Lock! Risk-Free Arbitrage in the Japanese Racetrack Betting
MarketMasahiro ASHIYA +January 2013This paper finds that arbitrage was possible in two out of 175 Japanesethoroughbred races even after taking account of (a) the size of the minimum
betting unit and (b) the negative effect of arbitrage on the odds. The guaranteed
profits in these two races were 5,120 yen (about $64) and 340 yen.

JEL Classification Codes: G14.
Keywords: arbitrage; pari-mutuel; market efficiency; betting; wagering.

+ Faculty of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe, 657-8501, Japan;+81-78-803-6840, ashiya@econ.kobe-u.ac.jp

11. Introduction
The racetrack betting market shares many characteristics with the stock market,including uncertain future earnings, a large number of participants, and potential
access to widely available rich information sets. Furthermore, Thaler and Ziemba(1988, p.162) point out a couple of advantages of the racetrack betting market: (1)each bet has a well-defined termination point at which its value becomes certain,and (2) quick and repeated feedback facilitates learning. On these grounds,
racetrack betting markets should have a better chance of being efficient comparedto the stock market or other financial markets.
Past literature, however, has found arbitrage opportunities in the racetrack
betting market, where risk-free profits are guaranteed. Such an opportunity iscalled a “lock”. To explain the “lock”, let us consider a hypothetical betting market
where one can buy an arbitrary amount of betting ticket without changing theodds. Suppose that n horses contest a race and that a successful unit bet on horse ito win, WINi, returns Oi units including the original stake. Also suppose that aunit bet on EXACTAij returns Oij units when horses i and j finish first and secondin correct order. Then, to secure a return of one unit when horse i wins, one must
bet either (a) 1⁄Oi units on WINi or (b) 1⁄Oij units on EXACTAij for every j ≠ i.
Consequently, one can make a risk-free profit when the following inequality issatisfied:
11iij∑ni=1 min { , ∑j≠i } ≤ 1.
OOIf one can bet on TRIFECTAijk (first three in order), risk-free arbitrage is possible
when1

11iijijkP ≡ ∑ni=1 min {O , ∑j≠i min {O , ∑k≠i,j O }} ≤ 1.

(1)

In general, a typical pari-mutuel betting market offers a variety of bet types suchas Win, Exacta, Quinella (first two in either order), and Trifecta. Since each bet
type has its own pool, the odds will be different among them. Arbitrage is possible
if the difference is large enough.
Nevertheless, the opportunities for the lock in pari-mutuel betting markets
are considered to be extremely rare because of the following two reasons.1 Firstly,arbitrage is self-destructive. Suppose that one buys WINi as a part of thearbitrage strategy. Then the return Oi decreases as a result, and it becomes moredifficult to satisfy the inequality (1). Secondly, the above illustration ignores theexistence of the minimum betting unit. In reality, one must buy at least one unitof the betting ticket even if an infinitesimal amount is enough for the arbitrage.
Therefore the chance for arbitrage is much smaller than it seems to be at first2glance.
These arguments give rise to one question: how often is the lock happening?
The answer of past literature is not satisfactory. Hausch and Ziemba (1990a)
construct a lock in the show pool of the 1979 Alabama Stakes at Saratoga, butthey do not investigate the frequency of arbitrage opportunities. Edelman andO’Brien (2004) find that a lock is possible in 31 out of 2667 Australianthoroughbred races in early 2000. Their analysis, however, neglects theself-destructive effect of the arbitrage on the odds and the size of the minimum
betting unit. Hence their result shows only the upper limit of the frequency oflocks.
We shed a new light on this issue with the help of a unique sales data ofJapanese thoroughbred races operated by a local municipality (NationalAssociation of Racing), which contains the sales volume of all bet types. We
incorporate the size of the minimum betting unit into the analysis, and we checkevery possible strategy of arbitrage given the available bet types (including
Quinella and Trifecta). Furthermore, we calculate the exact odds after executionof the arbitrage. This enables us to judge whether the arbitrage is ex-postprofitable.
The main findings in the next section are as follows. Firstly, a lock is possible
in two out of 175 races (1.14%). Secondly, the guaranteed profits of these races are
5,120 yen (about $64) and 340 yen (about $4.25). In light of the frequency ofarbitrage opportunities, the amount of the guaranteed profit is not large enoughto cover the opportunity cost of time. Thirdly, if one does not take account of theminimum betting unit and the negative effect of the arbitrage on the odds, theinequality (1) is satisfied for 35 out of 175 races (20.0%). This figure demonstrates
that we must give proper consideration to these two factors in order to evaluate
the true profitability of the arbitrage.
2. Data and results
We analyze published final sales data and final odds data of Japanesethoroughbred races operated by Arao city in the period of September 30 throughDecember 23 of year 2011. There are 175 races, and the average total sales
amount is 6,436,444 yen (approximately $80,000) per race. The average pool sizeof Win, Brackets Quinella,2 Quinella, Exacta, and Trifecta are 118,800 yen,193,264 yen, 480,807 yen, 740,348 yen, and 3,893,970 yen respectively. The
average number of starters is 8.35.
As a preliminary step, we compute the value of P in equation (1) for every
175 races. There is no race with P < 0.7. There are three races with 0.7 ≤ P < 0.8,3

six races with 0.8 ≤ P < 0.9, and 26 races with 0.9 ≤ P ≤ 1. In total, there are 35
races (20.0% of 175 races) that satisfy the inequality (1). This percentage is muchhigher than that of Edelman and O’Brien (2004) (1.16% of 2667 races). This might
be due to the peculiarity of the Japanese local thoroughbred races.3
Next, we try to construct an arbitrage strategy for these 35 races with P ≤ 1.4
In doing so, we must take account of the minimum betting unit and the negativechange of the relevant odds. To take a simple example, let us examine Table 1, thefinal odds tables of Race 11 at Arao on November 25, 2011. 5 Suppose that one
needs to secure a payout of 10,000 yen regardless of the order of arrival. Then
Table 2 shows the best strategy when the odds are fixed at the level of Table 1.
Since the minimum betting unit is 100 yen (about $1.25), one needs to buy 22
units of WIN1, one unit of WIN3, and so on. If Horse No. 1 wins, the payout is10,120 yen. If Horse No.2 wins, the payout is from 10,040 yen (when Horse No. 8
finishes second) to 71,490 yen (when Horse No.7 finishes second). The total cost of
this strategy is 8,200 yen (82 units of betting tickets), and the minimum payout is
10,040 yen. Hence the guaranteed net profit is 1,840 yen.
In reality, however, the odds will change from Table 1 if one buys thecombination of the betting tickets specified in Table 2. The odds (including theoriginal stake) of WINi, Oi , is computed in the following manner in the Japanese
local thoroughbred races (National Association of Racing):
S

Oi = max {0.1 × INT [1 + 7.38 S ] , 1},i(2)where INT[x] gives the largest integer not exceeding x, S is the total amount bet
in the win pool, and Si is the amount bet on WINi.6 For example, if one buys 22
additional tickets of WIN1, O1 decreases from 4.6 to 4.4. Table 3 shows the best
strategy to secure a payout of 10,000 yen after taking account of this negativechange of the odds. The total cost is now 8,500 yen (85 units of betting tickets),and the minimum payout is 10,000 yen (when Horse No. 4 wins). Hence theguaranteed net profit is reduced to 1,500 yen.
We have done the same calculation for other races, and have found that theguaranteed net profit is positive in two out of 175 races. In Race 11 on November25, the guaranteed net profit is maximized by the betting strategy of Table 4. The
total cost is 45,200 yen, and the minimum payout is 50,320 yen. Thus theguaranteed net profit is 5,120 yen. In Race 5 on October 20, the maximum of theguaranteed net profit is 340 yen.7 These profits would be too small for an
arbitrager to survive given the low frequency of arbitrage opportunities.
Finally, let us discuss the practicality of the arbitrage. Suppose that thearbitrager is free to bet once all other bettors have placed their bets. Then she can4

calculate back S and Si from the public information, namely equation (2) and theodds tables (the Appendix explains the method). Therefore she can compute theamount of ex post profit of her bets in advance. As Gramm et al. (2012) and others
point out, however, one can only attempt to be one of the last bettors in practice.
Mori and Hisakado (2009) find that almost half of the win tickets are bought
within the last ten minutes in the races of the Japan Racing Association (JRA).
Since this inflow of late money might eliminate the discrepancy of the odds among
different bet types, the above-mentioned strategy is not risk-free in a strict sense.83. Conclusions
This paper has investigated the true frequency of arbitrage opportunities in theJapanese thoroughbred races. It is the first attempt to take account of thefollowing three factors simultaneously: the size of the minimum betting unit, thenegative effect of the arbitrage on the odds, and the existence of various exoticbets. It has found that arbitrage is possible in two out of 175 races. The
guaranteed profits in these races are 5,120 yen (about $64) and 340 yen. Given therequired investment of time and effort, the frequency of arbitrage opportunities
and the level of the guaranteed profits are too low for professional arbitragers.
This might be the reason why the apparent inefficiency still exists in the market.
Notes
1. As for other types of betting markets, Hausch and Ziemba (1990b) demonstratethat an optimal cross-track betting yields a guaranteed profit in seven out of ten
Triple Crown races between 1982 and 1985. Pope and Peel (1989), Shin (1993)
and Marshall (2009) find arbitrage opportunities between bookmakers.
Vlastakis et al. (2009) and Franck et al. (forthcoming) find arbitrage
opportunities among bookmakers and exchange markets.
2. This is a unique bet type existing only in Japan. One bets on the brackets
numbers, not the horse numbers.
3. Note that we use the data of Trifecta in calculation of P whereas Edelman andO’Brien (2004) do not.
4. It is impossible to construct a lock for the races with P > 1.
5. The odds table of Trifecta is available upon request.
6. The odds of EXACTAij and TRIFECTAijk are calculated in the same way.
7. The total cost of this arbitrage is 13,100 yen. The odds tables and the exact
betting strategy are available upon request.
8. Rosenbloom (1992) describes other risks of the lock.

5

Appendix: Derivation of the sales amount from the odds table
Table 1 shows that the largest odds of Exacta is 1906.3 (EXACTA36), and thesecond largest odds is 1429.7 (EXACTA35). First suppose that S36 = 1, that is, only
one unit is bet on EXACTA36. Then S35 ≥ 2 because S35 must be larger than S36 .
If (O36 , S36 ) = (1906.3,1) is substituted in (2), S becomes 2582.9. If (O35 , S35 ) =(1429.7,2) , however, S becomes 3874.3 and a contradiction occurs. Secondly,suppose that S36 = 2. Then S35 ≥ 3. If (O36 , S36 ) = (1906.3,2), S must be 5165.9.
On the other hand, if (O35 , S35 ) = (1429.7,3) , S becomes 5811.4 and acontradiction occurs again. Thirdly, suppose that S36 = 3 . Then S35 ≥ 4 . If(O36 , S36 ) = (1906.3,3), S = 7748.8. If (O35 , S35 ) = (1429.7,4), S = 7748.5 and it isconsistent with the above calculation. Finally, if (S, S36 ) = (7749,3) is substitutedin (2), O36 = 1906.3 and it is consistent with Table 1. These arguments
demonstrate that S must be 7749 (or its multiples). The published sales data has
confirmed that S was indeed 7749.
Acknowledgments
This work was supported by JSPS Kakenhi Grant No. 24530197.
References
Edelman, D.C., O’Brian, N.R. (2004) “Tote Arbitrage and Lock Opportunities in
Racetrack Betting,” European J. of Finance, 10(5), 370-378.
Franck, Egon, Erwin Verbeek, and Stephan Nüesch (forthcoming) “Inter-Market
Arbitrage in Betting,” Economica, published online on December 17, 2012.
Gramm, C. Marshall, Nicholas McKinney, and Douglas H. Owens (2012)“Efficiency and Arbitrage across Parimutuel Wagering Pools,” Applied
Economics, 44(14), 1813-1822.
Hausch, D. B., Ziemba, W. T. (1990a) “Locks at the Racetrack,” Interfaces, 20,41-48.
Hausch, D. B., Ziemba, W. T. (1990b) “Arbitrage Strategies for Cross-Track
Betting on Major Horse Races,” J. of Business, 63(1), 61-78.
Marshall, Ben R. (2009) “How Quickly Is Temporary Market InefficiencyRemoved?” Quarterly Review of Economics and Finance, 49, 917-930.
Mori, Shintaro and Hisakado, Masato (2009) “Emergence of Scale Invariance andEfficiency in a Racetrack Betting Market,” mimeo.
Pope, P., and Peel, D. (1989) “Information, Prices and Efficiency in a Fixed-Odds
Betting Market,” Economica, 56, 323-341.
Rosenbloom, E. S. (1992) “Picking the Lock : A Note on ‘Locks at the Racetrack’,”
Interfaces, 22(2),15-17.
6

Shin, H. S. (1993) “Measuring the Incidence of Insider Trading in a Market forState-Contingent Claims,” Economic J., 103(420), 1141-1153.
Thaler, R. and Ziemba, W. (1988) “Parimutuel Betting Markets: Racetracks andLotteries,” Journal of Economic Perspectives, 2, 161-174.
Vlastakis, Nikolaos, Dotsis, George, and Markellos, Raphael N. (2009) “HowEfficient Is the European Football Betting Market? Evidence from Arbitrage
and Trading Strategies,” Journal of Forecasting, 28, 426-444.

7

Table 1. The final odds of Race 11 at Arao on November 25, 2011.
(a) Exacta
OijOijOijOij

1-1

…

2-1

11.9

3-1

714.9

4-1

5.4

1-2

11.9

2-2

…

3-2

1143.8

4-2

16.6

1-3

238.3

2-3

357.5

3-3

…

4-3

197.2

1-4

3.5

2-4

13.7

3-4

301.0

4-4

…

1-5

12.8

2-5

46.9

3-5

1429.7

4-5

16.8

1-6

197.2

2-6

336.4

3-6

1906.3

4-6

178.8

1-7

336.4

2-7

714.9

3-7

1906.3

4-7

260.0

1-8

12.6
Oij

2-8

50.2
Oij

3-8

1143.8
Oij

4-8

14.3
Oij

5-1

30.5

6-1

519.9

7-1

714.9

8-1

28.8

5-2

84.1

6-2

635.5

7-2

1143.8

8-2

90.8

5-3

381.3

6-3

1429.77-3

1429.7

8-3

635.5

5-4

39.8

6-4

301.0

7-4

301.1

8-4

34.7

5-5

…

6-5

714.9

7-5

1906.3

8-5

73.4

5-6

440.0

6-6

…

7-6

817.0

8-6

381.3

5-7

817.0

6-7

1143.8

7-7

…

8-7

953.2

5-8

59.6

6-8

635.5

7-8

953.2

8-8

…

12

3

4

5

6

78

4.6

1.3

133.7

4.2

29.4

80.2

120.3

37.6

(b) WinOi

8

Table 2. The best strategy to secure a payout of 10,000 yen when the odds are
fixed at the level of Table 1.
Odds of Table 1

Units Payout (Yen)

WIN1

4.6

2210,120

WIN3133.7

113,370

WIN44.2

2410,080

WIN529.4

411,760

WIN680.2

216,040

WIN7120.3

112,030

WIN837.6

311,280

EXACTA2111.9

910,710

EXACTA23357.5

135,750

EXACTA24

13.7

810,960

EXACTA2546.9

314,070

EXACTA26336.4

133,640

EXACTA27714.9

171,490

EXACTA2850.2

210,040

Total

829

Table 3. The best strategy to secure a payout of 10,000 yen after taking account ofthe negative change of the odds.
Ex-Post Odds

Units Payout (Yen)

WIN14.4

2310,120

WIN3124.7

112,470

WIN44.02510,000

WIN527.7

411,080

WIN673.4

214,680

WIN7113.4

111,340

WIN835.7

310,710

EXACTA21

11.7

910,530

EXACTA23337.6

133,760

EXACTA24

13.5

810,800

EXACTA2546.0

313,800

EXACTA26318.8

131,880

EXACTA27637.6

163,760

EXACTA2849.1

314,730

Total

8510

Table 4. The betting strategy that maximizes the guaranteed net profit.
Ex-Post Odds

Units Payout (Yen)WIN13.7

13650,320

WIN3104.0

552,000

WIN43.414850,320

WIN523.2

2251,040

WIN663.3

850,640

WIN791.0

654,600

WIN829.8

1750,660

EXACTA23

320.5

264,100

EXACTA24

12.6

4050,400

EXACTA25

43.1

1251,720

EXACTA26303.7

260,740

EXACTA27

641.0

164,100

EXACTA28

46.2

1150,820

TRIFECTA213

319.6

263,920

TRIFECTA214

23.0

2250,600

TRIFECTA215

73.8

751,660

TRIFECTA216

370.6

274,120

TRIFECTA217

544.4

154,440

TRIFECTA218

69.2

855,360

Total

452

11