Keskustelualoitteita #60 Joensuun yliopisto, Taloustieteet

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1 Keskustelualotteta #6 Joensuun ylopsto, Talousteteet Unfar games, subectve probabltes, and favourte-longsot bas n Fnns orse track race track Nko Suonen Mkael Lnden ISBN ISSN no 6

2 UNFAIR GAMES, SUBJECTIVE PROBABILITIES, AND FAVOURITE-LONGSHOT BIAS IN FINNISH HORSE RACE TRACK Nko Suonen*, & Mkael Lnden * December 28 Abstract Te paper dscusses decson makng under rsk and uncertanty n te framework of gamblng. Especally, te well-known anomaly of favourte-longsot bas s consdered. We argue tat gamblng beavour can be seen ust as a consumpton type actvty. Tus, gamblng beavour s more tan ust a rsky coce, and t s reasonable to assume tat te prce of gamblng s te take-out rate or te excepted loss. We llustrate wt some elementary gambles tat te gamblng markets consttute an envronment were rsky coces can be measured only wt te probabltes. A typcal gamblng beavour (e.g. orse bettng) s modelled by te approac tat ncludes attractveness of rsk, subectve probablty, and utlty of money. We tested te favourte-longsot bas wt a large dataset from Fnns orse race tracks tat ncludes more tan 95, races over a seven-year perod. Te emprcal results ndcate tat Fnns gamblers beavour s based: tey gamble too less on favourtes and too muc on longsots. Te results confrm te unversalty of te favourte-longsot bas found n te Western Europe, Australa and te USA. Keywords: Rsk, favourte-longsot bas, subectve probablty, attractveness of rsk ) Correspondng autor. E-mal: nko.suonen@oensuu.f *) Economcs and Busness Admnstraton Unversty of Joensuu, Fnland We tank Mka Kortelanen, Jan Saastamonen, and Tuukka Saarmaa for ter elp n andlng data, as well as for valuable comments and suggestons. Suonen gratefully acknowledges te fnancal support from Yrö Jansson Foundaton and Fnns Cultural Foundaton.

3 . Introducton Wy gamble? In economcs, durng te past ffty years decson makng under rsk as been modelled matematcally wtn te framework of te expected utlty teory (EUT). However, teory cannot unambguously explan te ndvdual s beavour n te gamblng markets: t s not ratonal tat a rsk aversve person partcpates n gambles, were te expected return s not postve. In spte of ts, gamblng markets are ncreasng all te tme. Tus, gamblng markets are nterestng and tey provde coce based data from an autentc envronment. Te data can be used to analyze and test decson makng models and teores. Moreover, nvestgatng tese markets gves nformaton for a wder descrptve approac of decson makng under rsk tan EUT can gve. Altoug te gamblng markets are an nterestng topc for a decson teorst, tey are also a remarkable entertanment busness tself. Terefore, tere are many good reasons to analyze markets from te bettor s pont of vew (e.g. consumpton actvty and enoyment), as well as from te bookmaker s pont of vew (e.g. competton and take-out rate). On te oter and, te gamblng markets ave also negatve externaltes suc as gamblng addctons. Te understandng of operaton of market and bettors beavour gve us a tool to analyze polcy alternatves, market regulaton, and ncentve desgns. In ts paper, our frst am s to brefly dscuss te most famlar rsk teores (EUT and te prospect teory) n te contexts of gamblng beavour. Second, we try to model te gamblng beavour and te evdent game anomales by a novel model tat ncludes elementary factors of decson makng, suc as attractveness of rsk, subectve probablty, and utlty of money or wealt. Fnally, we test te unversalty of te well-known favourte-longsot bas wt a large data from Fnns orse race tracks. Te paper proceeds as follows. Secton 2 dscusses sortly te development of decson makng teores. In Secton 3, we scrutnze te famlar matematcal propertes of an elementary gamble. Next, te paper consders bettor s utlty and prce of gamblng. Secton 5 presents te basc propertes of te most famlar fallaces n gamblng markets. In Secton 6 we derve a model tat explans gamblng wt uncertanty functon. Fnally, n Secton 7, we test emprcally favourtelongsot bas. Te last secton concludes te paper.

4 TP PT For 2. Sort revew on decson makng under rsk Snce 94s, decson makng under rsk as been modelled by EUT by von Neuman & Morgenstern (944). Te man aspects of EUT are preferences and axoms, wc determne a ratonal decson under rsk. However, EUT as been under eavy attacks snce te early 95s. Often te crtcsm as motvated by experments n wc decson makers often systemcally volate te ratonalty assumptons. Te most famous anomaly s Allas paradox (Allas 953). Terefore, new descrptve teores based on evdence ave been developedtp PT. Contrary to expectatons, none of tese new teores as replaced EUT. However, te prospect teory by Kaneman & Tversky (992, 979) as been used n economcs n some extenson. For example, Kanbur et al (27) and De Meza & Webb (24) used te prospect teory to model a basc prncple-agent problem. Also te gamblng markets ave been explaned by prospect teory (e.g. Jullen & Salané 2). 3. Basc propertes of elementary gamble Consder a Bernoull game n wc te fee of te gamble s one euro and te wnnng probablty s p (,). Assume tat te return s a two valued random varable Y wt PY ( = r) = p and PY ( = ) = p. Te game s far wen te expected return s zero,.e. EY [ ] = pr ( f ) + ( p)( ) = prf = rf =, () p were f represents te far game. In te far game, te probablty and te return depend on eac oter: wen te probablty decreases, te return ncreases and vce versa. Furtermore, assume tat te gambler can coose one bet tcket among several bet tckets k. In ts case, we can wrte te prospect or gamble n a followng vector form as q = ( r, p ;...; r, p ), (2) f f kf k were probabltes k p = and =, 2,..., k. = a revew, see Starmer (2). 2

5 TP PT Naturally, s Tus, f te wnnng probablty pbb very small, ten te return can be very g. Now we can defne for some r f n prospect q f = ( rf, p;...; rkf, pk) tat entals tat r f =, wen p. (3) p However, most gambles are not far for te gambler. We can defne te unfar game for q were expected return s negatve (.e. expected loss), and constant EY [ ] = pr ( u ) + ( p)( ) = τ ru = ( τ ), (4) p were u represents te unfar game. As a consequence, we can nterpret τ as an organzer s (te 2 bookmaker) take-out rate for a one-euro bettp PT. Let us defne odd O to every probablty p n te case of unfar game. We know tat te odds are nverses of probabltes tat depend on te take-out rate, tus ( τ ) = O p. (5) By Eq. (4), return can be represented by r u = O. (6) Moreover, from Eq. (5) = ( τ ) = ( τ ) = O k p k k = k =. (7) = O 2 te expected return or loss s not a rate. However for a one-euro bet tey are equvalent. Terefore we proceed to use te term expected loss n parallel wt te term take-out rate. 3

6 In oter words, f we ave nformaton on te odds, we can calculate te take-out rate and te wnnng probabltes. As a result, only varables tat affect a gambler s decson are te wnnng probabltes of dfferent gamble alternatves and take-out rate. In te case of unfar game t s not ratonal to gamble because eventually gambler wll lose s or er money 3. However te decson makers usually do not beave lke matematcal macnes. Typcally return presents only one dmenson of gamng. Terefore we consder also te utlty of gamblng. 4. Prce and utlty of gamblng Gambler s beavour as been explaned and modelled bot by normatve and descrptve teores. For example, Fredman & Savage (948) argued tat utlty functon s not globally concave and tere are some convex segments. Fsburn (98) and Decdue et al (24) buld a model n wc gamblng decsons are determned by two utlty functons. However tese models ave not found an unambguous answer to gamblng beavour. Usually, we understand rsk as unpleasant and we try to avod t. Despte ts, te gamblng markets do well. We can argue tat te utlty of gamblng s consumpton smlarly to any oter actvty tat as notng to do wt rsk. For example, Asc & Quandt (99) state as follows: A day at te racetrack or casno may be smply a consumpton type actvty for wc one s prepared to pay a prce tat ncludes te track take or te ouse advantage. (p. 423). Let assume tat gamblng s entertanment as Asc & Quandt (99) stated above. Now, wat s te prce of te gamblng? In te prevous secton, te fee of te elementary gamble was one euro. However, t s a qute obvous tat te fee s not te correct value or prce of te gamble because ) gambles were te gamblers cannot wn anytng at all are very rare, 2) casnos do not compete wt decreasng te fee, and 3) gamblers try to fnd te bookmaker tat offers te smallest take-out rate. Tus, te take-out (or expected loss) must see as a varable part of te prce of te gamble and bookmakers or casnos compete wt t. More formally te bookmaker s competton s sown n Appendx. 3 We assume tat gamblng cas s fnte. 4

7 TP TP From te gamblers pont of vew, we can sum up te prce of te gamble for a one euro bet and te results from te Secton 3: () τ =, te gamble s free of carge for te gambler, () τ <, te gambler s pad for te gamblng and () τ >, te gambler as to pay for te gamblng. As a result, wen te gamblers partcpate n te gamble tat as a negatve expected return, t s smlar consumpton smlarly as buyng a concert tcket. If te take-out s too g for te enoyment, te gambler wll not partcpate. Next, we consder te gamblng and consumpton from EUT pont of vew. Let us assume tat te 4 ndvdual s a ratonal rsk aversve person and tres to maxmze er utltytp PT. Tus, er utlty functon s globally concave, u (.) <. Moreover, we assume tat te gamble s unfar and te expected loss for a one euro bet s τ. Now, we can wrte te expected utlty as ( u ) V( q ) = pu r + W + ( p) u( + W) < u( W τ ), (8) u were W s gambler s ntal wealt level. On te rgt sde n Eq. (8) te expected loss or te takeout can be nterpreted as a cost of gamblng tat te gambler s prepared to pay for te enoyment and, tus, te alternatve decson of non-gamblng not only depends on te ntal wealt level but 5 also on te cost of gamblngtp PT. We can see tat ratonal gamblers do not partcpate n te gamble. To llustrate te fact assume tat te take-out rate s τ =.2, te gamblng cas s W =, and te utlty functon for a rsk aversve person s ux ( ) = ln(.). Because unfar game s returns are dependent on probabltes and take-out rate, a prospect tat can be wrtten n form q = ( r, p ;...; r, p ). Ts can be, for nstance, a par-mutuel orse race. Te gambler wll u u ku k coose only one orse from k orses and e nvests one euro bet for te orse. Terefore, te expected utlty of gamblng, wen te gambler cooses orse, can be obtaned as 4 PTWe ntend tat a ratonal ndvdual acts as orderng, transtvty, contnuty, and ndependence axoms assume. uw ( τ ) s te utlty from te wealt adusted wt te expected loss. 5 PT 5

8 TP PT In ( ) V( q ) = p ln r + W + ( p )ln( + W) < ln( W τ ). (9) u u Now we can analyze ow te expected utlty beaves wen probabltes canges. In Fgure s sown te expected utlty n dfferent probabltes (orses). u(w) u(w-tau) Utlty u(w-tau) u(w) EU.5 Probablty Fgure. Expected utlty for te rsk-aversve person n te unfar game. We can clearly see tat te utlty of a rsk-aversve person does not exceed te constant, uw ( τ ), by any values of probabltes. However, a rsk premum s smaller n contrast to te case were te gambler s not prepared to pay te prce of gamblng, uw ( ). Moreover, wen te take-out rate s negatve ( τ < ) t does not nfluence te gambler s decson, because t wegts te bot sdes of 6 Eq. (8).TP PT Fgure also sows tat te gambler dslkes partcularly gambles wc ave a g return and a low wnnng probablty as s assumed n EUT. Tus, even wen te prce of gamblng s te expected loss, te gambler naturally never partcpates n te gamble. However, te gamblng ouses are alve and well. 6 fact, as we assume tat te prce of gamblng s te take-out, t transforms te unfar game nto te far game for te gambler for all values τ <. 6

9 Te prospect teory can be seen to dverge from EUT at least n te followng ways. A) Te utlty or te value functon as tree man caracterstcs. a) defned on te bass of devatons from te reference pont, b) concave for gans and convex for losses, and c) steeper n te doman of losses. More formally, te functon can be presented by a two-part power functon α x wen x, vx ( ) = α λ( x) wen x <, () were λ ndcates loss averson and α s a sape parameter of utlty. Kaneman & Tversky (992) estmated values of te parameters. Te medan exponent of te value functon was.88 for bot losses and gans, n accord wt dmnsng senstvty. Te medan λ was 2.25 (loss averson). B) Te probabltes are wegted wt subectve probablty functon tat over-wegts low probabltes and under-wegts g probabltes. Te wegtng functon s a one-parameter functon γ p w( p) =, () γ γ / γ ( p + ( p) ) were γ was a sape parameter tat s dfferent for gans and losses. Estmated medan value of γ for gans was.6 and for losses t was.69 (Kaneman & Tversky 992). Consequently, n te frame of te prospect teory t s possble tat te ndvdual partcpates te gamble, partcularly wen te wnnng probabltes are low (.e. over-wegted). However, te decson depends only on te probablty and take-out rate wen r ( ) u = τ, and te values of p parameters of wegtng and value functons. Altoug we can explan te gamblng markets by te prospect teory, te loss averson wegts affect also te decson. 7

10 5. Anomales n gamblng markets Te gambler s fallacy or te Monte Carlo fallacy s a belef n negatve autocorrelaton for a nonautocorrelated random sequence. For example, we trow repeatedly a far con. After tree eads, we beleve tat te next trow wll be tals wt a probablty more tan.5. Ts bas can be found, for nstance, n casnos. In many casnos, tere are electronc dsplays besde te roulette table tat sows te prevous outcomes of te weel. Many gamblers make ter coces based on te dsplay nformaton. However, a roulette weel does not ave a memory. Consecutve numbers n te game are ndependent of eac oter and te lkelood of every number s te same n te next turn. Te anomaly s wdely documented n an emprcal researc. For nstance, Clotfelter & Cook (99) notced tat n te lotto gamble, gamblers rarely cose te number wc as rolled up n te prevous round. Croson & Sundal (25) conducted te feld experment n casnos (roulette) and found te evdence tat supports te assumpton of te gambler s fallacy. Te second fallacy s te so-called te ot and bas. Te name of bas stems from te basketball: ndvduals presuppose tat a player scores wt a ger probablty f e as succeeded n te prevous trow. However, Glowc et al (985) reected te result. Tus, te ot and fallacy s an erroneous belef n te postve autocorrelaton wen te true process s non-autocorrelated random sequence. Note tat f te even gambler s and ot and fallacy seem very opposte, ts s not te case. Te gambler s fallacy s connected to te assumpton of cange of probablty and te ot and fallacy s related to te skll of te gambler. Te most famlar bas s te so-called favourte-longsot bas. In te gamblng markets, especally n orse race bettng, t as notced tat gamblers over-bet longsots and under-bet favortes more tan t s ratonal n te terms of expected return. In practce, te bas mples tat gamblers overestmate low probabltes and underestmate g probabltes. Te bas was frst dscovered by Grfft (948), and McGlotln (956). Snce ten, t as been examned by Wetzman (965), Al (977), Hausc et al (98), and te latest results are documented by Jullen & Salané (2), Wnter & Kukuk (26) and Wenbac & Rodney (28). For a sort revew, see Coleman (24). Altoug te bas s well-documented n general, tere are some exceptons. For nstance, Buce & Hall (988), and Hauc & Zemba (995) dd not fnd te evdence for te favourte-longsot n Asa orse race tracks. Instead, tey even found some evdence for te opposte. 8

11 Because te bas conflct te assumptons of EUT, t as been explaned n several ways. Jullen & Salané (25) gve a sort revew on te explanatons. Frst, t s argued tat gamblers ave a based vew of probabltes (Grfft, 948). Note tat te prospect teory takes ts as a fact. Second, t s possble tat gamblers are rsk lovers and, tus, tey lke to gamble orses wt low probabltes. Trd, Sn (993, 992, 99) argued tat bookmakers ave to protect ter profts from te adverse selecton problem by gvng te odds tat equalze te bets. However, n suc gambles as par-mutuel orse bettng, ts s not a relevant queston, snce te odds are determned by te gamblers. Weztman (965) started te seres of papers tat estmate and explan emprcally te bas wt rsk-love utlty functon. After tat, for nstance, Al (977), Kanto et al (992), and Wnter & Kukuk (26) tred to fnd te answers wt same metods. However, for nstance, Jullen & Salané (2) conclude tat rsk-love does not explan te bas suffcently. In spte of all, te man problem s tat we cannot separate te rsk-love and te subectve probablty as Jullen & Salané (25) note: Tere s n fact notng n te data tat allows te econometrcan to dstngus between te two nterpretatons. (p. 8). 6. Modellng gambles wt uncertanty functon Te prospect teory s based on one parameter subectve probablty functon. We argue tat n te case of gamblng t s sutable to use a two-parametrc functon. One type of tese functons can be wrtten as γ δp w( p) = γ γ δp + ( p), (2) were te scale parameter δ llustrates prmarly te level of te functon and te parameterγ prmarly te curvature, respectvely. Note tat te functon s vald wen w ( ) = and w ( ) =. Moreover, we assume tat w ( p) + [ w( p)] = smlarly as n te obectve probabltes. Gonzalez & Wu (999) used Eq. (2) functon to model prospects were te consequences or returns were only postve. Te functon type was a varant of te functon by Lattmore et al 9

12 (992). Note tat curvature parameter reflects te decson maker s vew of probabltes, and te sape parameter reflects te attractveness of te gamble. Ts approac s unusual because normally te attractveness of te gamblng s caractersed by te utlty functon. We take te Gonzalez & Wu (999) approac one step aead. Assume tat a functon ncludes all aspects of attractveness of rsk, and subectve probablty exsts as above. We call t te uncertanty functon. Ts functon takes nto account all te uncertantes tat we run nto. Tus, tere s not a separate utlty functon for a rsk averson or rsk-love. Ts approac s based on te followng arguments: ) Emprcally we cannot separate attractveness of rsk and subectve probablty. 2) We can represent te gamble only by te functon of te probablty and te take-out rate (.e. Eqs. -4, 8-9 and Fgure ). 3) Fnally, we can smply assume tat an ndvdual beaves accordng te dea of decreasng margnal utlty of money, wc ave notng to do wt te rsk. Assume, as defned above, a prospect tat can be wrtten n form q u = ( r u, p;...; rku, pk) and te gambler wll coose only one orse from k orses and e nvests one euro bet for te orse. Moreover, we also assumed tat te ndvdual as a decreasng margnal utlty for money, u ( x) > and u ( x) <. Tus, te utlty functon of money does not ave anytng to do wt te rsk averson or rsk-love. Note tat ts approac ncludes utlty of rsk (attractveness) and te subectve probablty smlarly as mentoned above n Eqs. 8- but t s augmented wt money utlty functon. Terefore, te expected utlty of gamblng, wen te gambler cooses orse, can be wrtten as V( q ) = w( p ) u( r + W) + [ w( p )] u( W ), (3) u u were W s a wealt level or a gamblng cas. We can llustrate te stuaton by a followng example. Suppose tat te expected loss (.e. take-out) s τ =.2, te gamblng cas s W =, and te utlty functon of money s u ( x) = ln(.). Te r = / p ( τ ) s uncertanty functon s te same as Eq. (2). Ten te expected utlty for u ( )

13 TP PT Of γ γ δ p δ p V( qu ) = ln ( ) W ln( W ) ln( W ) γ γ τ + + τ γ γ >. δ p ( p) p + δ p + ( p) (4) Notce, tat te only varables tat nfluence te decson are te probablty and te expected loss. Te only unknown elements are te parameters. Moreover, te gambler partcpates n te gamble f and only f te left and sde of Eq. (4) (te expected utlty of gamblng) exceeds te rgt and sde (te utlty for certan revenue mnus te expected loss,.e. consumpton part of game). As te uncertanty functon and return depends only on te probablty, we can analyse ow te 7 expected utlty beaves wen probablty cangestp PT. In Fgures 2 and 3 s sown te expected utlty n dfferent probabltes (orses). In Fgure 2 s sown te nfluence of te attractveness parameter to te expected utlty wen te curvature parameter s a constant.7 and te attractveness parameter devate from.4 to.6. In bot fgures, te devaton unt s a. and te cross lne llustrate te utlty of certan revenue mnus consumpton part smlarly as n Fgure. Utlty u(w-tau) u(w-tau).5 Probablty Fgure 2. Influence of te attractveness parameter to expected utlty. 7 course, we ave to assume some values for te parameters of te uncertanty functon.

14 Respectvely, te nfluence of te subectve probablty s sown n Fgure 3. Te curvature parameter devates from.6 to.8 and te second parameter s a constant.5. Note tat all values of te parameters ave cosen to correspond to gamblers beavour. Utlty u(w-tau) u(w-tau).5 Probablty Fgure 3. Influence of te curvature parameter to expected utlty. We can nterpret Fgures 2 and 3 as follows. If te expected utlty s more tan a constant, gambler wll partcpate n te gamble, oterwse not. In oter words, even te expected return s te same; gamblers are preferrng orses, wc ave a small probablty and a large return. Ts s, ndeed, te favourte-longsot bas. Moreover, by ts model, t s also possble to explan te gambler s fallacy and te ot and bas. For nstance, gambler s fallacy reflects te gamblers beavour by te curvature parameter and te ot and by te attractveness parameter. Furtermore, n te uncertanty functon can explan or model te gamblng addcton: te sape parameter wll ncrease wle a gambler contnues to gamble. To sum up, f we assume tat gamblng s ust a consumpton type actvty for wc te gamblers are prepared to pay, ten t s moderate to explan te gamblng beavour wt te uncertanty functon. Furtermore, te represented approac ncludes bot te crucal concepts: attractveness of gamblng and te based vew of probabltes but t s augmented wt te utlty functon of money. 2

15 TP 7. Emprcal evdence from Fnns orse track Te emprcal part of paper studes ow Fnns gamblers beave n a par-mutuel orse race ( wn bets ). In ts game gamblers mpose te bets for te orse, wc tey assume to wn te race. After gamblers ave put ter bets, te odds are based on ts nformaton and returns are mpled by te odds. Evdently te odds are te gamblers vews of te wnnng probabltes. Te appled return odds are te representatve gambler s prospects. Tus, te subectve probabltes can be calculated by te odds. On te oter and, f we ave a storcal data on orse records, ten we can estmate te obectve probablty. We ave used te Fnns par-mutuel orse race track data from te Suomen Hppos (te Fnns Trottng and Breedng Assocaton). Tey ave a monopoly by te law to organze te orse bettng markets n Fnland. Te assocaton organzes races n te 43 tracks around te country more tan fve tmes per week. Te data as been collected from te Suomen Hppos (27) WebSte by a computer programme. Te tme perod was It ncludes races wt startng orses. Obvously te same orses appear n dfferent races qute often. Te bookmaker as te take-out rate for every bet. Typcally t s around 2 %. In te data, te average take-out rate was 2,97 %. In practce, ts means tat a gambler wll lose for a bet 2,97 %, n average. Tus, te expected return s negatve and te gamble s unfar for te bettor. Our man am s to calculate te subectve probablty for a representatve bettor, to estmate te obectve probablty, and to compare tese concepts. Frst, from te data we get te odds for every 8 orse by race to racetp PT. Te wrtten as odds are derved from te gamblers bets. Tus, O for te orse can be O b = ( τ ), (5) n b = were b s te all bets for te orse, = take-out rate. Due to propertes of te elementary gamble, we can wrte n b s te sum of bets for all n orses, and τ s te 8 PTe odds are rounded downwards, e.g., te odd.56 wll be.5. Ts nfluences te results,.e. te favourte bas decreases (Coleman 24). 3

16 n ( τ ) =. (6) = O As we know te take-out rate, te nverse of observed odd (probablty) for every orse s O ( τ ) = ρ. (7) Note tat take-out rate sould not nfluence te subectve probabltes, snce te take-out rate s leved on all pad returns wt te same proporton and t does not affect ow te orses run. Tus take-out rate fulfl te (tax) neutralty condton. Secondly, we ave to estmate te obectve probablty. Te lterature as used dfferent metods n ts context. For nstance, Jullen & Salané (2) calculated probabltes by dvdng te odds n to te segments (groups) n every race. Furtermore, f te odd of te orse was nsde te nterval, t was placed n ts odd group. Ts routne as, owever, one potental problem: t s possble tat nsde te same segment tere are more tan one orse, even toug only one can wn. Terefore, we construct te estmaton of te obectve probablty wt Al s metod (977). Frst, we dvde orses n groups () n every race: a favourte orse s group one, and so on. Overall, we wll create ten groups from all races. Suppose tat π s an obectve wnnng probablty to a orse, wc was ncluded n te group. Te obectve probablty for te group one (te favourte) s calculated by dvdng all wnnng cases of te group one wt all races. Formally, t can be represent by π m Y = =,... m = (8) were Y =, wen a orse n group wns te race, and oterwse zero. A number of races s m. 4

17 Notce tat π s Bernoull varable. Because te races are ndependent, π s Bnomally dstrbuted. Moreover, te estmates of Bnomal dstrbuton ave te followng unbased propertes: EY ( ) = π and Var( Y ) = π ( π )/ m. Furtermore, te subectve probablty n eac group can be solved by m ρ ρ =, =, 2,...,. (9) = m Te test for gamblng bases s H : ρ π = for all =, 2,...,,.e. are te subectve and obectve probabltes n dfferent groups equal. We can approxmate te Bnomal dstrbuton by te Normal dstrbuton f te sample of observaton s large enoug. By te Central Lmt Teorem, we wrte as z = ρ π π ( π ) / m N(,) (2) Te emprcal results are presented n Table. Table. z-test for te equalty of subectve and obectve probabltes. Group () ( ρ π ) π SE( π ) ρ SE( π ),364,58,328-23,3** 2,8,26,73-6,8** 3,26,9,8-7,56** 4,92,95,88-4,5** 5,69,83,69 -,57 6,5,72,55 4,5** 7,4,64,44 6,43** 8,33,59,36 4,9** 9,25,52,29 6,26**,8,43,23,5** ** Denotes te % sgnfcance level. 5

18 Te subectve and obectve probabltes are statstcally dfferent n all groups except te group fve. Moreover, te results ndcate tat bettors gamble for te favourtes too lttle and, respectvely, too muc for longsots. Te results do not reect te favourte-longsot bas. Fnns bettors beave as ter counterparts n te Western Europe, Australa and USA. 8. Concluson It s evdent tat te gamblng beavour s more tan ust rsky coces. Several reasons ndcate tat te gamblng beavour s a consumpton type actvty smlarly as enoyng a teatrcal performance, for nstance. Terefore, t s reasonable to assume tat te prce of gamblng s te take-out rate or te excepted loss of game. Te take-out vares between of dfferent gambles or even nsde te gamble (e.g. favourte-longsot bas). Tus some gambles or even coces are more attractve tan oters and te gamblers are prepared to pay te ger prce for tese games. We llustrated wt smple elementary gamble example and wt par-mutuel wagerng example tat te gamblng markets consttute an envronment weren rsky coces are only measured by probabltes. Terefore, te gamblng beavour can be andled wt te approac tat ncludes attractveness of rsk, subectve probablty, and utlty of money. However, te sutablty of ts approac to te oter rsk nvolved stuatons remans to be analysed. For nstance, te nsurance markets (e.g., te prce of a far/unfar nsurance) are very smlar to te gamblng markets. Tus more teoretcal researc s needed ere. Te results of te emprcal part of ts paper ndcate tat Fnns gamblers beave smlarly as te favourte-longsot bas predcts: gamble too lttle on favourtes and too muc on longsots n comparson to a ratonal rsk aversve person. Te results confrm te unversalty of te favourtelongsot bas. Next step s to consder te anomaly more closely,.e. try to fnd some elements tat systematcally explan gambler s beavour n te smlar fason as, for nstance, Wenbac & Rodney (28). However, a more precsely emprcal researc requres ndvdual nformaton and data from te gamblng decson. We ope tat ts wll be possble n te near future. 6

19 References Al M. M. (977) Probablty and Utlty Estmates for Racetrack Bettors. Journal of Poltcal Economy 85, pp Allas, M. (953) Le Comportement de l Homme Ratonnel devant le Rsque: Crtcue des Postulats et Axomes de l Ecole Amercane. Econometrca 2, pp Asc, P. & R. E. Quandt (99) Rsk Love. Journal of Economc Educaton 2, pp Busce, K. & C. D. Hall (988) An excepton to te Rsk Preference Anomaly. Journal of Busness 6, pp Clotfelter, C. T. & P. J. Cook (99) Te Gambler s Fallacy n Lottery Play. Workng Paper No Natonal Bureau of Economc Researc. Cambrdge, USA Coleman, L. (24) New Lgt on te Longsot Bas. Appled Economcs 36, pp Croson, R. & J. Sundal (25) Te Gambler s Fallacy and Hot Hand: Emprcal Data from Casnos. Te Journal of Rsk and Uncertanty, 3 pp De Meza, D.& D. C. Webb (24) Prncpal Agent Problems Under Loss Averson. An Applcaton to Executve Stock Optons. FMG Dscusson Papers dp478, Fnancal Markets Group Decdue, E., Scmdt U. & P. P. Wakker (24) Te Utlty on Gamblng Reconsdered. Te Journal on Rsk and Uncertanty 29, pp Fsburn, P. C. (98) A Smple Model for te Utlty of Gamblng. Psycometrca 45, pp Fredman, M. & L. J. Savage (948) Te utlty analyss of coces nvolvng rsks. Journal of Poltcal Economy 56, pp Glovc, T., R. Vallone & A.Tversky (985) Te Hot Hand n Basketball: On te Mspercepton of Random Sequences. Cogntve Psycology 7, pp Gonzalez, R. & G. Wu (999) On te Sape of te Probablty Wegtng Functon. Cogntve Psycology 38, pp Grfft, R. M., (949) Odds Adustment by Amercan Horse-Race Bettors. Amercan Journal of Psycology 62, pp Hausc, D. B., W.T. Zemba & M. Rubnsten (98) Effcency n te Market for Racetrack Bettng. Management Scence 27, pp Hausc, D. B. & W.T. Zemba (995), Effcency of Sports and Lottery Bettng Markets. Handbooks n Operaton Researc and Management Scence Jarrow R. et al (ed.), Elsever Scence Jullen, B. & B. Salané (25) Emprcal Evdence on te Preferences of Racetrack Bettors. Artcle for te book: Effcency of Sports and Lottery Markets, Hausc, D. & W. Zemba (ed.). Handbook n Fnance seres (2) Estmatng Preferences Under Rsk: Te Case on Racetrack Bettors. Journal of Poltcal Economy 8, pp Kaneman, D. & A. Tversky (979) Prospect Teory: An Analyss of Decson under Rsk. Econometrca 47:2, pp (992) Cumulatve Prospect Teory: An Analyss of Decson under Uncertanty. Journal of Rsk and Uncertanty 5 pp Kanbur, R., Prttlä, J. & M. Tuomala (27) Moral azard, taxaton, and prospect teory. Fortcomng Scandnavan Journal of Economcs Kanto, A.J., Rosenqvst G. & A. Suvas (992) On Utlty Functon Estmaton of Racetrack Bettors. Journal of Economc Psycology 3, pp Lattmore, P. K. & J. R. Baker & A. D. Wtte (992) Te Influence of Probablty on Rsky Coce: A Parametrc Examnaton. Journal of Economc Beavor and Organzaton 7, pp

20 McGlotln, W. H. (956) Stablty of coces among uncertan alternatves. Amercan Journal of Psycology 69, pp Sn, H. S. (99) Optmal Odds Aganst Insder Traders. Economc Journal, pp (992) Prces of State-Contngent Clams wt Insder Traders, and te Favorte- Longsot Bas. Economc Journal 2, pp (993) Measurng te Incdence of Insder Tradng n a Market for State-Contngent Clams. Economc Journal 3, pp Starmer, C. (2) Developments n Non-Expected Utlty Teory: Te Hunt for a Descrptve Teory of Coce under Rsk. Journal of Economc Lterature 38, pp Suomen Hppos (27), ttp:// (Readed ) von Neumann, J. & O. Morgenstern (944) Te Teory of Games and Economc Beavour. Prnceton Unversty Press, Prnceton Wenbac, A. P. & P. J. Rodney (28) Te Lnk Between Informaton and te Favorte-longsot Bas n Par-mutuel Wagerng Markets. HTe Journal of Gamblng Busness and EconomcsH, pp Weztman, M. (965). Utlty Analyss and Group Beavor: An Emprcal Study. Journal of Poltcal Economy 73, pp.8 26 Wnter, S. & M. Kukuk (26) Rsk Love and te Favorte-Longsot Bas: Evdence from German Harness Horse Racng. Scmalenbac Busness Revew 58, pp Appendx Let assume two bookmaker companes and. Tey offer bookmaker servce for te customers and tey compete by take-out rate τ and τ. Moreover, te total money, wc gamblers are gamblng s S = s + s. Te bookmaker companes are dentcal and offer a omogenous servce. We assume tat consumers are coosng te ceaper servce. Tus, we can wrte demands of te companes and for te bettngs as τ > τ s( τ, τ ) = /2 s( τ) τ = τ S τ < τ, Te companes profts can be wrtten S τ > τ s( τ, τ ) = /2 s( τ ) τ = τ τ < τ., =,2 were c and π ( τ, τ ) = τ s ( τ, τ ) cs ( τ, τ ), π ( τ, τ ) = τ s ( τ, τ ) c s ( τ, τ ), * * * * c are unt costs. Nas equlbrum for te companes are {,, s, s} τ τ, so tat * * * * * * * * * * * * s = s( τ, τ ), s = s( τ, τ ) and π ( τ, τ ) π( τ, τ ), π ( τ, τ ) π( τ, τ ) for all ( τ, τ ). Because te demand functons are not contnuous, we proceed as follows. 8

21 * * * * Clam: If c = c = c, ten τ = τ, s = s = /2 s( c). Proof:. τ < c and τ < c, cannot be equlbrum, because proft would be negatve. 2. τ > τ > c and τ > τ > c, eter cannot be equlbrum, because te company wc as a ger prce wll decrease t. 3. τ = τ > c, cannot be equlbrum, because bot companes ave an ncentve to decrease t. 4. τ > τ = c or τ > τ = c, cannot be equlbrum, because te company wc as τ = c t s proftable to ncrease te prce for a bgger proft. 5. τ = τ = c, s equlbrum, because for neter company t s not proftable to cange te strategy. 9