mathcal {C}}\).$$\begin{aligned} u_i\left( {\widetilde{s}}_K, {\widetilde{s}}_{{\mathcal {C}}}({\widetilde{s}}_K), s_{{\mathcal {B}}\setminus {\mathcal {C}}}\left( {\widetilde{s}}_K\right) \right) \le u_i\left( {\widetilde{s}}_K, s_{{\mathcal {B}}}\left( {\widetilde{s}}_K\right) \right) \end{aligned}$$
 2.For each \(k\in K\) and all bookmaker k’s strategy \({\widetilde{s}}_k\),$$\begin{aligned} u_k\left( {\widetilde{s}}_k, s_{K\setminus \{k\}}, s_{{\mathcal {B}}}\left( {\widetilde{s}}_k, s_{K\setminus \{k\}}\right) \right) \le u_k\left( s_K, s_{{\mathcal {B}}}\left( s_K\right) \right) . \end{aligned}$$
The first part of the definition states that no coalition of bettors has incentives coordinate in order to deviate in Step 2, even if the bookmakers did. The second part states that no bookmaker has incentives to deviate in Step 1.
3 Characterization of equilibria
In this section, we study the equilibrium payoff in each of the three bet markets and each attitude towards risk. We distinguish two possible scenarios: The monopolistic market and the competitive market. We say that the market is monopolistic when there exists a unique bookmaker. Remarkably, the results change drastically when we add a second one. In particular, the market becomes competitive with two bookmakers. There are no further changes in payoffs when adding a third, fourth, an so on. Hence, we define competitive market as that in which there are more than one bookmaker.
The monopolistic market does not only cover situations where there is an actual monopoly. The licensees in a particular country are offering bets continuously and the noncooperative game that we model can be seen as just a particular instance of a game that is repeatedly played. As it is wellknow from the theory of repeated games (Aumann and Shapley 1994; Rubinstein 1994; Joosten et al. 2003), almost any individual rational payoff is supported by subgame perfect equilibria. Hence, the bookmakers can eventually cooperate, even without forming a cartel, and ending up offering the bets of the monopolistic market.
3.1 Fixedodds bookmakers

If \(s_{i}\left( s_K\right) =D\), bettor i declines to bet (abstains) and her final payoff is zero.
 If \(s_{i}\left( s_K\right) =(k,H)\), bettor i bets for the home team at odd \(\pi ^{H}_k\) and her final payoff is$$\begin{aligned} u_{i}=\left( \frac{1}{\pi ^{H}_k}1\right) x_{i} + \left( 1\right) \left( 1x_{i}\right) . \end{aligned}$$
 If \(s_{i}\left( s_K\right) =(k,A)\), bettor i bets for the away team at odd \(\pi ^{A}_k\) and her final payoff is$$\begin{aligned} u_{i}=\left( 1\right) x_{i} + \left( \frac{1}{\pi ^{A}_k}1\right) \left( 1x_{i}\right) . \end{aligned}$$
The next result characterizes the (bettorstrong) subgame perfect equilibrium in the monopolistic case:
Proposition 1
 Riskadverse case:subject to$$\begin{aligned} \max \left( 1\upsilon \frac{1}{\pi ^H+\pi ^A} \right) \left( \int _{\pi ^{H}}^{1}f\left( t\right) dt+\int _{0}^{1\pi ^{A}}f\left( t\right) dt\right) \end{aligned}$$(3)$$\begin{aligned} \frac{1}{\pi ^{H}}\int _{\pi ^{H}}^{1}f\left( t\right) dt&=\frac{1}{\pi ^{A}}\int _{0}^{1\pi ^{A}}f\left( t\right) dt \\ \pi ^{H},\pi ^{A}&\in \left[ 0,1\right] , \pi ^{H}+\pi ^{A}\ge 1. \nonumber \end{aligned}$$(4)
 Riskneutral case:$$\begin{aligned} \pi ^H&\in {\arg \max }_{\pi \in (0,1]} \left( 1\upsilon \frac{q}{\pi } \right) \int _{\pi }^{1}f\left( t\right) dt \end{aligned}$$(5)$$\begin{aligned} \pi ^A&\in {\arg \max }_{\pi \in (0,1]} \left( 1\upsilon \frac{1q}{\pi } \right) \int _{0}^{1\pi }f\left( t\right) dt. \end{aligned}$$(6)
Proof
See “Appendix”. \(\square \)
From the previous result, we see that a bookmaker looks to balance the positive effect of a large volume (given by \(\int _{\pi ^{H}}^{1}f\left( t\right) dt\) and \(\int _{0}^{1\pi ^{A}}f\left( t\right) dt\)) against the negative effect of a big prize (given by \(\frac{1}{\pi ^{H}}\int _{\pi ^{H}}^{1}f\left( t\right) dt=\frac{1}{\pi ^{A}}\int _{0}^{1\pi ^{A}}f\left( t\right) dt\) in the riskadverse case, and by \(\frac{q}{\pi }\) and \(\frac{1q}{\pi }\) in the riskneutral case). A large volume is obtained by setting low \(\pi ^{H}\) and low \(\pi ^{A}\). A low prize is obtained by setting high \(\pi ^{H}\) and high \(\pi ^{A}\).
The effect of \(\rho \) (tax on profit) is irrelevant for the maximization problem. Hence, the optimal \(\pi ^{H}_k\) and \(\pi ^{A}_k\,\) are independent of the chosen \(\rho \). A different issue happens with \(\upsilon \), which gives less weight to the positive effect of a large volume. This suggests that the bookmaker would set a higher \(\pi ^{H}_k\) (and a higher \(\pi ^{A}_k\)) a larger \(\upsilon \) is, which means that the utility of bettors is reduced.
The next result characterizes the (bettorstrong) subgame perfect equilibrium in the competitive case:
Proposition 2
 Riskadverse case:
Equation (4) and \(\pi ^H + \pi ^A = \min \left\{ 2, \frac{1}{1\upsilon }\right\} \).
 Riskneutral case:
\(\pi ^H = \max \left\{ 1,\frac{q}{1\upsilon }\right\} \) and \(\pi ^A = \max \left\{ 1,\frac{1q}{1\upsilon }\right\} \).
Proof
See “Appendix”.\(\square \)
Again, the effect of \(\rho \) (tax on profit) is irrelevant. The optimal \(\pi ^{H}\) and \(\pi ^{A}\) are independent of the chosen \(\rho \). As opposed, a higher \(\upsilon \) increases the overround \(\pi ^H + \pi ^A\), which means that the utility of bettors is reduced.
3.2 Spread bookmakers

If \(s_{i}\left( s_K\right) =D\), bettor i declines to bet and her final utility is zero.

If \(s_{i}\left( s_K\right) =\left( k,H,\pi ^{H}\right) \), bettor i declares that she wants to bet in k for the home team at odd at most \(\pi ^{H}\).

If \(s_{i}\left( s_K\right) =\left( k,A,\pi ^{A}\right) \), bettor i declares that she wants to bet in k for the away team at odd at most \(\pi ^{A}\).
When \(s_{i}=\left( k,H,\pi _k\right) \) or \(s_{i}=\left( k,A,1\pi _k\right) \), we have two cases:
Example 1
Assume \(f(x) = 1\) for all \(i\in {\mathcal {B}}\), \(\left\ {\mathcal {B}} \right\ = 1\) and \(K=\{k\}\) and each bettor \(i\in {\mathcal {B}}\) announces \((k,H,x_i)\) if \(x_i > 0.5\) and \((k,A,1x_i)\) if \(x_i < 0.5\). Under these bets, \(\pi _k = 0.5\) clears the market, so that the ratio of Hbettors to Abettors should be 1. Moreover, \(\left\ {\mathcal {B}}^H_k \right\ = \left\ {\mathcal {B}}^A_k \right\ = 0.5 \) and \(\left\ \overline{{\mathcal {B}}}^H_k \right\ = \left\ \overline{{\mathcal {B}}}^A_k \right\ = 0\). Hence, there exists no excess of Hbettors nor Abettors. All bettors will be matched. In particular, the whole 0.5 volume of \((k,H,x_i)\)bettors matches the 0.5 volume of \((k,A,x_i)\)bettors.
Example 2
Assume \(\left\ {\mathcal {B}} \right\ = 1\) and \(K=\{k\}\) and the bets are D, (k, H, 0.4), (k, H, 0.6), (k, H, 0.8), (k, A, 0.2), (k, A, 0.4), and (k, A, 0.6) with volumes 0.2, 0.1, 0.3, 0.1, 0.1, 0.1, and 0.1, respectively, as shown in the first two columns of Table 2. Under these bets, \(\pi _k = 0.6\) clears the market, so that the ratio of Hbettors to Abettors should be \(\frac{0.6}{10.6}=\frac{3}{2}\). Moreover, \(\left\ {\mathcal {B}}^H_k \right\ = 0.1\), \(\left\ \overline{{\mathcal {B}}}^H_k \right\ = 0.3\), \(\left\ {\mathcal {B}}^A_k \right\ = 0.1\), and \(\left\ \overline{{\mathcal {B}}}^A_k \right\ = 0.1\). Since \(\frac{\left\ {\mathcal {B}}^H_k\cup \overline{{\mathcal {B}}}^H_k\right\ }{\pi _k} = \frac{0.4}{0.6} > \frac{0.2}{0.4} = \frac{\left\ {\mathcal {B}}^A_k \cup \overline{{\mathcal {B}}}^A_k\right\ }{1\pi _k}\), we are in Case 2 and there exists an excess of Hbettors that will not be matched. In particular, the whole 0.1 volume of (k, H, 0.8)bettors matches a \(\frac{0.2}{3}\) volume of (k, A, 0.6)bettors; a 0.05 volume of (k, H, 0.6)bettors matches the remaining \(\frac{0.1}{3}\) volume of (k, A, 0.6)bettors; finally, a 0.15 volume of (k, H, 0.6)bettors matches the remaining 0.1 volume of (k, A, 0.4)bettors. The remaining 0.1 volume of (k, H, 0.6)bettors, the 0.1 volume of (k, A, 0.2)bettors, and the 0.1 volume of (k, H, 0.4)bettors remain unmatched.
Example of a spread market
Bet  Volume  Matched 

D  0.2  No (abstain) 
(H, 0.4)  0.1  No 
(H, 0.6)  0.3  \(67\%\) 
(H, 0.8)  0.1  \(100\%\) 
(A, 0.2)  0.1  No 
(A, 0.4)  0.1  \(100\%\) 
(A, 0.6)  0.1  \(100\%\) 
Total  1  \(50\%\) 
The next result characterizes the bet volume in Step 2 for the spread bets case:
Lemma 1
Proof
See “Appendix”.\(\square \)
It follows from Lemma 1 that, as opposed to fixedodds, the spread bookmaker are not indifferent to which team will win the match. In fact, the bookmaker would always prefer the underdog (nonfavorite) to win the match, as next result shows:
Corollary 1
Let \(\pi \), \(1\pi \) be the odds that clear the market for some spread bookmaker with nonzero bet volume. If \(\pi >\frac{1}{2}\), then the bookmaker’s expost payoff is bigger when the away team wins. If \(\pi <\frac{1}{2}\), then the bookmaker’s expost payoff is bigger when the home team wins. If \(\pi =\frac{1}{2}\), then the spread bookmaker’s expost payoff is independent of which team wins.
Proof
See “Appendix”.\(\square \)
Intuitively, the explanation for this result is the following: The spread bookmaker has only one degree of freedom to modulate the actual thresholds that determine the bets. She can make the H and A bets volumes simultaneously larger or smaller, but not individually in order to equalize both scenarios. The worstcase scenario arises when the favorite team wins. Since commission is applied to prizes, when the favorite team wins, the bet volume is not high enough to compensate the low prize for winning a bet.
The next result characterizes the (bettorstrong) subgame perfect equilibria in the monopolistic case:
Proposition 3
 Riskadverse case:
\( \max _{c\in \left[ 0,1\right] } \left( \min \{1\pi ,\pi \}c  \upsilon \right) \gamma \).
 Riskneutral case:
\( \max _{c\in \left[ 0,1\right] } \left( (q+\pi 2q\pi )c  \upsilon \right) \gamma \).
Proof
See “Appendix”.\(\square \)
As c increases, the percentage of winners profits increase too, but this winner profit decreases because less bettors participate. Hence, the bookmaker looks to balance the positive effect of a big commission (hence big percentage of winnings) against the negative effect on the winnings (which decreases with c).
Like fixedodds bookmakers, the effect of \(\rho \) (tax on profit) is irrelevant for the maximization problem. Hence, the optimal c is independent of the chosen \(\rho \). Again, a different issue happens with \(\upsilon \), which penalizes the effect of a large volume. Hence, like fixedodds, the bookmaker would set a higher c, which means that the utility of the bettors is reduced.
The next result characterizes the (bettorstrong) subgame perfect equilibria in the competitive case:
Proposition 4
 Riskadverse case:
\( c = \min \left\{ 1, \frac{\upsilon }{\min \{1\pi ,\pi \}} \right\} \)
 Riskneutral case:
\( c = \min \left\{ 1, \frac{\upsilon }{q+\pi 2q\pi } \right\} \)
Proof
See “Appendix”.\(\square \)

if \({\widetilde{c}}_1 = c\), then \(s_i({\widetilde{c}}_K) = \left( 1,H,\pi \right) \) for all \(i\in {\mathcal {B}}\) such that \(x_i > \frac{\pi }{1(1\pi )c}\), \(s_i({\widetilde{c}}_K) = \left( 1,A,1\pi \right) \) for all \(i\in {\mathcal {B}}\) such that \(x_i < \frac{(1c)\pi }{1c\pi }\), and \(s_i({\widetilde{c}}_K) = D\) otherwise;

if \({\widetilde{c}}_1 \ne c\), then \(s_i({\widetilde{c}}_K) = D\) for all \(i\in {\mathcal {B}}\).
3.3 Parimutuel bookmakers

If \(s_{i}\left( s_K\right) =D\), bettor i declines to bet and her final payoff is zero.

If \(\left\ {\mathcal {B}}^{H}_k \right\ = 0\) or \(\left\ {\mathcal {B}}^{A}_k \right\ = 0\), bets are canceled for bookmaker k. The final payoff is zero for bookmaker k and bettors in \({\mathcal {B}}^{H}_k \cup {\mathcal {B}}^{A}_k\).
 If \(s_{i}\left( s_K\right) =(k,H)\), bettor i declares that she wants to bet for the home team in k. If \(\left\ {\mathcal {B}}^{H}_k \right\ >0\) and \(\left\ {\mathcal {B}}^{A}_k \right\ > 0\), her final payoff is:$$\begin{aligned} u_{i}=\frac{\left\ {\mathcal {B}}^{H}_k \cup {\mathcal {B}}^{A}_k\right\ }{\left\ {\mathcal {B}}^{H}_k\right\ }\left( 1c_k\right) x_{i} 1. \end{aligned}$$
 If \(s_{i}\left( s_K\right) =(k,A)\), bettor i declares that she wants to bet for the away team in k. If \(\left\ {\mathcal {B}}^{H}_k \right\ >0\) and \(\left\ {\mathcal {B}}^{A}_k \right\ > 0\), her final payoff is:$$\begin{aligned} u_{i} = \frac{\left\ {\mathcal {B}}^{H}_k \cup {\mathcal {B}}^{A}_k\right\ }{\left\ {\mathcal {B}}^{A}_k\right\ }\left( 1c_k\right) \left( 1x_{i}\right) 1. \end{aligned}$$
The next result characterizes the (bettorstrong) subgame perfect equilibria in the monopolistic case:
Proposition 5
Proof
See “Appendix”.\(\square \)
Proposition 5 uses bettorstrong subgame perfect equilibria. There are multiple subgame perfect equilibria, but they will involve an unreasonable coordination among bettors. For example, assume \(K=\{k\}\). Then, for any \(c^*\in \left[ 0,\frac{1}{2}\right] \), consider the following strategy profile: \(s_k = c^*\) and \(s_i({\widetilde{c}}_k) = D\) for all \(i\in {\mathcal {B}}\) when \({\widetilde{c}}_k \ne c^*\); when \({\widetilde{c}}_k = c^*\), \(s_i({\widetilde{c}}_k)\) is defined as in Proposition 5. This is a subgame perfect equilibrium. Hence, any \(c\in \left[ 0,\frac{1}{2}\right] \) is supported in a subgame perfect equilibrium.
The next result characterizes the bettorstrong subgame perfect equilibria in the competitive case:
Proposition 6
Proof
See “Appendix”.\(\square \)
Next result follows from Proposition 1, Proposition 2, Proposition 5 and Proposition 6:
Proposition 7
For any v and \(\rho \), riskadverse fixedodds and parimutuel yield the same payoff allocation in bettorsubgame perfect subgame equilibrium.
Proof
See “Appendix”.\(\square \)
4 Effect of taxation
We can now state our main results. These results hold for each of the three types of bookmakers: fixedodds, spread, and parimutuel. The first proposition is implied by the results presented in the previous section. It follows from the fact that \(\rho \) does not play any role in the characterization of the equilibria.
Proposition 8
In a monopolistic market, tax on profit (\(\rho \)) leaves odds, commissions and bettors’ utilities unaffected, and decreases linearly the bookmaker’s payoff. The maximum tax income is achieved for \(\rho = 1\).
Proof
See “Appendix”.\(\square \)
The second part of Proposition 8 simply says that the maximum tax income is achieved when the monopolistic bookmaker is a stateowned company.
As opposed, the role of \(\upsilon \) will depend on the particular distribution on the bettors. In general, one may expect that an increase in \(\upsilon \) would decrease the bet volume. Hence, the maximum utilitarian social welfare should be achieved for \(\upsilon = 0\). We will check it in a particular example after presenting the main result, which describes the effect of taxation in competitive markets.
Theorem 1
 a)
Taxes on profit (\(\rho \)) leave odds, commissions, tax income, and utilities unaffected.
 b)
Taxes on volume (\(\upsilon \)) increase odds and commission, and reduces the utility of bettors. The utility of bookmakers remains unaffected.
 c)
Maximum utilitarian social welfare is achieved for \(\upsilon = 0\).
 d)
Maximum tax income is achieved for some \(\upsilon \in \left( 0,\frac{1}{2}\right) \) in the riskadverse case, and \(\upsilon \in \left( 0,\max \{q,1q\}\right) \) in the riskneutral case.
Proof
See “Appendix”. \(\square \)
Theorem 1 provides a range of values where the tax income maximizer can be. The exact value of the maximizing \(\upsilon \) will depend on the distribution of bettors given by f. On the other hand, we have no complete counterpart for Proposition 1 in the monopolistic case, but we can still figure out how it behaviours for some particular function f and (for the riskneutral case) probability q.
For the riskneutral case, a natural choice for q is the one that agrees with f in the sense that odds \(q,1q\) will clear the market with maximum bet volume. Next lemma characterizes this q, that we denote as \(q^{*}\).
Lemma 2
Proof
See “Appendix”.\(\square \)
For example, when f is symmetric (i.e. \(f(x) = f(1x)\) for all \(x\in (0,1)\)) it is clear that \(q^{*} = \frac{1}{2}\). When \(f(x) = 2x\) for all \(x\in (0,1)\), then \(q^{*} = \frac{\sqrt{5}1}{2} \approx 0.618\).

Taxes on volume (\(\upsilon \)) increase odds and commission, and reduce the utility of both bettors and bookmaker.

Maximum utilitarian social welfare is achieved for \(\upsilon = 0\).

In the monopolistic case, maximum tax income is achieved for \(\rho = 1\) and \(\upsilon = 0\).
Effect of taxation when \(f(x)=2x\) and \(q = \frac{\sqrt{5}1}{2}\)
Bookmaker  Market  Risk  Max. income  Maximizer 

Fixed odds  Monopoly  Adverse  0.143087  \(\rho = 100\%,\upsilon = 0\%\) 
Fixed odds  Monopoly  Neutral  0.143852  \(\rho = 100\%,\upsilon = 0\%\) 
Spread  Monopoly  Adverse  0.110593  \(\rho = 100\%,\upsilon = 0\%\) 
Spread  Monopoly  Neutral  0.143828  \(\rho = 100\%,\upsilon = 0\%\) 
Parimutuel  Monopoly  –  0.143087  \(\rho = 100\%,\upsilon = 0\%\) 
Fixed odds  Competition  Adverse  0.143087  \(\upsilon = 25\%\) 
Fixed odds  Competition  Neutral  0.140669  \(\upsilon = 24\%\) 
Spread  Competition  Adverse  0.110531  \(\upsilon = 19\%\) 
Spread  Competition  Neutral  0.143828  \(\upsilon = 25\%\) 
Parimutuel  Competition  –  0.143087  \(\upsilon = 25\%\) 
Apart from the riskadverse spread case, where the bookmaker has no capability to adjust both equilibrium odds, the maximum tax income is similar in all the other markets.
5 Effect of taxation in the symmetric case
This case covers situations where there is no favourite team in the sport match, or when there exists a favourite but it has a handicap that makes the match even. Such handicap bets are quite common in online betting, and allow the bookmakers to assure that the volume of bets between home and away teams are balanced. In our model, this is particularly relevant for the spread bets bookmaker, since it makes her indifferent of who is the winning team (Corollary 1).
The next result characterizes the equilibrium payoffs and states that fixed odds, spread bets and parimutuel are equivalent in the symmetric case.
Proposition 9
 a)
Monopolistic case: \( \pi ^{*} = \arg \max _{\pi \in \left[ \frac{1}{2},1\right] } \left( 2(1\upsilon )  \frac{1}{\pi } \right) \int _{\pi }^1 f(t) dt. \)
 b)
Competitive case: \(\pi ^{*} = \min \left\{ 1,\frac{1}{2\left( 1\upsilon \right) }\right\} \).
Proof
See “Appendix”.\(\square \)
As a paradigmatic case, next proposition shows the effect of taxes when \(\alpha = 1\), i.e. the uniform distribution \(f(x) = 1\) for all \(x\in (0,1)\).
Proposition 10
 a)
Taxes on volume (\(\upsilon \)) increase odds and commission, and reduces the utility of bettors. In a monopolistic market, they also decrease the utility of the bookmaker when \(\rho < 1\).
 b)
Maximum utilitarian welfare is achieved for \(\upsilon = 0\).
 c)Maximum tax income is achieved as follows:
 c1)
In the competitive case, by \(\rho =1\) and \(\upsilon =0\).
 c2)
In the monopolistic case, by \(\upsilon =2\frac{\sqrt{2}}{2}\approx 29.3\%\).
 c1)
Proof
See “Appendix”.\(\square \)
For arbitrary \(\alpha \in (0,\infty )\), a simulation analysis shows that Proposition 10, parts a), b) and c1), hold in general, and the maximizing \(\upsilon \) in the competitive case [Proposition 10, part c2)] decreases with \(\alpha \). The \(\upsilon \) that maximizes tax income in the competitive case is represented in Fig. 1.
6 Concluding remarks
In this paper, we model three different online betting markets: those given by fixedodds, spread bets, and parimutuel, respectively. This allows us to analyse the effect of two different tax schemes: On volume (GBD) and on profit (GPT). In all these markets, odds (fixedodds) and commission (spread bets and parimutuel) are unaffected by GPT but they are by GBD. Hence, it should be expected that odds and commission to depend on the particular regulation. For example, Paddy Power Betfair, which includes one of the largest Internet spread betting companies, charges a different commission for spread bets on each country. This commission is 5% in the United Kingdom, Ireland, Italy, Gibraltar and Malta; 7% in Albany, Armenia, Croatia, Monaco, Serbia, Montenegro and Slovakia; and 6.5% in the rest of the countries, including Spain. Moreover, the company is restricted in Belgium, Greece, Germany,^{9} Turkey, Israel, France and Portugal, among other countries.
As opposed to other approaches in the literature, we do not need to assume the existence of an actual probability for the home (or away) team to win the match. Instead, the bettors are characterized by their subjective beliefs on this probability. An alternative interpretation is that each bettor is characterized by the the odd at which she is willing to bet, which includes the individual surplus of the act of betting itself. In this sense, a natural extension of the model, which does not change the results, is to assume that there are two subsets of bettors: one of them willing to bet for the home team, another willing to bet for the away team, and both characterized by the minimum odd they will bet.
As for the bookmakers, we cover two situations: either they are riskadverse and play a maximin strategy (i.e. they maximize profits under the worst match outcome scenario), or they are riskneutral because of a precise common estimation of the true probability of the match outcome. Assuming there is no such precise estimation, a more general decision criterium than maximin would be the Hurwicz’s rule, which uses a weighted average between both match outcomes. Checking the implications of using the Hurwicz’s rule is an open question. My own feeling is that the general results remain with a more elaborate characterization of the bet volume in equilibrium (as given by 4).
Another extension is to consider bettors betting on more than one event simultaneously. Of course, bettors decisions will become more elaborate when they have a limited budget and several matches to choose. Competition among different matches may arise. In fact, this situation is already partially covered by the model, because: (1) distribution f may depend on the existence of other potential matches, and (2) bettors’ strategies are not affected when they have no budget restrictions, so that they are able to bet in all the matches they find profitable.
Footnotes
 1.
Ley 13/2011, de 27 de mayo, de regulación del juego (in Spanish). Boletín Oficial del Estado 127, ref. BOEA20119280. Available at https://www.boe.es/buscar/pdf/2011/BOEA20119280consolidado.
 2.
In case the odds summed up less than 100%, it would be possible, by betting an appropriate amount of money on each possible match outcome, to win a positive amount irrespectively of the final match outcome.
 3.
The sum of the odds, called overround, provides a way to measure the operator advantage.
 4.
There are other posible decision criteria, as for example the Hurwicz’s rule (see Sect. 6). We study the maximin case (maximizing profits under the worst case scenario) due to its simplicity.
 5.
This is done in order to avoid meaningless strategies such as, for example, to bet iff \(x_i\) is a rational number.
 6.
Undominated strategies are required in order to avoid meaningless equilibria of the form “everybody chooses D”. This refinement is not needed for the bettorstrong subgame pefect equilibrium.
 7.
Again, undominated strategies are not required for the bettorstrong subgame equilibrium.
 8.
Tested on a sampling of 1000 instances of \(\upsilon \) uniformly distributed on [0, 1] in each market.
 9.
Betfair is only restricted in Germany for spread bets.
Notes
Acknowledgements
I am grateful to two anonymous referees for helpful comments. Usual disclaimer applies.
References
 Aumann RJ, Shapley LS (1994) Longterm competition–a gametheoretic analysis. In: Megiddo N (ed) Essays in game theory: in honor of Michael Maschler. Springer, New York, pp 1–15. doi: 10.1007/9781461226482_1
 Bag PK, Saha B (2011) Matchfixing under competitive odds. Games Econ Behav 73(2):318–344. doi: 10.1026/j.geb.2011.03.001 CrossRefGoogle Scholar
 Bag PK, Saha B (2016) Matchfixing in a monopoly betting market. J Econ Manag Stratagy. doi: 10.1111/jems.12172 Google Scholar
 Ferrari S, CribariNeto F (2004) Beta regression for modelling rates and proportions. J Appl Stat 31(7):799–815. doi: 10.1080/0266476042000214501 CrossRefGoogle Scholar
 Ficom Leisure (2011) Update on Italy’s online betting and gaming market. http://sectordeljuego.org/img_fckeditor/informeitalia.pdf
 Forbes C, Evans M, Hastings N, Peacock B (2011) Statistical distributions, 4th edn. Wiley, New JerseyGoogle Scholar
 Global Betting and Gaming Consultancy (2011) It’s all about the tax. http://www.gbgc.com/itsallaboutthetax/
 Hofmann J, Spitz M (2015) Germany. In: StewardJones H (ed) Gambling 2016: a practical crossborder insight into gambling law, chap. 11. Global Legal Group, pp 54–58. http://www.iclg.co.uk/practiceareas/gambling/gambling2016/germany
 Joosten R, Brenner T, Witt U (2003) Games with frequencydependent stage payoffs. Int J Game Theory 31(4):609–620. doi: 10.1007/s001820300143 CrossRefGoogle Scholar
 Levitt SD (2004) Why are gambling markets organized so differently from financial markets? Econ J 114(495):223–246. doi: 10.1111/j.14680297.2004.00207.x CrossRefGoogle Scholar
 National Audit Office (2005) Gambling duties. HM customs and excise. Report by the controller and Auditor General. HC 138 Session 2004–2005. http://www.nao.org.uk/publications/0405/gambling_duties.aspx
 Neyman A (2002) Values of games with infinitely many players. In: Aumann R, Hart S (eds) Handbook of game theory with economic applications, chap. 56, vol III. Elsevier Science, Amsterdam, pp 2121–2167Google Scholar
 Ottaviani M, Sørensen PN (2009) Surprised by the parimutuel odds? Am Econ Rev 99(5):2129–2134. doi: 10.1257/aer.99.5.2129 CrossRefGoogle Scholar
 Paton D, Siegel DS, Williams LV (2001) Gambling taxation: A comment. Aust Econ Rev 34(4):437–440. doi: 10.1111/14678462.00211 CrossRefGoogle Scholar
 Paton D, Siegel DS, Williams LV (2002) A policy response to the ecommerce revolution: the case of betting taxation in the UK. Econ J 112(480):296–314. doi: 10.1111/14680297.00045 CrossRefGoogle Scholar
 PwC (2011) Taxation and online sports betting in Germany: considering the relative merits of a tax on gross gaming revenue and a tax on stakes for the potential regulation of online sports betting stakes. https://gamblingcompliance.com/files/PwC%20Report%20German%20betting%20tax%202011.pdf
 Rubinstein A (1994) Equilibrium in supergames. In: Megiddo N (ed) Essays in game theory: in honor of Michael Maschler, chap. 2. Springer, New York, pp 17–27. doi: 10.1007/9781461226482_2
 Smith J (2000) Gambling taxation: public equity in the gambling business. Aust Econ Rev 33(2):120–144. doi: 10.1111/14678462.00143 CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.