# Estimating Probability Distributions by Observing Betting Practices

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3 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

5 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 N00 % of Customers to Estimate F Itervals F θ 0 0 θ 8 8 θ θ θ θ 4 F 3θ θ 4 θ θ { F 4θ F 5θ 0 0 θ 3 4 4θ θ θ 0 θ 4 + θ θ θ N500 % of Customers to Estimate F Itervals N500

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