Is There a Plausible Theory for Decision under Risk? A Dual Calibration Critique. By James C. Cox, Vjollca Sadiraj, Bodo Vogt, and Utteeyo Dasgupta*

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1 Is There a Plausble Theory for Decson under Rsk? A Dual Calbraton Crtque By James C. Cox, Vjollca Sadraj, Bodo Vogt, and Utteeyo Dasgupta* Expermental Economcs Center Georga State Unversty June 2010 * Fnancal support was provded by the Natonal Scence Foundaton (grant numbers IIS and SES ).

2 Is There a Plausble Theory for Decson under Rsk? A Dual Calbraton Crtque Abstract: Theores of decson under rsk that assume decreasng margnal utlty of money have been crtqued wth concavty calbraton arguments. Snce that crtque uses varyng payoffs and fxed probabltes, t cannot have mplcatons for calbraton of nonlnear probablty transformaton, whch s another way to model rsk averson. The concavty calbraton crtque also has no mplcaton for theores wth varable reference ponts. Ths paper ntroduces a new type of (varyng-probabltes, fxed-payoffs) calbraton that apples to nonlnear transformaton of probabltes. It also apples to theores wth constant or varable reference ponts. The two types of calbratons yeld dual paradoxes: a pattern of rsk averson that conforms to the (resp. dual) ndependence axom mples mplausble rsk averson for theores wth functonals that are lnear n payoffs (resp. probabltes). Functonals that are nonlnear n both payoffs and probabltes are subject to both types of calbraton crtque. The dual calbratons make clear why plausblty problems wth theores of decson under rsk may be fundamental. They are fundamental f ther emprcal relevance can be demonstrated. Ths paper reports seven experments that provde data on the emprcal relevance of the dual calbraton crtque of decson theory. (JEL C90, D81) Can promnent theores of decson under rsk ratonalze both small-stakes rsk averson and large-stakes rsk averson? How do loss averson and reference payoffs enter n the answer to ths queston? Can some exstng theores, but not others, ratonalze same-stakes (.e. smallstakes or large-stakes) rsk averson? We offer a theoretcal dualty approach that addresses these questons. We present two (dual) paradoxes n whch patterns of rsk averson that conform to one theory of decson under rsk mply mplausble rsk averson n the dual to that theory. One wonders then whether data conform to ether or both of the dual calbraton patterns for whch promnent theores mply mplausble rsk averson. We report seven experments that address that queston. Rabn (2000) sparked the lterature on concavty calbraton by dentfyng a varyngpayoffs pattern of small-stakes rsk averson that, through calbraton arguments, can be shown to mply mplausble large-stakes rsk averson for the expected utlty of termnal wealth model. Several subsequent authors extended Rabn s varyng-payoffs, concavty calbraton analyss to apply to a class of theores that assume decreasng margnal utlty of money. How fundamental s ths challenge to the plausblty of theores of decson under rsk? We address ths queston about fundamentalty both theoretcally and emprcally. Our theoretcal dscusson s based on dualty. We explan n ths paper that the varyngpayoffs patterns of small-stakes rsk averson used n calbratons by Rabn (2000) and all

3 2 subsequent authors conform to the dual ndependence axom (Yaar, 1987). Ths presents a paradox: patterns of rsk averson that characterze ratonal behavor for the dual theory of expected utlty (Yaar, 1987), wth constant margnal utlty of money for all rsk preferences, mply mplausble rsk averson for theores wth decreasng margnal utlty of money such as expected utlty theory and rank dependent utlty theores (Quggn, 1993, Tversky and Kahneman, 1992). One wonders then whether there are varyng-probabltes patterns of rsk averson that conform to the ndependence axom of expected utlty theory and yet mply mplausble rsk averson for theores wth nonlnear transformaton of probabltes such as the dual theory of expected utlty and cumulatve prospect theory. We explan that the answer to ths queston s yes. Ths presents a second (dual) paradox: patterns of rsk averson that characterze ratonal behavor for expected utlty theory, wth lnearty n probabltes for all rsk preferences, mply mplausble rsk averson for theores that represent rsk averse preferences wth nonlnear probablty transformatons (wth or wthout nonlnear transformaton of payoffs). Prevous lterature on concavty calbraton has focused on the nablty of some promnent theores to ratonalze both small-stakes rsk preferences and large-stakes rsk preferences. Ths leaves open the queston of whether calbraton arguments have mplcatons for decson theores ablty to ratonalze same-stakes (.e. small-stakes or large-stakes) rsk preferences. In other words, f a researcher s content to vew exstng applcatons of decson theores as havng only same-stakes mplcatons, then does (s)he escape crtcsm based on calbraton arguments? We explan that theores that represent rsk averson wth nonlnear probablty transformatons have mplausble mplcatons even for same-stakes rsk preferences. Of course the fundamentalty of the above theoretcal results rests on emprcal valdty of the patterns of rsk averson used n the calbraton propostons. To date, however, there has been only argument about the reasonableness of the calbraton suppostons but no data from real-payoff, controlled experments to nform the ssue. We explan why researchers encounter especally dffcult problems n conductng experments to test the emprcal valdty of suppostons n calbraton propostons and dscuss solutons to these problems that were mplemented n our experments. The paper reports seven experments conducted over several years n three countres (Inda, Germany, and the Unted States) wth dosyncratc opportuntes for mplementng a varety of expermental desgns and protocols coverng both varyng-payoffs

4 3 and varyng-probabltes calbraton patterns of rsk averson that have mplcatons for theores of decson under rsk. Prevous lterature reports varyng-payoffs calbraton patterns that apply to models defned on (a) termnal wealth or (b) ncome. Studes that focus on termnal wealth models nclude Rabn (2000), Nelson (2001), and Safra and Segal (2008). Rabn demonstrated results that apply to the expected utlty of termnal wealth model. Nelson showed that Rabn s concavty calbraton crtque apples to rank-dependent utlty of termnal wealth. Safra and Segal ntroduced a stochastc verson of Rabn s calbraton pattern that produces anomales for addtonal non-expected utlty models n whch preferences are defned on termnal wealth. None of the above calbraton propostons apply to theores that ncorporate loss averson because the reference ponts for termnal wealth models are amounts of ntal wealth not amounts of ncome (that defne losses and gans). Calbratons for models defned on ncome are reported by Barbers, Huang, and Thaler (2006), Cox and Sadraj (2006), and Rubnsten (2006). Barbers, Huang, and Thaler examned the mplcatons of calbraton for recursve utlty wth frst-order and second-order rsk averson. Cox and Sadraj looked at calbraton ssues for Tversky and Kahneman s (1992) cumulatve prospect theory and two expected utlty models that are alternatves to the termnal wealth model. Rubnsten took the concavty calbraton crtque to tme preferences under rsk. All of the prevous studes were bult on the same varyng-payoffs pattern of small-stakes rsk averson that frst appeared n Rabn (2000). As we explan, the varyng-payoffs calbraton patterns n prevous lterature have no mplausble rsk averson mplcatons for the dual theory of expected utlty (Yaar, 1987), an early alternatve to expected utlty theory that models rsk averson (solely) wth nonlnear transformaton of probabltes. As explaned by Wakker (2005), those calbraton patterns have no mplausble rsk averson mplcatons for recent versons of cumulatve prospect theory wth varable reference amounts of ncome. In ths paper, we ntroduce a varyng-probabltes pattern of rsk averson that has calbraton mplcatons for theores that ncorporate rsk averson wth nonlnear transformaton of probabltes (wth or wthout nonlnear transformaton of payoffs). The new calbraton does have mplausble rsk averson mplcatons for the dual theory of expected utlty and for cumulatve prospect theory wth varable reference amounts of ncome and loss averson. The new calbraton pattern mples same-stakes (as well as large-stakes vs. small-stakes) mplausble

5 4 rsk averson for theores that ncorporate nonlnear transformaton of probabltes. As a result, such theores are called nto queston even for applcatons that preserve the doman of payoffs. We report dual calbraton propostons and several corollares. Proposton 1 dentfes varyng-probabltes patterns of rsk averson that have mplausble (small-stakes vs. largestakes and same-stakes) rsk averson mplcatons for dual theory of expected utlty. Proposton 2 uses varyng-payoffs patterns, appearng n prevous lterature, that have mplausble small-stakes vs. large-stakes rsk averson mplcatons for expected utlty theory. Each proposton has a corollary that extends the calbraton to rank-dependent theores, ncludng cumulatve prospect theory, that model rsk preferences wth nonlnear transformatons of both probabltes and payoffs. In ths way, such theores are shown to be subject to both types of calbraton crtque. The new, varyng-probabltes calbraton also apples to theores wth nonlnear transformatons of probabltes and varable reference amounts of ncome (wth or wthout loss averson). I. Independence, Dual Independence, and Calbraton Patterns We start wth two examples that llustrate dual calbraton paradoxes. The frst example, known as Rabn s pattern, s a par of rsk preference statements that can be ratonalzed by the dual theory of expected utlty (DTEU) but cannot be ratonalzed by expected utlty theory (EUT). The second example ntroduces a new par of rsk preference statements that can be ratonalzed by EUT but cannot be ratonalzed by DTEU. These examples llustrate patterns n the dual calbraton propostons reported n sectons II and III. Both patterns have mplcatons for theores that ncorporate nonlnear transformatons of both payoffs and probabltes. A. An Example of Calbraton for Varyng Payoffs Consder a representatve example from prevous lterature (Rabn, 2000) consstng of Statement P.2 (a pattern of small-stakes rsk averson) and Statement Q.2 (a large-stakes lottery preference). Statement P.2 says that the agent rejects the 50/50 lottery wth loss of 100 or gan of 105 at all amounts of ntal wealth w between 100 and 300, Statement Q.2 says that the agent prefers the 50/50 lottery that pays 0 or 5 mllon to gettng 10,000 for sure at ntal wealth 290,000. Rabn shows that Statement P.2 s nconsstent wth Statement Q.2. So, the expected 1 Sectons III and V explore the mplcatons of varyng the sze of the payoff nterval over whch P.2 holds.

6 5 utlty of termnal wealth model cannot ratonalze both of these statements about rsk preferences; that s, ths model s nconsstent ether wth Statement P.2 or wth Statement Q.2. In contrast, DTEU can ratonalze rsk preferences that satsfy both Statements P.2 and Q.2. The Statement P.2 pattern of small-stakes rsk averson conforms to the dual ndependence axom: accordng to ths axom, a DTEU agent who rejects the 50/50 lottery wth payoffs of 100 or 105 for some value of ntal wealth w must reject the same lottery for all values of w. An easy way to see ths mplcaton s through the lnearty n payoffs property that characterzes the DTEU functonal (as a consequence of the dual ndependence axom). Paradoxcally, the pattern of small stakes rsk averson contaned n Statement P.2: (a) mples mplausble large-stakes rsk averson (negaton of statement Q.2) for EUT; but (b) conforms to ratonal behavor for DTEU because t conforms to the dual ndependence axom. It has no mplcaton of mplausble large stakes rsk averson for DTEU. B. An Example of Calbraton for Varyng Probabltes Here we ntroduce a par of rsk preference statements that cannot be ratonalzed by DTEU but can be ratonalzed by EUT. Consder an agent wth some ntal wealth w between 0 and 300,000 who (weakly) prefers 1 mllon for sure to a 50/50 lottery that pays 2.5 mllon or 0. Then t s a straghtforward mplcaton of lnearty n probabltes of the EUT functonal that EUT mples that ths agent prefers a three outcome lottery that pays 2.5 mllon or 1 mllon or 0, wth probabltes p 0.05 and 0.1 and 1 p 0.05, to a two outcome lottery that pays 2.5 mllon or 0, wth probabltes p and 1 p, for all p {0.05,0.1,...,0.9,0.95}. Although such rsk preferences conform to the ndependence axom of EUT they have mplausble rsk averson mplcatons for DTEU, as we shall now explan. Let Statement P.1 say that the agent rejects a lottery that pays 2.5 mllon or 0 wth probabltes p and 1 p n favor of a lottery that pays 2.5 mllon or 1 mllon or 0 wth probabltes p 0.05 and 0.1 and 1 p 0.05 for all p {.05,0.1,...,0.90,0.95}. 2 Statement Q.1 says that the agent prefers the 50/50 lottery that pays 0 or 58,665 to gettng 1,000 for sure. Proposton 1, below, shows that DTEU s nconsstent ether wth Statement P.1 or Statement Q.1. The mplausblty of the rsk preferences n Statement Q.1 s ncreasng wth the number of 2 Sectons II and V explore the mplcatons of varyng the sze of the probablty nterval over whch P.1 holds.

7 6 probablty sub-ntervals n Statement P.1. For example, f Statement P.1b s that an agent prefers the three outcome lottery to the two outcome lottery for all p {.01,0.02,...,0.98,0.99} then, accordng to DTEU, the agent wll prefer gettng 1,000 for sure to a 50/50 lottery that pays 0 or 633 bllon. The Statement Q.1b that s nconsstent wth Statement P.1b accordng to DTEU s: the agent prefers the 50/50 lottery that pays 0 or 633 bllon to gettng 1,000 for sure. In ths way, DTEU cannot ratonalze such rsk preferences. Paradoxcally, the Statement P.1 pattern of small-stakes rsk averson: (a) mples mplausble large-stakes rsk averson (the negaton of statement Q.1) for DTEU; but (b) conforms to ratonal behavor for EUT because t conforms to the ndependence axom. Theores such as cumulatve prospect theory wth functonals that exhbt both nonlnearty n payoffs and nonlnearty n probabltes are nconsstent wth slghtly modfed versons of both pars of statements, (P.1,Q.1) and (P.2,Q.2), as explaned n sectons II and III. II. Calbratons wth Varyng Probabltes (and Fxed Payoffs) We ntroduce a calbraton proposton for the dual theory of expected utlty and corollares that apply to theores wth functonals that are nonlnear n both probabltes and payoffs. The desgn of experments reported n secton VI s based on calbraton patterns dscussed here. A. Calbraton for Lnear Money Transformaton Functons { y, p ;...; y, p ; y }, denote an m-outcome lottery that pays y k wth probablty Let m m p k, for k 2,, m, and pays y 1 wth probablty 1 m pk. We use the conventon yk 1 yk, k 2 for k 1,, m 1. Whenever the smallest payoff s zero (.e., y 1 0 ), we use the smpler notaton { y, p ;...; y, p }. m m 2 2 Consder the 2n 1 pars of lotteres A { y, p }, and B { y, p ; x,2 }, where p /2n, 1/2n, and 1,2,,2n 1. In each par of lotteres, lottery B s constructed from lottery A by transferrng probablty mass 1/2n from both the hghest payoff y and the lowest payoff 0 to the ntermedate payoff x. Suppose that an agent prefers the three outcome lottery B to the two outcome lottery A, for all 1, 2,, 2n 1. Note that, by the ndependence axom, any expected utlty maxmzer

8 7 who prefers x for sure to the 50/50 lottery that pays y or 0 satsfes ths supposton. Proposton 1 shows that f the hgh outcome y s larger than twce the ntermedate outcome x then ths supposton mples mplausble rsk averson for DTEU agents. The followng standard notaton s used: ndcates weak preference; ndcates strong preference; and N denotes the set of postve ntegers. Defne m n j 1 (,, ) 1 ( 1) ( 1). Ktmn t t j1 1 Proposton 1. Let n N and y 2x 0be gven. Let p /2 n, 1/2 n and G K( y/ x, n, n). Consder the statements P.1 { y, p ; x, 2 } { y, p }, for all {1, 2,, 2n1} and Q.1 { zg,0.5} z, for some z 0. a. Any EUT agent who prefers x to { y,0.5} satsfes P.1. b. There are EUT agents who satsfy both P.1 and Q.1. c. There are no DTEU agents who satsfy both P.1 and Q.1. Proof: see appendx A.2. Note that G K( y/ x, n, n) for y/ x 2. Hence, the larger the value of n, the n more extreme the mplcatons of the P.1 pattern of rsk averson. Put dfferently, for any G, as bg as one chooses, there exsts n such that for weak preference for the three outcome lottery B over the two outcome lottery A, for all {1, 2,, 2n 1}, DTEU predcts a preference for z for sure over the rsky lottery { zg,0.5} for all z 0. Some numercal llustratons of Proposton 1 are reported n Table 1. In the table, C = y / x, the rato of the hghest payoff to the second hghest payoff n the three prze lottery. Wth C = 2.5 and n = 20 Proposton 1 tells us that for ths P.1 pattern DTEU predcts that the agent prefers 1,000 for sure to a lottery that pays 3.3 mllon or 0 wth even odds, as reported n the frst column and thrd row of Table 1. Wth C = 3.5 and n = 50 the predcton s preference for 23 1,000 for sure over a 50/50 lottery that pays 0 or Fnally, wth C 5 and n = 10, the predcton s preference for 1,000 for sure to the 50/50 lottery that pays 0 or 1 bllon. 3

9 8 B. Calbraton for Lnear and Nonlnear Money Transformaton Functons Proposton 1 s stated for the dual theory of expected utlty that s characterzed by a preference functonal that s lnear n payoffs for all rsk preferences. The generalzaton s straghtforward for a class of models wth nonlnear transformaton of decumulatve probabltes (referred to as NTDP) that represent rsk preferences wth lnear or nonlnear transformaton of payoffs ( ) as well as nonlnear transformaton of probabltes. Frst consder NTDP wth constant, zero-ncome reference pont, as n Tversky and Kahneman (1992). For ( ) subaddtve on postve payoffs one has: Corollary 1.1. For ( y) 2 ( x), there are no NTDP agents wth zero-ncome reference pont who satsfy both P.1 and Q.1 wth G K( ( y) / ( x), n, n). Proof: see appendx A.2. It can be verfed that for ( y)/ ( x) 2, lm G lm K( ( y) / ( x), n, n). Implcatons of Corollary 1.1 are gven n Table 1 for the (alternatve) defnton C ( y)/ ( x). For example, f the hgh payoff y s k tmes as large as the ntermedate (postve) payoff x and the concave value functon of (postve) payoffs s such that ( kx)/ ( x) 3 then mplcatons of n n calbraton pattern P.1 are gven by the C 3 column of Table 1, and so on. A reference-dependent theory such as prospect theory can ncorporate varable reference amounts of money payoff. Wakker (2005) argues that varable reference ponts can mmunze prospect theory to concavty calbraton arguments based on the small-stakes rsk averson pattern ntroduced by Rabn (2000). In contrast, the dual calbraton pattern ntroduced heren s robust to varable reference amounts of ncome. The reason for ths s straghtforward: the calbraton s constructed by varyng the probabltes for whch three or two payoffs are pad, not by varyng the payoff amounts. Hence t makes no dfference to the calbraton reported here whether the reference amount of payoff s or s not fxed at zero payoff. Here s a formal statement of the result. Let () 0 denote the value functon for negatve payoffs and defne R ( yx)/ ( x). For () sub-addtve on postve payoffs one has: 3 Note that ths proposton makes no explct assumpton on the curvature of the probablty transformaton.

10 9 Corollary 1.2. Let the reference pont be the ntermedate payoff x and ( y x) ( x). There are no loss averse NTDP agents who satsfy both P.1 and Q.1 wth G K( R 1, n, n). Proof: see appendx A.2. Note that for R 1, G K( R1, n, n) as n. Corollary 1.2 holds for both R nonlnear and pece-wse lnear value functons. Smlar corollares can be stated for cases n whch the reference pont s the hghest or lowest payoff rather than the ntermedate payoff. C. Calbraton for Proper Subsets of Dscrete Probabltes n [0,1] Proposton 1, Corollary 1.1, and Corollary 1.2 report the mplcatons of preference for the three-outcome lottery over the two-outcome lottery for all probabltes, p / 2 n [0,1] of the hgh payoff from 0 to 1 1/2n. But what f statement P.1 s not true for all p [0,1]? Perhaps the preference for the three outcome lottery over the two outcome lottery holds only for a proper subset of the [0,1] nterval of probabltes. Ths queston s addressed by Corollary 1.3. Some examples of questons addressed by Corollary 1.3 are the followng. What f statement P.1 s not true for all p but only for all p {0.50,0.51,...,0.98,0.99}? (Ths could occur for some patterns of non-eut ndfference curves n the Marschak-Machna trangle dagram that fan out.) Then, accordng to Corollary 1.3, DTEU predcts that the agent prefers a 50/50 lottery that pays 1 or 25,253 to a 25/75 lottery that pays 0 or 25,253, whch s clearly mplausble rsk averson. So statement Q.1 n ths case s preference for the 25/75 lottery wth outcomes 0 and 25,253 to the 50/50 lottery wth outcomes 1 and 25,253. Another example nvolves the case of statement P.1 beng satsfed only for all p {0.01, ,0.49,0.50}. (Ths could occur for some patterns of non-eut ndfference curves n the Marschak-Machna trangle dagram that fan n.) Then, accordng to Corollary 1.3, DTEU predcts that the agent prefers a 50/50 lottery that pays 0 or 1,000 to a 75/25 lottery that pays 0 or 25 mllon, whch s clearly mplausble rsk averson. Therefore, statement Q.1 n ths last example s preference for the 75/25 lottery that pays 0 or 25 mllon over the 50/50 lottery wth outcomes 0 and 1,000. Consder the class of models NTDP for whch C denotes: y / x for a functonal that s lnear n payoffs or ( y x) / x for a functonal that s pecewse lnear n payoffs, wth dscontnuous slope (loss averson) at x, or ( y) / ( x) for a functonal that s nonlnear n R

11 10 payoffs or ( yx) / ( x) 1 for a functonal that s nonlnear n payoffs wth dscontnuous slope at x. Wthout any loss of generalty let (0) 0. For ( ) sub-addtve on postve payoffs one has: Corollary 1.3. Denote G K( C, n/2, n/2), for an even n, and G' K( C, m, n) such that { mn and mn}. There are no NTDP agents who satsfy both: a. P.1 for all {1,, n,..., nm} and Q.1 wth G ' b. P.1 for all { n,..., 2n 1} and Q*.1: { zg,0.75} { zg,0.5; z}, for some z 0. c. P.1 for all {1,, n} and Q**.1: { zg,0.25} { z,0.5}, for some z 0. Proof: see Appendx A.2 Part a of the corollary states mplcatons for the case when the nterval of preference for the three outcome lottery over the two outcome lottery s a subset of (0,1) that ncludes (0,1/5]. Part b states results for the case where the nterval of preference s [0.5,1). Fnally, part c states results for the case when the nterval of preference for the three outcome lottery s (0,0.5]. In analyss of data from the experments, we wll also need to know the mplcatons of preference for the three-outcome lottery over the two-outcome lottery for only some of the probabltes n an experment desgn. Corollary 1.3 wll be appled n analyss of data n secton VI. D. Implausble Same-Stakes Rsk Averson Implausble rsk averson mplcatons of theores that transform probabltes are not lmted to large-stakes vs. small-stakes comparsons. Such theores also predct mplausble same-stakes rsk averson. There can be somewhat dfferent deas of what mght be meant by same-stakes. One natural defnton s that the payoffs n the lotteres n statement Q.1 are weakly between the hghest and lowest payoffs n the lotteres n statement P.1, that s, they are n the same payoff doman of applcaton of the theory. Proposton 1 and ts corollares mply such same-stakes mplausble rsk averson, as can be seen from the followng. Statement P.1 nvolves lotteres wth hgh payoff amount y, ntermedate payoff amount x, and low payoff amount 0. Calbraton mplcatons are derved for dfferent ratos of y / x C. Statement Q.1 says that for a suffcently large G, a 50/50 lottery that pays zg or 0 s preferred to z. As explaned n secton II.a and Table 1, the value of G can be set as large as one chooses by a sutable choce of sub-ntervals of the [0,1] nterval (as determned by the

12 11 choce of the nteger n ). Therefore, the lotteres n statements P.1 and Q.1 are same-stakes so long as y and zg are suffcently close. Ths last nequalty s satsfed by choosng z close to y / G, whch s always possble because P.1 places no restrcton on the sze of z. An example usng numbers reported n Table 1 may help to explcate ths pont. Consder the three payoff amounts 14, 4, and 0 (used n one of the experments run n Atlanta reported below). The theoretcal pont explaned here about same stakes rsk averson s robust to all choces of payoff scale such as dollars, or dollars dvded or multpled by any power of 10. Clearly, 14/4 = 3.5 (the value of C for DTEU). Suppose that the value functon for CPT s such that (14) / (4) 3, then the value of C for CPT s at least 3. In that case, the entry n the frst row and frst column of Table 1 (for C = 3) tells us that 1,000 for sure s preferred to the 50/50 lottery that pays 33,000 or 0 for a DTEU or CPT agent. But the entres n Table 1 are derved from the negaton of statement Q.1, wth small adjustment of the postve payoff to get a strct preference z { zg,0.5}, where G 33 f n 5 and C 3. To have the lotteres n statements P.1 and Q.1 be of the same-stakes set z 0.5. Then zg whch s comparable to the hgh payoff of 14 n the P.1 example. An mplcaton of ths P.1 example s then seen to be that a sure payoff of 50 cents s preferred to the 50/50 lottery that pays $16.50 or 0. Ths s an example of mplausble same-stakes rsk averson. It shows that DTEU and CPT are nconsstent wth the followng two plausble rsk preferences holdng smultaneously: Q.1.e A 50/50 lottery that pays or 0 s preferred to a sure payoff of 0.50; and P.1.e A three-outcome lottery that pays 14, 4 or 0, wth probabltes p 0.1, 0.2 and 1 p 0.1, s preferred to a two outcome lottery that pays 14 or 0, wth probabltes p and 1 p, for all p {0.1,0.2,...,0.8,0.9}. It s a straghtforward exercse to verfy that an expected utlty of ncome model wth CRRA preferences wth r = 0.5 s consstent wth both Q.1.e and P.1.e whereas CPT wth estmated parameters such as those reported by Tversky and Kahneman (1992) s nconsstent wth Q.1.e and P.1.e holdng smultaneously. 4 4 At least 75% of the subjects n the Atlanta 14/4 experment, reported n secton VI, revealed preferences consstent wth pattern P.1.e.

13 12 III. Calbratons wth Varyng Payoffs Calbraton propostons for theores wth nonlnear utlty of money payoffs have been reported n several papers (cted above n the ntroducton). In order to provde a foundaton for our concavty calbraton experments, we report a calbraton proposton for expected utlty theory and a corollary that apples to rank-dependent theores. Desgn of experments reported n secton VII s based on the calbraton patterns dscussed here. A. Calbraton for Lnear Probablty Transformaton Functons We now revst the large stakes rsk averson mplcatons of postulated patterns of small stakes rsk averson for expected utlty theory ntroduced nto the lterature by Rabn (2000). These mplcatons hold for all three expected utlty models dscussed n Cox and Sadraj (2006), the expected utlty of termnal wealth model, the expected utlty of ncome model, and the expected utlty of ntal wealth and ncome model. For bounded ntervals of ncome, Proposton 2 states a concavty calbraton result for expected utlty theory wth weakly concave utlty of money payoff functon u( ). 5 Let the varable x denote amounts of certan money payoff, nterpreted ether as ntal wealth or exogenous ncome. Consder bnary lotteres wth gan amount g and loss amount that yeld payoffs x g or x to the agent. Let x denote the smallest nteger larger than x and f () be the transformaton functon of decumulatve probabltes. Defne and N *() r 2 ln2/ln() r. 2 2K K A(, rk) r r r Proposton 2. Let postve g and nteger N (1 g/ ) N* ( / g) be gven. Denote N M N( g) and (*) J N K 1 ( / g) A( / g, K), for nteger K( N* ( / g), N). Consder statements P.2 x { x g,0.5; x }, for all x [ mm, ] Q.2 { z J( g ),0.5; m} z, for z mk( g ), for some K. a. Any DTEU agent who rejects { g,0.5; } satsfes P.2. b. There are DTEU agents who satsfy both P.2 and Q.2. c. There are no (u-concave) EU agents who satsfy both P.2 and Q.2. Proof: See appendx A.1 and appendx A.3. 5 See Rabn (2000) and Cox and Sadraj (2006) for concavty calbratons on unbounded domans.

14 13 Note that: K 2 ln 2 / ln( / g) mples A 0. Hence, for any gven m and z, the fourth term on the rght hand sde of statement ( ) ncreases geometrcally n M. Ths mples that for any amount of gan G, as bg as one chooses, there exsts a large enough nterval n whch preference for x over a rsky lottery { x g, 0.5; x }, for all ntegers x from the nterval [ mm, ], mples a preference for z for sure to the rsky lottery { G,0.5; m }. We use statements () and Q.2 n Proposton 2 to construct the llustratve examples n Table 2. Suppose that an agent prefers the certan amount of ncome x to the lottery { x110, 0.5; x 100}, for all ntegers x [100, M ], where values of M are gven n the Rejecton Intervals column of Table 2. In that case all three expected utlty (of termnal wealth, ncome, and ntal wealth and ncome) models predct that the agent prefers recevng the amount of ncome 3,000 for sure to a rsky lottery { G,0.5;100}, where the values of G are gven n the frst column of Table 2. For example, f [ mm, ] [100, 50000] then 13 G for all three expected utlty models. Accordng to the entry n the second column and M = 30,000 row of Table 2, expected utlty theory predcts that f an agent prefers certan payoff n amount x to lottery { x90,0.5; x 50}, for all ntegers x between 100 and 30,000, then such an agent wll prefer 3,000 for sure to the 50/50 lottery wth postve outcomes of 100 or B. Calbraton for Nonlnear Probablty Transformaton Functons The followng corollary to Proposton 2 apples to rank dependent theores ncludng cumulatve prospect theory (CPT) wth zero-ncome reference pont (Tversky and Kahneman, 1992). Defne rt ( ) (1 t) / tg. Let h( ) denote the probablty transformaton functon for the probablty of the hgh payoff n a bnary lottery. One has: Corollary 2.1. Let q r( h(0.5)) and nteger K 2 ln 2 / ln q, be gven. Let the value functon be (weakly) concave on postve doman. There are no CPT agents who satsfy both P.2 and Q.2. Proof: See appendx A.3. Recall that for expected utlty theory, wth functonal that s lnear n probabltes, Proposton 2 reveals mplausble large-stakes rsk averson f g. In the corollary, ths mplcaton holds when h(0.5) g [1 h(0.5)]. That s, for such lotteres, for any gven m and z, the thrd term on the rght hand sde of nequalty ( ) ncreases geometrcally n M because

15 14 qr( h(0.5)) 1. Ths mples that f h(0.5) g [1 h(0.5)] then for any amount of gan G, as bg as one chooses, there exsts a large enough nterval n whch preference for x over a rsky lottery { xg,0.5; x }, for all ntegers x from the nterval [ mm, ], mples a preference for z for sure to the rsky lottery { G,0.5; m }. Examples that llustrate the mplcatons of Corollary 2.1 are smlar to those n Table 2. IV. Emprcal Interpretaton of Calbraton Propostons Prevous calbraton lterature has been controversal. Some scholars (e.g., Rabn and Thaler, 2001; Wakker, 2005), appear to beleve that t s obvous that ndvduals rsk preferences conform to the type of small-stakes rsk averson pattern supposed n Rabn s (2000) calbraton. Others dsagree, and argue that Rabn s supposed small-stakes rsk averson pattern s tself mplausble, and that hs calbraton proposton has no emprcal relevance. For example Watt (2002) noted, correctly, that Rabn s small-stakes rsk averson supposton s nconsstent wth the Arrow-Pratt relatve rsk averson measure beng less than 170 for expected utlty theory. He ctes volumnous lterature reportng emprcal estmates of relatve rsk averson measures wth values much smaller than 170. Smlar crtcsms of the emprcal relevance of Rabn s (2000) calbraton proposton were stated by Palacos-Huerta and Serrano (2006). The small-stakes rsk averson pattern used n our varyng-probabltes calbraton has elcted opnons of acceptance and rejecton. Some economsts fnd preference for the threeoutcome lotteres supposed n our calbraton to conform to ther opnon. Others do not. And some have advanced crtques that follow the approach used by Watt and Palacos-Huerta and Serrano to crtque Rabn s calbraton. Some dscusson of the emprcal nterpretaton of calbraton propostons seems warranted. A. Interpretaton of Proposton 2 and ts Corollary How does one nterpret Proposton 2 and Corollary 2.1? Frst, they tell us that statements P.2 and Q.2 conform to the dual ndependence axom that characterzes the dual theory of expected utlty. Second, they tell us that statements P.2 and Q.2 are nconsstent for expected utlty theory (EUT) and cumulatve prospect theory (CPT) wth zero-ncome reference pont. Hence, these theores predct: f P.2 (the certan amounts x are preferred to the stated small-stakes lotteres) then Q.2 (large-stakes lotteres lke those n Table 2 are rejected). Equvalently, the theores predct that f not Q.2 (large-stakes lotteres lke those n Table 2 are

16 15 not rejected) then not P.2 (the certan amounts x are not preferred to the small-stakes lotteres). There s general agreement that ndvduals would accept extremely favorable large-stakes lotteres lke those n Table 2 and that theores of decson under rsk are consstent wth such acceptance. In that case, the calbraton proposton and corollary tell us that EUT and CPT wth zero-ncome reference pont must be nconsstent wth the small-stakes rsk averson pattern contaned n statement P.2. 6 But ths leaves us wth an emprcal queston: wll ndvduals actually reject the small-stakes lotteres n statement P.2, as predcted by EUT and CPT, or wll they accept them? Ths s the emprcal queston addressed by the experments we report n secton VII. B. Interpretaton of Proposton 1 and ts Corollares How does one nterpret Proposton 1 and ts corollares? Frst, they tell us that statements P.1 and Q.1 conform to the ndependence axom that characterzes expected utlty theory. Second, they tell us that statements P.1 and Q.1 are nconsstent for the class of models NTDP that ncludes the dual theory of expected utlty and varants of cumulatve prospect theory wth constant or varable reference amounts of payoff. Hence, these theores predct: f P.1 (the threeoutcome lotteres are preferred to two-outcome lotteres then Q.1 (large-stakes lotteres lke those n Table 1 are rejected). Equvalently, the theores predct that f not Q.1 (large-stakes lotteres lke those n Table 1 are not rejected) then not P.1 (the three-outcome lotteres are not preferred to two-outcome lotteres). It appears that everyone agrees that ndvduals would accept extremely favorable large-stakes lotteres lke those n Table 1 and that theores of decson under rsk are consstent wth such acceptance. In that case, the calbraton proposton and corollares tell us that dual theory and cumulatve prospect theory must be nconsstent wth the rsk averson pattern contaned n statement P.1 n Proposton 1. Agan, ths leaves us wth an emprcal queston: wll ndvduals actually reject the two-outcome lotteres n P.1 n favor of the three-outcome lotteres, as predcted by dual theory and cumulatve prospect theory, or wll they choose the two-outcome lotteres? Ths s the emprcal queston addressed by the experments we report n secton VI. 6 Ths statement for EUT s smlar to Rabn and Thaler s (2002) response to Watt (2002).

17 16 V. Experment Desgn Issues We here dscuss ssues that arse n desgnng experments wth the calbraton patterns contaned n statements P.1 and P.2. A. Power vs. Credblty wth Varyng-Probabltes Calbraton Experments Table 1 llustrates the relatonshp between the rato C of hgh payoff to ntermedate payoff n the three-outcome lottery and the dfference between probabltes n adjacent terms n 1 the calbraton (determned by the value of n n ). The desgn problem for varyngprobablty calbraton experments s nherent n the need to have small enough sub-ntervals of 2n 2n the [0,1] nterval for the calbraton pattern n Proposton 1 to lead to the mplcaton of clearly mplausble rsk averson. There are two problems wth bg values of the parameter n (.e., large numbers of subntervals). Frst, a subject s decsons may nvolve trval fnancal rsk because the dfferences between all of the moments of the dstrbutons of payoffs for the three-outcome lottery { y, p ; x,2 } and the two-outcome lottery { y, p } become nsgnfcant as n becomes large. Consder, for example, the case of y = $100, x = $25, and n 500. In ths case, the dfference between expected values of the two-outcome and three-outcome lotteres s 5 cents (for all ). The dfference between standard devatons of payoffs for the two-outcome and three-outcome lotteres, at = 500, s 4 cents. The second problem wth large n s that adjacent probabltes dffer by only 1/2n whle the subject s decson task s to make 2n 1 choces. For example, for n 500 adjacent probabltes dffer by and the subjects decson task s to make 999 choces. In such a case, the subjects may not be senstve to the probablty dfferences and the payoffs may not domnate decson costs because of the huge number of choces needng to be made. In contrast, f the length of each sub-nterval s 1/10 (.e. n 5) then the dfference n expected payoffs between the two-outcome and three-outcome lotteres s $5 for the above values y $100 and x $25, and for = 5 the dfference n standard devatons s $4.17; furthermore, the subjects decson task s to make only 9 choces. The calbraton mplcatons of n = 5 are less spectacular than for n = 500, as shown n Table 1, but the resultng experment can credbly be mplemented. In our experments, we use relatvely low values of the parameter n.

18 17 B. Affordablty vs. Credblty wth Varyng-Payoffs Calbraton Experments Table 2 llustrates the relatonshp between the sze of the nterval [ mm, ] n the left-most column, used n the supposton underlyng a utlty of money payoff calbraton, and the sze of the hgh gan G n the result reported n the other columns of the table. Varyng-payoff calbraton experments nvolve tradeoffs between what s affordable and what s credble, as we shall next explan. As an example, consder an experment n whch subjects were asked to choose between $x for sure and the bnary lottery {$ x $110, 0.5;$ x $100} for all x between m = $1,000 and M = $350,000. Suppose the subject always chooses the certan amount $x and that one of the subject s decsons s randomly selected for payoff. Then the expected payoff to a sngle subject would exceed $175,000. Wth a sample sze of 30 subjects, the expected payoff to subjects would exceed $5 mllon, whch would clearly be unaffordable. But why use payoffs denomnated n U.S. dollars? After all, Proposton 2 s dmenson nvarant. Thus, nstead of nterpretng the fgures n Table 2 as dollars, they could be nterpreted as dollars dvded by 10,000; n that case the example experment would cost about $500 for subject payments and clearly be affordable. So what s the source of the dffculty? The source of the dffcult tradeoff for experment desgn becomes clear from nspecton of Proposton 2: the unt of measure for m and M s the same as that for the loss and gan amounts and g n the bnary lotteres. If the unt of measure for m and M s $1/10,000 then the unt of measure for and g s the same (or else the calbraton doesn t apply); n that case the bnary lottery has hgh and low payoffs n amounts $0.0001x + $0.011 and $0.0001x - $0.010, whch nvolves trval fnancal rsk of 2.1 cents. The desgn problem for concavty calbraton experments wth money payoffs s nherent n the need to calbrate over an [ mm, ] nterval of suffcent length for the calbratons n Proposton 2 and Corollary 2.1 to lead to the mplcaton of mplausble rsk averson n the large. There s no way to avod ths problem; the desgn of any varyng-payoffs calbraton experment wll reflect a tradeoff between affordablty of the payoffs and credblty of the ncentves. In our experments, we address ths problem n two ways by: (a) conductng some experments n Inda, where we can afford to use [ mm, ] ntervals of rupee payoffs that are

19 18 suffcently wde for calbraton to have bte; and (b) conductng an experment n Germany, partly on the floor of a casno, whch makes use of large contngent euro payoffs affordable. VI. Experments wth Varyng Probabltes We ran four varyng-probabltes calbraton experments n Germany, Inda, and the Unted States. We explan the common desgn features and dosyncratc lotteres n these experments and present a more detaled dscusson of one experment to provde a representatve example. We begn wth the example. A. Experment Desgn: An Example Subjects n one experment parameterzaton were asked to make choces for each of the nne pars of lotteres shown n Table 3. The fractons n the rows of the table are the probabltes of recevng the przes n the two outcome (opton A) and three outcome (opton B) lotteres. Each row of Table 3 shows a par of lotteres ncluded n the experment. The subjects were not presented wth a fxed order of lottery pars, as n Table 3. Instead, each lottery par was shown on a separate (response form) page. Each subject pcked up a set of response pages that were arranged n ndependently drawn random order. He or she could mark choces n any order desred. On each decson page, a subject was asked to choose among a two outcome lottery (opton A n some row of Table 3), a three outcome lottery (opton B n the same row of Table 3), and ndfference ( opton I ). B. Experment Desgn: Alternatve Parameterzatons and Protocols We conducted four experments on emprcal valdty of the calbraton pattern P.1 postulated n Proposton 1. One experment parameterzaton uses pars of two outcome and three outcome lotteres Aj { y, p }, and B j { y, p 0.1; x,0.2}, for j 1,2, 9, and y 14, j x 4 as shown n Table 3. We also ran experments wth the parameterzatons ( y, x) (40, 10) and (400, 80). The experments were conducted n Magdeburg (Germany), Atlanta (U.S.A.) and Calcutta (Inda) wth payoffs, respectvely, n euros, U.S. dollars, and Indan rupees. The experments used the followng parameters: Magdeburg 40/10: y 40 euros, x 10 euros. Atlanta 40/10: y 40 dollars, x 10 dollars. Atlanta 14/4: y 14 dollars, x 4 dollars. Calcutta 400/80: y 400 rupees, x 80 rupees. Economc sgnfcance of the rupee payoffs s dscussed n secton VII.C. The payoff protocol used random selecton of one decson for j

20 19 payoff, whch s a standard procedure used n testng theores of decson under rsk wth or wthout the ndependence axom. Expermental tests of random selecton have generally reported consstency wth the solaton effect of subjects focusng on ndvdual decson tasks (Camerer, 1989; Starmer and Sugden, 1991; Beatte and Loomes, 1997; Cubtt, Starmer, and Sugden, 1998; Hey and Lee, 2005a, b; Laury, 2006; Lee, 2008). An appendx avalable from the authors reports subject nstructons (n Englsh), response forms (or pages), and detaled nformaton on the protocol used n all of the experments. C. Data Provde Support for Calbraton Pattern P.1 In testng for the presence of choces that satsfy the calbraton pattern, we aggregate choces of opton B wth (the very small number of) choces of opton I (ndfference) because statement P.1 n Proposton 1 nvolves weak preference for B over A. Aggregated choces of B and I are reported as B I. Subjects choce patterns are recorded as sequences of nne letters, ordered accordng to the probablty of the hgh outcome. For example, the pattern [A, B I, B I, A, B I, B I, B I, B I, A] would ndcate that a subject chose A (a two outcome lottery) when the probablty of the hgh outcome was 1/10, 4/10 and 9/10 - ndexed as j 1, 4, and 9 - and chose B or I for all other values of the ndex j. For the experment wth the parameterzaton as shown n Table 3, ths choce pattern would mean the subject chose opton A on (randomly ordered) pages wth the lottery pars n rows 1, 4, and 9 n the table and chose opton B or opton I on all other pages. We use error-rate analyss for statstcal nferences on the proporton of subjects who made choces consstent wth the calbraton patterns. 7 Choce probabltes are assumed to devate from 1 or 0 by an error rate, as n Harless and Camerer (1994). Thus f B I s preferred to A then Prob(choose B I ) = 1 and f B I s not preferred to A then Prob(choose B I ) =, where 0.5. The error rate model postulates that a subject wth real preferences for B I (respectvely A) over A (respectvely B I ) n all nne lottery pars could nevertheless be observed to have chosen the other opton n some rows. For example, accordng to ths model a subject wth underlyng preferences [B I, B I, B I, B I, B I, B I, B I, B I, B I ] could, nstead, be observed to 7 We are grateful to Nathanel Wlcox for generous advce about ths approach to data analyss and for supplyng SAS code. See Wlcox (2008) for dscusson of econometrc methods for analyss of data from bnary dscrete choce under rsk.

21 20 choose a dfferent pattern such as [B I, B I, A, B I, A, B I, B I, B I, B I ], an event wth probablty (1 ) 7 2. Stochastc choce Model I contans only the choce pattern wth a sequence of nne B I n the category calbraton pattern and ts dual ( mrror ) mage wth a sequence of nne A n the other pattern. Accordng to Proposton 1, ths calbraton pattern mples that 1,000 for sure s preferred to the 50/50 lottery that pays 98,000 or 0 for the Atlanta 14/4 experment, as reported n the top-most of the shaded rows n Table 4. For the Calcutta 400/80 experment, Proposton 1 mples that 1,000 for sure s preferred to the 50/50 lottery that pays 1 mllon or 0, as reported n the shaded row for the Calcutta 400/80 lstng n Table 4. Model I s overly conservatve n ts specfcaton of calbraton patterns because other data patterns can be calbrated to mply mplausble rsk averson. Stochastc choce Model II ncludes two patterns n the category calbraton patterns : the pattern wth choce of B I for ndex j 1, 2,,8 and the all B I pattern (that s, j 1,2,,9 ). The mrror mages of these two patterns comprse the other patterns for Model II. Applcaton of Corollary 1.3 demonstrates that these two calbraton patterns of no A except for ndex j 9 mply that 1,000 for sure s preferred to the 50/50 lottery that pays 81,000 or 0, as reported for the Atlanta 40/10 experment lstngs n Table 4. We also consder Model III whch ncludes the patterns no A except for ndexes j 8 and/or 9 n the category of calbraton patterns. The mrror mages of these four patterns comprse the other patterns for Model III. An mplcaton of Corollary 1.3 for these four calbraton patterns n case of n = 5 and C = 4 s preference for 1,000 for sure to the 50/50 lottery that pays 27,000 or 0, as shown n the Atlanta 40/10 and Magdeburg 40/10 lstngs n the table. Table 4 reports results from maxmum lkelhood estmaton of the proporton of subjects who exhbt the calbraton patterns for Models I, II and III. Estmatons are reported for a sngle error rate for all choces, for two dfferent error rates (one error rate for choces wth ndex j =1,..,4 and another one for choces wth ndex j =5,..,9), and three dfferent error rates (one error rate for choces wth ndex j 1, 2,3, another error rate for choces wth ndex j 4,5,6, and another one for choces wth ndex j 7,8,9 ). 8 8 The three error rate models can capture subjects dfferent senstvtes to hgh, ntermedate and low probabltes of the hgh outcome.

22 21 The frst row of Table 4 shows results for the Atlanta 14/4 experment data. For Model I wth one error rate the estmated proporton of subjects who exhbted the calbraton pattern s The Wald 90 percent confdence nterval s (0.55, 0.93). The 0.74 estmate s sgnfcant at one percent (as ndcated by **). The other columns n the frst row of Table 4 report the estmated proportons of subjects whose choce patterns n Atlanta 14/4 conform to calbraton patterns wth the 1 error, 2 error, and 3 error rate versons of Models I, II, and III. These estmates vary between 0.74 and 0.90, and they are all sgnfcant at one percent. The entres n bold font ndcate the model that s selected by lkelhood rato tests; that s, wth data from Atlanta 14/4, Model I wth 1 error or 3 errors and Models II and III wth 1 error, 2 errors, or 3 errors are all rejected n favor of Model I wth 2 error rates. The second through fourth rows of Table 4 show the estmated proportons of subjects whose choces are consstent wth calbraton patterns n experments Atlanta 40/10, Magdeburg 40/10, and Calcutta 400/80. Dependng on the model and number of errors, the estmated proporton of subjects wth data consstent wth the calbraton patterns n Atlanta 40/10 vares from 0.56 to 0.63, all sgnfcant at one percent. The estmates for data from Magdeburg 40/10 vary from 0.37 to 0.41, all sgnfcant at one percent. Estmates wth data from experment Calcutta 400/80 le between 0.72 and 0.74, all are sgnfcant at one percent. The entres n bold font ndcate the model that s selected by lkelhood rato tests over all other models n that row. VII. Experments wth Varyng Payoffs We ran three experments wth calbraton patterns for payoff transformaton theores dentfed n Proposton 2 and Corollary 2 n Inda and Germany. We explan the common features and dosyncratc lotteres used n these experments after presentng a detaled dscusson of one experment to provde a representatve example. A. Experment Desgn: An Example Subjects n one experment parameterzaton were asked to make sx choces between a certan amount of money x and a bnary lottery { x 30,0.5; x 20} for values of x from the set {100, 1K, 2K, 4K, 5K, 6K}, where K = 1,000. Subjects were asked to choose among opton A (the rsky lottery), opton B (the certan amount of money), and opton I (ndfference). The choce tasks gven to the subjects for ths parameterzaton are presented n Table 5. Each row of Table 5 shows a certan amount of money and pared lottery n a choce task ncluded n the experment. The subjects were not presented wth a fxed order of decson tasks, as n Table 5.

23 22 Instead, each par of sure payoff and lottery was shown on a separate (response form) page. Each subject pcked up a set of response pages that were arranged n ndependently drawn random order. He or she could mark choces n any order desred. B. Experment Desgn: Alternatve Parameterzatons and Protocols We conducted three experments on emprcal valdty of the calbraton pattern P.2 n Proposton 2. These experments used the random decson selecton payoff protocol. Calcutta 30 / 20 : bnary lotteres { x30,0.5; x 20} and sure payoffs x from the set {100, 1K, 2K, 4K, 5K, 6K}, where K = 1,000; payoffs n rupees. Calcutta 90 / 50 : bnary lotteres { x 90, 0.5; x 50} for values of x from the set {50, 800, 1.7K, 2.7K, 3.8K, 5K}, where K = 1,000; payoffs n rupees. Magdeburg 110 / 100 : bnary lotteres { x110,0.5; x 100} for values of x from the set {3K, 9K, 50K, 70K, 90K, 110K}, where K = 1,000; payoffs n contngent euros. An appendx avalable from the authors reports the subject nstructons (n Englsh), the response forms (or pages), and detaled nformaton on the protocol used n all of the experments. Before presentng data, we dscuss economc sgnfcance of the rupee payoffs n Calcutta experments and the meanng of contngent euro payoffs n the Magdeburg experment. C. Economc Sgnfcance of the Rupee Payoffs At the tme of the frst experment n Calcutta (2004), data collected show most student subjects ncomes ncluded only scholarshps that pad stpends of 1,200-1,500 rupees per month n addton to the standard tuton waver that each receved. Ths means that the hghest certan payoff used n the Calcutta 30 / 20 experment (6,000 rupees) was equal to four or fve months stpend for the subjects. The daly rate of pay for the students was 40 to 50 rupees. Hence the amount at rsk n the Calcutta 30 / 20 experment lotteres (the dfference between the hgh and low payoffs) was greater than or equal to a full day s pay. The amount at rsk n the Calcutta 90 / 50 experment (140 rupees) was almost three tmes as large. A sample of commodty prces n Calcutta at the tme of the frst experment conducted there (Calcutta 30 / 20 ) s reported n an appendx avalable from the authors. Prces of food tems were reported n number of rupees per klogram. There are about 15 servngs n a klogram