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[World Scientific Series in Finance] Stochastic Programming Volume 4 (Applications in Finance,...
[World Scientific Series in Finance] Stochastic Programming Volume 4 (Applications in Finance, Energy, Planning and Logistics)  Stochastic Programming and Optimization in Horserace Betting
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المجلد:
10.1142/84
عام:
2013
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english
DOI:
10.1142/9789814407519_0009
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October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Chapter 9 Stochastic Programming and Optimization in Horserace Betting1 William T. Ziemba∗ Summary Racetrack betting is simply an application of portfolio theory. The racetrack oﬀers many bets that involve the results of one to about ten horses. Each race is a special ﬁnancial market with betting, then a race that takes one or a few minutes. Unlike the ﬁnancial markets, one cannot stop the race when one is ahead, or have the market going almost 24/7. There is a welldeﬁned end point. Like standard portfolio theory, the key issues are to get the means right. In this case, it is the probabilities of, say, two, three or more horses ﬁnishing ﬁrst, second, third, etc., in a given order, and to bet well. For the latter, the Kelly capital growth criterion is widely used and that maximizes the expected logarithm of ﬁnal wealth. Transaction and price pressure odds changes ﬁt well into the stochastic programming models. Professional syndicates or teams have been successful as hedge funds with gains approaching one billion over several years for the most successful. In the modern era, there are two features used extensively. First, there are rebates for large bettors of the track take similar to discounts at Costco. So instead of facing a 13–30% transaction cost, it’s more like 10%. So to win, the bettors must make back this 10% disadvantage before proﬁts ensue. And this is not easy as the markets are quite eﬃcient. Also over half the betting is not recorded in the pools until the race is being run. This is because monies are bet near the start of the race and come from many oﬀ track sites which are combined with the ontrack bets into the track pool. All this takes time. So estimates of future ∗ Alumni Professor of Financial Modeling & Stochastic Optimization (Emeritus), University of Britis; h Columbia, Vancouver, BC, Canada. ICMA Centre, University of Reading, UK. Visiting Professor, Sabanci University, Istanbul,Turkey, Luiss — Guido Carlo University, Rome, Italy, Korea Advanced Institute for Science and Technology, University of Cyprus, wtzimi@mac.com. 1 Some of the material in this paper is also being published simultaneously in the book Ziemba (2012). 221 October 2, 2012 222 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. prices are crucial. Secondly, betting exchanges such as Betfair in London allow short as well as standard long bets. This allows for more arbitrage and the ability to take advantage of known biases. I have been involved in this research since the late 1970s with six books and a number of articles. In this paper I relate the theory, computations and examples of real races and experiences for various bets such as win, place and show, exactas, triactors, superfectas, super hi ﬁve, place pick all, double, pick 3, 4, 5 and 6. In the Halifax presentation I showed two of the greatest races ever by two undefeated female horses, one that was still then running (Zenyatta) and one retired (Personal Ensign) in 1988. The previous US undefeated horse was Colin in 1907! These and other great races can be seen free on the website chefderace.com. 1 Introduction Racetrack betting is simply an example of portfolio analysis. The investment horizon is short with betting for a period and then a race for about 1–2 minutes. Some of the bets involve multiple horses in a given race while others involve multiple races. The wagers are basically of two types: high probability of winning low payoﬀ bets and low probability high payoﬀ bets. The latter can return a million dollars or more. Table 1 describes a number of the bets. Examples of some of these follow. In all cases I made these actual bets. More detail on them with charts, etc. are in Ziemba (2012). Investing in traditional ﬁnancial markets has many parallels with racetrack and lottery betting and much of the analysis is similar. Behavioral anomalies such as the favoritelongshot bias are pervasive and also exist and are exploitable in the S&P500 and FTSE100 futures and equity puts and calls options markets. This is not discussed in this paper but I use this in personal and private managed accounts and in an oﬀshore hedge Table 1 Common US and Canadian Racetrack Wagers. High probability low payoﬀ Low probability high payoﬀ One horse is involved Two horses are involved Three horses are involved Four horses are involved Two races are involved Three races are involved Four races are involved Five races are involved Six races are involved N races are involved win pick 3–6 win place, exacta show, triactor superfecta double pick 3 pick 4 pick 5 pick 6 place pick all October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. SP and Optimization in Horserace Betting b1392ch09 223 fund. Biases there favor buying high probability favorites and selling low probability longshots just like the high probability low payoﬀ racing wagers. But in complex low probability high payoﬀ exotic wagers such as the Pick 6, the bias reverses to overbet the favorite so one must include other value wagers in the betting program. Fundamental information such as breeding is important and is especially useful for the Kentucky Derby and Belmont Stakes where horses have never run that far before. The idea is that more stamina is needed to win these races from the sires in the horses lineage. Since the horses have never run this distance before, a forecast of how they might do from their breeding is helpful. See Hausch, Bain and Ziemba (2006) and Gramm and Ziemba (2008, 2012) who study this by merging the odds (prices) with expert opinion (breeding measured by dosage). A horse named Stay Thirsty in race 12 on Travers Day (see section 9) has a dosage proﬁle of 461600. That is 4 brilliant points (pure speed); 6 intermediate points, 16 classic points, zero solid and zero professional points. These are categories on the speedstamina space. You can think of this as a discrete probability distribution. Now each chefderace stallion (that is a stallion which breeds consistent characteristics in their oﬀspring) in the horse’s pedigree counts: 16 for ﬁrst generation sires, 8 each for the second generation, 4 each for the four third generation and ﬁnally 2 for the eight fourth generation sires. So each generation is equally important. If a chef is in two categories, the points are split. Some generations may have no chefs. The dosage index is then DI = Brilliant + Intermediate + 1/2 Classic Solid + Professional + 1/2 Classic Despite its simplicity and crude weighting, the index does seem to work. An example of the pedigree and dosage of the 2005 Belmont Stakes winner, Aﬂeet Alex, is in Tables 2 and 3. Parenthesis in Table 2 show categories for chefderace stallions: B = Brilliant, C = Classic. A dual qualiﬁer is a horse whose dosage index is 4.00 or lower, which is the limit suggested for maximum speed for Kentucky Derby winner, and within 10 pounds of the top 2 year old horse on the experimental free handicapping ratings. Gramm and Ziemba (2008, 2012), Hausch, Bain and Ziemba (2006) and the website chefderace.com show that such horses have superior performance in the Kentucky Derby and Belmont Stakes. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Applications in Finance, Energy, Planning and Logistics 224 Table 2 Pedigree for the 2005 Belmont Stakes Winner Aﬂeet Alex. Afleet Mr. Prospector Raisea Native (B) (B/C) Gold Digger Venetian Jester Polite Lady Friendly Ways Northern Afleet Northern Dancer (B/C) Nureyev (C) Special Nuryette Tentam Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Stellarette Square Angel Roberto (C) Silver Hawk Gris Vitesse Hawkster Chieftain Strait Lane Level Sands Maggy Hawk Utrillo II Hawaii Ethane Qualique Sensitivo Dorothy Gaylord Gaylord’s Touch Table 3 Dosage Index Calculation for 2005 Belmont Stakes Winner Aﬂeet Alex. Generation Sire 1 Northern Afleet 2 Brilliant Intermediate Classic Solid Professional 0 0 Afleet Hawkster 3 Mr. Prospector 2 2 Nureyev 4 Silver Hawk Hawaii 4 Raise a Native 2 Venetian Jester Northern Dancer 1 1 Tentam Roberto 2 Chieftain Utrillo II Sensitivo Total 5 0 Note:Dosage Index = (5 + 0 + 9/2)/(0 + 0 + 9/2) = 2.11. 9 b1392ch09 October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Average Dosage Index SP and Optimization in Horserace Betting 5.5 5 4.5 4 3.5 3 2.5 2 1.5 b1392ch09 225 Belmont stakes 1½ miles Kentucky Derby 1¼ miles 6 8 10 Distance (Furlongs) Figure 1 12 Dosage indices. Source: Steve Roman. Figure 1a shows the average dosage index or speed over stamina for the average winner of the 1 14 mile Kentucky Derby and 1 12 mile Belmont stakes as well as a large number of high quality races at diﬀerent distances. It was compiled years ago from race data of Steve Roman. Steve has an updated graph of this on the chefderace website which has similar conclusions. Observe that the winners of longer races have lower dosages which means more stamina and less speed. Steve Roman has updated the graphs individually for the Kentucky Derby (Figure 1b), the Preakness (Figure 1c), and the Belmont (Figure 1d) up to the current races. Observe that the breed is moving more towards speed as the regression line is positive and a number of horses with dosage indices above the historical 4.0 cutoﬀ have been winners. But, observe that in the long 1 21 mile Belmont all the recent winners have had low dosage. And 2012 was no exception as the Dual Qualiﬁer Union Rags with a dosage index of 2.14 won the race and the favourite Dullahan with a 4.20 dosage ﬁnished seventh. There have been horses with dosage above 3.5 who won the Belmont Stakes such as Commendable (5.00 in 2000) and Sarava (4.50 October 2, 2012 1:24 9in x 6in b1392ch09 Applications in Finance, Energy, Planning and Logistics 226 in 2002). However, the last ten winners from 2012 back to 2003 had dosage indices of 2.14, 2.56, 1.75, 2.56, 2.43, 3.00, 3.00, 2.11, 1.77, and 1.88. Even the great Sunday Silence with a 3.8 dosage got crushed in the Belmont. It is not clear how useful Steve’s regression line is here as there is a lot of noise in the system. It is believed that the breed has changed toward speed and that shows up more in the Kentucky Derby and less in the Belmont. Gramm and Ziemba (2008, 2012) study this as well. Kelly and fractional Kelly betting is used extensively in racetrack betting. Full Kelly is the maximization of the expected logarithm of ﬁnal wealth subject to constraints, see some speciﬁc formulations below. That means: take an expected utility approach with u(w) = log w. Log with very low ArrowPratt risk aversion −u (w)/u (w) = 1/w ∼ = 0 is very risky short term despite wonderful long term growth properties; see MacLean, Thorp and Ziemba (2011) for an extensive treatment of the key ideas and major papers. Fractional Kelly is simply the idea to blend cash with the Kelly strategy. This, under lognormal asset assumptions, amounts to a less risky negative power utility function rather than log which is the most risky utility function one would ever want to use. Fractional Kelly leads usually to less growth and more security and a less violent wealth path. Half Kelly is a frequently used strategy. It has 75% of the full Kelly growth but the security, measured by the probability of breaking even rises from 87% with full Kelly to 95.4% with half Kelly. For lognormal assets this is u(w) = −1/w and this is approximate otherwise. To show this visually, see Figure 2(a) for the Kentucky Derby from 1934 to 2005 and Figure 2(b) with the dosage ﬁlter to eliminate horses that cannot run 1 14 miles on the ﬁrst Saturday in May of their three year old career. These use the Dr Z system discussed in section 4. The system that bets on 18000 2 16000 14000 Half Kelly Bets Flat $200 Bets on Favorite 12000 10000 8000 6755 6000 4562 4000 2000 0 1930 480 1940 1950 Figure 2 1960 1970 1980 1990 2000 Kelly, HalfKelly and Flatbelt Wealths Full Kelly Bets Kelly, HalfKelly and Flatbelt Wealths Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Applications in Finance, Energy, Planning and Logistics 1.8 1.6 Full Kelly Bets Half Kelly Bets 16,861 Flat $200 Bets on Favorite 1.4 1.2 1 0.8 6,945 0.6 0.4 0.2 0 1930 480 1940 1950 1960 1970 1980 1990 Wealth History of Some Kentucky Derby bets, 1934–2005. 2000 October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 227 the favorite turns $2500 into $480 so is a loser; while the full and half Kelly Dr Z systems have gains. In all cases the strategy to win is the same as in the ﬁnancial markets: Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 1. get the mean right: thus one must have accurate probabilities of various outcomes; 2. use the actual odds and a betting model such as the Kelly criterion to optimize the bet sizes, that is, the allocations. For situations with not many wagers, the Kelly capital growth maximizing expected logarithm or, its safer version, fractional Kelly, is useful as a decision tool, especially with many repeated bets. Then one has a stochastic program to maximize the expected utility using a logarithmic utility function of ﬁnal wealth subject to various constraints. The Kelly strategy bets more on the attractive situations. In wagers where one makes hundreds of bets, it is often better to use a tree approach where many of the bets are of equal value. Besides being more convenient to make these multiple bets, this gets around integer problems as the wagers will be integers that can easily be bet. Whereas the Kelly optimization needs modiﬁcations to produce integer wagers. Also this approach can be computerized to print out the tickets — see the examples later in the paper, and the higher probability wagers can be bet more to approximate a Kelly strategy. 2 The importance of getting the mean right Table 4 and Figure 3 show that getting the mean right is the most important aspect of any portfolio decision problem. Chopra and Ziemba (1993) discuss that and look at the eﬀect of errors in means, variances and covariances using the cash equivalent of the approximate versus exact optimal solutions. Basically it is in the ratio 20:2:1 for errors in means, Table 4 Average Ratio of CEL for Errors in Means, Variances and Covariances. t Risk Tolerance Errors in Means vs Covariances Errors in Means vs Variances Errors in Variances vs Covariances 25 50 75 5.38 22.50 56.84 ↓ Error Mean 20 3.22 10.98 21.42 ↓ Error Var 2 1.67 2.05 2.68 ↓ Error Covar 1 Source: Chopra and Ziemba (1993). October 2, 2012 228 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics CashEquivalent Loss (%) 11 10 9 8 7 6 5 4 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 3 2 1 0 0 0.05 0.10 0.15 0.20 Magnitude of Error (k) Means Variances Covariances Figure 3 Typical Relative Importance of Errors in Means, Variances and Covariances in Terms of Certainty Equivalent. Source: Chopra and Ziemba (1993). variances and covariances in terms of error impact. We measure risk aversion by the ArrowPratt risk aversion index RA (w) = −u (w)/u (w), where primes denote diﬀerentiation of the utility of wealth function u. In investment practice, risk tolerance t = 100/(0.5 ∗ RA ) is typically used. Referring to Table 4, we see that low risk aversion utility functions such as log with RA = 1/w ∼ = 0, the eﬀect of the errors is more like 100:3:1 so getting the mean right is even more important. And for horse racing, that is the probabilities for horses coming ﬁrst, second, third, etc. 3 The favoritelongshot bias The favoritelongshot bias is the tendency in horseracing, sports betting, and ﬁnancial options for the most likely outcome to be underbet and the less likely outcomes overbet. So people tend to like junk and dislike the best possibilities. This bias has been well known to Irish and other bookmakers who actually create the bias with the bets they oﬀer for the last 100+ years. Figure 4 shows the 1949 study by Griﬃn for 1386 races in 1947 for races at Churchill Downs, Belmont and Hialeah. In this graph, there are the number of entries, winners and winners times odds for every odds group. The axes show the odds, equivalent to the subjective probabilities, versus the actual number of winners, the objective probabilities. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 229 1000 WINNERS X ODDS CORRECTED FOR LOSS TO TRACK TOTAL ENTRIES 900 WINNERS X ODDS 800 WINNERS 700 600 500 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 400 300 200 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 ODDS Figure 4 Griﬃth’s 1949 Study on the Favoritelongshot Bias; see Hausch, Lo and Ziemba (2008) for the Reprinted Paper. Figure 5 shows the eﬀective track payback less breakage for various odds levels in California and New York, more than 300,000 races over various years and tracks, as of 1986 as reported in Ziemba and Hausch (1986). There actually was a small proﬁt, about 3%, in betting horses to win at US odds of 3–10 (UK odds of 1.30 or less), but at odds of 1001, the fair odds are about 7001 so that such bets were worth about 13.7 cents per dollar bet. The California and New York graphs diﬀer slightly because of diﬀerent track takes. There are approximately three piecewise linear segments, small proﬁts on extreme favorites with favorites underbet and longshots overbet, more and more losses as the odds lengthen and extremely poor returns at high odds levels, like lotto tickets. Since 1986 there have been a number of developments that have tended to inﬂuence the betting and have shifted these graphs, such as: 1. There are no longer separate pools for individual races at diﬀerent racetracks. All the races now have pooled betting which now comes in late with about half the bets not recorded in the pools until the race is already running; 2. There are rebates where tracks send a signal with the results and the rebate shops and the track share the track commission between the October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Applications in Finance, Energy, Planning and Logistics 230 120 QCALF = .8467 Slight profits for extreme favourites QNY = .83 Subtract = .01 for breakage Effective Track Payback in Percent These bets return more than average 100 .83 80 NEW YORK CALILFORNIA 60 40 These bets return less than average Extremely poor returns when odds pass  151 at 1001 13.7c/s bet 20 9/2 /1 100 1 20/ 1 30/ 1 50/ 1 10/ 2/1 3/1 4/1 5/1 3/5 4/5 1/1 2/5 0 1/2 1/10 0 1/5 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. b1392ch09 Track Odds (Log Scale) Figure 5 The Eﬀective Track Payback Less Breakage for Various Odds Levels in California and New York (more than 300,000 Races over Various Years and Racetracks). Source: Ziemba and Hausch (1986). rebate shop, the tracks and the bettors. So instead of facing 13–30% transaction costs, large bettors are actually charged about 10% net; and 3. Betting shops such as Betfair, oﬀer long and short bets on racing and many other events such as political campaigns, etc. Online internet betting of this sort is legal in Canada, the UK and many other countries but it is not legal in the US. Figure 6a–c look more closely at the extreme favorites in the US and the UK. Figure 6c shows that in Britain, bookies construct odds, creating the favoritelongshot bias to clear the market and equilibrate bettor demand. Figure 7 shows that the bias curve may be diﬀerent for diﬀerent types of races. It shows the bias for the Kentucky Derby for 1903–1986. Other higher quality races like the Derby may well have ﬂatter biases. See also Tompkins, Ziemba and Hodges (2008) who demonstrate similar biases in the S&P500 and FTSE100 index futures options. I use such ideas in an oﬀshore hedge fund and in personal and private investment accounts. Figure 8 and its data in Table 5 and Figures 9ab show the bias based on more recent data. Observe that the favorites are no longer underbet October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics 1.034 1.000 0.976 0.962 0.929 0.919 3/5 4/5 1/1 Track Odds Expected Value Per Dollar Bet 2/5 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Break Even Expected Value Per Dollar Bet Expected Value Per Dollar Bet SP and Optimization in Horserace Betting 1.046 1.000 0.993 231 1.06 1.000 0.94 0.87 1/10 6/5 b1392ch09 3/10 2/5 3/5 Track Odds 7/10 1/1 Break Even 0.92 1/100 2/5 4/9 3/5 Track Odds 6/11 1/1 Figure 6 Extreme Favorites, Small Proﬁts: see Ziemba and Hausch (1986) for these References. enough to turn a proﬁt betting them and the ﬂatness of the curve until you get to fairly long priced horses. You can still short longshots on Betfair and make a proﬁt if you are careful. Additional discussion and results are in Hausch and Ziemba (2008). This bias forms part of the KahnemanTversky (1979) prospect theory where low probability events are overestimated and high probability events are underestimated. This also forms a part of the behavioral ﬁnance literature. Thaler and Ziemba (1988) discuss reasons for the bias as do Ziemba and Hausch (1984). These include the fact that there are more bragging rights from picking longshots than from favorites: 501, wow was I smart while 2–5 is an easy pick. Transactions costs are another factor: bet $50 to win $10 is hardly worth the eﬀort. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics 232 Percent +20 +10 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 0 Actual (Kentucky Derbies) −10 Actual (Snyder Study) −20 −30 1 2 3 4 5 7.5 10 18 20 30 TRACK ODDS (log scale) Figure 7 Expected Return Per Dollar bet with and without the Track Take Deducted for Diﬀerent Odds Levels in the Kentucky Derby 1903–1986 and in 35,285 Races Run During 1947–1975, from Data in Snyder (1978). Source: Ziemba and Hausch (1987). 4 Place and show optimization with transactions costs The Dr Z system, codeveloped with Donald Hausch with some early help from Mark Rubinstein, presents a winning method for betting on underpriced wagers. The idea of the system is simple: use the data from a simple market, in this case the win probabilities to fairly price bets in the more complex markets, such as place and show. For example, with ten horses, there are 720 possible ﬁnishes for show. Then one searches for mispriced place and show opportunities. This is a weak form violation of the eﬃcient market hypothesis based solely on prices. How much to bet depends on how much the wager is out of whack and it is a good application of the Kelly betting system. The formulation below shows such an optimization. There is a lot of data here on all the horses and not much time at the track. So a simpliﬁed approach is suggested. Don and I solved thousands of such models with real data and estimated approximation regression equations October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 233 120 Effective Track Payback Percent 0.8467 80 60 40 20 9/2 1 /1 100 50/ 20/ 1 30/ 1 1 10/ 3/1 4/1 5/1 2/1 3/5 4/5 1/1 2/5 0 1/2 1/10 0 1/5 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 100 Track Odds (Log Scale) Figure 8 Eﬀective Track Payback Less Breakage for Various Odds Levels in California. Source: Ziemba and Hausch [1986] and Hausch and Ziemba (2008). that only involve four numbers, namely, the amounts bet to win in the total pool and the horse under consideration for a bet, plus the total place or show pool and the place or show bet on the horse under consideration. These equations appear below. In our books Ziemba and Hausch (1984, 1986, 1987) and papers Hausch, Ziemba and Rubinstein (1981) and Hausch and Ziemba (1985), we study this in various ways, including diﬀerent track takes, multiple bets for place and show on the same horse and how many can play the system before the edge is gone. This system revolutionized the way racetrack betting was perceived, viewing it as a ﬁnancial market, not just a race. This led to pricing of wagers and the explosion of successful betting by syndicates in the US,Hong Kong and elsewhere; see, for example, my joint books referenced here and Hausch, Lo and Ziemba (1994, 2008) and Hausch and Ziemba (2008). The eﬀect of transactions costs which is called slippage in commodity trading is illustrated with the following place/show horseracing optimization formulation; see Hausch, Ziemba and Rubinstein (1981). Here qi is the probability that i wins, and the Harville probability of an ij ﬁnish October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics 234 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Table 5 Data for the Eﬀective Track Payback Less Breakage for Various Odds Levels in California. Source: Ziemba and Hausch [1986] and Hausch and Ziemba (2008). Figure 9 Rate of Return at Diﬀerent Odds. Source: Snowberg and Wolfers (2008). q q i j is 1−q , etc. That is, qj /(1 − qi ) is the probability that j wins a race that i does not contain i, that is, comes second to i. Q, the track payback, is about 0.82 (but is about 0.90 with professional rebates). The players’ bets are to place pj and show sk for each of the about ten horses in the race out n of the players’ wealth w0 . The bets by the crowd are Pi with i=1 Pi = P n and Sk with k=1 Sk = S. The payoﬀs are computed so that for place, the ﬁrst two ﬁnishers, say i and j, in either order share the net pool proﬁts once each Pi and pi bet’s cost of say $1 is returned. The show payoﬀs are October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 235 computed similarly. The maximum expected utility model is P Q(P + n l=1 pl )−(pi +pj +Pij ) 2 pj i × pi p+P + pj +P n n i j n P qi qj qk Q(S+ n sl )−(si +sj +sk +Sijk ) l=1 log + max pi ,si 3 (1 − qi )(1 − qi − qj ) i=1 j=i k=i s s s j i k j=i k=i,j + sj +Sj + sk +Sk × si +S i +w0 − n l=i sl − nl=i pl Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. l=i,j,k s.t. n (pl + sl ) w0 , pl 0, sl 0, l=i,j l = 1, . . . , n, l=1 While the Harville formulas make sense, the data indicate that they are biased. But for place and show, the win favoritelongshot bias and the second and third ﬁnish bias tend to cancel so the corrected Harville formulas are not needed here. For other bets to correct for this, professional bettors adjust the Harville formulas, using, for example, discounted Harville formulas,2 to lower the place and show probabilities for favorites and raise them for the longshots; see papers in Hausch, Lo and Ziemba (1994, 2008) and Hausch and Ziemba (2008) and the discussion below on place pick all. Rebate is added to ﬁnal wealth inside the large brackets by adding the rebate rate times all the bets,winners and losers. This is a nonconcave program but it seems to converge when nonlinear programming algorithms are used to solve such problems. But a simpler way is via expected value regression approximation equations using 1000s of sample calculations of the NLP model. These are 2 The Ex Placei = 0.319 + 0.559 wi /w pi /p Ex Showi = 0.543 + 0.369 wi /w . si /s discounted probabilities come from qα qi∗ = Pni α i qi for α about 0.81 then one uses the qi∗ in the second place position. For third one uses α2 about 0.64. These empirical numbers vary over time and by track. This is more important for exacta pricing than place and show because for the latter the win bias from the favoritelongshot and the second and third biases tends to cancel. The favoritelongshot bias is the empirical observation that favorites are underbet and longshots overbet; see the discussion above. October 2, 2012 236 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. The expected value (and optimal wager) are functions of only four numbers — the totals to win and place for the horse in question and the totals bet. These equations approximate the full optimized optimal growth model. See Hausch and Ziemba (1985) for more on this plus additional features. This is used in Dr Z calculators. See the discussion in Ziemba and Hausch (1986) and Ziemba (2011) for a discussion of typical use at the ﬁrst Breeders’ Cup in 1994. An example is the 1983 Kentucky Derby. Here, Sunny’s Halo has about 1/6 of the show pool versus 1/4 of the win pool so the expected value is 1.14 and the optimal Kelly bet is 5.2% of one’s wealth. You might ask: does the system still work in 2012 and what is changed? The main new features are: 1. these days we bet at rebate shops by phone or electronically. The rebate is a sharing of the track take by the track, the rebater and the bettor. The eﬀect is to take all bets from a track take of 13–30% for various bets to about 10%; 2. betting exchanges in the UK and elsewhere allow for short as well as long wagers; and 3. there is a lot of cross track and last minute betting and this takes time to be sent to the pools at the racetrack. Hence, about 50% of the wagers don’t actually appear in the pools until after the horses are running. So one must estimate the ﬁnal odds (probabilities). Syndicates exist that break even on their wagers yet make millions on the rebate. Regarding the Dr Z system, John Swetye works with me and we wager with rebate searching for bets at 80 racetracks. Basically the system October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 237 still works but the task is not easy. One successful six month period in 2004 with a $5000 bankroll, the system lost 7%, received a 9% rebate. The total wagers were $1.5 million giving a 2% or $30,000 proﬁt. Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 5 The place pick all This is a less commonly used and known bet but it is available at Santa Anita, for example. The idea is to create a ticket with I horses over I races where you have either the winner or the second place horse in each race i, i = 1, . . . , I. The number of races I varies from about 7 to 12. The probability that a ticket with j the chosen horse in race i wins that race is the probability that j is ﬁrst plus the probability that j is second, namely qij pik for k = j pij + 1 − qik k=1,...,Ki where pα qik = ikα and α ∼ = 0.81 and is track dependent. pik These are discounted Harville formulas, see papers in Hausch, Lo and Ziemba (1994, 2008) for more on this. Then the chance that a given ticket with i = 1, . . . , I is a winner is qij pik pij + p̂ij = 1 − qik i=1,...,I 6 k=1,...,Ki Some stochastic programming formulations There are basically two strategies for the optimization: Kelly expected log optimization and the tree tickets approach that approximates the Kelly strategy. To supplement the rest of the paper, a few formulations appear here. The expected log problems are typically solved using CONOPT (Drud 1994) which has produced good results even though the problems have nonconcavity in them. The bets must be computed very fast as the odds are changing. While this may not be general but because of an epsilon optimality convergence criteria, the MINOS code (Murtagh and Saunders 1998) may converge to a nonoptimal strategy. Hence, CONOPT is safer. In general, the Kelly Elog optimization given modern computing can be used for essentially all the bets, even possibly Hong Kong’s triple trio, namely, getting the 123 ﬁnish in any order in three races with 14 horses in October 2, 2012 1:24 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 238 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics each race of which many horses are 2001 but they can still ﬁnish 3rd. This has 48 million combinations. I focus here on the US bets and suggest that we focus on Elog Kelly optimization for high probability low payoﬀ bets and the tickets approach for the low probability high payoﬀ events which can closely approximate the Kelly strategy and yields easily implemented tickets that are integers. The simplest bet is the exacta. To win you must get the winner plus the second place ﬁnisher. This uses elements of the place pick all formulation except it is just for one race and it is not ﬁrst or second but ﬁrst and Second uses the second. First is easy, it is just pi , the probability that i wins. qj pα i P where q = , where α ∼ discounted Harville formula so it is 1−q = 0.81 i pα i i pi qj So the probability of an ij ﬁnish is 1−qi . Let sij be our bet on an ij ﬁnish. So the Kelly optimization problem is max Ew log W = x∈K J I X X pi i=1 j=i qj 1 − qi “ ” 9 8 P P « X J J I X I X = < E + Ii=1 Jj=i xij „ X xij − × log W0 + r xij + Q xij ; : Eij + xij xij + Eij i=1 j=i i=1 j=i where r is the rebate percent payable on all bets, losers and winners, E is the total exacta bet by the crowd, with the Eij their bets on ij and Q is the track payback. The constraints can include a maximin bet on any combination ij as well as on each i and on the total bet. The wealth is ﬁnal wealth plus rebate plus proﬁts minus the bets and Exp is the expected value. Other high probability low payoﬀ bets are similar. Let’s now consider a tickets model as well as an expected log model. I supervised an unpublished MSc thesis on this at the Oxford Math Department, see Assamoi (2010). Some of the theory is there but no calculations. Before we consider this, let us do it for the place pick all. I am not aware of any published research or calculations, real or simulated, on this bet. Let xij be the bet that either the chosen horse j will win or come second in race i. The probability of winning is p̂ij as given above. So the Kelly formulation is max Exp log W = x∈K J I X X p̂ij i=1 j=i ” “ 8 9 P P « X J J I X I X < = P L+ Ii=1 Jj=i xij „ x X ij × log W0 + r xij + Q xij − : ; Eij +P Lij xij +P Lij i=1 j=i i=1 j=i October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 239 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. which is very similar to the exacta formulation where P L is the place pick all pool with P Lij bet on ij, r is the rebate percent and Q is the track payback percent. The ticket formulation breaks the picks, the js, into categories I, II and III for each race j. I’s are high value and high probability of winning horses. II’s are major contenders and III’s are longer odds horses who could upset the favorites. Suppose there are eight races so I = 8, where nij is the number of horses in category ij. I II III 1 2 N1I N1II N1III N2I N2II N2III 3 4 5 6 7 8 Given the probabilities and other factors, the horses are put into I, II or III or not considered in each race i. The score 8 tickets have all I’s and they have the most money on them. There are 8i=1 NiI of these. The score 9 have 7 ones and one II. There are 8 such tickets with lower bets. The score 10 tickets have 6 I’s and 2 twos or 7 I’s and one III with even lower bets. There are 8 2 and eight of these with the lowest bets. One might go to score 11 and have bet sizes to approximate a Kelly strategy. As you can see, the number of tickets gets very large here, as does the cost. We use a computer program to generate these tickets. For a sample printout, see section 11. 7 The Pick3 and Pick4: Theory of pricing the bets Example: the 9/11 Pick3: don’t trust odds from newspaper stories On September 11, 2011, the tenth anniversary of the attacks on New York and Washington, the ﬁrst three races at Belmont had winners 911. The Pick3 paid $18.60. The parlay on the three track takes versus one for the Pick3 paid 4.20 ∗ 4.20/2 ∗ 6.20/2 = $30.5, which is more than the Pick3 payoﬀ without even factoring in the two extra track takes. So the 911 Pick3 was overbet just like popular numbers in lotteries are. The track take for the Pick3 is 26% and for win 16%. So the fair value of the Pick3 with zero track take is $18.60/0.74 = $25.14 and for the parley 3 1 = $51.34, 30.5 0.84 October 2, 2012 240 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics so the numbers 911 were overbet. A newspaper article said the odds of such an outcome is a million to one. Actually by looking at the ﬁnal odds on the charts for these three races, we can estimate that. Recall, the probability of winning is Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Q + ∆Q(odds) . odds + 1 Referring to a table in Ziemba (2012) for the ∆Qs and the payoﬀs below, the odds of 911 occurring are about one divided by the probability of this outcome which is 0.84 − .02315 0.84 − .02315 0.84 − .0345 2.1 2.1 3.45 = 0.3890 ∗ 0.3890 ∗ 0.2335 = 0.0353 since the payoﬀs for win were $4.20, $4.20 and $6.90 for each $2 bet. The odds against the 911 outcome then are found to be approximately (1 − 0.0353)/0.0353 = 27.32 to 1, not 1 million to 1. 8 The Pick4 This bet has no consolation prize so you need to win all four races to collect. The bet is usually oﬀered twice daily, early and later with the late Pick4 covering the top races on the card. Like the other multiple race bets, it is hard to win as the probability that you win is P 4 = P1 P2 P3 P4 , namely, the product of the probabilities that you win each of the four races. For example, if each Pi is 1/2, so you have a 50% chance of winning each race, 4 1 = 16 = 6.25% so once every 16 times you would win on then P 4 = 12 average. And getting to 50% in each race is challenging. The bet is $1 and sometimes 50 cents so you can take many multiple combinations to get P 4 to an acceptable level. As always, try to bet with an advantage so focus on horses of three types: good value and good chance to win and possible bombs to give a large payoﬀ. When facing the Pick4, determine a strategy to play the bet. If there are one to three standouts and some wide open races then you can have a simple ticket that singles the standouts and spreads in the contentious races. If the standouts are very strong, then singling make sense otherwise make the standout a I and have some backups as IIs. I was at Santa Anita on Saturday April 9, 2011 for the Santa Anita Derby, a major prep race for the three year olds before the Kentucky Derby. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. SP and Optimization in Horserace Betting b1392ch09 241 1 There was a standout, Premier Pegasus, who won the San Felipe 1 16 (the previous major race) by seven lengths, but in a soft time — he won so easily that his time (1:41:23) was not fast. He towered over the ﬁeld but unfortunately had a minor injury and was scratched. So was Uncle Mo, another Kentucky Derby hopeful, who disappointed in the Wood Memorial at Aqueduct, ﬁnishing third at 1–10 win odds. But he was a dual qualiﬁer and became the favorite. But he too was scratched. The rest of the ﬁeld looked weak and it was hard to separate them. So spreading was suggested and I took seven horses in that 10th race — all except two very longshots. who were at over 40 to 1 odds. The other three races all had standouts and I singled all three. One was First Dude in race 8 who had never won a race but was second in last years Preakness and third in the Belmont and in the money in many Grade II races. His trainer, Bob Baﬀert, put him in a $56k allowance against weaker competition to try to get that ﬁrst win. He got it at 45, winning easily, paying $3.60 for a $2 win ticket. It was good that I spread in the Santa Anita Derby race 10 as Midnight Interlude, trained by Bob Baﬀert at 13.90 to 1, was the winner, paying $29.80 for a $2 win ticket. Baﬀert has a knack of winning these top races. Race 9’s standout was Cambina, who also won at the essentially even money odds of 1.1 to 1 to pay $4.20 for a $2 win ticket. Race 11’s standout, Hey Maria, again at close to even money, won at 1.30 to 1 paying $4.60 for a $2 win ticket. The Pick4 was a 50 cent bet but I bet $10 on it costing 7 · 1 · 1 · 1· $10 = $70. The Pick4 paid $99.20 for the 50 cents so I collected $1995 which was a good gain. Like the Pick6 below, singling top standouts and spreading in tough races often is a very good strategy with low cost that can have a large payoﬀ. 9 Example of Pick4 with embedded Pick3’s and Doubles: Travers Day at Saratoga Saturday, August 27, 2011 was the Travers Stakes at Saratoga. This is the mid summer Derby, the top race for three year old colts between the triple crown races in May and June and the Haskell at Monmouth Park on July 31st and the Breeders’ Cup in November. This is a 1 14 mile race with a $1 million purse. A strategy for the $1 million guaranteed Pick4 which was races 9–12 is to accumulate the picks of various handicappers and then cover the bets well. To be a Kelly bettor, you must bet more on the top favorites and less on the longer priced horses. In addition to the handicappers at Saratoga who discuss the races on TV for free and the ones with subscription services, October 2, 2012 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 242 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics there is betfair.com in London which provides bidask spreads on the races to give an idea of the ﬁnal odds as they allow shorts as well as long bets. A drawback of betfair.com is that the spreads are wide but they close up near race time. Another drawback for US bettors is that you cannot use it in the US but Canadian and British bettors can use it. The strategy is to establish bets on the Pick4 races 9–12 and have as backup the races 9–11 and 10–12 Pick3’s and the races 9–10, 10–11 and 1112 doubles. Thus if you get the P4 right you also win the two P3s and the three P2s. The strategy is to have multiple tickets, they can be through the score 8 system with I’s, II’s and III’s or through straight tickets where you pick several horses in each race and box them all. That means that you win if any one of the horses in each race you selected wins, assuming you win all the races. These boxed tickets put more weight on the weaker horses than the multiple tickets approach. The multiple tickets approach is more complex but fully manageable. For a Pick5 or Pick6 we use a computer program to generate the tickets as there can be many of them. For a Pick4 there are 17 tickets if you do a score 8 down to IIIIIIII. Examples follow. Both these approaches are discussed below. Here are the races: Race 9 was the 9th running of the Victory Ride, a Grade III, $100,000 race for three year old ﬁllies. The past performances follow. • Saratoga handicapper Mike Watermaker chooses one horse per race and here he chose #6 Hot Summer. • The TV handicappers Dan Ulman and Mike Beer chose 2,5/6,8,9 and 6/5,8,9. This means double keying 2 and 5 as I’s and 6,8,9 as II’s. This is set up for the score tickets approach, see below. • I use three additional services plus two of my colleagues have their picks, but one of them was rained out on that day by the great hurricane Irene. Our handicapper, who selects for superfecta as well so picks the best ﬁve horses, liked 2,5,6,8,9. One service ranked the horses 82913, the other 291813. The third, which are the Equiform pace numbers and ﬁnal speed ﬁgures for the best ﬁnal time in the last four races, #2 74 43 twice; #5 74; #6 73 43 ; #8 71 41 and #9 76. So the suggested picks were 2,5,6,9 because I skipped 8 so the ticket would not be too large. But in the multiple score tickets, 8 must be a II. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 243 The result was 619. A $16.40 win for a $2 bet was a good start. Watermaker had it but as we will see that was it for his picks, the other three did not win. This is an important point — none of the handicappers will get it all right very often so it is important to use several outside handicappers plus your own analysis. Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Race 10 was the 33rd running of the Ballerina, a Grade I race with a $250,000 purse for three year old ﬁllies. The past performances follow. • Watermaker picked #7 Sassy’s Image who was the favorite but did not win. • Ulman and Beer chose 6/1, 2/3, 5, 7 and 6, 7/2. As we will see, #3 Hilda’s Passion won and they basically threw her out because of the last race — a real clunker. But in the previous race to that, she ran a Beyer 107! Ranking her as a III meant that with a score 8 ticket 3 of the 8 spots are used up so one can still win with 2 II’s and one I but this score 8 ticket will have the minimum bet $1 or 50 cents depending on the track rules. • Our handicapper liked 3, 6 and 7. One service ranked them 36275, the other 175264. Equiform’s pace service got it right and as we will see, got the next two races correct as well for one third pick and three top ones. A very good performance. But that will not happen all the time. Equiform had #2 77 41 and 77 21 ; #3 79 43 ; #6 78 and 77 21 and #7 77 21 . So my picks were 2,3,6,7 and #3 Hilda’s Passion was a short price and won paying $7.20. Race 11 was the 27th running of the Foxwoods King Bishop, a 7 furlong Grade I race with a purse of $250,000. The race featured the return to the track of the two year old champion Uncle Mo who had been sidelined since being scratched just prior to the Kentucky Derby where he was favored to win. He had a variety of ailments but had good recent workouts. The big issue was, is this the old Uncle Mo? It was a tough challenge. • Watermaker favored #3 Run Flat Out. • The TV handicappers favored #1 Flashpoint with one double keying #4 Dominoes, the favorite in the Grade 2 Jim Dandy run on July 30 at 1 18 at Saratoga. He lost that race ﬁnishing third to Stay Thirsty, ran a Beyer 106 and was in Race 12 at the Travers. So Ulman and Beer had 1/2, 5, 7 and 1, 4/2, 6/7. Arguing that the return was too tough for Uncle Mo, they downgraded him to a II and a III, respectively. • Our handicapper favored 4, 5 and 7. The two services had 1274 and 175264. The Equiform pace analysis had #1 76 (but double top); #2 79, #4 75, #5 75, and Uncle Mo #7 78 43 but a long time ago. So this pointed to Caleb’s Posse, #2. October 2, 2012 1:24 244 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. I rather liked Caleb’s Posse with the 79 at Saratoga in the Grade II Amsterdam on August 1 with a Beyer of 105 which was higher than Flashpoint’s 104 at Monmouth. The 105 for Caleb’s Posse was way above his previous high Beyer of 92 so that made it a risk. The only horse to win a Grade I was Uncle Mo who won two as a two year old, the Champaign and Breeders’ Cup Juvenile, the latter with a 108 Beyer in the race he got the 78 43 speed ﬁgure. My picks were 1,2 and 7. The ﬁnish was 2764 with Caleb’s Posse nipping Uncle Mo at the wire. I had now won the ﬁrst Pick3 there and the ﬁrst two of the doubles. Race 12, the ﬁnal one in the Pick4 was the 142nd running of the Travers, a Grade I, $1 million purse race. This is the midsummer classic and one of the longest running top races in the US. • The TV handicappers felt that the winner here would come out of either the Jim Dandy run at Saratoga or the Haskell run at Monmouth. So that favored #7 Coil, #10 Shackleford and #9 Stay Thirsty. They chose 7,9,10/3 and 10/6,9. 3 mile • Watermaker liked Shackleford, a front running horse who won the 1 16 Preakness and was ahead in the Kentucky Derby late in the race, but faded to ﬁfth. His dosage proﬁle was 613902 for a dosage index of 3.62. This is under the usual Kentucky derby cutoﬀ of 4.00 but is high. Stay Thirsty with 461600 (2.25), Coil 20910 (1.18) and Ruler on Ice, the winner of the Grade I Belmont Stakes in the slop with 61900 (2.56) were more suited for this 1 41 mile race. But he still could win going wire to wire. • Our handicapper favored 7,9,10 with 4 should there be rain and mud which there was not. The two handicapping services had the following rankings: 9274106 and 927468310. So they favored Stay Thirsty and did not like Coil much but gave him third and basically threw out Shackleford. They rather liked #2 Rattlesnake Bridge rated second and the mud runner Ruler on Ice, fourth with two wins on sloppy tracks and the rest mediocre. Rattlesnake did not look competitive to me with three Beyer’s of 91, 90 and 91 with one win. Equiform’s top picks were #6 75 43 ; #7 74 41 , 74, 77; #9 77 41 and #10 76 21 . So they suggested #9, Stay Thirsty. My picks were 6,7,9,10 on the bigger ticket and 7,9,10 on the smaller ticket. The ﬁnish was 9284. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 245 My tickets were Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 6 3 2 9 2,5,,9/2, ,6,7/1, ,7/6,7, ,10 3 2 9 2,5,9/,6,7/1, ,7/7, ,10 for $384 less rebate, and for $162 where circles indicate the winners So the big ticket won the Pick4, the two Pick3s and the three daily doubles. I made some win, place and show and superfecta bets. Some won, some lost. The Pick4 paid $1453.00, the two Pick3s $471 and $195 and the three doubles $86.50, $58.00, and $48.80 per $2 bet. So there was a good gain. The Pick3 tickets I played were 6 3 2 2,5,,9/2, ,6,7/1, ,7 3 2 9 2,,6,7/1, ,7/6,7, ,10 3 2 2,5,9/2,,6,7/1, ,7 3 2 9 ,6,7/1, ,7/7, ,10 $96 for $2, to collect $86.50 $96.00 for $2, to collect $195 cost $72.00, a loser for $2 $54.00, a winner for $2, to collect $195 The doubles were 9/10 10/11 11/12 6 3 2,5,,9/2, ,6,7 3 2,5,9/,6,7 3 2 2,,6,7/1, ,7 3 2 ,6,7/1, ,7 2 9 1,,7/7, ,10 winner loser winner winner winner $32 for $2 to collect $86.50 $18 for $2 $24 for $2 to collect $58.00 $18 for $2 to collect $58.00 $18 for $2 to collect $48.00 plus rebate on all the bets, winners and losers I did not play a score ticket approach but it would have had all these winners. It would have cost more but could have returned more since scores 8,7,6,5 and 4 would likely have $20, $10, $5, $3, and $1 tickets. Rather than speculate, I will discuss the tickets approach in other situation where I actually made such bets. Rebate always help. 10 The Pick6: Theory of pricing the bets 2001 Breeders Cup insurance bets for SCA In 2001, a racing colleague, Cary Fotias, creator of the Equiform ratings, and I were hired by SCA, a Dallas based sports insurance company, to help insure the Breeders’ Cup which was at Belmont near New York City. We insured the $2 to $3 million part. So the insurance company would guarantee a pool of $3 million. For example, if $2.15 million was bet, they would be liable for October 2, 2012 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 246 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics $850,000. We studied and proposed to bet a random amount if needed. The idea being to get to $3 million and return the insurance company’s money by winning Pick 6s and Pick 5/6s. It was risky as September 11 had just occurred and all the Arab owners such as Sheik Mohammed of Dubai were not in attendance. Their horses and trainers were though. It turned out to be a glorious day so the crowd sent the Pick 6 pool well over $4 million. Our client, himself the world’s most famous bridge player, Bob Hamman, said you two can just play about $25–30,000 of the tickets. So we had a $2000 ticket twice and what we call a gorilla ticket for $28,000. We had some 5/6’s and got most of the money back. The Pick 6 paid $250,000. The race we lost was the sprint. Squirtle Squirt, which my handicapping colleague did not like at 91 beat the front running ﬁlly, Extra Heat at 141, who we had, and who had led all the way until the ﬁnish. So if she had won we would have had three about $450,000 Pick 6s plus more 5/6s. Squirtle Squirt had run at Belmont and had the very top jockey Jerry Bailey and was trained by the recently deceased legendary trainer Broadway Bobby Frankel. I should have overruled and included Extra Heat on the ticket, adding a bit of extra cost. Too bad. This was another example of a lot of operational risk involved in Pick6 tickets. It is very easy to have an error leading to a loss. Various models and research yields the horses to include. Then the optimization creates a good ticket. But it was fun! But the next week we won a similar case at Santa Anita, while guaranteeing a $1 million Pick 6, collecting $240,000 for the client and a nice bonus for us. 11 The one that got away: the hittable $2 million Pick 6 at the 2009 Breeders’ Cup The Breeders’ Cup is now up to 14 major races over two days3 and was held again at Santa Anita on Saturday, November 6 and 7, 2009. I went in 2008 and it is fun to see it live. This year on wide screen high deﬁnition TV it was wonderful to watch. Being at home, the handicapping and betting is a lot easier. There are many opinions. That’s what makes a horse race. The spreads on Betfair are fairly tight and it is easy to bet from Canada and you frequently get better odds there than at the track. We don’t actually bet at the track but bet through rebate shops that give back part of the track take. That’s easy to do on the phone or by email. The rebaters take their cut and the track gets more easy, low expense business to up their revenues. The big race was as usual the $5 million classic. It is no longer the world’s richest race. The $10 million Dubai World Cup has that honor. But 3 In 2011 there were 15 Breeders Cup races. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. SP and Optimization in Horserace Betting b1392ch09 247 it is the most important race in the world and frequently determines the horse of the year. The two candidates for horse of the year are were both female. The 3yearold Rachel Alexandra won all her 2009 races. She beat the top females in the top female races by 20 lengths. I was at Churchill Downs to see this in the Kentucky Oaks held on the ﬁrst Friday in May, the day before the Kentucky Derby. In four races against males, she beat them handily. So she would normally be an almost sure bet for horse of the year. But Zenyatta, a ﬁve year old mare, had won all 13 of her races but always against females. She has a dynamite kick and just cruises by the other horses near the buzzer to win easily. Some of her races were in slow times (76 area on the Equiform scale that I follow) and some in fast times (81 area). To put this scale in perspective, the highest I ever saw was four 84’s by Ghostzapper. One of my most treasured but small bets was on Ghostzapper’s Breeders’ Cup win. There was a top ﬁlly in that race, Azeri — I had watched her at Saratoga getting beat by females in a 1 14 race so the fact that she had numerous wins at short distances I was pretty sure she would not be in the top 4 in this male dominated 1 14 race. So a $20 superfecta bet in 2004 boxing for $5 Ghostzapper (1or 2) with Roses in May (1 or 2) with the two next leading horses, Pleasantly Perfect and Perfect Drift (3 or 4) and (3 or 4) came in to provide a $5000 payoﬀ. The big mistake was not betting more! Ghostzapper ran the 1 41 race in 1:59:02, faster than Secretariat’s record setting Kentucky Derby 1:59:40. The 52 odds were quite generous given Ghostzapper’s brilliant record. The 13/13 of Zenyatta is historic since only Personal Ensign (13/13) in 1988 and Colin in 1907 had undefeated records in the US. Tesio, the great Italian trainer in the 1930s had the other three of the ﬁve undefeated horses since 1900, among horses with at least ten major races at major race tracks. Tesio was a great anomaly — a person looking at many many generations to bread cheap to cheap to get great champions. He did this without computers with a lot of help from his wife. Currently, the way to check such matings is to go to Steve Roman who does this for ten generations back. See Steve’s website, www.chefderace.com, for much valuable information. The Europeans and many other handicappers were pushing for Rip Van Winkle, trained by Irish legend Aidan O’Brien, who was the pick since he had won his two races in 2009 when he did not face the superstar Sea the Stars. The three times Sea the Stars beat Rip Van Winkle he was close behind and well ahead of the competition. Unfortunately, the Sea the Stars connections preferred to hedge and cash in on this horse reputed to be the best in Europe in ten years by retiring him to stud duties. He had won October 2, 2012 1:24 9in x 6in b1392ch09 Applications in Finance, Energy, Planning and Logistics 248 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. Applications in Finance, Energy, Planning and Logistics the Epson Derby, the 2000 Guineas and the ARC. Perhaps they recalled the last “best horse of the decade”, Dancing Brave, who “could not lose” but ﬁnished fourth in the classic. The polytrack at Santa Anita — not dirt or the European’s grass — could have been a factor too. Getting back to Zenyatta and Rip Van Winkle. The Betfair and track odds showed the local biases. You could get better odds on Zenyatta in Europe on Betfair and Rip Van Winkle in the US at the track. It was not possible to do an arbitrage here. I just concentrated on better odds on Zenyatta on Betfair. My assumption was despite the fact that her running times were not super outstanding she had the will to win. And indeed she did with Rip Van Winkle ﬁnishing out of the money. She could have been retired as the only undefeated mare who beat males in the toughest race in the world. But let’s discuss the Pick 6. In the Pick 6, to win you must have all six winners who share 75% of the net pool with the 5/6 sharing the remaining 25%. The Pick 6 was races 4–9. There were three standouts but they were deﬁnitely not certain winners. The other three races seemed wide open. So you could play the Pick 6 in the following very simple way: I thought about doing this but did not — it was a $2 million mistake. You have a single ticket with about 10 horses in the three wide open races and single the three standouts. That would cost about 10*10*10*1*1*1*$2=$2000, not a large Pick 6 ticket. You only win if all three standouts win and they did. The payoﬀs for $2 win tickets were as follows Race Race Race Race Race Race 4 5 6 7 8 9 Dancing in Silks Value of York Goldikova Furthest Land Conduit Zenyatta 52.60 63.20 4.80 44.60 3.80 7.60 the ﬁrst standout the second standout the third standout Goldikova was the winner of the mile grass race in 2008 and was arguably better this year. Her connections were the same as those of Miesque, also a two time winner of this race. The second standout, Conduit, was also the winner last year on the 1 12 mile turf. The Pick 6 paid $1,838,305.20 for one winning 6/6 ticket and the 3*9=27 Pick 5/6 tickets (of the 10 losers in the three wide open races) paid 27*$4822.40 for a total of $130,204.80 plus the 6% rebate on the $2000 of $120 for a grand total of $1,968,630.00. It is not quite $2 million but as Johnnie Hooker played by Robert Redford in the Sting said: “it’s not enough but it’s close”. Of course, taxes would take 25% at the track and October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 249 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. be sorted out later when ﬁling and my winning would depress the Pick 6 and Pick 5/6 prizes. The big question is would I have gotten these three winners from my 10 picks in these races? Of course, more than 10 was possible. So let’s look at these three races. I use about 56 handicapping services plus my own analysis of the daily racing form and the Equiform pace numbers. So the idea is to handicap the handicappers. In many bets this is computerized. Race 4. The Sprint, 6 furlongs Handicapper #1 had 3156 (the winner, Dancing in Silks) 84 Handicapper #2 had 53816. So at 121 that’s one of the 10 for sure especially when you observe that its last race Beyer speed rating at 106 was the highest of any horse in the race and it was right there at Santa Anita. Three of the horses ran higher Beyers than #6 but not in their last race. Actually there were only nine horses in this race so likely I would have taken them all including the 201 shot #2 and the 301 shot #7, neither of which listed above. #2 had a Beyer of 110 in a Grade I at Santa Anita so must be used. #7 looked greatly outclassed. Race 5 Juvenile 2yearolds on polytrack with 13 horses The winner Vale of York (#7) was racing in the UK and in Italy. He always had short odds and had two wins in ﬁve races and was close in the other three races. The morning line odds were 201. Handicapper #1 had 51349108 (so no 7 but he was a foreign shipper so likely not rated by this US service). Handicapper #2 had 134956. Only handicapper James Quinn with 13811576 had 7 anywhere. These are twoyear olds so there is a lot of noise. There were four other longshot horses one at 201, one at 151, one at 301 and one at 50–1. So going with all 13 horses called all is suggested. Race 7 Dirt Mile #2 Furthest Land won with 10 in the ﬁeld. #1 had 437192 (the winner). Handicapper #2 had 3728. The pace numbers are competitive. So 2 at 201 must be used. Summary: all three of the wide open races had winners that were competitive horses. So they would be on our ticket. But even if we bet on all horses, these three races, the ticket only costs 9*12*10*1*1*1=$2340. This ticket made a lot of sense so I should have played it.4 It would have had 8+11+9=28 Pick 5/6s. Oh well, there is always next year. 4 Another way to play this is to have three sets of tickets in which you assume that at least two of the three standouts will win. So you have N1i , N2i , N3i , N4i , 1, 1 combinations, i = 1, 2, 3 all at $2 each. So depending on the Nji you likely have a larger ticket than the three singles approach. You might win more than one Pick 6 and more Pick 5/6s, but you might miss the Pick 6 as well unless the tickets are well spread. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics 250 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. An example of the multiple ticket to approach to the Pick 6: Santa Anita, March 6, 2002 with $202,790 carryover from Sunday’s wagers. Score 9: you win if the score is 9 or less, here 3 I’s, 2 II’s, total 9 so we won. • This was the behavioral key. • Filigree, the third choice in the morning line went off at 81. • The 3rd and 5th races back, he ran faster than the favorite Love at Noon ran in his last two races. So he had a chance to win and he did. • Love at Noon went off at 15 and had most of the P5 money October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics SP and Optimization in Horserace Betting b1392ch09 251 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. In the following Equiform numbers, the top number is ﬁnal speed number, other numbers are pace within the race. Normally we approximate Kelly with more money on higher probability wagers, but in the following bets we made equal $2 bets. October 2, 2012 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 252 12 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics Professional racetrack betting syndicates I had a hand with several of the major syndicate hedge fund teams through the Beat the Racetrack books Ziemba and Hausch (1984, 1986, 1987) and Hausch, Lo and Ziemba (1994, 2008) and other contacts. Hausch and I both talked to Bill Benter early in his Hong Kong career. He had started betting but had not put together a successful syndicate yet. So he quizzed us on the Dr Z system and other ideas in phone calls. We did help him a bit but as he said “we were academics spreading knowledge and he was a businessman so could not pay us”. He did have other paid consultants on factor models and he pioneered successfully using 80+ factor models of two types: 1. predict the fair odds probabilities of various horses outcomes and compare these to the public’s odds, and 2. bet most likely with the Kelly criterion. I do not know if he picked up the Kelly from Ziemba and Vickson (1975) or from Ed Thorp’s blackjack writings. Benter had been a blackjack player and Thorp introduced Kelly betting there as Fortune’s formula so that may be where he learned it. A key early Thorp paper is in Ziemba and Vickson. The second type of model is to include the track odds as one of the variables to get even better probability estimates. Benter pioneered the use of such models. I had a bit of a hand in there as the major paper on this was published while I was the Management October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. SP and Optimization in Horserace Betting b1392ch09 253 Science departmental editor for ﬁnance and I processed and accepted it for publication. That’s the Bolton and Chapman (1986) paper which along with the only paper Benter published are reprinted in Hausch, Lo and Ziemba. Chapman (1994) using Hong Kong data is in our book. I met Benter in 1993 at the INFORMS meeting in Phoenix where I organized the ﬁnance sessions and helped on the racing sessions. I recall correcting Benter’s (1994) paper in Hausch, Lo and Ziemba, which had one good new development. As discussed above, in the Dr Z method, the biases to win and being second and third tend to cancel so in the work I did with Donald Hausch, we did not need to make any changes except say that because of approximations, bets should not be made to place or show unless the expected value was signiﬁcantly above break even. We suggested 1.10 for the best races at the best tracks and 1.14 and 1.18 for lesser races. This worked well for US place and show betting. Benter and others found that the Dr Z system did not really work well in Hong Kong as the biases there were diﬀerent. Also, he discovered how to correct the second, third, etc biases through the discounted Harville formulations that are discussed above. Victor Lo did his PhD thesis in Hong Kong, directed by statistician John BaconShone on this problem and much of his research is in Hausch, Lo, Ziemba along with papers by others on this. BaconShone has a joint paper in Hausch and Ziemba (2008) with the late Alan Woods who had his own small team in the Philippines after he left Benter. Benter’s real contribution is shown in Figure 10. Namely, he made it all work and in the process became a very rich man with total proﬁts in the one billion area. His paper in Hausch, Lo and Ziemba plus the other papers made our book a cult item with originals selling for $2000 up to $12,000 on EBay and Amazon. Originals are still trading at high prices, about $600. I sold one for $1400 to one of the copycat syndicates in Australia who I was consulting for. Another syndicate wanted to buy up all the Hausch, Lo and Ziemba books and burn them keeping one for their research. I decided to make a second edition which was published with a new preface in 2008 along with the sports and lottos handbook (Hausch and Ziemba, 2008). The gains in Hong Kong by Benter’s team and others were in a market without rebates and high commissions. But they utilized several advantages. 1. Hong Kong Chinese bettors favor and dislike certain numbers from their culture, which makes horses with these numbers diﬀer from the true odds. 2. All the horses are the same ones: mostly Australian geldings running in almost all the races on just two racetracks so prediction was easier than in the US. October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics 254 0 0 Log (wealth / initial wealth) 1 2 3 4 4 log (wealth / initial wealth) 1 2 3 4 RESULTS Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 0 500 1000 1500 roces bet Figure 10 2000 2500 0 1000 2000 3000 Races 4000 5000 6000 Benter’s Hong Kong Racing Syndicate Returns. 3. Data feeds were every 12 seconds and later every minute giving access to pool odds which could be successfully used. 4. The market was deep with huge betting so the price impact was low and lastly, they could bet electronically into the pools. Since the mainland takeover of Hong Kong in 1997, there have been some changes. But the syndicates continue and trade in many markets today in 2012 such as Japan and Korea as well as in the US, Canada and Europe. My personal experience consulting extensively for one other syndicate is that the setup cost for the research and computer implementation is a major time and ﬁnancial undertaking. Like most markets it was easy earlier and much more diﬃcult now. The syndicates with many workers and good experience have an edge on new ones, as I can report from our own activity on this. 13 Conclusion Racetrack betting remains a very active set of markets. The basic betting problems are various versions of portfolio management. The problems are stochastic programs, usually one period but with nonconcave objective functions because of the fractional functions inside the objective function. But the problems are easily solved and for many situations there are simpliﬁed strategies. The objective is usually the Kelly expected log criterion but in cases of low probability high payoﬀ bets there can be hundreds or thousands of separate tickets and the bets must be integers. So a tickets network tree approach is useful and the Kelly strategy to bet more on the higher probability outcomes can be approximated. The racetrack market is small compared to the ﬁnancial markets such as currency and stock markets but there’s enough there for a number of October 2, 2012 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. SP and Optimization in Horserace Betting b1392ch09 255 syndicates in the US, Australia and Hong Kong and elsewhere to make ﬁfty to a hundred million dollars per year. It is not an easy market to enter at a high level as the setup costs are high and the competition ﬁerce. Take a grass race at a mile with seven horses: one has not run in a year but did well then; one has only run on dirt; one has never run past 6 furlongs (three quarters of a mile); one was racing in France long distances 1 21 miles plus on grass losing consistently and the others have run similar distances on grass but not on this racetrack. Add in jockey and trainer changes and you see why “it is a supreme intellectual challenge” as argued by Andy Beyer, a noted racetrack writer. The models try to bypass this with probabilities and optimization. References Assamoi, K. V. Optimal investment strategies with Kelly capital growth criterion. Thesis: MSc in Mathematical Finance, Christ Church, University of Oxford, 2010. BaconShone, J. and A. Woods. Modeling money bet on horse races in Hong Kong. In D.B. Hausch and W.T. Ziemba (eds.), Handbook of Sports and Lottery Markets, pp. 17–25. North Holland, 2008. Benter, W. Computer based horse race handicapping. In D. B. Hausch, V. Lo, and W. T. Ziemba (eds.), Eﬃciency of Racetrack Betting Markets, pp. 173–182. Academic Press, 1994. Bolton, R. N. and R. G. Chapman. Searching for positive returns at the track: A multinomial logit for handicapping horse races. Management Science 32, 1040–1059, 1986. Chapman, R. G. Still searching for positive returns at the track: empirical results from 2000 Hong Kong races. In D. B. Hausch, V. Lo, and W. T. Ziemba (eds.), Eﬃciency of Racetrack Betting Markets, pp. 173–181. Academic Press, 1994. Chopra, V. K. and W. T. Ziemba. The eﬀect of errors in mean, variance and covariance estimates on optimal portfolio choice. Journal of Portfolio Management, 19 : pp. 6–11, 1993. Drud, A. S., CONOPT — A LargeScale GRG code. INFORMS Journal on Computing, 6: pp. 207–216, 1994. Gramm, M. and W. T. Ziemba. The dosage breeding theory for horse racing predictions. In D. B. Hausch and W. T. Ziemba (eds.), Handbook of Sports and Lottery Markets, pp. 307–340. North Holland, 2008. Gramm, M. and W. T. Ziemba. Update on the dosage theory of horse race predictions for the triple crown races, mimeo, 2012. Harville, D. A. Assigning Probabilities to the Outcomes of MultiEntry Competitions. Journal of the American Statistical Association, 68: pp. 312–316, 1973. Hausch, D. B., R. Bain, and W. T. Ziemba. An application of expert information to win betting on the Kentucky Derby, 1981–2001. European Journal of Finance, 12(4): pp. 283–302, 2006. October 2, 2012 Stochastic Programming Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 08/28/17. For personal use only. 256 1:24 9in x 6in Applications in Finance, Energy, Planning and Logistics b1392ch09 Applications in Finance, Energy, Planning and Logistics Hausch, D. B., V. Lo, and W.T. Ziemba (eds.). Eﬃciency of Racetrack Betting Markets. Academic Press, 1994. Hausch, D. B., V. Lo, and W. T. Ziemba (eds.). Eﬃciency of Racetrack Betting Markets (2nd edition). Singapore: World Scientiﬁc, 2008. Hausch, D. B. and W. T. Ziemba. Transaction costs, extent of ineﬃciencies, entries and multiple wagers in a racetrack betting model. Management Science, 31: pp. 381–394, 1985. Hausch, D. B. and W. T. Ziemba (eds.). Handbook of Sports and Lottery Markets. North Holland, 2008. Hausch, D. B., W. T. Ziemba, and M. E. Rubinstein. Eﬃciency of the market for racetrack betting. Management Science XXVII, pp. 1435–1452, 1981. Kahneman, D. and A. Tversky. Prospect theory: an analysis of decisions under risk. Econometrica, 47 (2): pp. 263–92, 1979. Lane, D. and W. T. Ziemba. Arbitrage and risk arbitrage in team jai alai. In D. Hausch and W. T. Ziemba (eds.), Handbook of Sports and Lottery Markets, pp. 253–271. North Holland, 2008. MacLean, L. C., E. O. Thorp, and W. T. Ziemba. The Kelly Capital Growth Investment Criterion. Singapore: World Scientiﬁc, 2011. Murlagh, B. A. and M. A. Saunders. MINOS 5.5 User’s Guide. Technical Report SOL 8320R. Stanford University. Revised 1998. Snowberg, E. and J. Wolfers. Examining explanations of a market anomaly: preferences or perceptions? In D.B. Hausch and W. T. Ziemba (eds.), Handbook of Sports and Lottery Markets, pp. 103–136. North Holland, 2008. Thaler, R. H. and W. T. Ziemba. Anomalies: parimutuel betting markets: racetracks and lotteries. Journal of Economic Perspectives, 2 : pp. 161–174, 1988. Tompkins, R., W. T. Ziemba, and S. Hodges. The favoritelongshot bias in the S&P500 and FTSE100 index futures options: the return to bets and the cost of insurance. In D. B. Hausch and W. T. Ziemba (eds.), Handbook of Sports and Lottery Markets, pp. 161–180. North Holland, 2008. Ziemba, W. T. Eﬃciency of racetrack, sports and lottery betting markets? In D. B. Hausch and W. T. Ziemba (eds.), Handbook of Sports and Lottery Markets, pp. 183–222. North Holland, 2008. Ziemba, W. T. Place and show. Wilmott Magazine (November), pp. 32–42, 2011. Ziemba, W. T. Exotic Betting at the Racetrack. In progress, 2012. Ziemba, W. T. and D. B. Hausch. Beat the Racetrack. Harcourt, Brace and Jovanovich, 1984. Ziemba, W. T. and D. B. Hausch. Betting at the Racetrack. Dr Z Investments, San Luis Obispo, CA, 1986. Ziemba, W. T. and D. B. Hausch. Dr Z’s Beat the Racetrack. William Morrow, 1987. Ziemba, W. T. and R. G. Vickson (eds.). Stochastic Optimization Models in Finance. New York: Academic Press, 1975. Ziemba, W. T. and R. G. Vickson (eds.). Stochastic Optimization Models in Finance (2nd edition). Singapore: World Scientiﬁc, 2008.