Deal or No Deal? Decision Making under Risk in a Large-Payoff Game Show

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1 Deal o No Deal? Decision Making unde Risk in a Lage-Payoff Game Show Thiey Post, Matijn J. van den Assem, Guido Baltussen and Richad H. Thale * Published in the Ameican Economic Review, Mach 2008 (98:1), Abstact We examine the isky choices of contestants in the popula TV game show Deal o No Deal and elated classoom expeiments. Contay to the taditional view of expected utility theoy, the choices can be explained in lage pat by pevious outcomes expeienced duing the game. Risk avesion deceases afte ealie expectations have been shatteed by unfavoable outcomes o supassed by favoable outcomes. Ou esults point to efeence-dependent choice theoies such as pospect theoy, and suggest that path-dependence is elevant, even when the choice poblems ae simple and well-defined, and when lage eal monetay amounts ae at stake. JEL: D81 A WIDE RANGE OF THEORIES OF RISKY CHOICE have been developed, including the nomative expected utility theoy of John von Neumann and Oska Mogensten (1944) and the desciptive pospect theoy of Daniel Kahneman and Amos Tvesky (1979). Although isky choice is fundamental to vitually evey banch of economics, empiical testing of these theoies has poven to be difficult. Many of the ealiest tests such as those by Mauice Allais (1953), Daniel Ellsbeg (1961), and the ealy wok by Kahneman and Tvesky wee based on eithe thought expeiments o answes to hypothetical questions. With the ising populaity of expeimental economics, isky choice expeiments with eal monetay stakes have become moe popula, but because of limited budgets most expeiments ae limited to small stakes. Some expeimental studies ty to cicumvent this poblem by using small nominal amounts in developing counties, so that the subjects face lage amounts in eal tems; see, fo example, Hans P. Binswange (1980, 1981) and Steven J. Kachelmeie and Mohamed Shehata (1992). Still, the stakes in these expeiments ae typically not lage than one month s income and thus do not povide evidence about isk attitudes egading pospects that ae significant in elation to lifetime wealth. 1

2 Nonexpeimental empiical eseach is typically plagued by what amounts to joint hypothesis poblems. Reseaches cannot diectly obseve isk pefeences fo most eal-life poblems, because the tue pobability distibution is not known to the subjects and the subjects beliefs ae not known to the eseache. Fo example, to infe the isk attitudes of investos fom thei investment potfolios, one needs to know what thei beliefs ae egading the joint etun distibution of the elevant asset classes. Wee investos eally so isk avese that they equied an equity pemium of 7 pecent pe yea, o wee they supised by an unexpected numbe of favoable events o woied about catastophic events that neve occued? An additional complication aises because of the possible diffeence between isk and uncetainty: eal-life choices aely come with pecise pobabilities. In ode to cicumvent these poblems, some eseaches analyze the behavio of contestants in TV game shows, fo example Cad Shaks (Robet H. Getne, 1993), Jeopady! (Andew Metick, 1995), Illinois Instant Riches (Philip L. Hesch and Geald S. McDougall, 1997), Lingo (Roel M. W. J. Beetsma and Pete C. Schotman, 2001), Hoosie Millionaie (Connel R. Fullenkamp, Rafael A. Tenoio and Robet H. Battalio, 2003) and Who Wants to be a Millionaie? (Roge Hatley, Gauthie Lanot and Ian Walke, 2006). The advantage of game shows is that the amounts at stake ae lage than in expeiments and that the decision poblems ae often simple and bette defined than in eal life. The game show we use in this study, Deal o No Deal, has such desiable featues that it almost appeas to be designed to be an economics expeiment athe than a TV show. Hee is the essence of the game. A contestant is shown 26 biefcases which each contain a hidden amount of money, anging fom 0.01 to 5,000,000 (in the Dutch edition). The contestant picks one of the biefcases and then owns its unknown contents. Next, she selects 6 of the othe 25 biefcases to open. Each opened biefcase eveals one of the 26 pizes that ae not in he own biefcase. The contestant is then pesented a bank offe the oppotunity to walk away with a sue amount of money and asked the simple question: Deal o No Deal? If she says No Deal, she has to open five moe biefcases, followed by a new bank offe. The game continues in this fashion until the contestant eithe accepts a bank offe, o ejects all offes and eceives the contents of he own biefcase. The bank offes depend on the value of the unopened biefcases; if, fo example, the contestant opens high-value biefcases, the bank offe falls. This game show seems well-suited fo analyzing isky choice. The stakes ae vey high and wide-anging: contestants can go home as multimillionaies o pactically empty-handed. Unlike othe game shows, Deal o No Deal involves only simple stop-go decisions ( Deal 2

3 o No Deal ) that equie minimal skill, knowledge o stategy, and the pobability distibution is simple and known with nea-cetainty (the bank offes ae highly pedictable, as discussed late). Finally, the game show involves multiple game ounds, and consequently seems paticulaly inteesting fo analyzing path-dependence, o the ole of ealie outcomes. Richad H. Thale and Eic J. Johnson (1990) conclude that isky choice is affected by pio outcomes in addition to incemental outcomes due to decision makes incompletely adapting to ecent losses and gains. Although Deal o No Deal contestants neve have to pay money out of thei own pockets, they can suffe significant pape losses if they open high-value biefcases (causing the expected winnings to fall), and such losses may influence thei subsequent choices. (Thoughout this study we will use the tem outcomes to indicate not only monetay pay-offs, but also new infomation o changed expectations.) We examine the games of 151 contestants fom the Nethelands, Gemany and the United States in The game oiginated in the Nethelands and is now boadcast aound the wold. Although the fomat of Deal o No Deal is geneally simila acoss all editions, thee ae some notewothy diffeences. Fo example, in the daily vesions fom Italy, Fance and Spain, the banke knows the amounts in the biefcases and may make infomative offes, leading to stategic inteaction between the banke and the contestant. In the daily edition fom Austalia, special game options known as Chance and Supecase ae sometimes offeed at the discetion of the game-show poduce afte a contestant has made a Deal. These options would complicate ou analysis, because the associated pobability distibution is not known, intoducing a laye of uncetainty in addition to the pue isk of the game. Fo these easons, we limit ou analysis to the games played in the Nethelands, Gemany and the United States. The thee editions have a vey simila game fomat, apat fom substantial vaiation in the amounts at stake. While the aveage pize that can be won in the Dutch edition is oughly 400,000, the aveages in the Geman and US edition ae oughly 25,000 and 100,000, espectively. At fist sight, this makes the pooled dataset useful fo sepaating the effect of the amounts at stake fom the effect of pio outcomes. (Within one edition, the stakes ae stongly confounded with pio outcomes.) Howeve, coss-county diffeences in cultue, wealth and contestant selection pocedue could confound the effect of stakes acoss the thee editions. To isolate the effect of stakes on isky choice, we theefoe conduct classoom expeiments with a homogeneous student population. In these expeiments, we vay the pizes with a facto of ten, so that we can detemine if, fo example, 100 has the same subjective value when it lies below o above the initial expectations. 3

4 Ou findings ae difficult to econcile with expected utility theoy. The contestants choices appea to be diven in lage pat by the pevious outcomes expeienced duing the game. Risk avesion seems to decease afte ealie expectations have been shatteed by opening high-value biefcases, consistent with a beak-even effect. Similaly, isk avesion seems to decease afte ealie expectations have been supassed by opening low-value biefcases, consistent with a house-money effect. The othodox intepetation of expected utility of wealth theoy does not allow fo these effects, because subjects ae assumed to have the same pefeences fo a given choice poblem, iespective of the path taveled befoe aiving at this poblem. Ou esults point in the diection of efeence-dependent choice theoies, such as pospect theoy, and indicate that path-dependence is elevant, even when lage eal monetay amounts ae at stake. We theefoe popose a vesion of pospect theoy with a path-dependent efeence point as an altenative to expected utility theoy. Of couse, we must be caeful with ejecting expected utility theoy and embacing altenatives like pospect theoy. Although the standad implementation of expected utility theoy is unable to explain the choices of loses and winnes, a bette fit could be achieved with a nonstandad utility function that has convex segments (as poposed by, fo example, Milton Fiedman and Leonad J. Savage, 1948, and Hay Makowitz, 1952), and depends on pio outcomes. Theefoe, this study does not eject o accept any theoy. Rathe, ou main finding is the impotant ole of efeence-dependence and path-dependence, phenomena that ae not standad in typical implementations of expected utility, but common in pospect theoy. Any plausible explanation of the choice behavio in the game will have to account fo these phenomena. A theoy with static pefeences cannot explain why vaiation of the stakes due to the subject s fotune duing the game has a much stonge effect than vaiation in the initial stakes acoss diffeent editions of the TV show and expeiments. The emainde of this pape is oganized as follows. In Section I, we descibe the game show in geate detail. Section II discusses ou data mateial. Section III povides a fist analysis of the isk attitudes in Deal o No Deal by examining the bank offes and the contestants decisions to accept ( Deal ) o eject ( No Deal ) these offes. Section IV analyzes the decisions using expected utility theoy with a geneal, flexible-fom expo-powe utility function. Section V analyzes the decisions using pospect theoy with a simple specification that allows fo patial adjustment of the subjective efeence point that sepaates losses fom gains. This implementation of pospect theoy explains a mateial pat of what expected utility theoy leaves unexplained. Section VI epots esults fom classoom 4

5 expeiments in which students play Deal o No Deal. The expeiments confim the impotant ole of pevious outcomes and suggest that the isolated effect of the amounts at stake is limited compaed to the isolated effect of pevious outcomes. Section VII offes concluding emaks and suggestions fo futue eseach. Finally, an epilogue gives a synopsis of othe Deal o No Deal studies that became available afte ou study was fist submitted to the Ameican Economic Review in Octobe I. Desciption of the Game Show The TV game show Deal o No Deal was developed by the Dutch poduction company Endemol and was fist aied in the Nethelands in Decembe The game show soon became vey popula and was expoted to dozens of othe counties, including Gemany and the United States. The following desciption applies to the Dutch episodes of Deal o No Deal. Except fo the monetay amounts, the stuctue of the main game is simila in the Geman and US vesions used in this study. Each episode consists of two pats: an elimination game based on quiz questions in ode to select one finalist fom the audience, and a main game in which this finalist plays Deal o No Deal. Audience membes have not been subjected to an extensive selection pocedue: playes in the national lottey sponsoing the show ae invited to apply fo a seat and tickets ae subsequently andomly distibuted to applicants. Only the main game is the subject of ou study. Except fo detemining the identity of the finalist, the elimination game does not influence the couse of the main game. The selected contestant has not won any pize befoe enteing the main game. The main game stats with a fixed and known set of 26 monetay amounts anging fom 0.01 to 5,000,000, which have been andomly allocated ove 26 numbeed and closed biefcases. One of the biefcases is selected by the contestant and this biefcase is not to be opened until the end of the game. The game is played ove a maximum of nine ounds. In each ound, the finalist chooses one o moe of the othe 25 biefcases to be opened, evealing the pizes inside. Next, a banke ties to buy the biefcase fom the contestant by making he an offe. Contestants have a few minutes to evaluate the offe and to decide between Deal and No Deal, and may consult a fiend o elative who sits neaby. 1 The emaining pizes and the cuent bank offe ae displayed on a scoeboad and need not be memoized by the contestant. If the 1 In the US vesion and in the second Geman seies, thee o fou fiends and/o elatives sit on stage neaby the contestant. In the Dutch vesion and in the fist Geman seies, only one peson accompanies the contestant. 5

6 contestant accepts the offe ( Deal ), she walks away with this sue amount and the game ends; if the contestant ejects the offe ( No Deal ), the game continues and she entes the next ound. In the fist ound, the finalist has to select six biefcases to be opened, and the fist bank offe is based on the emaining 20 pizes. The numbes of biefcases to be opened in the maximum of eight subsequent ounds ae 5, 4, 3, 2, 1, 1, 1, and 1. Accodingly, the numbe of pizes left in the game deceases to 15, 11, 8, 6, 5, 4, 3, and 2. If the contestant ejects all nine offes she eceives the pize in he own biefcase. Figue 1 illustates the basic stuctue of the main game. [INSERT FIGURE 1 ABOUT HERE] To povide futhe intuition fo the game, Figue 2 shows a typical example of how the main game is displayed on the TV sceen. A close-up of the contestant is shown in the cente and the oiginal pizes ae listed to the left and the ight of the contestant. Eliminated pizes ae shown in a dak colo and emaining pizes ae in a bight colo. The bank offe is displayed at the top of the sceen. [INSERT FIGURE 2 ABOUT HERE] As can be seen on the scoeboad, the initial pizes ae highly dispesed and positively skewed. Duing the couse of the game, the dispesion and the skewness geneally fall as moe and moe biefcases ae opened. In fact, in the ninth ound, the distibution is pefectly symmetic, because the contestant then faces a 50/50 gamble with two emaining biefcases. Bank Behavio Although the contestants do not know the exact bank offes in advance, the banke behaves consistently accoding to a clea patten. Fou simple ules of thumb summaize this patten: Rule 1. Bank offes depend on the value of the unopened biefcases: when the lowe (highe) pizes ae eliminated, the aveage emaining pize inceases (deceases) and the banke makes a bette (wose) offe. Rule 2. The offe typically stats at a low pecentage (usually less than 10 pecent) of the aveage emaining pize in the fist ound and gadually inceases to 100 6

7 pecent in the late ounds. This stategy obviously seves to encouage contestants to continue playing the game and to gadually incease excitement. Rule 3. The offes ae not infomative, that is, they cannot be used to detemine which of the emaining pizes is in the contestant s biefcase. Only an independent audito knows the distibution of the pizes ove the biefcases. Indeed, thee is no coelation between the pecentage bank offe and the elative value of the pize in the contestant s own biefcase. Rule 4. The banke is geneous to loses by offeing a elatively high pecentage of the aveage emaining pize. This patten is consistent with path-dependent isk attitudes. If the game-show poduce undestands that isk avesion falls afte lage losses, he may undestand that high offes ae needed to avoid tivial choices and to keep the game entetaining to watch. Using the same easoning, we may also expect a pemium afte lage gains; this, howeve, does not occu, pehaps because with lage stakes, the game is aleady entetaining. Section III gives desciptive statistics on the bank offes in ou sample and Section IV pesents a simple model that captues the ules of thumb noted above. The key finding is that the bank offes ae highly pedictable. II. Data We examine all Deal o No Deal decisions of 151 contestants appeaing in episodes aied in the Nethelands (51), Gemany (47), and the United States (53). The Dutch edition of Deal o No Deal is called Miljoenenjacht (o Chasing Millions ). The fist Dutch episode was aied on Decembe 22, 2002 and the last in ou sample dates fom Januay 1, In this time span, the game show was aied 51 times, divided ove eight seies of weekly episodes and fou individual episodes aied on New Yea s Day, with one contestant pe episode. A distinguishing featue of the Dutch edition is the high amounts at stake: the aveage pize equals oughly 400,000 ( 391,411 in episode 1 47 and 419,696 in episode 48 51). Contestants may even go home with 5,000,000. The fact that the Dutch edition is sponsoed by a national lottey pobably explains why the Dutch fomat has such lage pizes. The lage pizes may also have been pefeed to stimulate a successful launch of the show and to pave the way fo expoting the fomula aboad. Pat of the 51 shows wee ecoded on videotape by the authos and tapes of the emaining shows wee obtained fom the Dutch boadcasting company TROS. 7

8 In Gemany, a fist seies of Deal o No Deal Die Show de GlücksSpiale stated on June 23, 2005 and a second seies began on June 28, Apat fom the numbe of pizes, the two seies ae vey simila. The fist seies uses 20 pizes instead of 26 and is played ove a maximum of 8 game ounds instead of 9. Because these 8 ounds ae exactly equal to ound 2 9 of the egula fomat in tems of the numbe of emaining pizes and in tems of the numbe of biefcases that have to be opened, we can analyze this seies as if the fist ound has been skipped. Both seies have the same maximum pize ( 250,000) and the aveages of the initial set of pizes ae pactically equal ( 26,347 vesus 25,003 espectively). In the emainde of the pape we will conside the two Geman seies as one combined subsample. The fist seies was boadcast weekly and lasted fo 10 episodes, each with two contestants playing the game sequentially. The second seies was aied eithe once o twice a week and lasted fo 27 episodes, with one contestant pe episode, binging the total numbe of Geman contestants in ou sample to 47. Copies of the fist seies wee obtained fom TV station Sat.1 and fom Endemol s local poduction company Endemol Deutschland. The second seies was ecoded by a fiend of the authos. In the United States, the game show debuted on Decembe 19, 2005, fo five consecutive nights and etuned on TV on Febuay 27, This second seies lasted fo 34 episodes until ealy June The 39 episodes combined coveed the games of 53 contestants, with some contestants stating in one episode and continuing thei game in the next. The egula US fomat has a maximum initial pize of $1,000,000 (oughly 800,000) and an aveage of $131,478 ( 105,182). In the games of six contestants, howeve, the top pizes and aveages wee lage to mak the launch and the finale of the second seies. All US shows wee ecoded by the authos. US Dollas ae tanslated into Euos by using a single fixed ate of 0.80 pe $ (the actual exchange ate was within 5 pecent of this ate fo both the 2005 and 2006 peiods). Fo each contestant, we collected data on the eliminated and emaining pizes, the bank offes, and the Deal o No Deal decisions in evey game ound, leading to a panel dataset with a time-seies dimension (the game ounds) and a coss-section dimension (the contestants). We also collected data on each contestant s gende, age and education. Age and education ae often evealed in an intoductoy talk o in othe convesations duing the game. 2 An ealie edition called De MillionenDeal stated on May 1, The initial aveage pize was 237,565 and the lagest pize was 2,000,000. This edition howeve lasted fo only 6 episodes and is theefoe not included hee. 8

9 The level of education is coded as a dummy vaiable, with a value of 1 assigned to contestants with a bachelo degee level o highe (including students) o equivalent wok expeience. Although a contestant s level of education is usually not explicitly mentioned, it is often clea fom the stated pofession. We estimate the missing values fo age based on the physical appeaance of the contestant and infomation evealed in the intoductoy talk, fo example, the age of childen. Howeve, age, gende and education do not have significant explanatoy powe in ou analysis. In pat o in whole, this may eflect a lack of sampling vaiation. Fo example, duing the game, the contestant is pemitted to consult with fiends, family membes, o spouse, and theefoe decisions in this game ae in effect taken by a couple o a goup, mitigating the ole of the individual contestant s age, gende o education. Fo the sake of bevity, we will pay no futhe attention to the ole of contestant chaacteistics. Moeove, pio outcomes ae andom and unelated to chaacteistics and theefoe the chaacteistics pobably would not affect ou main conclusions about path-dependence, even if they would affect the level of isk avesion. Table 1 shows summay statistics fo ou sample. Compaed to the Geman and US contestants, the Dutch contestants on aveage accept lowe pecentage bank offes (76.3 pecent vesus 91.8 and 91.4 pecent) and play oughly thee fewe game ounds (5.2 vesus 8.2 and 7.7 ounds). These diffeences may eflect unobseved diffeences in isk avesion due to diffeences in wealth, cultue o contestant selection pocedue. In addition, inceasing elative isk avesion (IRRA) may help to explain the diffeences. As the Dutch edition involves much lage stakes than the Geman and US editions, a modest incease in elative isk avesion suffices to yield sizeable diffeences in the accepted pecentages. Futhemoe, the obseved diffeences in the numbe of ounds played ae inflated by the behavio of the banke. The pecentage bank offe inceases with elatively small steps in the late game ounds and consequently a modest incease in elative isk avesion can yield a lage eduction in the numbe of game ounds played. Thus, the diffeences between the Dutch contestants on the one hand and the Geman and US contestants on the othe hand ae consistent with modeate IRRA. [INSERT TABLE 1 ABOUT HERE] Coss-County Analysis Apat fom the amounts at stake, the game show fomat is vey simila in the thee counties. Still, thee ae some diffeences in how contestants ae chosen to play that may 9

10 ceate diffeences in the contestant pool. In the Dutch and Geman episodes in ou sample thee is a peliminay game in which contestants answe quiz questions, the winne of which gets to play the main game we study. One special featue of the Dutch edition is the existence of a bail-out offe at the end of the elimination game: just befoe a last, decisive question, the two emaining contestants can avoid losing and leaving empty-handed by accepting an unknown pize that is announced to be woth at least 20,000 (appoximately 5 pecent of the aveage pize in the main game) and typically tuns out to be a pize such as a wold tip o a ca. If the moe isk-avese pe-finalists ae moe likely to exit the game at this stage, the Dutch finalists might be expected to be less isk avese on aveage. In the United States, contestants ae not selected based on an elimination game but athe the poduce selects each contestant individually, and the selection pocess appeas to be based at least in pat on the appeaance and pesonalities of the contestants. (The Web site fo the show tells pospective contestants to send a video of themselves and thei poposed accompanying fiends and elatives. The show also conducts open casting calls.) Contestants (and thei fiends) thus tend to be attactive and lively. Anothe concen is that iche and moe isk-seeking people may be moe willing to spend time attempting to get onto lage-stake editions than onto small-stake editions. To cicumvent these poblems, Section VI complements the analysis of the TV shows with classoom expeiments that use a homogeneous student population. III. Peliminay Analysis To get a fist glimpse of the isk pefeences in Deal o No Deal, we analyze the offes made by the banke and the contestants decisions to accept o eject these offes in the vaious game ounds. Seveal notable featues of the game can be seen in Table 2. Fist, the banke becomes moe geneous by offeing highe pecentages as the game pogesses ( Rule 2 ). The offes typically stat at a small faction of the aveage pize and appoach 100 pecent in the late ounds. The stong similaity between the pecentages in the Dutch edition (panel A), the Geman edition (panel B) and the US edition (panel C) suggest that the banke behaves in a simila way acoss the thee editions. 3 The numbe of emaining contestants in evey ound clealy shows that the Dutch contestants tend to stop ealie and accept elatively lowe bank offes than the Geman and US contestants do. Again, this may eflect the substantially lage stakes in the Dutch edition, o, altenatively, unobseved diffeences in isk avesion due to 3 A spokesman fom Endemol, the poduction company, confimed that the guidelines fo bank offes ae the same fo all thee editions included in ou sample. 10

11 diffeences in wealth, cultue o contestant selection pocedue. Thid, the contestants geneally exhibit what might be called only modeate isk avesion. In the US and Geman sample, all contestants keep playing until the bank offe is at least half the expected value of the pizes in the unopened biefcases. In ound 3 in the Nethelands, 20 pecent of the contestants (10 out of 51) do accept deals that aveage only 36 pecent of the expected value of the unopened biefcases, albeit at stakes that exceed 400,000. Many contestants tun down offes of 70 pecent o moe of amounts exceeding 100,000. Fouth, thee can be wide discepancies, even within a county, in the stakes that contestants face. In the Dutch show, contestants can be playing fo many hundeds of thousands of Euos, down to a thousand o less. In the late ounds, the contestant is likely to face elatively small stakes, as a consequence of the skewness of the initial set of pizes. [INSERT TABLE 2 ABOUT HERE] It is not appaent fom this table what effect the paticula path a playe takes can have on the choices she makes. To give an example of the decisions faced by an unlucky playe, conside poo Fank, who appeaed in the Dutch episode of Januay 1, 2005 (see Table 3). In ound 7, afte seveal unlucky picks, Fank opened the biefcase with the last emaining lage pize ( 500,000) and saw the expected pize tumble fom 102,006 to 2,508. The banke then offeed him 2,400, o 96 pecent of the aveage emaining pize. Fank ejected this offe and play continued. In the subsequent ounds, Fank delibeately chose to ente unfai gambles, to finally end up with a biefcase woth only 10. Specifically, in ound 8, he ejected an offe of 105 pecent of the aveage emaining pize; in ound 9, he even ejected a cetain 6,000 in favo of a 50/50 gamble of 10 o 10,000. We feel confident to classify this last decision as isk-seeking behavio, because it involves a single, simple, symmetic gamble with thousands of Euos at stake. Also, unless we ae willing to assume that Fank would always accept unfai gambles of this magnitude, the only easonable explanation fo his choice behavio seems to be a eaction to his misfotune expeienced ealie in the game. [INSERT TABLE 3 ABOUT HERE] In contast, conside the exhilaating ide of Susanne, an extemely fotunate contestant who appeaed in the Geman episode of August 23, 2006 (see Table 4). Afte a seies of vey lucky picks, she eliminated the last small pize of 1,000 in ound 8. In ound 9, she then 11

12 faced a 50/50 gamble of 100,000 o 150,000, two of the thee lagest pizes in the Geman edition. While she was concened and hesitant in the ealie game ounds, she decidedly ejected the bank offe of 125,000, the expected value of the gamble; a clea display of iskseeking behavio and one that poved fotuitous in this case as she finally ended up winning 150,000. [INSERT TABLE 4 ABOUT HERE] Thus both unlucky Fank and lucky Susanne exhibit vey low levels of isk avesion, even isk-seeking, wheeas most of the contestants in the shows ae at least modeately isk avese. Fank s behavio is consistent with a beak-even effect, a willingness to gamble in ode to get back to some peceived efeence point. Susanne s behavio is consistent with a house-money effect, an inceased willingness to gamble when someone thinks she is playing with someone else s money. To systematically analyze the effect of pio outcomes such as the exteme ones expeienced by Fank and Suzanne, we fist develop a ough classification of game situations in which the contestant is classified as a lose o a winne and analyze the decisions of contestants in these categoies sepaately. Ou classification takes into account the downside isk and upside potential of ejecting the cuent bank offe. A contestant is a lose if he aveage emaining pize afte opening one additional biefcase is low, even if the best-case scenaio of eliminating the lowest emaining pize would occu. Using x fo the cuent aveage, the aveage emaining pize in the bestcase scenaio is: (1) BC = n x n x 1 min whee n stands fo the numbe of emaining biefcases in game ound = 1, L, 9 and fo the smallest emaining pize. Similaly, winnes ae classified by the aveage emaining pize in the wost-case scenaio of eliminating the lagest emaining pize, max x : min x (2) WC = n x n x 1 max 12

13 Moe specifically, we classify a contestant in a given game ound as a lose if belongs to the wost one-thid fo all contestants in that game ound and as a winne if WC belongs to the best one-thid. 4 Game situations that satisfy neithe of the two conditions (o both in ae occasions) ae classified as neutal. Of couse, thee ae numeous ways one could allocate playes into winne and lose categoies. The esults we show ae obust to othe classification schemes, povided that the classification of winnes accounts fo the downside isk of continuing play: the house-money effect a deceased isk avesion afte pio gains is weak if incemental losses can exceed pio gains. Fo example, patitioning on just the cuent aveage ( x ) does not distinguish between situations with diffeent dispesion aound that aveage, and theefoe takes no account of the downside isk of continuing play. Table 5 illustates the effect of pevious outcomes on the contestants choice behavio. We see that, compaed to contestants who ae in the neutal categoy, both winnes and loses have a stonge tendency to continue play. While 31 pecent of all Deal o No Deal choices in the neutal goup ae Deal in the Dutch sample, the Deal pecentage is only 14 pecent fo loses despite the geneous offes they ae pesented ( Rule 4 ). The low Deal pecentage fo loses suggests that isk avesion deceases when contestants have been unlucky in selecting which biefcases to open. In fact, the stong loses in ou sample geneally exhibit isk-seeking behavio by ejecting bank offes in excess of the aveage emaining pize. The low Deal pecentage could be explained in pat by the smalle stakes faced by loses and a lowe isk avesion fo small stakes, o inceasing elative isk avesion (IRRA). Howeve, the loses geneally still have at least thousands o tens of thousands of Euos at stake and gambles of this magnitude ae typically associated with isk avesion in othe empiical studies (including othe game show studies and expeimental studies). Also, if the stakes explained the low isk avesion of loses, we would expect a highe isk avesion fo winnes. Howeve, isk avesion seems to decease when contestants ae lucky and have eliminated low-value biefcases. The Deal pecentage fo winnes is 25 pecent, below the 31 pecent fo the neutal goup. BC 4 To account fo the vaiation in the initial set of pizes within an edition (see Section II), BC and BW ae scaled by the initial aveage pize. 13

14 Inteestingly, the same patten aises in all thee counties. The oveall Deal pecentages in the Geman and US editions ae lowe than in the Dutch edition, consistent with modeate IRRA and the substantially smalle stakes. Within evey edition, howeve, the loses and winnes have elatively low Deal pecentages. These esults suggest that pio outcomes ae an impotant deteminant of isky choice. This is inconsistent with the taditional intepetation of expected utility theoy in which the pefeences fo a given choice poblem do not depend on the path taveled befoe aiving at the choice poblem. By contast, path-dependence can be incopoated quite natually in pospect theoy. The lowe isk avesion afte misfotune is eminiscent of the beak-even effect, o decision makes being moe willing to take isks due to incomplete adaptation to pevious losses. Similaly, the elatively low Deal pecentage fo winnes is consistent with the house-money effect, o a lowe isk avesion afte ealie gains. Obviously, this peliminay analysis of Deal pecentages is athe cude. It does not specify an explicit model of isky choice and it does not account fo the pecise choices (bank offes and emaining pizes) the contestants face. Futhemoe, thee is no attempt at statistical infeence o contolling fo confounding effects at this stage of ou analysis. The next two sections use a stuctual choice model and a maximum-likelihood methodology to analyze the Deal o No Deal choices in geate detail. [INSERT TABLE 5 ABOUT HERE] IV. Expected Utility Theoy This section analyzes the obseved Deal o No Deal choices with the standad expected utility of wealth theoy. The choice of the appopiate class of utility functions is impotant, because pefeences ae evaluated on an inteval fom cents to millions. We do not want to estict ou analysis to a classical powe o exponential utility function, because it seems too estictive to assume constant elative isk avesion (CRRA) o constant absolute isk avesion (CARA) fo this inteval. To allow fo the plausible combination of inceasing elative isk avesion (IRRA) and deceasing absolute isk avesion (DARA), we employ a vaiant of the flexible expo-powe family of Atanu Saha (1993) that was used by Mohammed Abdellaoui, Caolina Baios and Pete P. Wakke (2007) and by Chales A. Holt and Susan K. Lauy (2002): 14

15 (3) 1 exp( α ( W u ( x) = α + x) 1 β ) In this function, thee paametes ae unknown: the isk avesion coefficients α and β, and the initial wealth paamete W. The classical CRRA powe function aises as the limiting case whee α 0 and the CARA exponential function aises as the special case whee β = 0. Theoetically, the coect measue of wealth should be lifetime wealth, including the pesent value of futue income. Howeve, lifetime wealth is not obsevable and it is possible that contestants do not integate thei existing wealth with the payoffs of the game. Theefoe, we include initial wealth as a fee paamete in ou model. We will estimate the thee unknown paametes using a maximum likelihood pocedue that measues the likelihood of the obseved Deal o No Deal decisions based on the stop value, o the utility of the cuent bank offe, and the continuation value, o the expected utility of the unknown winnings when ejecting the offe. In a given ound, B x ) denotes the bank offe as a function of the set of emaining pizes x. The stop value is simply: ( (4) sv x ) = u( B( x )) ( Analyzing the continuation value is moe complicated. We elaboate on the continuation value, the bank offe model and the estimation pocedue below. Continuation Value The game involves multiple ounds and the continuation value has to account fo the bank offes and optimal decisions in all late ounds. In theoy, we can solve the entie dynamic optimization poblem by means of backwad induction, using Richad E. Bellman s pinciple of optimality. Stating with the ninth ound, we can detemine the optimal Deal o No Deal decision in each peceding game ound, accounting fo the possible scenaios and the optimal decisions in subsequent ounds. This appoach assumes, howeve, that the contestant takes into account all possible outcomes and decisions in all subsequent game ounds. Studies on backwad induction in simple altenating-offes bagaining expeiments suggest that subjects geneally do only one o two steps of stategic easoning and ignoe futhe steps of the backwad induction pocess; see, fo example, Johnson et al. (2002) and 15

16 Ken Binmoe et al. (2002). This pleads fo assuming that the contestants adopt a simplified mental fame of the game. Ou video mateial indeed suggests that contestants geneally look only one ound ahead. The game-show host tends to stess what will happen to the bank offe in the next ound should paticula biefcases be eliminated and the contestants themselves often comment that they will play just one moe ound (although they often change thei minds and continue to play late on). We theefoe assume a simple myopic fame. Using this fame, the contestant compaes the cuent bank offe with the unknown offe in the next ound, and ignoes the option to continue play theeafte. Given the cuent set of pizes ( x ), the statistical distibution of the set of pizes in the next ound ( x + 1) is known: n (5) P [ x + 1 = y x ] = = p n fo any given subset y of n + 1 elements fom x. In wods, the pobability is simply one divided by the numbe of possible combinations of n + 1 out of n. Thus, using Χ ( x ) fo all such subsets, the continuation value fo a myopic contestant is given by: (6) cv ( x ) = u( B( y)) p y Χ( ) x Given the cognitive buden of multi-stage induction, this fame seems the appopiate choice fo this game. Howeve, as a obustness check, we have also eplicated ou estimates using the ational model of full backwad induction and have found that ou paamete estimates and the empiical fit did not change mateially. In the ealy game ounds, when backwad induction appeas most elevant, the myopic model undeestimates the continuation value. Still, the myopic model geneally coectly pedicts No Deal, because the expected bank offes usually incease substantially duing the ealy ounds, so even the myopic continuation value is geneally geate than the stop value. In the late game ounds, backwad induction is of less impotance, because fewe game ounds emain to be played and because the ate of incease in the expected bank offes slows down. Fo contestants who each ound nine, such as Fank and Susanne, the decision poblem involves just one stage and the myopic 16

17 model coincides with the ational model. The low popensity of loses and winnes in late game ounds to Deal is theefoe equally puzzling unde the assumption of full backwad induction. Bank Offes To apply the myopic model, we need to quantify the behavio of the banke. Section I discussed the bank offes in a qualitative manne. Fo a contestant who cuently faces emaining pizes x and pecentage bank offe b in game ound = 1, L, 9, we quantify this behavio using the following simple model: (7) B ( x ) + 1 = b+ 1x+ 1 (8) b + 1 b + (1 b ) = ρ (9 ) whee ρ, 0 ρ 1, measues the speed at which the pecentage offe goes to 100 pecent. Since myopic contestants ae assumed to look just one ound ahead, the model pedicts the offe in the next ound only. The bank offe in the fist ound needs not be pedicted, because it is shown on the scoeboad when the fist Deal o No Deal choice has to be made. B ( x = x and b 1 efe to the pize in the contestant s own biefcase. 10 ) = The model does not include an explicit pemium fo loses. Howeve, befoe misfotune aises, the continuation value is diven mostly by the favoable scenaios and the pecise pecentage offes fo unfavoable scenaios do not mateially affect the esults. Afte bad luck, the pemium is included in the cuent pecentage and extapolated to the next game ound. Fo each edition, we estimate the value of ρ by fitting the model to the sample of pecentage offes made to all contestants in all elevant game ounds using least squaes egession analysis. The esulting estimates ae vey simila fo each edition: fo the Dutch edition, fo the fist Geman seies, fo the second Geman seies and fo the US shows. The model gives a emakably good fit. Figue 3 illustates the goodness-of-fit by plotting the pedicted bank offes against the actual offes. The esults ae highly compaable fo the thee editions in ou study and theefoe the figue shows the pooled esults. Fo each individual sample, the model explains well ove 70 pecent of the total vaiation in the individual pecentage offes. The explanatoy powe is even highe fo monetay offes, with an R-squaed of oughly 95 pecent fo each sample. Aguably, accuate 17

18 monetay offes ae moe elevant fo accuate isk avesion estimates than accuate pecentage offes, because the favoable scenaios with high monetay offes weigh heavily on expected utility. On the othe hand, to analyze isk behavio following the elimination of the lagest pizes, accuate estimates fo low monetay offes ae also needed. It is theefoe comfoting that the fit is good in tems of both pecentages and monetay amounts. In addition, if ρ is used as a fee paamete in ou stuctual choice models, the optimal values ae appoximately the same as ou estimates, futhe confiming the goodness. [INSERT FIGURE 3 ABOUT HERE] Since the pinciple behind the bank offes becomes clea afte seeing a few shows, the bank offe model (7) (8) is teated as deteministic and known to the contestants. Using a stochastic bank offe model would intoduce an exta laye of uncetainty, yielding lowe continuation values. Fo loses, the bank offes ae hadest to pedict, making it even moe difficult to ationalize why these contestants continue play. Maximum Likelihood Estimation In the spiit of Godon M. Becke, Mois H. DeGoot and Jacob Maschak (1963) and John D. Hey and Chis Ome (1994), we assume that the Deal o No Deal decision of a given contestant i = 1, L, N in a given game ound = 1, L, 9 is based on the diffeence between the continuation value and the stop value, o cv x ) sv( x ), plus some eo. The ( i, i, eos ae teated as independent, nomally distibuted andom vaiables with zeo mean and standad deviation σ i,. Aguably, the eo standad deviation should be highe fo difficult choices than fo simple choices. A natual indicato of the difficulty of a decision is the standad deviation of the utility of the outcomes used to compute the continuation value: (9) δ ( xi, ) = ( u( B( y)) cv( x y Χ ( x i, ) i, )) 2 p We assume that the eo standad deviation is popotional to this indicato, that is, σ = δ x ) σ, whee σ is a constant noise paamete. As a esult of this assumption, the i, ( i, simple choices effectively eceive a lage weight in the analysis than the difficult ones. We 18

19 also investigated the data without weighting. The (unepoted) esults show that the oveall fit in the thee samples deteioates. In addition, without weighting, the estimated noise paametes in the thee editions stongly divege, with the Dutch edition having a substantially highe noise level than the Geman and US editions. The incease in the noise level seems to eflect the highe difficulty of the decisions in the Dutch edition elative to the Geman and US editions; contestants in the Dutch edition typically face (i) lage stakes because of the lage initial pizes and (ii) moe emaining pizes because they exit the game at an ealie stage. The standad deviation of the outcomes (9) picks up these two factos. The deteioation of the fit and the divegence of the estimated noise levels povide additional, empiical aguments fo ou weighting scheme. Given these assumptions, we may compute the likelihood of the Deal o No Deal decision as: (10) l ( x i, cv( xi, ) sv( xi, ) Φ if No Deal = δ ( xi, ) σ ) sv( xi, ) cv( xi, ) Φ if Deal δ ( xi, ) σ whee Φ ( ) is the cumulative standad nomal distibution function. 5 Aggegating the likelihood acoss contestants, the oveall log-likelihood function of the Deal o No Deal decisions is given by: i (11) ln( L) = ln( l( xi, )) N R i= 1 = 2 whee R i is the last game ound played by contestant i. To allow fo the possibility that the eos of individual contestants ae coelated, we pefom a cluste coection on the standad eos (see, fo example, Jeffey M. Wooldidge, 2003). Note that the summation stats in the second game ound ( = 2). The ealy Geman 5 This eo model allows fo violations of fist-ode stochastic dominance (FSD). The pobability of Deal is pedicted to be lage than zeo and smalle than unity, even when the bank offe is smalle than the smallest outcome ( No Deal dominates Deal ) o lage than the lagest outcome ( Deal dominates No Deal ). As pointed out by an anonymous efeee, a tuncated eo model can avoid such violations of FSD. In ou dataset, howeve, the bank offe is always substantially lage than the smallest and substantially smalle than the lagest outcome, and violations of FSD do not occu. 19

20 episodes with only eight game ounds effectively stat in this game ound and in ode to align these episodes with the est of the sample, we exclude the fist ound ( = 1) of the editions with nine game ounds. Due to the vey consevative bank offes, the choices in the fist ound ae always tivial (no contestant in ou sample eve said Deal ); including these choices does not affect the esults, but it would falsely make the ealy Geman episodes look moe noisy than the est of the sample. The unknown paametes in ou model (α, β, W, and σ) ae selected to maximize the oveall log-likelihood. To detemine if the model woks significantly bette than a naïve model of isk neutality, we pefom a likelihood atio test. Results Table 6 summaizes ou estimation esults. Apat fom coefficient estimates and p- values, we have also computed the implied cetainty equivalent as a faction of the expected value, o cetainty coefficient (CC), fo 50/50 gambles of 0 o 10 z, z = 1,,6. These values help to intepet the coefficient estimates by illustating the shape of the utility function. Notably, the CC can be intepeted as the citical bank offe (as a faction of the expected value of the 50/50 gamble) that would make the contestant indiffeent between Deal and No Deal. If CC = 1, the contestant is isk neutal. When CC > 1, the contestant is isk seeking, and as CC appoaches zeo, the contestant becomes extemely isk avese. To help intepet the goodness of the model, we have added the hit pecentage, o the pecentage of coectly pedicted Deal o No Deal decisions. In the Dutch sample, the isk avesion paametes α and β ae both significantly diffeent fom zeo, suggesting that IRRA and DARA ae elevant and the classical CRRA powe function and CARA exponential function ae too estictive to explain the choices in this game show. The estimated wealth level of 75,203 significantly exceeds zeo. Still, given that the median Dutch household income is oughly 25,000 pe annum, the initial wealth level seems substantially lowe than lifetime wealth and integation seems incomplete. This deviates fom the classical appoach of defining utility ove wealth and is moe in line with utility of income o the type of naow faming that is typically assumed in pospect theoy. A low wealth estimate is also consistent with Matthew Rabin s (2000) obsevation that plausible isk avesion fo small and medium outcomes implies implausibly stong isk avesion fo lage outcomes if the outcomes ae integated with lifetime wealth. Indeed, the estimates imply nea isk neutality fo small stakes, witness the CC of fo a 50/50 gamble of 0 20

21 o 1,000, and inceasing the wealth level would imply nea isk neutality fo even lage gambles. Rabin s point is einfoced by compaing ou esults fo lage stakes with the laboatoy expeiments conducted by Holt and Lauy (2002) using the lowe stakes typical in the lab. Holt and Lauy s subjects display significant isk avesion fo modest stakes, which, as Rabin notes, implies exteme isk avesion fo much lage stakes behavio ou contestants do not display. Indeed, contestants with Holt and Lauy s paamete estimates fo the utility function would geneally accept a Deal in the fist game ound, in contast to the actual behavio we obseve. We conclude, ageeing with Rabin, that expected utility of wealth models have difficulty explaining behavio fo both small and lage stakes. The model also does not seem flexible enough to explain the choices fo loses and winnes simultaneously. The estimated utility function exhibits vey stong IRRA, leading to an implausibly low CC of fo a 50/50 gamble of 0 o 1,000,000. Indeed, the model es by pedicting that winnes would stop ealie than they actually do. If isk avesion inceases with stakes, winnes ae pedicted to have a stonge popensity to accept a bank offe, the opposite of what we obseve; witness fo example the Deal pecentages in Table 5. Howeve, stong IRRA is needed in ode to explain the behavio of loses, who eject geneous bank offes and continue play even with tens of thousands of Euos at stake. Still, the model does not pedict isk seeking at small stakes; witness the CC of fo a 50/50 gamble of 0 o 10,000 oughly Fank s isky choice in ound 9. Thus, the model also es by pedicting that loses would stop ealie than they actually do. Inteestingly, the estimated coefficients fo the Geman edition ae quite diffeent fom the Dutch values. The optimal utility function educes to the CARA exponential function (β = 0) and the estimated initial wealth level becomes insignificantly diffeent fom zeo. Still, on the obseved domain of pizes, the two utility functions exhibit a simila patten of uneasonably stong IRRA and high isk avesion fo winnes. Again, the model es by pedicting that loses and winnes would stop ealie than they actually do. These eos ae so substantial in this edition that the fit of the expected utility model is not significantly bette than the fit of a naive model that assumes that all contestants ae isk neutal and simply Deal wheneve the bank offe exceeds the aveage emaining pize. Contay to the Dutch and Geman utility functions, the US utility function appoximates the limiting case of the CRRA powe function (α 0). The CC is again vey high fo small stakes. Fo lage stakes, the coefficient deceases but at a slowe pace than in the othe two counties, eflecting the elatively low popensity to Deal fo US contestants 21

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