5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad, Galway carolielych@uigalwayie Prof D Barry Uiversity of Limerick vpa@ulie Abstract A bookmaker takes bets o a two-horse race, attemptig to miimise expected loss over all possible outcomes of the race Profits are cotrolled by maipulatio of customers bettig behaviour; i order to do this, we eed some iformatio about the probability distributio which describes how the customers will bet We examie what iformatio iitial customers bettig behaviour provides about this probability distributio, ad cosider how to use this to estimate the probability distributio for remaiig customers Keywords EM Algorithm, bookmaker, horse race, Markov decisio process Itroductio A bookie takes bets o a cotest for which there are oly two possible outcomes, which we will label as A ad B The bookie wishes to maximise his miimum expected profit purely by maipulatio of customers bettig practice A gambler eters the bookie s shop seekig to place a wager o this cotest Let p deote the gambler s probability that outcome A will occur The bookie quotes odds of O agaist outcome A ad of O agaist outcome B This meas that a wiig wager of oe uit o outcome A produces a retur of O + while a wiig wager of oe uit o outcome B produces a retur of O + Hece a wager o outcome A will be attractive to the gambler if or equivaletly po + p O + θ Similarly a wager o outcome B will be attractive to the gambler if p O + θ or equivaletly p θ The quatities θ ad θ are called the bookie s quoted probabilities for outcome A ad B respectively Hece the strategy for a idividual gambler is simple - he places a wager o ay outcome for which his probability exceeds that quoted by the bookie It should be oted that the quoted probabilities, θ ad θ, described above, are ot probabilities, i the sese that their sum will geerally be greater tha I fact, they may more properly be described as upper probabilities, as defied i [3]It ca be show that it is ever to the advatage of the bookie to have these upper probabilities sum to less tha, as i Lemma of [4] Thus, for the remaider of this paper, we shall assume that θ + θ We also assume that the quoted odds, O ad O, are positive This follows aturally from the requiremet that a wager of uit leads to a retur of O + o outcome A or O + o outcome B, ad the customer is ulikely to wager more tha the expected retur By the defiitios of θ ad θ, this meas that θ ad θ, i tur, are positive These coditios o θ ad θ esure the coherece of the upper probabilities i this case We idealise the bookie s shop by assumig that the bookie sells two types of tickets - oe which guaratees a retur of oe uit should outcome A occur ad costs θ, ad oe which guaratees a retur of oe uit should outcome B occur ad costs θ This avoids sure loss, as the customer s oly optios are to bet o A or B, idividually; to bet o both would esure a loss for the customer, as he would be required to bet a amout θ + θ, greater tha his guarateed retur of We also assume that the bookie kows, before opeig the book, that N customers will cosider a wager o the cotest ad that their probabilities p, p,, p N of outcome A occurrig behave like a radom sample from a probability
distributio Fially, we assume that customers ca buy at most oe of each type of ticket ad that the bookie is free to alter the quoted probabilities after each customer leaves The bookie seeks to maipulate customers bettig behaviour as best suits himself; this depeds, however, o kowig the probability distributio from which the customers probabilities p, p,, p N derive After first cosiderig the optimal procedure whe the distributio is kow, we will cosider how the bookie may estimate this probability distributio usig iformatio derived from customers bettig practices Distributio Kow Assumig the distributio of customers probabilities to be kow, the optimal algorithm for the bookie to follow is the Dyamic Programmig Algorithm, as described i Barry & Hartiga[] This iterative algorithm depeds o kowledge of the customers probability distributio, F, the umber of customers left to come,, ad the curret state of the book, ie the amout of the bookie s profit o outcome A, deoted a, ad o outcome B, deoted b, if the book was closed at that istat, ie o more bets were take Assumig kowledge of these quatities, the algorithm is the give as follows; where P d P d + max θ,θ R a, b a + b + P d { [-F θ ][θ + P } d P d] +F -θ [θ + P d + P d] with P 0 d d ad d a b Here, R a, b deotes the expected value of the bookie s fial miimum profit betwee both outcomes The algorithm gives the bookie a method for derivig optimal quoted probabilities for the ext customer, give the curret state of the book, customers left to go ad F kow The above equatio for P d describes how it depeds o the previous value, P d, the adds a term, maximised over θ ad θ, which describes the profit accruig if the customer bets o A, with probability F θ, ad if the bet is o B - with probability F θ ; the oly two possible bets 3 Strategy for Distributio Ukow Havig determied a strategy for F kow, we must cosider how to estimate F whe it is ukow We subdivide the iterval [0,] ito r subitervals of equal width - the choice of the value of r will be discussed i Sectio 4 We the estimate F by meas of a histogram, with r itervals For each of these r itervals, the height of the histogram will be determied by the probability assiged to that iterval, π j This probability will be determied by the bettig behaviour of the customers, as described hereafter F θ may the be determied by the formula F θ rθπ 0 θ r π + rθ π r θ r π + π + rθ π 3 r θ 3 r j i π j i + rθ jπ j+ r θ j+ r r i π r i + rθ r + π r r θ 3 Estimatio of F This ivolves the EM Algorithm; we have a estimatio, ad a maximisatio, step 3 Estimatio Each customer s bettig patter gives us iformatio about their value of p, as follows; Bet o Horse A θ p Bet o Horse B 0 p θ No Bet θ < p < θ We deote the lower limit of the rage i which p falls by a k for customer k ad the upper limit by a k such that a k a k We also deote the lower ad upper limit of each of the subitervals of [0,], I j, by [L j, R j ], with R j L j+ We have a idicator fuctio, X jk, defied as follows; X jk { p [Lj, R j ] 0 otherwise I this case, the log likelihood fuctio is give by l k j X jk log π j Give the customer s behaviour, we have a rage for the customer s probability - ie a k p a k Let us call this iformatio Y k
We will seek to maximize the Expected value of the log likelihood, give this iformatio, ie El Y k j EX jk Y k log π j EX jk Y k P X jk Y k We have P p [L j, R j ] p [a k, a k ] P p [L j, R j ] [a k, a k ] P p [a k, ak ] P p [L j, R j ] [a k, a k ] r i P p [L i, R i ] [a k, ak ] l jk P p [L j, R j ] [a k, a k ] π j R j L j where l jk is the legth of [L j, R j ] [a k, a k ] ad is give by 0 R j a k R j a k L j a k R j a k a l jk k a k L j a k a k R j R j L j a k L j R j a k a k L j a k L j a k R j 0 L j a k 3 Maximisatio Next, we seek to maximise the expected value of the log likelihood fuctio The Maximum Likelihood Estimate for π j is give by N k ˆπ j EX jk Y k N Each of the subitervals of [0,] was assiged a iitial probability, πj For simplicity, this iitial probability was the same for each subiterval, assumig the Uiform distributio, so that, with r subitervals, the iitial values of πj are give by πj πj r, j j This iitial probability was the updated by observig each customer s behaviour As will be see i this, ad the ext, subsectio, we will ow divide our customers ito three groups; the very first will be used to iitialise the iformatio matrix, the secod group will be used for the purpose of maximisig the iformatio we may obtai, leavig us with the third ad fial group for maximisig profit, oce F has bee satisfactorily estimated Oe of the questios with which we will be cocered is how may customers should be allocated to each group 3 Early Customers For the first few customers, the odds are chose so as to maximise the iformatio obtaied We derive the iformatio matrix, I, usig the formula [ ] I ij E l ˆπ i ˆπ j where l is the log likelihood, defied as before As described i the previous sectio, we decided to divide the iterval [0,] ito a umber of subitervals, each of which was assiged a probability, π j, which was updated by observatio of customers behaviour As before, we may express the log likelihood fuctio as l k j [ r k X jk log π j j X jk log π j + X rk log r N X jk log π j j k k N + X rk log as π r r j π j Hece, we fid that ad l π j k [ Xjk π j r π j j ] X rk r j π j r ] π j j N l k X N jk k π i π j πj δ ij X rk r j π j, where δ ij { if i j, 0 otherwise} Thus, we have [ ] E l Nδ ij N + π i π j π j r j π j The etries are added for each successive customer Havig calculated the iformatio matrix, we use it to choose the odds for each of the customers before F
is determied Firstly, both θ ad θ are set at r, for coveiece of programmig The iformatio matrix is recalculated for each combiatio of θ ad θ, each beig icremeted i steps of r Fially, that combiatio of odds which maximises the determiat of the iformatio matrix is used for the ext customer, so log as it satisfies the coditio θ + θ I practice, this coditio was satisfied by every optimal combiatio of odds This procedure is repeated for each of the customers i tur The optimal umber of customers used to estimate F is foud by ispectio This procedure is described later After each of these customers bets, our estimate of F is updated usig the EM Algorithm, as described previously Fially, we must iitialise the iformatio matrix 33 Iitialisatio 33 Odds for the Iitial Customers I order to iitialise the iformatio matrix, the theta-values for the first few customers are chose accordig to the followig pla; If we divide the iterval [0,] ito r equally-spaced subitervals, placig the theta-values o the divisios of these subitervals will give us precise iformatio about the distributio of probability withi these subitervals The optimal value of r is foud by ispectio, ad is described subsequetly We do ot eed to set either a theta-value equal to, which guaratees o bets, or equal to 0, which guaratees a bet from ay customer Bearig these poits i mid, we set the theta-values for the first customer as θ θ r r We the take each theta-value dow by a value r i tur for each of the ext few customers 33 No of Customers i this Group As customers bet o Horse A with probability F θ, the value of θ will provide us with iformatio about the probabilities of the subitervals above θ Thus, this value provides us with iformatio about the subitervals at the upper ed of the iterval [0,] Similarly, customers bet o Horse B with probability F θ Thus, the value of θ provides us with iformatio about the probabilities of the subitervals below θ, ad thus provides us with iformatio about the subitervals at the lower ed of the iterval [0,] So the theta-values for the very first customer tell us somethig about the probability i the first, ad last, subitervals Each successive customer s set of theta-values tells us about a additioal subiterval Fially, we oly eed iformatio about r subitervals, as we kow that the probabilities sum to i total Altogether, this tells us that we eed r customers i the first group, to iitialise the iformatio matrix 4 Choice of No of Subitervals ad No of Customers to Use i Estimatio Firstly, as discussed previously, we divide the iterval [0,] ito r subitervals, to each of which is assiged a probability, so as to estimate F We ow eed to determie the optimal umber of subitervals, ad also the optimal umber of customers, as a percetage of the total assumed kow, whose odds we should use i order to maximise the iformatio matrix, as described i the previous sectio This umber is i additio to the r customers used i the begiig to iitialise the iformatio matrix These were estimated simultaeously, by calculatig profits for the same Dyamic Programmig profit fuctio for a variety of combiatios of umbers of subitervals ad umbers of customers used i estimatio of F, ad choosig the combiatio which proved best overall The state of the book for a particular outcome deotes the bookie s profit if that outcome occurs Let A deote the state of the book for outcome A, ad B the state of the book for outcome B, whe of the N customers remai I the strategy which our bookie uses, we have A B θ which are chose to maximise ad θ A B, mi{e[a 0 A a, B b], E[B 0 A a, B b]}
This gives the fuctio to be maximised as mi { d θ [ F θ ] + θ F θ, d } + θ [ F θ ] θ F θ, where d a b This is the objective fuctio which was used i the simulatio study, a summary of whose results follows It assumes that the quoted probabilities remai costat for all remaiig customers, ad calculates the fial expected profit if A occurs as the icome from those customers who bet a amout θ o B, with probability F θ, less the outgoig retur of uit to those who bet a amout θ o A, with probability F θ A similar calculatio determies the fial expected profit if B occurs The algorithm ivolves the calculatio of the miimum of these two expected fial profits, give the curret state of the book It will be oted that this is a differet algorithm to the optimal oe discussed i Sectio ; as discussed i Barry & Hartiga[], this algorithm provides a easier method for calculatio of the quoted probabilities, without excessive pealty i terms of the bookie s profit The measure of which combiatio of ad proved best was provided by obtaiig the mea profit, over fifty replicatios i each case, for each idividual combiatio The differece betwee each of these values ad the maximum value over all combiatios was the obtaied for each distributio This was repeated for each of N 00, 500 ad 500 customers, ad for each of five distributios; amely, Uiform ie F θ θ We foud the maximum differece, for each combiatio of umber of subitervals ad percetage of customers, over the five distributios This represets the maximum loss per customer Thus, we use the combiatio which provides the smallest value of maximum loss The maximum loss per customer over all distributios is show i the followig tables Maximum Differece over Five Distributios % of Customers to Estimate F Itervals 0 007 0075 0074 00358 00357 00356 3 009 006 0059 4 0059 008 0095 5 00396 00364 0036 6 003305 003005 0095 7 00339 003453 003369 8 00353 003939 00378 0 003357 003355 0036 N00 % of Customers to Estimate F Itervals 0 005 005 005 0009 0009 0009 3 00049 00054 00054 4 00035 00034 00036 5 00044 00057 00066 6 00049 00047 0005 7 0005 0005 00048 8 0004 0004 0004 0 0004 0004 0004 3 4 5 F θ 0 0 θ 8 8 θ 5 8 5 8 θ θ 8 0 0 θ 4 F 3θ θ 4 θ 3 4 3 4 θ { F 4θ F 5θ 0 0 θ 3 4 4θ 3 3 4 θ θ 0 θ 4 + θ 3 4 4 θ 3 4 3 4 θ N500 % of Customers to Estimate F Itervals 0 0005 00049 00049 0003 0003 0003 3 0006 0008 0008 4 0005 0005 0005 5 0005 0009 000 6 0009 0009 0009 7 0009 000 0007 8 0005 0005 0005 0 0005 0004 0004 N500
From these tables, we may see that the optimal % of customers for maximisatio of the iformatio matrix i all cases is %; higher percetages are ot show here, as they led to greater loss We also see that the optimal umber of subitervals ito which to divide the iterval [0,] is 4 for N 00 ad 500, ad 7 itervals for N 500 The optimal combiatio is that which miimises the differece show i the above tables We may further see from these tables, however, that the maximum differece over all distributios decreases, for each combiatio, as the total umber of customers icreases- demostratig that, for larger umbers of customers, there is reduced loss i usig a o-optimal combiatio 5 Summary ad Coclusios I summary, the method described i this paper provides us with a meas of estimatig the overall distributio of customers probabilities, based solely o the bettig practices of relatively few iitial customers, which provide us with iterval estimates of these probabilities This proves a highly useful tool whe distributios are ukow Further work o this topic might iclude the examiatio of whether it is possible to icorporate a elemet of profit maximisatio ito the stage where F is beig determied Aother obvious extesio of the work is to the case where there are more tha two possible outcomes; however, each extra outcome leads to multiple extra possibilities for the customer, who may bet o ay idividual outcome, or possibly o a combiatio of them As well as leadig to a much more complicated model, this gives rise to the possibility of icoherece, ad to the icurrece of sure loss; care eeds to be take i this sceario [4] Bellma, R 967 b Itroductio to the Mathematical Theory of Cotrol Processes Academic Press, New York [5] Blackwell, D 976 The Stochastic Processes of Borel Gamblig ad Dyamic Programmig Aals of Statistics 4 370-374 [6] Dubis, L E & Savage, L J 965 How to Gamble if you Must McGraw-Hill, New York [7] Heery, R J 984 A Extreme-value Model for Predictig the Results of Horse Races Appl Statist 33 5-33 [8] Heery, R J 985 O the Average Probability of losig Bets o Horses with give Startig Price Odds J R Statist Soc A 48 34-349 [9] Hoerl, A E & Falli, H K 974 Reliability of subjective evaluatios i a high icetive situatio J R Statist Soc A 37 7-30 [0] Plackett, R L 975 The Aalysis of Permutatios Appl Statist 4 93-0 [] Rieder, U 976 O optimal policies ad martigales i dyamic programmig J Appl Prob 3 507-58 [] Whittle, P 98 Optimizatio Over Time: Dyamic Programmig ad Stochastic Cotrol Academic Press, New York [3] Walley, P 99 Statistical Reasoig with Imprecise Probabilities Chapma ad Hall [4] Barry, D & Lych, C 006 The Miimax Bookie: The Two-Horse Case Adv i Appl Prob Vol 38 No 4 Ackowledgemets This paper is based o part of the first author s PhD thesis, completed at the Uiversity of Limerick uder the supervisio of Prof D Barry, whose help ad support have prove ivaluable Refereces [] Aoki, M 967 Optimizatio of Stochastic Systems Academic Press, New York [] Barry, D & Hartiga, J A 996 The Miimax Bookie J Appl Prob 33 093-07 [3] Bellma, R 957 a Dyamic Programmig Priceto Uiversity Press, Priceto, NJ