On the effect of taxation in the online sports betting market

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1 On the effect of taxation in the online sports betting market Juan Vidal-Puga Research Group in Economic Analysis Departamento de Estatística e IO Universidade de Vigo, Spain June 8, 2 We analyze the effect of taxation on the online sport betting market. Abstract This market is characterized by its negligible marginal costs. Taxation can be on volume (General Betting Duty) or on gross profit (Gross Profit Tax). We model the two most popular online sport betting bets: fixed-odds and spread, as compared with other traditional sport betting: parimutuel. We show that from the bettor s point of view taxation on gross profit is always preferable to taxation on volume. Keywords: taxation, online betting market. Work in progress. Latest version at: Financial support by the Spanish Ministerio de Ciencia y Tecnología and FEDER through grant ECO C2-/ECON and Xunta de Galicia through grant PXIB362299PR is gratefully acknowledged. vidalpuga@uvigo.es. Address: Facultade de Ciencias Sociais. Campus A Xunqueira. 365 Pontevedra. Spain.

2 Introduction Over the last few 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. Some examples: the United Kingdom applied a 6.75% tax on GBD until October 2, when it was replaced by a 5% tax in GPT ([6]). Italy applies a 2%-5% tax on GBD ([2]) since March 2. France applies a 8.5% tax on GBD ([3]) since 2. Cyprus applied a % tax on GPT until March 2, when it was replaced by a 3% tax on GBD. Recently, Spanish authorities approved a law that applies a 25% tax on GPT for some types of bets and a 5-22% tax on GBD for others, plus a general,% tax on GBD ( [5]). In the next table we summarize the data: GBD GPT UK 6.75% (until 2) 5% (since 2) Italy 2-5% - France 8.5% - Cyprus 3% (since 2) % (until 2) Spain 25%(parimutuel)+.%(general) 5-22% (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: Fixed-odds and spread. In a fixed-odd market, the operator sets the odds for each possible outcome of the match, and the bettors decide whether they accept or not these odds. 2

3 In a spread market, the operator acts as a intermediator among the users, whom bargain the odds. For sport matches, a bet of monetary unit on a particular team yields a return of monetary units in case the team wins the match, and otherwise. In this context, an odd (, ) is defined as the probability assigned by the market. Notice that any risk-neutral bettor would find it profitable to bet at odd when its private probability estimation is higher than. In parimutuel and spread bets, the operator s profit comes from a commission on either the amount at risk or the winning amount (typically a 5% in online spread operators). In fixed-odds bets, the operator s profit comes from the odds, which should sum up more than % 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 (fixed-odds 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. As opposed, in the parimutuel market GPT and GBD provide the same effect. These results agree with the ones presented by Smith ([]) and Paton et al. ([9][8]), whom analyze the effect of the different taxation schemes in Australian, UK and USA betting markets. These results, however, are more 2 In case the odds summed up less than %, 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. 3

4 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 concentrate on parimutuel markets: Ottaviani and Sørensen ([7] and references herein) 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 3.3) that the spread bet operator would get a higher profit if the underdog wins the game. An immediate intuition suggests that bettors have incentives to bet for the favorite in spread markets. Hence, the existence of arbitrators betting in different online operators would provide an alternative explanation for the underdog bias in parimutuel markets 4. Other works concentrate on fixed-odd markets. For example, Bag and Saha ( [] and references herein) study the externalities due to bribery in sports; and Levitt ([4]) 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. In Section 2 we provide the model. In Section 3, we study the equilibrium payoffs in each of the three markets and provide the main results. 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 B of bettors are interested in betting, but only if the odds are attractive; a finite set N of bookmakers that offer odds; and a regulator (Government) that decides on taxes. We assume that bettors are risk neutral and try to maximize their ex- 4 Work in progress. 4

5 pected profit. The bookmakers, on the other side, are risk adverse and try to maximize their monetary profit under the worst possible outcome of the match. Finally, the regulator tries to maximize social welfare (via tax income). Each bettor i B is characterized by its individual belief (i.e. the probability) x i that the home team wins ( x i is the probability that the away team wins); x i is distributed following an absolutely continuous cdf F with full support over (, ). There are three types of bookmakers: Fixed-odd bookmakers, spread bookmakers and parimutuel bookmakers. Fixed-odd bookmakers decide odds, A [, ] such that any bettor that bet on the home (away) team receives ( A ) in case of home (away) win, and in case of away (home) win. Spread bookmakers decide a commission c on the profit of any winning bettor. Parimutuel bookmakers decide a commission c on the stake of any bettor. Let N =,..., n} be the set of bookmakers. Each bookmaker k is characterized by its type (fixed-odds/spread/parimutuel) and a subset B k B of bettors. We assume that B,..., B n } is a partition of N. We assume that the conditional distribution of F is also continuous in each B k with full support over (, ). 3 The non-cooperative game The study the effect of taxation in the online market. Assume the regulator announces a tax, that could be a percentage υ on volume (GBD) or a percentage ρ on gross profit (GPT). The non-cooperative game has two steps: Step : Bookmakers observe υ or ρ and announce their odds (fixed-odds) or commission (spread/parimutuel). Step 2: bettors observe odds and commission on their respective bookmakers and choose to participate or not. 5

6 3. Fixed-odds bookmakers Bookmaker chooses odds and A. Bettors observe the odds and (simultaneously) choose s i D, H, A}. is is If s i = D, bettor i declines to bet (abstains) and its final payoff is zero. If s i = H, bettor i bets for the home team at odd and its final payoff ( ) u i = x i + ( x i ) ( ). If s i = A, bettor i bets for the away team at odd A and its final payoff ( ) u i = x i ( ) + ( x i ). A Let B H = i B : s i = H} and B A = i B : s i = A}. The bookmaker s final payoff is } ( υ) ( BH + B A ) u k = ( ρ) min H B H, ( υ) ( B H + B A ) A B A }} = ( ρ) ( υ) ( B H + B A ) max B H, B A. A Theorem 3. Given υ and ρ, there exists a (strong) subgame perfect equilibrium. Moreover, the odds in equilibrium are characterized by the maximization problem max ( ρ) ( υ) [ + subject to A ] } A = () A, A [, ], + A Proof. (sketch) Assume w.l.o.g. B k =. In Step 2 of the game, it is optimal for any bettor i B k with x i > to announce s i = H. Analogously, it is optimal for any bettor i B k with x i > A to announce s i = A. Hence, B H = and B A = A. 6

7 Equality () comes from } the fact that the bookmaker wants to minimize max B H, A B A, that is the maximum bettors winnings when either the home team ( B H ) or the away team ( A B A ) wins. Hence, in equilibrium both amounts should be equal. For a fixed A, there exists a unique that satisfies (). To see why, notice that f H () = is a strictly decreasing function on (, ) with f H ( + ) = + and f H ( ) = +, whereas f A () = is a strictly increasing function of (, ) with f A ( + ) = + and f A ( ) = +. Hence, for each A, there exists a unique with f H ( ) = f A ( A ). Moreover, the largest is, the largest A is. Let Π A : (, ) (, ) be the function that assigns to each its corresponding A. This function is well-defined and strictly increasing. Notice that the bet volume is given by B H + B A. Since tax v applies on volume, and ρ applies on profit, the bookmaker would maximize ( ρ) ( υ) [ B H + B A ] } B H which is the desired maximizing function. Furthermore, it is straightforward to check that there exists at least one maximizing (, ). From the previous result, we see that the bookmaker looks to balance the positive effect of a large volume (given by + A ) against the negative effect a big prize (given by = A A ). A large volume is obtained by setting low (and hence low A ). A low prize is obtained by setting high (and hence high A ). The effect of ρ (tax on profit) is irrelevant for the maximization problem. Hence, the optimal and A are independent of the chosen ρ. A different issue happens with υ, which gives less weight to the positive effect of a large volume. Hence, the bookmaker should set a high means that the utility of the bettors is reduced: (and high A ), which Corollary 3. Assume there exists a unique (strong) subgame perfect equilibrium. Then, 7

8 a) Taxes on volume (υ) decreases the bookmakers utility, increases the over-round and decreases the volume. Bettors utility is decreased. b) Taxes on profit (ρ) decreases the bookmakers utility, but maintains the over-round and the volume. Bettors utility remains unchanged. 3.2 Spread bookmakers Bookmaker chooses commission c (, ). Bettors observe the commission and (simultaneously) choose s i D} H, A} [, ]. If s i = D, bettor i declines to bet and its final utility is zero. If s i = (H, ), bettor i declares that it wants to bet for the home team at odd at most. If s i = (A, A ), bettor i declares that it wants to bet for the away team at odd at most A. The bookmaker matches (H, )-bettors with (A, A )-bettors that satisfy A with odds, such that: and A. The matching is done so that each volume (H, )-bettors is matched with a volume of (A, A )-bettors. The reason is that, in case home team wins, a volume of money is transferred from (A, A )-bettors to (H, )-bettors, so that each (H, )-bettor receives a gross winning (profit + bet): + = hence granting their request to bet for the home team at odd at least. Analogously, in case away team wins, a volume of money is transferred from (H, )-bettors to (A, A )-bettors, so that each (A, A )-bettor receives a gross winning (profit + bet): + = A hence granting their request to bet for the away team at odd at least A. Hence, is chosen so that Γ H + ΓH ΓA Γ A + ΓA ΓH 8

9 where Γ H = i B k : s i = (H, ), > } Γ H = i B k : s i = (H, )} Γ A = i B k : s i = (A, A ), A > } Γ A = i B k : s i = (A, )} If s i = (H, ) with >, bettor i bets for the home team at odd and its final payoff is u i = x i ( c) ( ) + ( x i ) ( ). (2) If s i = (A, A ) with A >, bettor i bets for the away team at odd and its final payoff is u i = x i ( ) + ( x i ) ( c) ( ). (3) If s i = D, or s i = (H, ) with <, or s i = (A, A ) with A <, bettor i does not bet and its final payoff is zero. When s i2 =, we have two cases: Γ H Case : Γ A. If s i = (H, ), then bettor i bets for the home team and his final payoff is (2). If s i = (A, ), then bettor i bets for the away team with probability u i = Γ H Γ A and its final payoff is ΓH [ ( )] x ΓA i ( ) + ( x i ) ( c). Γ H Case 2: Γ A. If s i = (A, ), then bettor i bets for the away team and his final payoff is (3). If s i = (H, ), then bettor i bets for the home team with probability u i = Γ A Γ H and its final payoff is ΓA [ ( ) ] x ΓH i ( c) + ( x i ) ( ). 9

10 Let Γ = Γ H Γ H Γ A Γ A. The final payoff for the bookmaker is c ( ( } ) u k = ( ρ) ) Γ A ΓH Γ min + min, Γ H c ( ( } ) ) Γ A + min, Γ H υ Γ Γ Γ A A ( ΓH + min } ) } ΓH, ΓA υ Γ = ( ρ) c min ( ΓA + min } ) ΓA, ΓH υ Γ Theorem 3.2 Given υ and ρ, there exists a (strong) subgame perfect equilibrium with undominated strategies 5. Moreover, the commission and odds in equilibrium are characterized by the maximization problem max, } ( c) c c( ) c( ) max ( ρ) c ( c (,) υ + ) ( c) c subject to c( ) c( ) =, c [, ] ( c) c (4) Proof. (sketch) Assume w.l.o.g. B k =. In Step 2 of the game, it is optimal for any bettor i B k to bet for the home team at odds when is such that which is equivalent to: ( ) x i ( c) + ( x i ) ( ) > < x i ( c) x i c x i > c ( ). 5 Undominated strategies are required in order to avoid meaningless equilibria of the form everybody chooses D. This requirement is not needed for strong subgame perfect equilibrium though.

11 Analogously, it is optimal for any bettor i B k to bet for the away team at odd when is such that which is equivalent to: x i ( ) + ( x i ) ( c) > x i < x i c + x i c ( c) c. ( ) > Given c (, ), we study the subgame that begins in step 2. Let c be the unique odd that satisfies c c c( c ) = c ( c) c c. c A (strong) subgame perfect equilibrium strategy is given as follows. Let i B k. Then, If x i > c c( c ), announce s i = (H, c ) If x i < c ( c) c c, announce s i = (A, c ) Otherwise, announce s i = D. In this equilibrium, the odds are c, c. Moreover: } c Γ H = i B k : x i > c ( c ) Γ H = } Γ A = i B k : x i < c ( c) c c Γ A =.

12 The payoff for the bookmaker is: ( ΓH + min }) Γ H, Γ A υ Γ u k = ( ρ) c min ( ΓA + min }) ΓA, ΓH υ Γ ( ) c υ Γ c( = ( ρ) c min c ) ( ) c ( c) c c υ Γ c = ( ρ) c c c c( max c ) c( c ) c ( c) c c ( υ + c ( c) ) c c c c( c ) } It is straightforward to check that the bigger c is, the bigger is. As c increases, the percentage of winners profits increase too, but this winner profit decreases because less bettors bet. 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 fixed-odds bookmakers, the effect of ρ (tax on profit) is irrelevant for the maximization problem. Hence, the optimal c is independent of the chosen ρ. Again, a different issue happens with υ, which penalizes the effect of a large volume. Hence, like fixed-odds, the bookmaker would set a higher c, which means that the utility of the bettors is reduced: Corollary 3.2 Assume there exists a unique (strong) subgame perfect equilibrium. Then, a) Taxes on volume (υ) decreases the bookmakers utility, increases the commission and decreases the volume. Bettors utility is decreased. b) Taxes on profit (ρ) decreases the bookmakers utility, but maintains the commission and the volume. Bettors utility remains unchanged. As opposed to fixed-odds, the spread bookmaker is not indifferent to which team will win the match. In fact, the bookmaker would always prefer the underdog (non-favorite) to win the match, as next result shows: 2

13 Corollary 3.3 Let, be the odds in equilibrium. If >, then the 2 bookmaker s profit is bigger when the away team wins. If <, then bookmaker s profit is bigger when the home team wins. If =, then the spread 2 2 bookmaker s profit is independent on which team wins. Proof. From Theorem 3.2, v H = c( ) is the volume of bettors that bet for the home team, whereas v A = ( c) c is the volume of bettors that bet for the away team. When the home team wins, the gross profit of the bookmaker is c ( ) v. When the away team wins, the gross profit of the bookmaker is c ( ) v A. Moreover, we have v = v A. Hence, the bookmakers prefers the home (away) team to win when v H < v A (v H > v A ). Equivalently, the bookmaker prefers the home (away) team to win when v H va < ( v H va > ). Since v H va =, it only happens when is straightforward. ( > ), i.e. < 2 ( > 2 ). The result for = Parimutuel bookmakers < Bookmaker chooses commission c (, ). Bettors observe the commission and (simultaneously) choose s i D, H, A}. Let B H = i B : s i = H} and B A = i B : s i = A}. If s i = D, bettor i declines to bet and its final utility is zero. If s i = H, bettor i declares that it wants to bet for the home team. Its final payoff is: u i = x i B H + B A B H ( c). If s i = A, bettor i declares that it wants to bet for the away team. Its final payoff is: u i = ( x i ) B H + B A B A The bookmaker s final payoff is ( c). u k = ( ρ) c ( υ) ( B H + B A ). 3

14 Theorem 3.3 Given υ and ρ, in any (strong) subgame perfect equilibrium, each bettor i bets for the home team if x i > and for the away team if x i < A for some thresholds, A [, ]. Moreover, the commission and thresholds in equilibrium are characterized by the maximization problem [ A ] max ( ρ) ( υ) c + c (,) subject to = A A + B = c A, B. A Proof. (sketch) Assume (s i ) i B is an equilibrium strategy. Then: ( x i ) x i + A + A A These A and B are characterized by: c x i c x i A. + A A + A A = = c c which implies A A =. 4

15 and also + + A A = = + A = + A = c c c c. The existence of an equilibrium is straightforward. Corollary 3.4 Assume there exists a unique (strong) subgame perfect equilibrium. Then, taxes on volume (υ) and on profit (ρ) are equivalent; both decrease bookmaker s utility, but maintains the commission and the volume. Bettors utility remains unchanged. References [] Bag P.K. and Saha B. (2) Match-fixing under competitive odds. Games and Economic Behavior. Forthcoming. [2] Ficom Leisure Advisory & Investments (2) Update on Italy s Online Betting & Gaming Market. Available at fckeditor/informeitalia.pdf [3] Global Betting & Gaming Consultants (2) It s all about the tax. Available at [4] Levitt S.D. (24) Why are gambling markets organised so differently from financial markets? The Economic Journal 4, [5] Ley 3/2, de 27 de mayo, de regulación del juego (in Spanish) (2). Boletín Oficial del Estado 27(I), Available at 5

16 [6] National Audit Office (25) Gambling Duties. HM Customs and Excise. Report by the comtroller and Auditor General. HC. 38 Session Available at duties.aspx [7] Ottaviani M. and Sørensen P.N. (29) Surprised by the Parimutuel odds? American Economic Review 99(5), [8] Paton D., Siegel D.S. and Williams L.V. (2) Gambling taxation: a comment. The Australian Economic Review 34(4), [9] Paton D., Siegel D.S. and Williams L.V. (22) A policy response to the e-commerce revolution: the case of betting taxation in the UK. The Economic Journal 2, F296-F34. [] Smith J. (2) Gambling taxation: Public equity in the gambling business. Australian Economic Review 33,

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