Staying Ahead and Getting Even: Risk Attitudes of Experienced Poker. Players

Transcription

1 Saying Ahead and Geing Even: Risk Aiudes of Experienced Poker Players David Eil George Mason Universiy Inerdisciplinary Cener for Economic Sciences Jaimie W. Lien Tsinghua Universiy School of Economics and Managemen Version: March 19 h, 2013 Absrac 1 We sudy he behavior of frequen online poker players who have exensive experience calculaing probabiliies and expeced values. Such players migh be expeced o behave as expeced uiliy maximizers, in he sense ha small shocks o heir wealh would no change heir risk preferences (Rabin, 2000). By conras, he predicion of reference-dependen loss aversion (as in Prospec Theory) (Koszegi and Rabin, 2006; Kahneman and Tversky, 1979) is ha risk aversion decreases as a player s wealh ravels away from he reference poin in eiher direcion. In erms of coninuing o play, as well as via a more aggressive playing syle, we find srong evidence for he break-even effec, he increased willingness o ake on risk as a player is losing wihin a session. However, we do no find evidence for he house money effec, he increasing willingness o ake on risk in he gains domain. Raher, experienced poker players behavior appears o be more consisen wih exising evidence on reference-dependen labor supply, wih players ending o reduce effor and risk-aking in response o being ahead. Taken ogeher, our findings provide evidence for reference-dependen labor supply in a flexible, high-skilled seing, under condiions of well-undersood moneary risk. Keywords: Decision-making under uncerainy, reference-dependence, experience, risk JEL classificaion: D81 1 deil@gmu.edu ; jwlien@sem.singhua.edu.cn ; We would like o hank Vincen Crawford and Julie Cullen for heir generous advice hroughou he implemenaion of his projec. We are graeful o Nageeb Ali, James Andreoni, Michael Bauer, Micheal Croeau, Gordon Dahl, Uri Gneezy, Daniel Houser, Brad Humphreys, Ryan Lim, John Lis, Juanjuan Meng, Craig McKenzie, Paul Niehaus, Jusin Rao, Raymond Sauer, Charles Sprenger, and Li Zhou for helpful conversaions and encouragemen. We also hank paricipans a he Workshop on Gambling Economics a Universiy of Albera, Edmonon, and he Workshop in Behavioral and Decision Sciences a Nanyang Technological Universiy. The UCSD Insiue for Applied Economics and Tsinghua Universiy provided research funding suppor. All errors are our own. 1

2 Inroducion The shape of an individual s uiliy of money funcion is an imporan deerminan of economic behavior across many domains. Neoclassical uiliy of money funcions ake only he decision maker's final wealh saes as inpus so he marginal uiliy of an exra dollar is fully deermined by how much wealh he decision maker currenly possesses, bu no he amouns he may have possessed in he pas, or he amouns he could have possessed in he presen in differen saes of he world. This assumpion has implicaions for a myriad of imporan economic decisions, including hose explicily or implicily involving risk, such as labor supply choices. A predicion of he neoclassical model is ha wealh shocks ha are small relaive o lifeime income should have no effec on an individual s risk aversion. By conras, he main compeing model, reference-dependen loss aversion (Koszegi and Rabin, 2006; Kahneman and Tversky, 1979), does predic ha risk preferences are affeced by small changes in wealh, and in a paricular paern: individuals become relaively more willing o accep risk afer oucomes which move oal oucomes away from heir reference poin, in eiher direcion. In his paper we sudy he behavior of experienced and on-average successful online poker players in order o es wheher reference-dependence persiss under experise and well-developed knowledge of probabiliies. Individuals in our sample play regularly and earn money doing so - on average, hey play abou 10 hours a week and earn nearly $40 per hour, ypically playing several ables a once. We find ha alhough he amouns of money won and los wihin a siing are small compared o heir lifeime wealh, hese ransiory gains and losses do significanly affec heir propensiy o ake on risk. We documen hese paerns using wo differen measures of a player s willingness o ake risk. Firs, we esimae a player s likelihood of coninuing o play. Presuming ha whaever aciviy hey would do insead involves less risk han playing poker, coninuing o play represens more risk-loving preferences han does disconinuing play. Second, we esimae a player s likelihood of folding wihou puing any money ino he po in a given hand. Folding incurs a cerain payoff while no folding induces some disribuion of moneary oucomes. Therefore we consider no folding o be represenaive of more risk-loving preferences han folding. In a manner consisen wih reference-dependen loss aversion, players indeed exhibi a break-even effec, in which losses make hem more risk-loving. This resul is consisen wih oher evidence from he effec of shocks on risk preferences. For example, Pos, Van den Assem, Balussen and Thaler (2008) considers he behavior of conesans on he game show Deal or No Deal, in which conesans make a series of decisions under uncerainy. They find ha losers in he game, who have been unluckier han he average conesan, are less risk averse han neural conesans. Pos e al also find ha, again consisen wih loss aversion, winners, who have been luckier han average, are also somewha less risk averse, bu he effec is smaller. This is he house money effec, wherein individuals become relaively less risk averse he more money hey win. However we find ha experienced poker players do no generally exhibi he house money effec. Insead, we find ha gains acually make players more conservaive. This lack of house-money effec is robus o specifying he reference poin as individual expeced winnings as proposed by Koszegi and Rabin (2006). Fundamenally, risk preferences are deermined by an individual's valuaion of a dollar gained relaive o a dollar los. A closely relaed choice is ha of labor supply, in which an individual rades dollars gained no agains dollars los, bu agains cosly effor. Since we sudy a player s willingness o coninue playing, we are 2

3 considering he valuaion of dollars gained agains boh he valuaion of dollars los and he expendiure of effor. In his sense, decisions on how long o play resemble a labor supply choice, a leas for skilled players, for whom expeced profis are posiive. The lieraure on labor supply in response o wage shocks is hus also closely relaed o our quesion. Using deailed daa on axicab rip records, Camerer, Babcock, Loewensein and Thaler (1997), as well as Crawford and Meng (2011) find evidence for daily income (and hours) argeing among drivers, a labor supply behavior which can be accouned for wih reference-dependence, bu no neoclassical life-cycle labor supply models. Fehr and Goee (2007) run a field experimen on bicycle messengers, esing wheher hey reduce effor in response o a posiive ransiory shock o wages. They find ha reduced effor is prevalen and correlaed wih laboraory-syle measures of loss aversion. In earlier lieraure, Dunn (1996) finds survey evidence for loss aversion among differen worker ypes by racing income-leisure indifference curves. Rizzo and Zeckhauser (2003) find evidence for loss aversion among physicians, by using survey responses regarding adequae income as he reference poin. A main conribuion of our sudy is o he lieraure ha sudies differences in levels of loss aversion beween novices and expers. Lis (2003, 2004) has presened evidence ha reference dependen behavior is no exhibied by expers, and goes away wih markeplace experience. Lis found ha professional card dealers were less likely han novice card show aendees o exhibi he endowmen effec, ofen explained as an arifac of loss aversion, drawing ino quesion wheher empirical evidence from laboraory experimens carries over o seings in which decision-makers are expers. Koszegi and Rabin (2006) show ha his could resul no from differen degrees of loss aversion, bu differen expecaions - in he case of Lis's expers, hey expec o sell he hings hey ge. Therefore selling he good does no feel like a loss o hem, since hey never expeced o keep i. Crawford and Meng (2011) also find less loss aversion among he more experienced cab drivers in heir sample. Pope and Schweizer (2011) documen evidence of loss aversion in professional PGA our golfers. Our daa comes from experienced and successful players playing wih large sakes. On average, hey play 300 hours over he seven monhs of our daa, and earn almos $40 an hour. Moreover, he expeced value calculaions required o make profiable decisions as a successful player give hem he compuaional skills knowledge of probabiliy needed o make raional decisions. Ye, we sill find ha daily profis have a significan impac on heir decisions. Our sudy also offers a deailed analysis he dynamics of risk preferences in a whie-collar, high skilled seing. 2 Sudying risk-aking behavior in a whie-collar seing is a poenially imporan addiion o he curren exising lieraure, due o he possibiliy ha here may exis cerain feaures of previously considered physical labor seings which could make hem more prone o reference-dependence. For example, he risk associaed wih coninuing pas one s reference poin may increase in a paricularly unappealing manner in he case of manual labor seings such as driving or consrucion, due o safey concerns and human physical limiaions. In such occupaions here may be a prevalen philosophy of no working more han you have o jus for he sake of money. When individuals coninue o play online poker, he risk players perceive due o any decrease in performance or concenraion owards he ask a hand, is moneary raher han physical. 3 A furher advanage of online poker is ha for profiable players, i is a mehod of earning money wih no 2 See Levi and Miles (2011) which invesigaes wheher poker is primarily a game of luck or skill, and finds srong empirical evidence ha poker is a game of skill. 3 The differen naure of demands on workers employed in manual or physically inensive labor versus compuer and desk work is recognized and refleced in he exempion rules of he Fair Labor Sandards Ac. 3

4 explicily imposed consrains on working hours, and no fixed coss of working on a daily basis. 4 No cerificaion or raining is required so in order o survive a his job in he long run, players need o develop or research ino heir own sraegies on when o sop, since hey receive no official educaion on his maer. Thus we believe wha we observe in online poker players behavior is really heir naural response o heir prior oucomes, and no any arifac of insiuional consrains or advice. Wih subsanial numbers of people working a home or aking on enrepreneurial projecs wih various associaed risks on heir own free ime, our resuls may be indicaive of he risk aiudes and behaviors of workers in oher freelance labor supply seings. 5 A sudy which is close o ours in erms of he naure of he work examined is Coval and Shumway (2005) which finds evidence of loss aversion among proprieary raders a he Chicago Board of Trade (CBOT). Like financial markes raders, individuals who regularly play poker for real money are well-versed in dealing wih risk, and have demonsraed compeence in numeracy and probabiliy. Poker players, however, have he advanage of never seeing heir marke close. Tables say open 24 hours a day, 7 days a week. In addiion, he impac of winning and losing on fuure oucomes is less of a confounding facor in poker han in rading. The mos obvious hypohesis in he case of poker is ha earnings would be posiively auo-correlaed losing early in he day predics losing laer in he day. This would sugges ha players should qui earlier and fold more when hey re losing, which would be he opposie of wha we find. Also, we can remove a leas one source of auo-correlaion in earnings by examining he effec on earnings a he nh able of earnings a he oher n-1 ables. As shown in Secion 5, his analysis does no change our esimaes a all, indicaing ha he change in behavior is coming from he uiliy funcion, no beliefs. We are no he firs o look a he game of poker o deec evidence for he break-even and house money effecs. Smih, Levere and Kurzman (2009) use high sakes online poker daa o evaluae playing syle before and afer paricularly large wins and losses, and comparing wha oher possible behavioral biases migh accoun for player behavior. They resric heir analysis o aggressive versus conservaive playing syle wihin paricular hands, raher han overall playing ime behavior, and focus on daa from a high sakes level. Our findings are consisen wih heirs in ha hey find players are less conservaive and more aggressive in heir play afer losing an especially large po, while becoming more conservaive and passive afer winning an especially large po. While hey ake his as concree evidence of he break-even effec, hey refrain from speculaing or invesigaing in deail abou he lack of house money effec. Combined wih recen progress in he lieraure on reference-dependen labor supply, by examining when players choose o end poker playing sessions, our findings sugges a possible labor-supply reason why Smih e al (2009) did no find much of a house money effec in he playing syle domain. In Secion 5, we replicae heir main findings on our daa using probi analysis. Throughou our analysis, we will be reaing player choices as decision problems, raher han sraegic problems. Of course poker is a sraegic game, and i is possible ha wha we are observing is par of a sraegy designed o increase long-erm winnings raher han changing risk preferences. I is also possible ha 4 One can imagine online poker as a markeplace for experience, in which one ype of player (leisure ype) paricipaes primarily for he leisure uiliy of playing, and is willing o pay a cos for he experience of doing so. Anoher ype of player (profiable ype) paricipaes primarily o earn money hrough he losses of leisure players, while supplying playing experience o leisure players. Thus, while online poker may no be considered a convenional labor marke per se, ransacions analogous o a labor marke seing ake place indirecly. 5 Some examples could include selling iems on Ebay or oher online cusomer-o-cusomer reail sies, day rading in he financial markes, or individuals uilizing paricular personal skills such as ars, eaching or programming o earn money on a job-by-job basis. 4

5 behavior reflecs a belief in changes in ohers risk preferences, even while no player s preferences acually change. We have several jusificaions for his sraegy. Firs, as we will discuss, he player pool is quie large, and players play on many ables a once. This means ha execuing a negaive expeced-value sraegy early on in hopes of creaing a bigger posiive expeced-value sraegy laer on agains ha same player is risky, since he chances of playing a big po agains ha player are relaively small. Oher players a he able can observe play, bu given ha mos of hem play muliple ables, hey are unlikely o concenrae closely on hands in which hey are no hemselves involved. Second, while he equilibrium for his ype of poker is no known and even if i were we would no expec all players o follow i, a sraegy of giving up money early o win back more money laer would be ou of equilibrium. Mos imporanly, we see no signs of his behavior in he daa. Player performance does no seem o depend on previous winnings in he same session, alhough his daa is quie noisy. In our probi analysis in Secion 5, we also include in our regressions conrol variables such as sack size and number of players ha could be imporan sraegically. We also find ha all ypes of winnings affec risk preferences equally. As discussed in Secion 5.1, boh winnings from luck (which canno resul from sraegy) and skill (which migh) affec risk preferences in he same way. If wha we consider changes were acually a sraegic decision, we would insead find effecs only for winnings from skill. Finally, as discussed a he end of Secion 5, winnings from he oher n-1 ables have an effec on a player s acions a a given nh able. Since players a able n will observe he proagonis s acions a he oher n-1 ables only very rarely, here is lile reason o believe his change in playing syle would be an effecive sraegy. All of hese facs are, however, fully consisen wih changing risk preferences. The remainder of he paper is organized as follows: Secion 2 discusses loss aversion and is predicions for poker players' playing ime and risk-aking behavior; Secion 3 describes he daa se used and player characerisics; Secion 4 explains he empirical sraegy and describes resuls concerning ime spen playing as a funcion of ne winnings relaive o various reference poins. Our specificaions in his secion include boh individual-level and pooled esimaes of a duraion model. We also consider alernaive specificaions of he reference poin as suggesed by players expeced winnings, and recen winnings. Secion 5 deails findings on how poker playing syle changes wih ne winnings, an issue invesigaed for large sakes decisions in Smih, Levere and Kurzman (2009). We use a probi approach, conrolling for several relevan variables which we observe in he daa, and compare our resuls o heirs; Secion 6 esimaes he coss of he break-even effec o he online poker players in our sample; Secion 7 concludes. 2. Prospec Theory As discussed above, prospec heory specifies a value funcion ha akes as is argumen no he final wealh sae of he individual, bu a change relaive o a reference poin. Koszegi and Rabin (2007) wrie down he following uiliy funcion for money which combines boh consumpion uiliy and gain-loss uiliy : U(x) = m(x) + μ(m(x)-m(r)) where x is some cerain wealh oucome and r is he reference poin. A commonly used special case is for m(x) o be linear. In fac, for amouns of a few hundred dollars, as discussed above, i mus be he case ha m(x) is linear. A common specificaion for he gain-loss uiliy value funcion μ is as follows: 5

6 μ(x-r) = -1 {x-r < 0} *λ*( x-r ) α + 1 {x-r > 0} (x-r) α Where 1 A is an indicaor funcion equaling 1 under even A and 0 oherwise. This funcion saisfies he hree condiions on he value funcion saed in Kahneman and Tversky (1979), ha i be defined on he deviaion from he reference poin, be concave for gains and convex for losses ( diminishing sensiiviy ), and be seeper in losses han gains ( loss averse ). I also saisfies he similar condiions (A1-A4) saed in Koszegi and Rabin (2007). Firs consider when α is one, so ha he funcion is piecewise linear. This corresponds o assumpion A3' in Koszegi and Rabin (2007). Figure 1 shows a picure of his value funcion. Consider an individual's risk preferences saring a various wealh posiions relaive o he reference poin. Fixing a loery A, afer he individual has experienced a large gain or loss, and is herefore far away from he reference poin, mos of he payoffs in A will sill leave he individual on he same side of he reference poin. Therefore he secion of he uiliy funcion over which A is evaluaed is mosly linear, largely avoiding he firs-order concaviy of he kink. Pu anoher way, all risk aversion is driven by he fac ha a dollar gained is less pleasing han a dollar los is painful. For a loss-averse individual, his difference is maximized a he reference poin. When he individual is already in he loss domain, he difference is decreased because he benefi of a dollar gained increases, since now i offses a loss, o which he individual is paricularly averse. As he individual goes deeper ino he gains domain, he difference is again decreased, his ime because he pain of a dollar los has decreased, since now i is simply he reducion of a gain, no an acual loss. Willingness-o-pay for a given gamble hen follows a V-shaped paern as a funcion of disance from he reference poin, wih is minimum a he reference poin. This piecewise linear model, wih α equal o one, is he specificaion frequenly used in applicaions. 6 For α less han one, curvaure miigaes he break-even effec and exacerbaes he house money effec. The reason is ha in each case, as he agen's wealh posiion moves farher away from he reference poin, he uiliy funcion becomes more linear. When i is gains ha are increasing, his decreases risk aversion, since he gains porion of he uiliy funcion is concave and less concaviy resuls in less risk aversion. However in he loss porion, his decreases risk lovingness, since he loss porion is convex, and less convexiy resuls in less risk lovingness, working agains he break-even effec. This may be couner-inuiive, since inroducing 6 For insance, Heidhues and Koszegi (2005, 2008) use he reference dependen model o predic pricing sraegies of firms. Gill and Prowse (2012) esimae loss aversion coefficiens in an effor provision experimen. In boh seings, he piecewise linear version of he uiliy funcion is employed. 6

7 concaviy in gains creaes more risk aversion in gains, and convexiy in losses inroduces risk lovingness in losses. However his is only relevan for comparing gains o losses. Tha is, for α < 1, an individual will be more risk-loving afer losing $100 han afer winning $100. However, compared o α=1, an individual wih α<1 would have a smaller decrease in risk aversion when moving from $100 o $200 in losses, and a bigger decrease in risk aversion when moving from $100 o $200 in gains. In heir original ouline of reference dependen preferences, Kahneman and Tversky were (perhaps deliberaely) vague abou wha he reference poin migh be. Candidaes include he saus quo, expeced values, or he oucomes of ohers. Koszegi and Rabin (2007) specifies ha he reference poin should be he individual's recen beliefs. In our analysis, we will sar ou by assuming ha a player's reference poin is heir wealh a he sar of heir session, so ha session profis are exacly equal o deviaion from he reference poin. However we will also ry relaxing ha by allowing he reference poin o equal heir wealh plus he amoun ha hey win in an average session. A relaed quesion is he lengh of ime before a change in wealh becomes inernalized, and becomes he new reference poin. When does he house's money become he gambler's own? This lengh of ime is referred o as he bracke of an individual's decision. Once he bracke has closed, any gains are losses are inernalized. While wihin he bracke, he individual can gain money o offse a loss before booking i, and likewise lose money o offse a gain. Here we will sar wih an assumpion ha he player's bracke is a he session level. Tha is, he player sars every session in a new bracke, bu is no forced o book gains or losses menally unil he session ends. For example, if a player loses $200 in he firs hour of play and hen makes $400, he will code his as a gain of $200. By conras, if he loses $200 in one hour, quis, and comes back he nex day and makes $400 in an hour, a ha poin he will consider himself o be up $400. We can use he reference dependen model o inform our hinking abou a poker player's decisions. Firs, le us consider a player's decision of how long o play. A each poin in ime, players face a decision of wheher o end heir session or coninue playing. If hey coninue playing, hey may gain some uiliy from playing iself (i.e., unaffeced by he amoun won or los in he hand), and pay some cos in he effor required o make he bes decisions possible. They also gain or lose uiliy based on he money ha hey win or lose from he game. In paricular, consider a player who has played minues and is considering wheher or no o coninue for anoher d. If he sops, he will have a uiliy of: V ( w0, ) = U ( w0, π ( s) ds c( )) (1) 0 where U(w,π) represens he players uiliy of wealh funcion, w 0 represens he player's iniial wealh, π() represens he player's profis in ime, and c() represens a combinaion of he posiive uiliy from gambling and he effor cos paid o play he game, which are boh assumed independen of he amoun won or los in he session. For wha follows, we will simplify noaion by using Π() o denoe π ( s) ds, he cumulaive session profis up unil ime, while sill using he lowercase π() o refer o he profis arriving a ime. If insead he coninues o play an addiional d minues, he will have a expeced uiliy of: d V ( w + = Π + + 0, d) E[ U ( w0, ( ) π ( z) dz)] c( + d) (2) Wheher V(w 0,) or V(w 0, + d) is higher depends on he cos of addiional effor, he disribuion of 7 0

8 profis he player faces in he nex d minues, and he marginal uiliy of hese gains and losses. Clearly he player coninues if (2) - (1) is posiive, and sops oherwise. We will assume ha he marginal cos of effor is increasing over a leas some range, so ha he player always sops evenually, and ha i is separable from he uiliy of money. This same model can be used o hink abou he player's decision o fold or no. Folding guaranees a profi of zero. 7 Coninuing in he hand, eiher by raising or by calling, gives he player some disribuion of profis. 8 Coninuing may also give he player some fixed uiliy from gambling, and require some cos in making furher decisions in laer being rounds, again boh capured by c(). We will be ineresed in he effec of session profis up unil ime, Π(), on hese decisions. Due o Rabin's calibraion heorem, we can consider he neoclassical expeced uiliy case o be one where uiliy is linear in he amouns of money under consideraion. In his case, (2) - (1) reduces + d o E [ π ( s) ds] ( c( + d) c( )). 9 Π(), he amoun earned up unil ime $$ in he session, can ener his expression solely hrough is effec on beliefs regarding profis over he res of he session. In absence of any such effec, we should expec expeced uiliy maximizers o exhibi no sysemaic effec of session winnings on coninuaion probabiliy. Now consider he predicions of prospec heory, using he piecewise linear value funcion specified above. We will sar off by considering he reference poin o be he individual's wealh going ino he session, w 0. Then (2) - (1) > 0 reduces o: P + + d π ( z) dz > Π( ) E[ Π( ) + > Π + π ( z) dz π ( z) dz ( )] + d d λ P + d + d + d π ( z) dz < Π( ) E[ Π( ) + π ( z) dz π ( z) dz < Π( )] 1 Π > Π ( ) + λ 1 Π < Π( )) > c( + d) c( ) (3) ( { ( ) 0} { ( ) 0} where P(A) represens he probabiliy of even A occurring. Le H(Π()) represen he LHS of (3). The op wo lines show he expeced uiliy from aking he risky opion, divided ino wo pars: he op line represens he par of expeced uiliy ha comes from he gains. The second line represens he par ha comes from losses. The boom line subracs off he opporuniy cos, or he uiliy value of no aking he gamble. H(Π()) is hen compared o c( + d)-c(), he marginal effor of aking he gamble. This framework allows us o evaluae risk preferences a differen values of Π(). Recall ha he player acceps he gamble whenever H(Π()) > c( + d)-c() and oherwise akes he fixed amoun (eiher by folding or leaving he able, depending on he decision problem considered). Since he righ hand side is independen of H(Π()) for a given session lengh, a higher 7 Unless he player is in one of he blinds, in which case folding guaranees a loss of he amoun of he blind. 8 Clearly his disribuion is condiional on he cards he player is deal. Bu since he disribuion of hese cards is independen of Π(), here is no reason o believe ha he cards would somehow dicae ha a player should be more or less likely o fold based on Π(). 9 Sricly speaking, here we have se U(x) = x, which is more resricive han lineariy. However, for any linear uiliy funcion U ˆ ( x ), we can creae a cos funcion cˆ ( ) such ha + d + d E[ w + Π + s ds c + d = E Uˆ 0 ( ) π ( ) ] ( ) [ ( w0, Π( ) + π ( s) ds] cˆ( + d), so ha here is no loss of generaliy by simply assuming U(x) = x here. 8

9 H(Π()) means ha he player is more likely o coninue playing. Firs le us consider H(0), evaluaing he coninuaion decision when he individual is a he reference poin: + d + d + d H ( 0) = P π ( z) dz > 0 E[ π ( z) dz π ( z) dz > 0] + + d + d + d λ P π ( z) dz < 0 E[ π ( z) dz π ( z) dz < 0] If H(0) > c( + d)-c(), i mus be ha eiher +d E [ π ( z) dz] is much greaer han zero, so much so ha i would be posiive even when he losses are weighed wice as heavily as he gains, or c'() < 0. Now consider when Π() is much less han zero, some value Π(), so much so ha H(Π()) reduces o: H ( Π( )) = λ E[ Π( ) + + π ( z) dz] λ Π( ) d P > Π +d π ( z) dz ( )) is zero. Then + d = λ E [ π ( z) dz] Clearly H(Π())>H(0), since + d + d + d π ( z) dz > 0) E[ π ( z) dz H ( Π( )) = H (0) + ( λ 1) P( π ( z) dz > 0]. This is he break-even effec, ha when he individual is far below her reference poin, she is more likely o engage in risky acion (here, H(Π()) is more likely o be above c( + d)-c()) han when she is a he reference poin. Finally, consider he case when Π() is much greaer han zero a some value Π () d ( Π, so ha P π ( z) dz < ( )) is zero. Then H ( Π ( ) ) reduces o simply E [ +d π ( z) dz]. Again, his is greaer + d + d + d han H(0), since H Π( )) = H (0) + ( λ 1) P( π ( z) dz < 0) E[ π ( z) dz ( π ( z) dz < 0]. This represens he house money effec - he individual is more likely o accep a given gamble when above her reference poin han when a i. We have given he inuiion for he break-even and house money effecs by comparing exreme cases, comparing very high and low Π() o Π()=0. In general, a loss averse individual's marginal uiliy from aking a given gamble is a funcion of he probabiliy wih which hey cross he reference poin and he amoun by which i is crossed. For gambles ha are approximaely symmeric around zero, such as he disribuion of profis from a given hand, his is a funcion of he disance beween Π() and he reference poin. We can also see ha H ( Π ( )) > H ( Π( ) ) as long as expeced profis are posiive. This reflecs he fac ha he marginal uiliy of money relaive o effor is higher in he loss porion han in he gain porion of he value funcion. Figure 2 summarizes hese effec by graphing H(Π()) where d is se o be long enough o play exacly one hand. For illusraive purposes, we have used for his graph he disribuion of profis for jus one 9

10 player in our daa and assumed λ=2.the graph would look qualiaively similar regardless of which player's hands we had used, and any λ>1, as he differences in he disribuion of profis for a given hand are relaively small across players. Figure 2: Example: Expeced Uiliy Gain for earnings from one hand (Player 12) Figure 2 shows he amoun of effor, in uils, ha he player would be willing o expend in order o draw he profis from a random hand. Figure 2 (dashed line) also shows he same relaionship under a non-linear μ wih diminishing sensiiviy, i.e., α < 1. We see he same V-shaped paern of risk preferences, bu wih some difference in he slopes. Diminishing sensiiviy increases he lef-hand side derivaive a zero, break-even effec for small losses, since he convexiy in losses creaes risk-lovingness. However i causes he break-even effec o decline a a faser rae, as he value funcion is becoming less and less convex as losses moun. 10 Conversely for gains, he decrease in risk aversion is iniially smaller han in he linear case, so he righ-hand side derivaive a zero is smaller. This is because more of he gamble is covered by he concave gains porion of he value funcion and no he convex losses porion. However farher ou in he gains domain, he house money effec is enhanced, so ha H(Π(x)) has a second derivaive closer o zero. This is a resul of he individual moving o a more linear secion of he value funcion. We could also consider he reference poin o be saring wealh plus he amoun ha he player expecs o win when hey si down, as in he KR model. In his case, he player's risk aversion would increase wih heir session profis unil hey go o heir expeced winnings, a which poin hey would hen sar o decrease. Tha is, he graph above would jus shif o he righ, placing he minimum a expeced winnings insead of a zero. 3. Our Daa Se Our daa consiss of 9,120,559 No Limi Hold 'Em poker hands played online on he Full Til Poker sie played beween March and Sepember All of hese hands were played a cash ables wih blinds of $2 10 For large enough losses, he slope can urn posiive, so ha more losses make he individual more risk averse. 11 Full Til is one of he wo large online sies which accep US players, he oher being Poker Sars. We used Full Til because gahering daa is easier on his sie. Alhough Full Til was one of several major online poker companies o have heir gambling license emporarily revoked in lae 2011, his did no affec he gahering or accuracy of our daase which was colleced wo years prior. 10

11 12, 13, 14 and $4, wih a maximum of nine possible players seaed a he able. The amouns of money being won and los a he $2/$4 level are significan enough ha he bes players could use heir winnings as heir sole source of income. The mean hourly profi among our players is $ While $2 and $4 may sound like rivial amouns, he amoun of variance each player faces is subsanial due o he unlimied being srucure of No Limi Hold 'Em. Even hough he po sars relaively small wih jus $6, hands in which over $500 ransfers from one player o anoher are no uncommon. Furhermore, players are able o play many ables a once in order o increase heir produciviy. Players in our sample ypically play beween six and welve ables a once. Thus he hourly variance in winnings is quie large. The mean sandard deviaion in winnings is $22.80 per hand. Since player play hundreds of hands per hour, his ranslaes o an average hourly sandard deviaion of $ Our daa se includes approximaely sixy percen of all nine-player-maximum, $2/$4 Hold 'Em hands played on Full Til during his ime period. 15 We conduc our analysis on he 100 players wihin his sample who played he mos number of hands. The firs reason for our sample selecion is pracical, in ha hese are he players on whom we cerainly have enough daa o do an individual-level analysis. Furhermore, hese are also he players who should be leas likely o exhibi some kind of bias in heir decision-making, since playing as ofen as hey do, hey can fairly be considered expers. The players in our sample are for he mos par, making profiable decisions in heir poker play. They also should have enough experience o know rules of basic probabiliy, including calculaing he probabiliies of differen cards being deal, or he likelihood of anoher player holding a paricular hand given heir acions. Players commiing sysemaic errors in calculaions of his sor would be unlikely o remain profiable over he large ime frame ha we have covered in our daa. We acquired our daa from a sie ha colleced i for he purpose of selling i o players. Because of server space limiaions of his sie, hey gahered only abou sixy percen of he oal hands played during his ime period. The main effec of missing fory percen of he daa is o creae aenuaion bias, so ha he esimaes we presen here ac as a lower bound on he effecs we describe (Li and Ryan, 2004). Our daa follows a given subse of ables coninuously, and hen randomly swiches o anoher subse, independenly of he winnings of he players in he sample. Given he number of ables ypically running a once a he $2/$4 sakes level, and he number of ables being played a a ime by high-frequency players, he chance of a high-frequency player playing a session ha is nowhere in our daa is very low. However, since a any given ime we only observe some fracion of he hands played by a given player, we do ge a noisy esimae of ha player's winnings. This noise biases our coefficien esimaes owards zero. An advanage of using daa from he $2/$4 ables is ha sakes are small enough ha here are many ables running all he ime, so ha here is no shorage of daa o be gahered. This also means ha he player pool is quie large, so ha we have less worry ha players are leaving he game due o oher paricularly bad players hemselves enering and leaving. A larger games, his is quie common. For example, an enire $10/$20 game could sar because one bad player sis down. Once ha player leaves, all he res of he players 12 Cash games are easier o analyze han ournamen games, since in a cash game, a player who is risk neural over money should also be risk neural over chips. This is no necessarily he case in a ournamen, for a number of reasons. 13 Blinds refer o he required bes ha wo players a he able mus conribue in order o play he curren hand - in his case $2 is he ``small blind and $4 is he ``big blind. Blinds roae around he able so ha each player akes urns conribuing his required amoun o he po. 14 Typically he ables are full or close o full. The average number of players per hand was For Hold 'Em rules, see 11

12 will also leave, no waning o play each oher. A $2/$4, he raio of regulars o casual players is much lower, so ha he regulars can always find profiable ables should hey wan o play. Full Til also hoss wo-player and six-player maximum games, and a variey of differen blind levels. $2/$4 is he highes blind level a which here are many ables running around he clock. The sie also has oher ypes of poker, such as Po Limi Omaha and Limi Hold 'Em. 16 However, players end o play only one game and make i heir regular game. Since we do no have comprehensive daa from hese oher games, we canno conclusively prove his o be he case using our daa se. However conversaions wih online poker players, commens on online message boards, and aggregae player daa from sies which monior individual players sugges srongly ha his is he case. 17 The raionale is ha here is an invesmen required o learn each ype of game, including adaping o differen syles of play a differen sakes levels of he same game. As players experience long erm gains or losses, hey may decide o move up or down in sakes, bu a a given poin in ime, hey ypically play jus one class of game. This allows us o claim wih confidence ha he 100 players we use in our sample are $2/$4 No Limi Hold Em regulars, and ha when we observe a player o have sopped playing a he $2/$4 nine-person No Limi Hold 'Em ables, he has ended his poker-playing session. Table 1 presens summary saisics on he mos frequen 100 players in our daa se. These are he players on whom we will focus all of our analysis. As seen in Table 1, hese players invesed significan ime in playing, wih an average of over 300 hours over he course of 7 monhs. They were also rewarded handsomely for his effor, wih an average hourly wage of nearly 40 dollars, or roughly wice he hourly wage an undergraduae could expec in an experimenal lab. As such, hese players are well-incenivized expers, capable of consisenly making decisions ha produce expeced profis when analyzing risky choices. 4. The Break-Even Effec, Locking in he Win Effec, and he House Money Effec: Sopping Decisions 4.1 Correlaion Coefficiens The principal finding of our analysis of sopping decisions is ha a player's winnings in a session are negaively correlaed wih his likelihood of ending he session. Tha is, when he has los money in he session, he more he has los, he more likely he is o coninue o play in an effor o ge back o even (he break-even effec ). When he has won money, he more he has won, he more likely he is o sop playing in order o preserve he win. 16 Limi Hold 'Em has he same srucure as No Limi Hold 'Em, excep ha bes mus be made in fixed incremens. For Omaha rules see 17 See for insance The oher worry is ha players may have muliple accouns, so ha when ending a session under one screen name, hey coninue o play under anoher name. Again we canno rule his ou, bu he sie, as well as he poker communiy, srongly discourages his muli-accouning, and akes seps o preven i. 12

13 For our analysis, we consider a session o have ended when a player does no play any poker hand for 6 hours consecuively. Alhough our choice of 6 hours as a precise cuoff poin is arbirary, we consider 6 hours a reasonable minimum inerval of ime afer which players are likely o hink of heir playing as a new session once siing down a he ables again. We have also run he analysis requiring a 10 hour break o end a session, wih no qualiaive impac on resuls. This is mainly because here are no very many breaks beween 6 and 10 hours long. Players ofen ake smaller breaks, in which hey do no play a hand for fifeen minues or an hour a a ime. In hese cases, for breaks longer han 5 minues bu less han six hours, we subrac he break ime from he lengh of a player's session, bu do no consider he session o have ended. So for example, if a player plays from 6:00 am o 9:00 am, hen from 11:00 am o 1 pm, hen from 10 pm o midnigh, he has played wo sessions, he firs lasing five hours, he second lasing wo hours. Our assumpion, from a Prospec Theory poin of view, is ha he player brackes around each session. Tha is, he considers gains and losses wihin each session, hen sars each session afresh a zero. His gain-loss uiliy in one session is herefore unaffeced by his profis in an earlier session. In his sense, preferences are separable across sessions. The firs way we invesigae he relaionship beween hese gains and losses is by looking a he correlaion beween session lengh (in minues) and session winnings. Since 96 of he 100 players in our sample are winning players (i.e., heir oal winnings are posiive), if here were no relaionship beween a player's likelihood of quiing and heir wihin-session winnings, we would expec he correlaion beween session winnings and session lengh o be posiive. In fac, many of he players, even some of he bigges winners, have a negaive correlaion beween session winnings and session lengh. The average correlaion across all 100 players is slighly posiive, a To evaluae wheher his is more or less han one migh expec if session sopping decisions were independen of winnings, we also calculaed correlaion coefficiens using simulaed daa. To do his, for each player, we replaced he winnings from each hand wih winnings from a hand drawn randomly from all of ha paricular player's hands. Because we sample from he enire disribuion randomly while keeping he session lenghs he same as in he acual daa, he quiing decisions in our simulaed daa canno be based on how much he player is up or down wihin he simulaion. Figure 3 shows a scaer plo of he mean of 1000 simulaions for each player on he x-axis, along wih he acual correlaion coefficiens found in he daa on he y-axis. Mos of he poins lie below he 45-degree line in black, indicaing ha he correlaion beween session winnings and lengh in he daa is lower han in he simulaed daa. The red line indicaes he regression line. The consan is negaive and saisically significan. 13

14 Fifeen players have correlaion coefficiens significanly less han heir simulaed values a he five percen level, while only four have a correlaion significanly above heir simulaed value. The average -saisic across all 100 players is The probabiliy of drawing an average his far from zero on 100 draws from he sandard normal disribuion is 0.001%. This indicaes eiher ha players end o coninue heir sessions longer when hey are down and cu hem shor when hey are up, or ha players end o do poorly when hey play oo long, or boh. In he nex subsecion we show ha i is he former effec ha drives his resul. 4.2 Cox Likelihood Analysis In order o analyze quiing decisions in more deail, we esimae a hazard model for each player. Uilized frequenly in he biomedical lieraure, hazard models have been previously uilized o model indusry exi of new firms (Audresch and Mahmood, 1995), employee aendance and absences (Johansson and Palme, 2005), and he effec of labor marke programs on unemploymen duraion (Lalive, Van Ours and Zweimuller, 2008). Since we are ineresed in he effec of winnings on he likelihood of quiing more han any parameric relaionship beween ime iself and quiing, we implemen he Cox proporional hazard (Cox, 1972), which has he advanage ha i does no require specifying an underlying hazard rae wih respec o ime, and is ideal for a primary ineres in he relaionship of he covariaes o quiing han he hazard rae iself. We focus primarily on individual player-level esimaes because we have sufficienly rich daa, and o avoid concerns abou unobserved individual heerogeneiy (Elbers and Ridder, 1982 ; Heckman and Singer, 1984). The Cox model allows for he effec of ime on duraion o be arbirary, as long as we assume ha i is muliplicaive in he overall hazard funcion, and idenical across sessions for a given player (noe ha since our regressions are a he individual player level, across-player heerogeneiy is fully allowed). Given ha our objecive is o pick up direcions of deparure from he case of neoclassical expeced uiliy model as a funcion of curren wealh, we find his assumpion reasonable for our purposes. Non-proporional hazard models on he oher hand, require some specific assumpions abou how he passing of ime srucurally affecs he hazard rae (such as he frequenly used Weibull disribuion). We found hese assumpions o be oo sringen for our daa, as we esimaed significan coefficiens using he simulaed daa described in he 14

15 previous secion when none should have been found in heory. This lead us o conclude ha he frequenly used forms of parameric hazard funcion are likely misspecified in our conex. The Cox model, by conras, did no esimae significan coefficiens in our random simulaions, allowing confidence ha effecs deeced in esimaions using our acual daa are real. In order o consruc he likelihood funcion, we order a player's sessions 1,2,...,n,...,N from shores o longes. We have a vecor of ime-varying variables X(n,) ha we hypohesize affecs he hazard rae of he session. Now le us suppose ha session 1 ends a ime T(1). Given ha a session did end a ime T(1), we can model he likelihood of session 1 being he session ending a ha ime as: exp( X N j = n n, T ( n) exp( X β ) g( T (1)) j, T ( n) β ) g( T (1)) The g(t(1)) funcion, which represens he effec ha ime has on he hazard rae (ofen modeled as a Weibull disribuion in oher applicaions) cancels, and we are lef wih jus funcions of he covariaes of ineres. While his allows us flexibiliy, he cos is loss of daa. To see why, assume ha he firs session ends a = 1 and he second session ends a = 3. While we may have daa on many sessions for = 2, we are unable o conclude anyhing from hese coninuaion decisions, since we canno say if hese decisions were due o effecs from he covariaes in X, or somehing o do wih g(2), i.e., he effec of ime. I could be ha here is some srong impac o coninue precisely a = 2, which is independen of he covariaes in X. As a resul we have o essenially disregard all daa from coninuaion decisions made a duraions where no oher session ends. 18 The enire likelihood funcion for a player is consruced by muliplying he above raios for each session we observe. Le T i (s) be he lengh of session s for player i. Le S i be he oal number of sessions played by player i. Le Π i,s, be he winnings in session s for player i a ime. A player's enire likelihood funcion is hen: L ( β ) i i S = s 1 = i i exp( X S j= s i, s, T ( s) exp( X i i, j, T β i ) i ( s) β i ) X s,, he vecor of independen variables, will vary according o he specificaion a hand, bu in general will be some funcion of Π i,s,. Noice ha we esimae coefficiens separaely for every player. I is possible o esimae a kind of pooled Cox regression, where n would index all player sessions. However his would assume no only ha he effec of ime is proporional, bu ha i is uniform across players, which may no be a very realisic assumpion. There is also considerable possibiliy of, and ineres in, heerogeneiy in he β i coefficiens, which is los in a pooled regression. These facs, ogeher wih he compuaional difficuly of a pooled regression due o he amoun of daa, led us o working wih he individual-level esimaes. In Secion 4.3 we address he possibiliy of classifying players by ypes. Thus he coefficiens in β i will indicae how he differen variables in X s, affec he likelihood of a session ending for player i. Noe ha because our players are playing several ables a once, and we conduc he hazard analysis wih ime as he uni of observaion, we canno meaningfully conrol for able-level covariaes such as sack size, posiion a he able, ec. We do however include hese covariaes of ineres in our analysis of playing syle a 18 In general when doing Cox likelihood regressions, one mus worry abou ies in duraions. However since our daa is a he seconds level, he ime pariioning is fine enough such ha we have no ies in lenghs of sessions played by he same player. 15

16 he hands level in Secion 5. Our sandard errors are obained via jackknife esimaion, wih omied observaions a he session level as implied by he Cox model. 19 Figure 4: Coefficiens on winnings and losses, by individual player Prospec heory predics boh a break even effec and a house money effec. Tha is, players should be less likely o qui as hey lose more, bu as hey win more, hey should coninue playing longer as well. To es his, we allow for differen coefficiens on wins and losses. For his specificaion, X i,s, = [ (Π i,s, ) + (Π i,s, ) - ] Prospec heory would predic ha going away from zero in eiher direcion should decrease risk aversion, and herefore decrease a player's propensiy o qui. If his were he case, hen he coefficien on gains would be negaive, while he coefficien on losses would be posiive. Insead boh coefficiens end o be posiive. Since he individual esimaes for each player and heir associaed sandard errors are cumbersome o display for all 100 players, we summarize our esimaion resuls in he bubble plos in Figure The bubble plos show he coefficiens on winnings and losses in our baseline specificaion, wih each bubble represening an individual player, he size of he bubbles reflecing saisical significance of he winnings coefficien (plo on lef), and he saisical significance of he losses coefficien (plo on righ). Colors of he bubbles represen 19 We chose he jackknife approach o circumven any concerns abou consisen esimaion of he Hessian marix in our maximum likelihood esimaion. Compared o a boosrap approach, jackknifing has he advanage of only requiring a se number of esimaing rounds o obain an esimae of he sandard error. 20 Exac numerical esimaes and sandard errors are available on reques, bu we omi hem here since heir main message is beer represened graphically by Figure 4. 16

17 esimaed classificaion groups, which we explain in deail in he nex secion (4.3). Our primary resul is ha he likelihood of quiing ends o increase in he amoun won so far in he session, regardless of wheher he amoun won is posiive or negaive. This can be seen in Figure 4 by he fac ha mos of he larger bubbles are concenraed in he upper righ quadran of he plos. The coefficien on losses is posiive for 60 players, 32 of hem saisically significan a he 5% level. The endency owards posiive coefficiens is even sronger for gains. 67 players have a posiive coefficien on gains, 35 of hese significan. Thus, while we do see a break even effec, as Prospec Theory predics, insead of a house money effec, where players gamble freely wih found money, here is insead a preserve he win menaliy, where once a player has won some money, he is more likely o qui and book a win, raher han coninue playing and risk falling back o even or even possibly o a loss. 4.3 Classificaion Analysis While he esimaes from hese individual-level regressions show a general endency owards he house-money effec and a lock-in-he-win effec, here is clearly heerogeneiy across he players. Anoher useful way of describing his heerogeneiy is by creaing differen groups of behavioral ypes, forcing coefficiens o be equal wihin each esimaed ype, and assigning each player o he group ha bes describes his behavior. We follow he approach of El Gamal and Greher (1990), which classifies individual belief updaing processes using experimenal daa. When allowing for N classificaions, we choose N parameer vecors such ha: * * * { β, β,..., β } = 1 2 N max { β, β,..., β } ( R ) Table 2 presens he daa for 2, 3, and 4 group classificaions: 1 2 N 2 N 100 max βi { β1, β2,..., β N } i 1 ln( L ( β )) i i When only wo groups are allowed, he firs exhibis boh a break-even effec and a lock-in-he-win effec, he principal endencies in he daa we have been discussing. The second group exhibis a weaker break-even effec and a weak house-money effec, he sandard paern for loss-averse behavior. Adding in a hird group creaes a caegory for players whose behavior is exacly conrary o he main endency in he daa. Tha is, for his caegorizaion of players, winning ends o lead hem o play longer, while losing leads hem o qui sooner. This is he paern one migh expec from a player who undersands resuls o be some kind of signal of heir expeced profis in fuure hands played in ha session. These players are a minoriy, including only 22 of he 100 players. These 22 players come almos exclusively ou of he loss averse caegory when only wo groups are allowed. The populaion of his ype drops from 61 o 42, whereas he populaion of he firs group says relaively consan (drops from 39 o 36). Also, when his caegory is 17