OPTIA EXCHANGE ETTING STRATEGY FOR WIN-DRAW-OSS ARKETS Darren O Shaughnessy a,b a Ranking Software, elbourne b Corresonding author: darren@rankingsoftware.com Abstract Since the etfair betting exchange launched in, sorts gamblers have had a gambling forum quite different from the traditional bookmaker. Three features of betting exchanges in articular require new analysis methods extending the Kelly criterion originated by John Kelly (1956): (i) The ability to lay (i.e., bet against) a team as well as back it (ii) Negotiation of odds, where one can set one s own odds and wait for another unter to match them, not just accet the market valuation at the time (iii) The bookmaker takes a fee as a fixed fraction of one s net rofit on a market, not as a hidden margin in each betting otion s rice In sorts where there are more than two ossible outcomes, such as soccer (football), usually the rosective gambler will find that if he/she wants to bet on one team using the Kelly criterion, the same criterion will advocate laying against the other team. asic Kelly betting offers no resolution to these correlated markets, and some unters at traditional bookmakers will instead seek a binary handica or draw-no-bet market in order to find rices that they can immediately understand. This aer derives the criterion one should use when investing in a win-draw-loss market, with the imortant feature that rofits are significantly higher by combining back and lay bets than by relying on one or the other. The draw-no-bet aroach is shown to be otimal only in a narrow band of cases, where the advantage of having the draw result untaxed outweighs the rofits to be gained by effectively backing it. Keywords: etting exchange, Kelly criterion, etfair, Football, Soccer Prerint submitted to ATHSPORT 1 Setember 17, 1
1. INTRODUCTION The Kelly criterion (Kelly, 1956) is a vital tool in the armoury of both ortfolio investors and gamblers. y maximising logarithmic utility simultaneously minimising the risk of ruin Kelly rovided the formula that gamblers with erfect robabilistic knowledge must use to grow their bank at the largest exected rate. With the exlosion in sorts betting around the world since the rise of the internet, several aers have been written exanding the Kelly criterion, for instance to account for multile simultaneous indeendent market investments (Thor, 1997; Insley et al, 4), sread betting (Chaman, 7), and back/lay comarison on betting exchanges (Walshaw, 1). arnett (11) alied the Kelly criterion to the game of Video Poker, where there are multile ossible outcomes but only a single bet to make on each hand. eanwhile, betting exchanges have rovided markets that are often more attractive than regular bookmakers, by allowing unters to match their money with eers. Soccer (association football) matches are some of the most oular and therefore most liquid markets, with over $1 million matched on the final of the 1 FIFA World Cu (etfair, 1). This aer addresses dilemmas that gamblers can feel in sorts such as soccer and cricket that have a three-otion market. Is it more rofitable to back the team that the unter believes is underrated by the market, or match someone else s money by laying the team that aears overrated? And is the draw no bet market worthwhile?. ETHODS The basic Kelly criterion for a single otion on a regular betting market gives the Kelly et as: 1 1 (1) where is the team s market rice and is the gambler s resumed robability of the team winning. is exressed as a ercentage of the bettor s bankroll, and a bet should be laced if >1. The formula is derived by maximising log(exected bank) with resect to the bet roortion. etfair, which comrises about 9% of the betting exchange economy worldwide (Sydney orning Herald, 6), does not build its rofit margin into each rice like a traditional bookmaker, but instead taxes each market winner on their net rofit once the event is resolved. A gambler could have several individual bets on the same market, even arbitraging a guaranteed rofit as the odds change, and only ay a fee on his/her net result on the winning otion(s). The level of tax t varies from 5% for low-volume gamblers down to % for those who have the largest betting history. This leads to an adjusted formula, where is the agreed rice for the bet: ( (1 ) ( (1 t) () It is immediately obvious that for a single bet, taking etfair odds is exactly equivalent to taking a slightly lower rice at a standard betting sho: 1+ ( (1 t) () e.g. for a 5% tax, etfair $ is equivalent to traditional $1.95 while etfair $1. is equivalent to $1.19 at a regular bookmaker. In lay betting, the unter risks ( by acceting a bet of size from an anonymous eer, having negotiated a rice. The Kelly et in this case is: ( (1 ) (4) In this aer, we limit our analysis to a combination of betting on one team and laying against the other, ignoring the market rice for the central otion (the draw). This is done without loss of generality if the market is fully saturated and has automated betmatching, which is usually the case for etfair soccer markets. Under this assumtion, the draw rice can be derived as D 1/[1 (1/ ) (1/ )] but the other two rices cature all necessary market information. We also do not consider the risks and benefits of setting our own odds and waiting for the market to match them, although this should be art of a ractical alication along with an assessment of the reliability of the unter s resumed robabilities. To find the otimal combination of bets {,} we must go back to first rinciles and maximise W, the log of the exected bankroll. At a traditional bookmaker who offers the equivalent of lay odds (usually called a second chance or win or draw market), this is easily solved.
W log(1 ( + (1 + Solving for W log(1 + ( W and ) )log(1 + ) + ) (6) (5) We find that the maximum bankroll growth is achieved when and ( (1 ( ( 1 (1 ) ( ( ) ( 1 (7) rovided both and are ositive. If only one is ositive, the unter should revert to (1). The situation with tax is more comlicated, requiring maximising of the function: W t + (1 + log(1 ( )log(1 + log(1 + ( ( ) H( ) + )) ( )) (8) where H(x) is the Heaviside ste function, indicating that the draw result is only taxed if the lay is larger than the bet.. RESUTS First, consider the simler situation described in equation (7) where a traditional bookmaker offers odds to bet on a win-or-draw market. In general, these rices tend to be unattractive as the bookmaker has built a substantial rofit margin for itself into them; however they are a useful demonstration of the method for the more comlex betting exchange situation. In this examle, City is laying United. Our hyothetical unter with a $1, bankroll believes that the true robabilities are 6% City wins, 15% United wins, and 5% the match will be drawn. The bookmaker is offering $. about City, and a draw or City market at $1., equivalent to laying United at $4.. Using the indeendent formula (1), the Kelly criterion would advise us to bet % or $ on City. The mean rofit is exected to be ($4 6% $) $4. Considering instead the draw or City market, the Kelly criterion advises a bet of size $5 (5%), equivalent to laying United for $15 $5 (1.. The mean rofit is exected to be ($455 85% $5) $6.75. To reconcile these two criteria, we must use formula (7) to find {.1571,.649}. I.e., bet $15.71 on City and simultaneously bet $14.9 on draw or City (equivalent to laying United for $64.9). For an exosure of $5 in this case the same as draw or City alone the unter has increased his exected rofit to $49.64, or 14.% of his outlay. Paradoxically, it should be noted that the unter has effectively taken a rice on the draw outcome that the Kelly criterion would advise has an exected loss. The unter believes that the draw is a 5% rosect, but the difference between the odds of $ (5%) and $1. (77%) is 7%, meaning that he is aying a remium for including this otion in his betting ortfolio. Additionally, if the match results in a draw he will still suffer a net loss of $71.4. However, as the goal is to minimise the risk of longterm ruin, the increased diversification to include the draw is the correct strategy. The etfair version of this roblem in equation (8) must take into account three different ossibilities: i. >, and the net rofit from a draw will be taxed ii. <, so the draw will be a net loser iii., draw no bet The log formula to be maximised is different in all three cases, so the zeroes in the derivatives must be examined for domain relevance and comared with each other..i. The Case > Solving (6) for W t in (8) with the taxed draw gives: 1 1 t[ t + + (1 [( rovided 1 > 1. ( ) + ( ( 1] (9) )]
.ii. The Case < Solving (6) for W t in (8) with the untaxed draw gives: + t + t[ (1 ) [( rovided <. (1 ) + ( 1] (1).iii. The Case ( draw no bet ) y eliminating the draw outcome, (6) is simly solved for the first and third terms of (8) with set to : ( + ) ] (11) This is the common boundary of the other two cases. Examles Returning to our City vs United examle, consider a market where the exchange rices are $.5 about City, and $4.5 for United. For an individual market otion, this is equivalent to the standard bookmaker s rices earlier in this section after a t.5 tax is factored in. Checking the three case functions, case ii is consistent and outerforms case iii, which dominates case i along the entire boundary (the maximum of W t occurs outside of the > domain). The formula recommends values {.1587,.466}. I.e., bet $158.7 on City and simultaneously lay United for $4.66 (lay exosure $149., total exosure $7.67). The exected net rofit on the market is $44., or 14.% of his outlay. His loss in the case of a draw is $115.71. Using the exchange tends to ush the successful strategy in the direction of betting on the favourite as oosed to laying against the underdog. (z >.7). While this method roduces suerficially credible sets of odds, a more recise simulation of soccer should use an acceted modelling aroach such as that recommended by Dixon (1998). Otimal ack/ay From $1, bankroll City Draw United Case ack ay () () $5 $11.7 $1.1 > $9 $498 $ $6.75 $1.5 > $1 $5 $1 $4.65 $1.45 > $41 $9 $7.5 $4.1 $1.6 > $5 $197 $6. $.75 $1.75 > $66 $166 $5. $.5 $1.95 > $79 $14 $4. $.5 $. > $98 $111 $.5 $.5 $.46 $14 $14 $.5 $.5 $.55 $14 $14 $. $.5 $.6 $1 $1 $.9 $. $.9 < $114 $9 $.6 $.5 $.5 < $18 $76 $.4 $.5 $.6 < $19 $67 $. $.5 $4.5 < $15 $57 $. $.5 $4.7 < $169 $47 $1.8 $.7 $5.75 < $19 $6 $1.6 $4.1 $7.6 < $19 $6 $1.4 $5. $11.5 < $59 $15 $1. $7.8 $6 < $9 $6 $1.1 $1.5 $6 < $91 $. Table 1: Effective Strategy for a Range of arkets To examine how the otimal back/lay roortion changes with varying odds, a series of rice sets was generated from the cumulative normal distribution with a fixed z difference of. between the market odds and the unter s robabilities. The central draw otion was given a width of.8 on the z scale to mimic real soccer draw odds in rofessional leagues. For examle, the market centred on z would have City and United both on a rice of $.9 (equivalent to 4.5% robability, or z <.4). The unter s belief is that City has a 46.% chance of winning (z <.1), comared to 4.% for United 4 Figure 1: Otimal ack and ay Fractions for a range of favourite s rices, showing a narrow central zone where draw no bet is the otimal strategy Table 1 shows the otimal strategy for a range of odds, and Figure 1 lots the function, clearly showing a kink for the narrow range of situations where the gambler should attemt not to make a
rofit on the draw, in order to avoid tax on that result. The function using standard bookmaker odds does not dislay such a discontinuity in the derivative. 4. DISCUSSION etting via an exchange has subtle and surrising reercussions for otimal gambling strategy. The way in which a bookmaker rofits is quite different from the situation at an exchange like etfair, which does not set the odds centrally but takes a fraction of each ayout when the market settles. This aer has extended the well-known Kelly criterion to the oular sort of soccer, and shown that use of this advanced strategy leads to a substantial increase in exected rofit greater than % in our City vs United examle. Naturally, unters in the real world ought to revisit the assumtions of section, articularly if the market is not as liquid as they might wish. In ractice, blindly using the full Kelly et fraction would be a risky rollercoaster for most unters, as they do not account for the error in the robabilities generated when they frame the market. A more comlete aroach might use a ayesian distribution of redicted market outcomes, taking into account a variety of known and unknown factors in the game then otimising a more comlex log-utility function. Some gamblers use a fractional Kelly system, which assigns an active bankroll to the Kelly formula that is only a fraction of the full bankroll. This works as an aroximation of the effect of having limited information, while leaving funds available for betting in other markets simultaneously. It is also wise to wager on the conservative side of the Kelly fraction as the enalty function for overbetting is stee. The equations derived here rovide a handy rule of thumb: in general, a unter who wants to back the favourite should ut most of his/her money into that otion, while backing the underdog is usually not as efficient in growing the bankroll as laying against the favourite articularly when the favourite s odds are close to 5/5. reasonable likelihood of neither team winning such as hockey and chess. It is also alicable to sorts such as Australian Rules Football, where the unter must decide how to allocate his funds to a head-to-head (win/loss) or line bet, which are the two most liquid markets. The range of outcomes between zero and the ublished line handica can be treated as the middle outcome in the formulas ublished here. ore generally, future work could extend the methodology to n ublished lines and numerically find the otimal W t for a betting ortfolio that would otentially be sread across a number of them. 6. ACKNOWEDGEENTS The author wishes to thank two anonymous reviewers for their imrovements and ointers to overlooked reference aers. References arnett, T. (11). How much to bet on Video Poker. CHANCE, 4 () Chaman, S.J. (7). The Kelly criterion for sread bets. IA Journal of Alied athematics, 7, 4-51 Dixon,.J. (1998). A irth Process odel for Association Football atches. The Statistician, 47, 5-58 Insley, R., ok,. and Swartz, T.. (4). Issues related to sorts gambling. The Australian and New Zealand Journal of Statistics, 46, 19-. Kelly, J.. (1956). A New Interretation of Information Rate. IRE Transactions on Information Theory,, 185-189 Sydney orning Herald (6). etfair out to conquer the world, February 6, 6. Available at htt://www.smh.com.au/news/business/betfair-out-toconquer-the-world/6//5/1197411158.html Thor, E.O. (1997). The Kelly Criterion in lackjack, Sorts etting, and the Stock arket, in Finding the Edge: athematical and Quantitative Analysis of Gambling (eds. Eadington, W.R. & Cornelius, J.A.) 16-14 Walshaw, Rowan. (1). Kelly Criterion Part a acking and aying ets with etfair. Available at htt://www.aussortsbetting.com/1/7/18/kellycriterion-backing-andlaying-bets-with-betfair/ 5. CONCUSIONS This aer has solved the Kelly criterion for the case of win-draw-loss markets. This is immediately alicable to soccer, cricket, and other sorts with a 5