How do bookmakers (or FdJ 1 ) ALWAYS manage to win?

Transcription

1 How do bookakers (or FdJ ALWAYS aage to w? Itroducto otatos & varables Bookaker's beeft eected value 4 4 Bookaker's strateges5 4 The hoest bookaker 6 4 "real lfe" bookaker 6 4 La FdJ 8 5 How ca we estate the bookaker's beeft?8 5 Marg estato 8 5 Eales (etracted fro actual odds 9 FdJ stads for Fraçase des Jeu, whch s the oly bookaker authorsed Frace Ths coay rus a gae ("Cote & Match" whch retty looks lke bookakers gaes

2 Itroducto We suose of course that the uesto relates to soccer lay betwee two teas, ad that outcoes are ossble : hoe w, draw or away w These results are usually referred to as,, (frech otato whch stads for,, teratoal otato I the les to coe, we wll suose that we address OE secal atch Ths assuto s ecessary to ake the elaatos ore sle (though stll tough, but the reasog etraolates easly to ay atch We wll thus derve the aalyss that bookakers ake (or should ake our sese to always w agast uters To aswer the uesto, we eed varous forato Frst, the bookaker ust have hs ow dea of results robabltes : utg s after all a cotest betwee the bookaker (who wats to r off layers ad the layers who do ot agree wth that urose, cosder theselves sarter tha the bookaker (or try to be, ad of course wat to get ther oey back at least ad eve ore f ossble Secod, t s ecessary to kow the bets reartto aog,, ossbltes ot oly the uber of the (ths s of course obvous but also the aout of oey reartto For eale, 50% of layers ay ut sall oey o a hoe w, whereas 0% ut bg oey o a dfferet result ( or Fally, ths last oe wll gather the ost oey, ad that's what effectvely couts for the bookaker whe t coes to the evaluato of hs eargs chaces Ths beg sad, we wll ow address the core uesto whch s bookaker trck to fool layers It wll reure soe atheatcal develoets (we do't kow how to ake t sler So the webaster would wat to war readers that fro ow o the tet ay be aful for soe of the Most courageous of you ca kee o readg (ake sure to have oe asr tube at had

3 otatos & varables Kow otatos eag By layers by bookakers by FdJ Total uber of uters o yes o uber of bets o result (, ou Average euro bet o result o yes o o yes o Bookaker's odds yes yes yes M α Probablty of result estated by the bookaker Probablty of result estated by the uters Total aout of euros bet o the atch Average bet er uter over all results (,, The bookaker's ta o bets (hs arg! o yes yes dffcult yes o o yes o o yes o o yes yes Recall : the bookaker odds are the ubers used to ultly the uter's bet to calculate uter's eargs f he redcted the good result Relatos betwee varables : varable Is eual to varable Is eual to M

4 4 coets : To decde f a varable s kow or ot recedg table, we looked for each kd of layer (bookaker, uter, FdJ f they have a ea, DURIG BETTIG PERIOD OF TIME, to kow ths varable It haes that both layer ad FdJ who ake ther bets or f ther odds before the bettg erod are soewhat eve The uter eve has a slght advatage over FdJ : f he s sart, he wll wat for the last utes of bettg erod to ake hs bet order to have as uch forato as ossble Bookakers o the cotrary ca eraetly adjust ther odds utl the bettg erod edg So they have a lot of very foratve data (esecally the aouts of bets er each result whch gves the a huge advatage over uters FdJ however has access to all data after bettg erod ad ca thus use ths eerece for et bettg days The gae s thus uch ufar tha t could see at frst sght FdJ has also the ossblty to cacel the bet o ay atch f t aears that ts rsks are too hgh (sae for bookakers, but of course deed to uters Accordg to ths frst aalyss o access to bettg data, bookakers have already ay eas to bas the gae ther favor But that's ot the oly ea they have avalable They have above all the oortuty to decde the odds levels, ad ths s a treedous weao to ake the balace shft ther way That's what we try to deostrate et aragrahs Bookaker's beeft eected value Ths value corresods to the average beeft the bookaker could ake f the atch cosdered was layed ay tes I fact, the atch s oly layed oce, but as there are ay atches, the coutato of the "eected value" s evertheless relevat Ths "eected value" s by coveto wrtte E (, where s the robablstc varable we look for, aely the bookaker's beeft here For stace, the eected value of the result of a roll of a 6 faces dce s 5; ths does't ea you'll get 5 whe you roll the dce (of course, or you have really werd dce but that you'll get ths average result f you roll the dce ay tes Frst, let's look at bookaker's eargs for each gae result : Match result Bookaker's earg (evetually egatve, e loss ( ( ( As the bookaker's redcto for,, results are,,, hs beeft eected value s fally :

5 5 ( ( ( ( ( ( ( E Ths ca be rewrtte usg, total uber of bets o the atch : ( ( ( ( E ( ( ( [ ] ( ( ( ( E ( E ad fally, as ( E [] Ths s the ost geeral forula for bookaker's beeft eected value, as t does ot clude aroatos or hyothess of ay kd 4 Bookaker's strateges I order to slfy the deostrato, we wll assue that average aouts of euros bet o each result are the sae The bookaker ca have the actual data, s ot oblged to go through ths aroato, ad ca udate the followg reasog as ofte as he wats We thus have : The relato [] becoes the a lttle sler : ( E [']

6 6 (because, as before, The bookaker, wth ths relato, ca ow develo hs strateges 4 The hoest bookaker Of course, ths varety of bookaker does ot est, but hs fcttous estece wll hel us ela further how "real lfe bookakers" always aage to w So for ths good ol' vrtual hlathrost, othg atters but akg the gae betwee h ad uters far To reach ths goal, he eeds : ( E 0 There's oly oe way to obta ths result, whch s to choose the odds sartly Whe lookg to relato ['], our ave bookaker ca observe that he has two obvous choces : or I the frst case, the relato ['] becoes fact : E( 0 whch s the objectve that our bookaker has fed to hself I the secod case, the result s the sae because et act syetrcally relato ['] But the frst soluto s uch ore terestg for the bookaker because he does't eve have to ake hs ow bets Effectvely, whatever the bookaker's values, eve f they are very badly estated, whe choosg the fal earg wll be the sae (w or loss eual to zero 4 "real lfe" bookaker Ths bookaker, that everyoe kows, has two secal characterstcs that ake h dffer fro the revous sece : he ust f hs odds BEFORE kowg the uters bets (so he does't kow the ad ust estate the he strogly whshes to have a ostve beeft "eected value" To solve hs frst roble, he as ot uch alteratves : all he ca do s to suose the uters to be as sart (or clueless as he s, ad assue that at the ed of bettg erod he wll have

7 7 (evertheless, he wll have the oortuty to sca the evoluto of ['] utl the ed, ad to udate the so as to have E( rea ostve Wth ths assuto, hs beeft "eected value" ['] becoes : E ( [ ] ad f he was the "hoest" bookaker, he would set But he wll ot, because two thgs bother h uch : Frst, he has o guaratee that uters wll have the sae redctos as h Ths gves brth to a very ebarrassg ucertaty o, because t ca ake the relato [''] ot relevat, ad relato ['] (whch s to be accouted for the ca be egatve Hece uleasat loss vew! Secod he s ot retty sure of hs redctos ether, ad would lke to decrease ths rsk So he wll "work" hs odds so as to guaratee a ostve earg Whch eas he wll odfy the order to get a arg o E( Fro ow o, the true bookakers ethods ca oly be guesses They ca, for eale, odfy the three odds o (,, the sae way ad calculate : α [] ( where α s chose as a fucto of the arg the bookakers wats for hself He wll have the oortuty durg the bettg erod to verfy that the devate too uch fro each other If they do, he wll be able to "re-coute" the odds accordg to α [ ] ( ad the do ot order to lower hs rsks (ths s the case, because the are kow eactly ad recsely Ths wll effectvely guaratee h a ostve beeft wth u rsk

8 8 4 Beeft "eected value" uder hyothess [] : ( E α α ( ( E( α α ( E ( α where α s the bookaker's ta o uters' bets [] 4 Beeft "eected value" uder hyothess ['] The bookaker ca greatly reduce hs rsks, as the are erfectly kow Hs beeft "eected value" ca the be calculated wth relato ['] whch does ot ake ay assuto o uters' bets ad s thus uch ore recse Ths eected value the wrtes : ( E α ( whch gves the sae eresso [] as for hyothess [], BUT wthout ay hyothess o Ths cosderably reduces the bookaker's rsk I fact, the bookaker ca refe eve ore hs strategy by usg relato [] whch DOES OT IMPLY AY HYPOTHESIS AT ALL He ca thus coute hs α args wth o error ad roose very attractve odds 4 La FdJ Ths bookaker ever udate hs odds, ad s oblged to lay accordg to odel [] Hs rsks are hgher, whch elas for a art why the odds are less attractve tha other ole bookakers 5 How ca we estate the bookaker's beeft? 5 Marg estato Let's suose that the bookaker uses relato [] As we kow hs odds, we ca easly fer the "bet ta ercetage" (ad thus access to hs "beeft eected value"

9 9 To do so, we ust aga assue a ufor average bet over all results (,, Fro [] we deduce α ad fro, we the get the forula to α calculate (ukow fro the (kow : α [4] 5 Eales (etracted fro actual odds Odds Frace Lgue, Jauary d to rd 005 FdJ : teas Metz Marselle 7 65 St Etee PSG Basta ce Cae Auerre 7 65 Istres Strasbourg Moaco Les 5 5 Rees Ajacco Sochau Bordeau Toulouse ates Llle Lyo 4 55 Bookaker : teas Metz Marselle St Etee PSG Basta ce Cae Auerre Istres Strasbourg Moaco Les Rees Ajacco Sochau Bordeau Toulouse ates 8 45 Llle Lyo We use relato [4] to get each bookaker's beeft estatos, whch gves atch er atch the two followg tables:

10 0 FdJ : Metz Marselle 99% St Etee PSG 99% Basta ce 99% Cae Auerre 99% Istres Strasbourg 99% Moaco Les 00% Rees Ajacco 00% Sochau Bordeau 97% Toulouse ates 99% Llle Lyo 96% Bookaker Metz Marselle 97% St Etee PSG 97% Basta ce 99% Cae Auerre 0% Istres Strasbourg 96% Moaco Les 99% Rees Ajacco 00% Sochau Bordeau 99% Toulouse ates 99% Llle Lyo 0% We have wth ths eale a guess : of the arg that ole bookakers ake o uters bets (>0%, whch s eorous seakg of reveue o oey you do't ow of the "ucertaty bous" that FdJ grats to herself (0% wrt bookaker, whch eas a cofortable 0%!!!