superbowl betting ISSUES RELATED TO

ISSUES RELATED TO superbowl betting

SummaryThis paper looks at various issues that are of interest to the superbowl gambler. First, anexpression is obtained for the distribution of the final bankroll using fixed wagers witha specified initial bankroll. Second, fixed percentage wagers are considered where theKelly method is extended to the case of simultaneous bets placed at various odds; acomputational algorithm is presented to obtain the Kelly fractions. Finally, we considerthe problem of determining whether a betting system is profitable based on the historicalresults of bets placed at various odds.Key words: gambler’s ruin, Kelly method, optimal wagering, comparative inference, Gibbssampling, meta-analysis.1. IntroductionDespite its illegality in many jurisdictions, betting on the outcome of sporting eventsis a common activity. For example, Crist (1998) states that Americans illegally wager∗Robin Insley is Senior Lecturer, Lucia Mok is an MSc graduate and Tim Swartz is Professor, De-partment of Statistics and Actuarial Science, Simon Fraser University, Burnaby BC, Canada V5A1S6.Swartz’s work was partially supported by a grant from the Natural Sciences and Engineering ResearchCouncil of Canada. The authors thank an Associate Editor and two referees whose comments lead to animprovement in the manuscript.1--------------------------------------------------------------------------------Page 2 over $100 billion annually on professional and college superbowl. On the 1999 Superbowl(the championship game of the National Football League), approximately $87 millionwas wagered legally in Las Vegas superbowlbooks (Ordine, 2000). In Australia, commercialbetting on racing alone (horses and greyhounds) resulted in individual losses of $1.6billion in 1997-1998 (Productivity Commission, 1999).superbowl betting has been increasing over the years (Productivity Commission, 1999)and it is likely that the trend will continue as more government bodies are eager to sharein the huge profits that have and can be made. One particular avenue for growth is superbowlbetting via the internet (Haywood, 2000). Whereas such internet sites are illegal in theUnited States and in Canada, they are legal and operational in various Caribbean andLatin American countries and in Australia. Internet betting yields various murky legalissues such as the determination of where the wager is actually placed (e.g. in one’s homein North America where the activity is illegal or in the offshore country).In this paper, we put ethics and legal issues aside and take a look at various practicalproblems associated with superbowl betting. We are primarily concerned with issues involvingthe performance of wagering systems. To a lesser extent, the results are also applicableto financial investments corresponding to an investor who engages in numerous transac-tions. Many fundamental probabilistic results concerning optimal systems for favourablegames are reviewed in Thorp (1969). We are also interested in the problem of identifyingprofitable betting systems based on past data.In section 2, we provide an introduction to pointspread wagering where we emphasizethat it is theoretically possible to develop winning systems. betting terminology isexplained and the main objective of a bookmaker is discussed. In section 3, we obtaindistributional results for the gambler’s current bankroll after placing a finite number ofbets. This is done in the context of both ‘large’ and ‘small’ initial bankrolls. Although the2--------------------------------------------------------------------------------Page 3 results involve only binomial calculations and a simple extension of the classical drunk-ard’s walk problem, they are practical and we have not seen them recorded in the superbowlbetting literature. Moreover, much of the work concerning the gambler’s ruin and re-lated problems was developed long ago with an emphasis on challenging mathematics andapproximate solutions. Our approach is based on the realization that today many peoplehave a computer on their desk. In section 4, we turn to fixed percentage wagering whereeach bet is a fixed percentage of the current bankroll. We provide some theory and anumerical algorithm to obtain optimal fixed percentages in the context of simultaneousbets placed at various odds. The results in this section are extensions of the Kelly system(Kelly, 1956). We also give some distributional results concerning the final bankroll underfixed percentage wagering. In section 5, we consider the historical results of bets placed atvarious odds for a proposed betting system. The practical question arises as to whetherthe historical data provides evidence that the betting system is profitable. Profitabilityentails more than simply checking whether a profit is made in a given year; this couldoccur by chance. We are instead interested in whether there is long-term profitability ina system. The problem raises inferential issues for which different approaches yield con-flicting results. The major inferential problem that arises is the testing of a hypothesisΩ0versus a hypothesis Ω1where both Ω0and Ω1are ‘small’ sets relative to the entireparameter space. We propose a non-standard Bayesian approach to the problem whichcalculates ‘distances’ from Ω0and Ω1as measures of evidence in favour of each hypoth-esis. In section 6, we provide a concluding discussion with advice applicable to typicalsuperbowlbook scenarios where upper limits on wagering are imposed.2. A primer on betting the pointspreadThere are many types of wagers that can be placed on sporting events (McCune,1989). Perhaps the most common, is a wager placed against the pointspread. For ex-3--------------------------------------------------------------------------------Page 4 ample, consider a contest between a strong team (Team A) and a weak team (Team B).Whereas popular sentiment may overwhelmingly favour Team A to win, there is typicallyno concensus on the magnitude of the victory. To facilitate interest in wagering on sucha match, a posted line may appear asTeam A −l −110(1)Team B +l −110 .The line (1) is based on American odds and stipulates that a wager of $110 placed onTeam A returns the original $110 plus an additional $100 if Team A wins by more thanl points. Alternatively, a wager of $110 placed on Team B returns the original $110 plusan additional $100 if Team B wins or if Team B loses by less than l points. In the casewhere Team A wins by exactly l points, the original bets are returned. The quantity l isreferred to as the pointspread and is determined by the bookmaker, i.e. the individual ororganization that posts the line and collects the bets.We point out a few variations in the above example. First, different odds may be given.For example, American odds of -120 stipulate that a winning wager of $120 returns theoriginal $120 plus an additional $100. When the American odds are positive, this suggeststhat an event is less likely. For example, American odds of +140 stipulate that a winningwager of $100 returns the original $100 plus an additional $140. Second, wagers canbe made in multiples or fractions of the amounts discussed. Third, a nearly equivalentexpression of (1) is based on European odds and appears asTeam A −l 1.91(2)Team B +l 1.91 .4--------------------------------------------------------------------------------Page 5 Here, a winning wager of x dollars returns x(1.91) dollars. In the case of a $110 wager,the return is $110(1.91) = $210.10 which is nearly equivalent to the $110 + $100 = $210situation described above.Having discussed the betting procedure, it is important to understand the objective ofthe bookmaker. In (2), suppose that a total of y dollars is wagered on Team A and a totalof y dollars is wagered on Team B. In this case, the bookmaker collects 2y dollars, andgiven that a winner is decided, the bookmaker pays out y(1.91) dollars. The bookmakerhas made a profit of y(0.09) dollars regardless of the winning team and the percentageprofit (vigorish) is calculated as y(0.09)/(2y) → 4.5%. Thus, in the case of the line in(2), the ‘safe’ strategy for the bookmaker is to attempt to select the pointspread l soas to balance the total bets placed on Team A and on Team B for this guarantees aprofit. It is this fact that suggests opportunities for the gambler since the bookmakeris not trying to achieve an optimal line from the point of view of prediction. Rather,the bookmaker is trying to assess public opinion (i.e. determine the pointspread l) so asto balance the bets. A gambler then ‘simply’ needs to (a) have more insight on realitythan the rest of the betting public and (b) win often enough to overcome the vigorish.Unlike most mechanical games (e.g. roulette), it is evident that the possibility existsto develop winning strategies when betting on the outcome of sporting events. Stern(1998) suggests that bookmakers are good at setting lines in the sense that actual winningpercentages closely resemble the probabilities implied by the lines. For example, he claimsthat it is reasonable to approximate the outcome of a National Football League game usingthe normal distribution with mean equal to the point spread and standard deviation equalto 13.5.A simple heuristic for wagering is to develop a procedure for constructing pointspreads.When one’s personal pointspread differs sufficiently from the pointspread in the posted5--------------------------------------------------------------------------------Page 6 line, this signals a condition to wager. The size of the wager may depend on the magnitudeof the departure. We note in passing that the stated heuristic of comparing one’s personalpointspread to a bookmaker’s pointspread is an application of subjective probability. Infact, it is difficult to imagine how such an application can be reconciled in terms of afrequentist notion of probability.Finally, we mention two other types of wagers commonly available in the same matchinvolving Team A and Team B. First, odds are usually posted for Team A winning outright(i.e. without a pointspread). Odds are also posted for Team B winning outright. Thispair of odds is known as the moneyline. Second, there is usually a total t associated withthe match together with under and over odds. An under wager wins if fewer than t totalpoints are scored in the match and an over wager wins if more than t total points arescored in the match. If exactly t total points are scored in the match, the wagers arerefunded.3. Fixed wagersIn fixed wagering, without loss of generality, a gambler bets 1 unit on the outcome ofeach match. Given a betting system with probability 0 <> 1 in each of the matches, we are interested in the bankroll Bmthat is realized after placing m bets.Consider first the simplest situation where the gambler has an initial bankroll B0exceeding m (i.e. the gambler bets within his means and is prepared to lose each of the mbets). In this case, given m independent bets, it is straightforward to write the binomialprobability for the bankroll BmasPr (Bm= B0+ j(θ − 1) − (m − j)) = (mj) pj(1 − p)m−jj = 0, . . ., m.(3)6--------------------------------------------------------------------------------Page 7 From the binomial distribution, we have E(Bm) = B0+m(θp−1) and var(Bm) = θ2mp(1−p). The normal approximation to the binomial then suggestsBmd= N B0+ m(θp − 1), θ2mp(1 − p)from which probabilities may be calculated. For example, the probability of realizing aprofit after m bets is Pr(Bm> B0) ≈ Φ m(θp − 1)/ θ2mp(1 − p) . We might also usea continuity correction in the calculation of Pr(Bm> B0).In practice, a gambler does not know the probability p of choosing winners. Althoughp may be estimated from historical data, perhaps the most useful applications of (3) andsubsequent formulae contained in this paper are sensitivity analyses. A gambler mayinvestigate the range of p necessary to produce worthwhile profits. A gambler may alsoinvestigate the size of the loss if the system yields values of p unable to overcome thevigorish. For example, with ‘even’ odds such as those in (2), the gambler might be willingto assume a worst case scenario of p = 0.5 for this corresponds to no knowledge whatsoever(i.e. choosing Team A or Team B strictly by the flip of a coin).We now turn to the more complicated situation where the gambler has an initialbankroll of B0, where 0 < i =" 1," i =" 1," i =" 1," j =" 0," j=" Pr" bi=" B0+">i−B0θ0j ≤i−B0θwhere Q0,0= 1 and Q−1,k= Qk,−1= 0 for all k. This forms a tree structure where Qj,i−jis the jth entry along the ith row from the top of the tree. In this case, we constructthe tree beginning with the top element Q0,0and work downward along rows where thelast row {Qj,m−j: j = 0, . . ., m} gives the probabilities of interest. We note that theprobability of ruin is available from the final row via Pr(ruin) = 1 −mj=0Qj,m−j.Whereas much of the early work corresponding to the drunkard’s walk concerneditself with approximate solutions and detailed mathematics, we note that the approachdescribed here is based on ready access to a computing environment; the computationalapproach is fast and involves straightforward programming. For example, with θ = 1.91,B0= 5, p = 0.56 and m = 500, we obtain Pr(ruin) = 0.428 and this was obtained nearlyinstantaneously running Fortran on a SUN workstation.4. Fixed percentage wagersIn simple fixed percentage wagering, a gambler bets a fraction 0 ≤ f ≤ 1 of thebankroll on the outcome of a single match. When the outcome of the match is determined,the gambler resumes fixed percentage wagering on the new balance. Assuming the infinitedivisibility of money, one of the immediate attractions of fixed percentage wagering is that8--------------------------------------------------------------------------------Page 9 the gambler never goes into debt.In the context of information theory, Kelly (1956) provided a neat result that is oftenquoted but misused in betting circles. Given a betting system with probability 0

1, Kelly showed that the ‘optimal’betting fraction isf∗=pθ − 1θ − 1(4)provided that p > 1/θ. For example, consider a system that historically picks winners 54%of the time (i.e. p = 0.54) with the standard European odds payout of θ = 1.91. In thiscase, the optimal betting fraction is 3.45% of the bankroll. The Kelly criteria is optimalfrom several points of view; for example, it maximizes the exponential rate of growth andit provides the minimal expected time to reach a preassigned balance (Breiman, 1961).Breiman (1961) investigated the properties of betting systems and considered a moregeneral wagering scenario than the simple situation discussed above. However, Breiman’swork did not provide the derivation of solutions nor sharp statements concerning theuniqueness of solutions under the general framework.We are concerned with restricted problems that are of real interest to the superbowl bettor.For example, it is likely that a superbowl bettor would like to bet on several matches in asingle day. If the bettor uses the Kelly fraction (4) on n such matches where nf∗> 1,then the total amount to bet would exceed the current bankroll. The problem then is todetermine fixed percentages that satisfy some optimality.We therefore consider the situation where on day j = 1, . . ., m, the gambler wishes toplace njiwagers on matches with European odds θiand where the probability of pickingwinners is pi, i = 1, . . ., k. For example, it is possible that a gambler has a system forbetting the pointspread and another system for betting totals (i.e. k = 2). The question9--------------------------------------------------------------------------------Page 10 arises as to what are the optimal betting fractions f∗j1, . . ., f∗jkon day j where njiwagersare placed with a fraction fjiof the bankroll, i = 1, . . .k? Given an initial bankroll B0,the bankroll at the completion of day j isBj=nj1xj1=0· · ·njkxjk=0(1 −ki=1njifji)Bj−1+ki=1xjiθifjiBj−1∆ji(5)where ∆ji= I(Xji= xji), i = 1, . . ., k, in which the random variable Xjidenotes thenumber of winning wagers of type i on day j. Therefore, only one term in the product(5) is not equal to 1. Note also that the first term in the outer parentheses is the balanceof the bankroll not bet in a given day and we requireki=1njifji< i =" 1," g =" G(fj1," andg =" Enj1xj1="0·" njkxjk="0∆jilog" ki="1njifji+ki="1xjiθifji="nj1xj1="0·" njkxjk="0ki="1njixjipxjii(1" ki="1fji(xjiθi−" k =" 3," nj1=" 2," nj2=" 3," nj3=" 3," p1=" 0.545," p2=" 0.565," p3=" 0.585" 1=" θ2="" 3=" 1.91." j1=" 0.0416," j2=" 0.0811" j3=" 0.1215.The" asbm=" (1" ki="1nmifmi)Bm−1+ki="1XmiθifmiBm−1="" b0mj="1(1" ki="1fji(Xjiθi−" xjid=" Bi(nji," j =" 1," i =" 1," m =" 162,k" pi=" 0.56," i=" 1.91" nji=" 1" ji=" 0.0761." b0=" 100.Figure" ki="1E(njifji)(piθi−" pb=" 1"> 0 → pA> 1/θA.Similarly, we should bet on Team B if pB> 1/θB. Recall that pAand pBare unknownand the inequality 1/θA+ 1/θB> 1 is due to the vigorish.Now we first test the profitability of a betting system in a restricted context. Supposethat we have the results of n historical matches where the proposed betting system wouldhave bet on Team A with European odds as in (9). Suppose further that x of the n betsare winning bets. Then we test for a profitable system by considering H0: pA≤ 1/θAversus H1: pA> 1/θA. The corresponding P-value is given by the binomial probabilityPr(X ≥ x) =ni=x(ni) (1/θA)i(1 − 1/θA)n−i(10)13--------------------------------------------------------------------------------Page 14 where a small P-value indicates evidence of a profitable system. This is a simple procedureto determine whether historical data provides evidence of long-term profitability. Forexample, if there are x = 60 winning wagers out of n = 100 matches with standard payoutθA= 1.91, then pA= 1/θA= 0.524 and we have mild evidence (P = 0.076) of a profitablesystem. Remarkably, tests such as these are rarely (perhaps never) discussed in the myriadof betting books available at the Gamblers Book Shop (www.gamblersbook.com), a keysource for information on betting.The situation becomes more complex when bets are placed at various odds. Wehave collected data arising from the 2000 Major League Baseball season for which aproposed betting system was established. Ignoring some of the details, personal oddswere constructed corresponding to Team A defeating Team B. These odds were based onlogistic regression using the pitchers’ earned run averages, overall team batting averagesand team winning percentages as covariates. When the personal odds differed from theposted odds at the website (www.) by more than 10%, this signaleda condition to wager.Table 1 gives the data and P-values for groups of bets placed at 24 differents sets ofodds. The P-values are obtained using (10). We observe that only one of the results aresignificant. The problem now is to determine whether the overall system is profitable.A standard approach is based on Fisher’s test (D’Agostino & Stephens, 1986 p. 357).Following the notation above, let H0k(H1k) denote the hypothesis that the kth type ofbet is unprofitable (profitable) and let qkdenote the corresponding P-value, k = 1, . . ., 24.Fisher’s test considers the overall null hypothesis Ω0=24k=1H0kversus the alternativehypothesis that at least one of the H0kis false. Applying Fisher’s test to the data in Table1, we obtain the overall P-value Pr(χ248> −224k=1log qk) = Pr(χ248> 52.767) = 0.295which does not allow us to reject the overall null hypothesis Ω0that all of the component14--------------------------------------------------------------------------------Page 15 types of bets are unprofitable.The approach above is not quite right. With respect to the data in Table 1, there is notenough evidence to reject Ω0, and even if there was, this does not imply that the bettingsystem is profitable. We define a profitable betting system as one for which all of theH1kare true. Therefore we should instead test Ω0versus the alternative Ω1=24k=1H1kand exclude regions of the parameter space that do not belong in Ω0∪ Ω1. We excludesuch regions because the betting system uses the same betting criteria regardless of theodds and it is therefore logical that if it is an unprofitable (profitable) betting system,then it should be an unprofitable (profitable) system for all component types of bets. Weemphasize that the relevant problem is to test a hypothesis Ω0which is a small set whereall component bets are unprofitable versus a hypothesis Ω1which is a small set where allcomponent bets are profitable.To test Ω0versus Ω1in a classical framework, it is natural to reject Ω0based on largevalues of the generalized likelihood ratio statisticΛ =supΩ0∪Ω1L(x)supΩ0L(x)where L(x) is the corresponding product probability mass function based on the data x.The exact discrete distribution of Λ is beyond reach since the sample space is so large.Also, we cannot appeal to a large sample χ2distribution for 2 log Λ since dim(Ω0) =dim(Ω1) which means that we have ‘zero’ degrees of freedom. In addition, the supremumsin Λ occur on the boundaries of the parameter space and this invalidates necessary largesample assumptions.We consider now a Bayesian approach to testing Ω0versus Ω1. The Bayesian modelcontains a little more structure due to the necessity of prior distributions. Given theEuropean odds (9), we assume that over many matches the odds are set such that bets15--------------------------------------------------------------------------------Page 16 Table 1: Betting results using a proposed betting system; θAis the posted odds forTeam A, θBis the posted odds for Team B, x is the number of wins by betting on TeamA and n is the total number of corresponding matches. The P-value is given and anasterisk indicates significance at level 0.05.θAθBxn P-value2.800 1.455 12 32 0.4822.700 1.498 11 25 0.2992.650 1.541 10 23 0.3572.600 1.5568 18 0.3842.550 1.571 13 34 0.6102.500 1.588 17 35 0.1932.450 1.606 13 38 0.8402.400 1.625 44 89 0.0842.350 1.645 19 43 0.4722.300 1.666 26 56 0.3762.250 1.690 28 59 0.3672.200 1.714 39 65 0.013∗2.150 1.740 22 43 0.3222.100 1.770 23 41 0.1762.050 1.800 26 47 0.2262.000 1.833 20 39 0.5001.910 1.9109 17 0.5781.833 2.000 12 20 0.3991.800 2.05047 0.6201.770 2.100 29 51 0.5381.740 2.15046 0.4921.690 2.25033 0.2071.645 2.35035 0.6961.625 2.40035 0.70916--------------------------------------------------------------------------------Page 17 on Team A and Team B are equally attractive (i.e. they yield the same expected return).Therefore, based on a unit bet,(θA− 1)pA+ (−1)(1 − pA) = (θB− 1)(1 − pA) + (−1)pA→ pA= θB/(θA+ θB)where θB/(θA+θB) represents the expected probability that Team A wins and the expec-tation is taken over many matches. In any particular match, the true probability pAmaybe something different than θB/(θA+ θB). Let Xkdenote the number of winning bets oftype k out of nkwagers and let p(0)k= θB/(θA+ θB) for the kth type of bet, k = 1, . . ., N.For the data in Table 1, N = 24. This suggests the hierarchical model(Xk pk)d= Bi(nk, pk)k = 1, . . ., N(pk m)d= β mp(0)k, m(1 − p(0)k)k = 1, . . ., Nmd= U(l, ∞)where l = maxk1/p(0)k, 1/(1 − p(0)k)(11)where the pkare conditionally independent. The beta prior for pkis reasonable from thepoint of view that pkis constrained to the interval (0, 1) and E(pk m) = p(0)kas arguedpreviously. The hyperparameter m controls the variance of the pk, and we assign a flatimproper prior for m. We include the lower limit on m as this forces concave densities forp1, . . ., pN.The hierarchical model (11) induces a (N +1) dimensional posterior distribution givenby (p, m x). Inference based on (p, m x) is straightforward using the Gibbs samplingalgorithm as the full conditional distributions of the pkare beta distributions and mcan be generated in closed form from its full conditional distribution via inversion. Toobtain the posterior probabilities of Ω0and Ω1, we simply calculate the proportion of thegenerated ps that fall into the two respective sets. For the data in Table 1, it turns out17--------------------------------------------------------------------------------Page 18 that both of these probabilities are essentially zero.We are therefore faced with the problem of assessing two hypotheses Ω0and Ω1, oneof which we believe to be true, when both hypotheses correspond to very improbable sets.This is a general problem of inference which goes beyond the application considered here.Our approach which borrows on ideas from Swartz (1999) and Evans et al. (1997) is togenerate p from the Gibbs sampling algorithm and then calculate its Euclidean distanceD0from Ω0and its Euclidean distance D1from Ω1. The smaller (larger) the quantityD0/D1, the more evidence p provides in favour of the hypothesis Ω0(Ω1). We thereforeconsider posterior probabilities Pr(D0/D1≤ t x) for different values of t. To put morereliance on the data we also calculate the Bayes factorBFD=Pr(D0/D1≤ t x)Pr(D0/D1> t x)Pr(D0/D1> t)Pr(D0/D1≤ t)where small (large) values give evidence in favour of Ω1(Ω0). There is one difficulty withthe calculation of BFDand this concerns the improper prior for m. With an improperprior, we are unable to generate from the joint prior distribution of (p, m) to obtainPr(D0/D1≤ t) and Pr(D0/D1> t). For the data in Table 1, we therefore set m = 2.98which is the posterior mean of m estimated from the output of the Gibbs sampling algo-rithm, and we then simulate the pks from the beta priors with this value of m. Table 2shows BFD, the posterior odds of D0/D1≤ t and the prior odds of D0/D1≤ t correspond-ing to the data in Table 1. As expected, we see that both the posterior odds and priorodds are increasing in t. However, the prior odds are larger and increase more rapidlythan the posterior odds. This causes the Bayes factor BFDto be small and approach zeroas t increases. We also observe that BFDhas its largest value when t is around 1.Now if you have confidence that the prior is realistic, then it is often argued thatinference should be based solely on the posterior as Bayes Theorem provides the recipe18--------------------------------------------------------------------------------Page 19 Table 2: BFD, the posterior odds of D0/D1≤ t, the prior odds of D0/D1≤ t for selectedvalues of t using the data in Table 1.tBFDposterior odds prior odds0.6 0.0090.0010.1140.7 0.0300.0080.2710.8 0.0350.0180.5220.9 0.0460.0400.8591.0 0.0510.0811.5911.2 0.0480.2034.2081.4 0.0360.41011.5001.6 0.0250.63125.3161.8 0.0140.99670.4292.0 0.0121.532124.0002.5 0.0063.219499.0003.0 0.0005.410∞4.0 0.00013.925∞6.0 0.00070.429∞8.0 0.000165.667∞19--------------------------------------------------------------------------------Page 20 for combining information from the data (i.e. the likelihood) and information from theprior. In this case, the posterior odds from Table 2 clearly suggest that the bettingsystem is profitable based on the data in Table 1. Note that using the posterior odds fort = 1.0 in Table 2, we obtain Pr(D0< b0=" $500" x1=" $400," x2=" $200," bygi="∂G∂fji="nj1xj1="0·" njkxjk="0ki="1njixjipxjii(1" ki="1fji(xjiθi−" andgi1i2="∂2G∂fji1∂fji2="nj1xj1="0·" njkxjk="0ki="1njixjipxjii(1" ki="1fji(xjiθi−" i2=" 1," halfspaceki="1njifji<"> 1/θi, i = 1, . . ., k, we obtain(a) Gi> 0 when fji= 0, i = 1, . . ., k,(b) Gi1i2< i2=" 1," aski="1njifji→"> 0.The shape and smoothness of G yields a simple algorithm which is guaranteed to findf∗j1, . . ., f∗jk. We begin by initializing an interior point fji= 1/ki=12nji, i = 1, . . ., k. Toobtain the root of G1along the first coordinate direction, bisection is carried out using thelower starting value f(l)j1= 0 and the upper starting value f(u)j1which intersects the planeki=1njifji= 1. After the first coordinate is updated, the procedure is repeated along thecoordinate directions 2, . . ., k. The loop in this sequential procedure is repeated until themovement in the point is sufficiently small. The maximum has then been obtained. Theproposed algorithm is a special case of Gauss-Seidel iteration (Thisted 1988, page 187)where success is based on the recognition that every step of bisection results in a moveup the hill and that there are no saddle points or minima of the function G.ReferencesBreiman, L. (1961). Optimal betting systems for favorable games. Proceedings of the FourthBerkeley Symposium on Mathematical Statistics and Probability, Jerzy Neyman, editor,65-78.Crist, S. (1998). All bets are off. superbowl Illustrated, 88 (3), 82-92.22--------------------------------------------------------------------------------Page 23 D’Agostino, R.B. & Stephens, M.A. (1986). Goodness-of-Fit Techniques, Marcel Dekker, NewYork.Evans, M., Gilula, Z., Guttman, I. & Swartz, T.B. (1997). Bayesian analysis of stochasti-cally ordered distributions of categorical variables. Journal of the American StatisticalAssociation, 92, 208-214.Feller, W. (1968). An Introduction to Probability Theory and its Applications, Volume 1, ThirdEdition, John Wiley and Sons, Inc.Haywood, H. (2000). BeatWebCasinos.Com: The Shrewd Player’s Guide to Internet betting,RGE Publishing, Oakland, California.Kelly, J.L. (1956). A new interpretation of information rate. Bell System Technical Journal,35, 917-926.McCune, B. (1989). Education of a superbowl Bettor, McCune superbowl Investments, Las Vegas,Nevada.Mok, L. (2001). Testing whether a betting system is profitable. MSc project, Simon FraserUniversity, Department of Statistics and Actuarial Science.Ordine, B. (2000). Super bowl Sunday means lots of action in Las Vegas. Seattle Times,January 23.Productivity Commission (1999). Australia’s betting industries (Report No 10), PC InquiryReport, December, H.S. (1998). How accurate are the posted odds?. In the column A Statistician Readsthe superbowl Pages, Chance, 10 (4), 17-21.Swartz, T.B. (1999). Nonparametric goodness-of-fit. Communications in Statistics: Theoryand Methods, 28, 2821-2841.23--------------------------------------------------------------------------------Page 24 Thisted, R.A. (1988). Elements of Statistical Computing: Numerical Computation, Chapmanand Hall, New York.Thorp, E.O. (1969). Optimal betting systems for favorable games. Review of the Interna-tional Statistical Institute, 37, 273-293.24--------------------------------------------------------------------------------Page 25 02000400060008000100000200400600Figure 1: Histogram of the final bankroll based on 1000 simulationsfinal bankrollfrequency25--------------------------------------------------------------------------------Page 26 24680204060Figure 2: Histogram of the logarithm of thefinal bankroll based on 1000 simulationslog(final bankroll)frequency26

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