SAVAGE IN THE MARKET. FEDERICO ECHENIQUE Caltech, Pasadena, CA 91125, U.S.A. KOTA SAITO Caltech, Pasadena, CA 91125, U.S.A.

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1 Econometrca, Vol. 83, No. 4 (July, 2015), SAVAGE IN THE MARKET FEDERICO ECHENIQUE Caltech, Paadena, CA 91125, U.S.A. KOTA SAITO Caltech, Paadena, CA 91125, U.S.A. The copyrght to th Artcle held by the Econometrc Socety. It may be downloaded, prnted and reproduced only for educatonal or reearch purpoe, ncludng ue n coure pac. No downloadng or copyng may be done for any commercal purpoe wthout the explct permon of the Econometrc Socety. For uch commercal purpoe contact the Offce of the Econometrc Socety (contact nformaton may be found at the webte or n the bac cover of Econometrca). Th tatement mut be ncluded on all cope of th Artcle that are made avalable electroncally or n any other format.

2 Econometrca, Vol. 83, No. 4 (July, 2015), SAVAGE IN THE MARKET BY FEDERICO ECHENIQUE AND KOTA SAITO 1 We develop a behavoral axomatc characterzaton of ubjectve expected utlty (SEU) under r averon. Gven an ndvdual agent behavor n the maret: aume a fnte collecton of aet purchae wth correpondng prce. We how that uch behavor atfe a revealed preference axom f and only f there ext a SEU model (a ubjectve probablty over tate and a concave utlty functon over money) that account for the gven aet purchae. KEYWORDS: Revealed preference theory, expected utlty, uncertanty. 1. INTRODUCTION THE MAIN RESULT OF THIS PAPER gve a revealed preference characterzaton of r-avere ubjectve expected utlty. Our contrbuton to provde a neceary and uffcent condton for an agent maret behavor to be content wth r-avere ubjectve expected utlty (SEU). The meanng of SEU for a preference relaton ha been well undertood nce Savage (1954), but the meanng of SEU for agent behavor n the maret ha been unnown untl now. R-avere SEU wdely ued by economt to decrbe agent maret behavor, and the new undertandng of r-avere SEU provded by our paper hopefully ueful for both theoretcal and emprcal purpoe. Our paper follow the revealed preference tradton n economc. Samuelon (1938) and Houthaer (1950) decrbe the maret behavor that are content wth utlty maxmzaton. They how that a behavor content wth utlty maxmzaton f and only f t atfe the trong axom of revealed preference. We how that there an analogou revealed preference axom for r-avere SEU. A behavor content wth r-avere SEU f and only f t atfe the trong axom of revealed ubjectve expected utlty (SARSEU). (In the followng, we wrte SEU to mean r-avere SEU when there no potental for confuon.) The motvaton for our exerce twofold. In the frt place, there a theoretcal payoff from undertandng the behavoral counterpart to a theory. In the cae of SEU, we beleve that SARSEU gve meanng to the aumpton 1 We than Km Border and Chr Chamber for npraton, comment, and advce. Matt Jacon uggeton led to ome of the applcaton of our man reult. We alo than emnar audence n Boccon Unverty, Caltech, Collego Carlo Alberto, Prnceton Unverty, RUD 2014 (Warwc), Unverty of Queenland, Unverty of Melbourne, Larry Epten, Edde Deel, Mar Machna, Mamo Marnacc, Fabo Maccheron, John Quah, Ludovc Renou, Kyoungwon Seo, and Peter Waer for comment. We are partcularly grateful to the edtor and three anonymou referee for ther uggeton. The dcuon of tate-dependent utlty and probabltc ophtcaton n Secton 4 and 5 follow cloely the uggeton of one partcular referee The Econometrc Socety DOI: /ECTA12273

3 1468 F. ECHENIQUE AND K. SAITO of SEU n a maret context. The econd motvaton for the exerce that SARSEU can be ued to tet for SEU n actual data. We dcu each of thee motvaton n turn. SARSEU gve meanng to the aumpton of SEU n a maret context. We can, for example, ue SARSEU to undertand how SEU dffer from maxmn expected utlty (Secton 6). The dfference between the SEU and maxmn utlty repreentaton obvou, but the dfference n the behavor captured by each model much harder to grap. In fact, we how that SEU and maxmn expected utlty are ndtnguhable n ome tuaton. In a mlar ven, we can ue SARSEU to undertand the behavoral dfference between SEU and probabltc ophtcaton (Secton 5). Fnally, SARSEU help u undertand how SEU retrct behavor over and above what captured by the more general model of tate-dependent utlty (Secton 4). The Supplemental Materal (Echenque and Sato (2015)) dcue addtonal theoretcal mplcaton. Our reult allow one to tet SEU nonparametrcally n an mportant economc decon-mang envronment, namely that of choce n fnancal maret. The tet doe not only dctate what to loo for n the data (SARSEU), but t alo ugget expermental degn. The yntax of SARSEU may not mmedately lend telf to a practcal tet, but there are two effcent algorthm for checng the axom. One of them baed on lnearzed Afrat nequalte (ee Lemma 7 n Appendx A). The other mplct n Propoton 2. SARSEU on the ame computatonal tandng a the trong axom of revealed preference. Next, we decrbe data one can ue to tet SARSEU. There are experment of decon-mang under uncertanty where ubject mae fnancal decon, uch a Hey and Pace (2014), Ahn, Cho, Gale, and Karv (2014), or Boaert, Ghrardato, Guarnachell, and Zame (2010). Hey and Pace, and Ahn et al. tet SEU parametrcally: they aume a pecfc functonal form. A nonparametrc tet, uch a SARSEU, eem ueful becaue t free the analy from uch aumpton. Boaert et al. (2010) do not tet SEU telf; they tet an mplcaton of SEU on equlbrum prce and portfolo choce. The paper by Hey and Pace ft our framewor very well. They focu on the explanatory power of SEU relatve to varou other model, but they do not tet how well SEU ft the data. Our tet, n contrat, would evaluate goodne of ft and, n addton, be free of parametrc aumpton. The experment by Ahn et al. and Boaert et al. do not ft the etup n our paper becaue they aume that the probablty of one tate nown. In an extenon of our reult to a generalzaton of SEU (ee Appendx B), we how how a veron of SARSEU characterze expected utlty when the probablte of ome tate are objectve and nown. Hence, the reult n our paper are readly applcable to the data from Ahn et al. and Boaert et al. We dcu th applcaton further n Appendx B. SARSEU not only ueful for tetng SEU wth extng expermental data, but t alo gude the degn of new experment. In partcular, SARSEU ugget how one hould chooe the parameter of the degn (prce and budget)

4 SAVAGE IN THE MARKET 1469 o a to evaluate SEU. For example, n a ettng wth two tate, one could chooe each of the confguraton decrbed n Secton 3.1 to evaluate where volaton of SEU come from: tate-dependent utlty or probabltc ophtcaton. Related Lterature The cloet precedent to our paper the mportant wor of Epten (2000). Epten etup the ame a our; n partcular, he aume data on tatecontngent aet purchae, and that probablte are ubjectve and unoberved but table. We dffer n that he focue attenton on pure probabltc ophtcaton (wth no aumpton on r averon), whle our paper on r-avere SEU. Epten preent a neceary condton for maret behavor to be content wth probabltc ophtcaton. Gven that the model of probabltc ophtcaton more general than SEU, one expect that the two axom may be related: Indeed we how n Secton 5 that Epten neceary condton can be obtaned a a pecal cae of SARSEU. We alo preent an example of data that are content wth a r-avere probabltcally ophtcated agent, but that volate SARSEU. Polon and Quah (2013) develop tet for model of decon under r and uncertanty, ncludng SEU (wthout the requrement of r averon). They develop a general approach by whch tetng a model amount to olvng a ytem of (nonlnear) Afrat nequalte. See alo Bayer, Boe, Polon, and Renou (2012), who tudy dfferent model of ambguty by way of Afrat nequalte. Nonlnear Afrat nequalte can be problematc becaue there no nown effcent algorthm for decdng f they have a oluton. Another tran of related wor deal wth objectve expected utlty, aumng obervable probablte. The paper by Green and Srvatava (1986), Varan (1983, 1988), and Kubler, Selden, and We (2014) characterze the data et that are content wth objectve expected utlty theory. Data et n thee paper are jut le our, but wth the added nformaton of probablte over tate. Green and Srvatava allow for the conumpton of many good n each tate, whle we focu on monetary payoff. Varan and Green and Srvatava characterzaton n the form of Afrat nequalte; Kubler et al. mprove on thee by preentng a revealed preference axom. We dcu the relaton between ther axom and SARSEU n the Supplemental Materal. The yntax of SARSEU mlar to the man axom n Fudenberg, Ijma, and Strzalec (2014) and n other wor on addtvely eparable utlty. 2. SUBJECTIVE EXPECTED UTILITY Let S be a fnte et of tate. We occaonally ue S to denote the number S of tate. Let Δ ++ ={μ R S ++ S =1 μ = 1} denote the et of trctly

5 1470 F. ECHENIQUE AND K. SAITO potve probablty meaure on S. In our model, the object of choce are tate-contngent monetary payoff, or monetary act. A monetary act a vector n R S +. We ue the followng notatonal conventon: For vector x y R n, x y mean that x y for all = 1 n; x<y mean that x y and x y; and x y mean that x <y for all = 1 n.theetofallx R n wth 0 x denoted by R n + and the et of all x Rn wth 0 x denoted by R n ++. DEFINITION 1: A data et a fnte collecton of par (x p) R S + RS ++. The nterpretaton of a data et (x p ) K =1 that t decrbe K purchae of a tate-contngent payoff x at ome gven vector of prce p and ncome p x. A ubjectve expected utlty (SEU) model pecfed by a ubjectve probablty μ Δ ++ and a utlty functon over money u : R + R. AnSEUmaxmzng agent olve the problem (1) μ u(x ) max x B(p I) S when faced wth prce p R S ++ and ncome I>0. The et B(p I) ={y RS + : p y I} the budget et defned by p and I. A data et our noton of obervable behavor. The meanng of SEU a an aumpton, the behavor that are a f they were generated by an SEU maxmzng agent. We call uch behavor SEU ratonal. DEFINITION 2: A data et (x p ) K =1 ubjectve expected utlty ratonal (SEU ratonal) f there μ Δ ++ and a concave and trctly ncreang functon u : R + R uch that, for all, y B ( p p x ) μ u(y ) μ u ( ) x S S Three remar are n order. Frt, we retrct attenton to concave (.e., ravere) utlty and our reult wll have nothng to ay about the nonconcave cae. In the econd place, we aume that the relevant budget for the th obervaton B(p p x ). Implct the aumpton that p x the relevant ncome for th problem. Th aumpton omewhat unavodable and tandard procedure n revealed preference theory. Thrd, we hould emphaze that there n our model only one good (whch we thn of a money) n each tate. The problem wth many good nteretng, but beyond the method developed n the preent paper (ee Remar 4).

6 SAVAGE IN THE MARKET A CHARACTERIZATION OF SEU RATIONAL DATA In th ecton, we ntroduce the axom for SEU ratonalty and tate our man reult. We tart by dervng, or calculatng, the axom n a pecfc ntance. In th dervaton, we aume (for eae of expoton) that u dfferentable. In general, however, an SEU ratonal data et may not be ratonalzable ung a dfferentable u; ee Remar 3 below. The frt-order condton for SEU maxmzaton (1)are (2) μ u (x ) = λp The frt-order condton nvolve three unobervable: ubjectve probablty μ, margnal utlte u (x ), and Lagrange multpler λ The 2 2 Cae: K = 2 and S = 2 We llutrate our analy wth a dcuon of the 2 2 cae, the cae when there are two tate and two obervaton. In the 2 2 cae, we can ealy ee that SEU ha two nd of mplcaton and, a we explan n Secton 4 and 5, each nd derved from a dfferent qualtatve feature of SEU. Let u mpoe the frt-order condton (2) on a data et. Let (x 1 p 1), (x 2 p 2) beadataetwthk = 2andS = 2. For the data et to be SEU ratonal there mut ext μ Δ ++, (λ ) =1 2, and a concave functon u uch that each obervaton n the data et atfe the frt-order condton (2). That, (3) μ u ( x ) = λ p for =,and = 1 2. Equaton (3) nvolve the oberved x and p, a well a the unobervable u, λ, andμ. One free to chooe (ubject to ome contrant) the unobervable to atfy (3). We can undertand the mplcaton of (3) byconderng tuaton n whch the unobervable λ and μ cancel out: u ( ) x ( ) 1 x ( ) 2 x ( ) 1 x 2 (4) u ( ) u x 2 u ( ) = μ u 1 x 1 μ 1 u ( ) μ u 2 x 2 μ 2 u ( ) = λ 1 p 1 x 1 λ 2 p 2 λ 2 p 2 λ 1 p 1 = p 1 p 2 p 2 p 1 Equaton (4) obtaned by dvdng frt-order condton to elmnate term nvolvng μ and λ: th allow u to contran the obervable varable x and p. There are two tuaton of nteret. Suppoe frt that x 1 >x 2 and that x 2 >x 1. The concavty of u mple then that u (x 1 ) u (x 2 ) and u (x 2 ) u (x 1 ). Th mean that the left-hand de of (4) maller than 1. Thu, (5) x 1 >x 2 and x 2 >x 1 p1 p 2 p 2 1 p 1

7 1472 F. ECHENIQUE AND K. SAITO In the econd place, uppoe that x 1 >x 1 whle x 2 >x 2 (o the bundle x 1 and x 2 are on oppote de of the 45 degree lne n R2 ). The concavty of u mple that u (x 1 ) u (x 1 ) and u (x 2 ) u (x 2 ).Thefarleftof(4) then maller than 1. Thu, (6) x 1 >x 1 and x 2 >x 2 p1 p 1 p 2 1 p 2 Requrement (5) and(6) are mplcaton of r-avere SEU for a data et when S = 2andK = 2. We hall ee that they are all the mplcaton of ravere SEU n th cae and that they capture dtnct qualtatve component of SEU (Secton 4 and 5) General K and S We now turn to the general etup and to our man reult. Frt, we hall derve the axom by proceedng along the lne uggeted above n Secton 3.1: Ung the frt-order condton (2), the SEU ratonalty of a data et requre that u ( x u ( ) = μ λ p x μ λ p ) The concavty of u mple omethng about the left-hand de of th equaton when x >x, but the rght-hand de complcated by the preence of unobervable Lagrange multpler and ubjectve probablte. So we chooe par (x x ) wth x >x uch that ubjectve probablte and Lagrange multpler cancel out. For example, conder x 1 >x 2 x 3 >x 1 3 and x 2 3 >x 3 By manpulatng the frt-order condton, we obtan that u ( ) x 1 u ( ) u ( ) x 3 x 2 u ( ) u ( ) x 2 3 x 1 3 u ( ) x 3 ) ( μ2 λ 1 = μ 1 λ 2 p 1 p 2 ( μ3 μ 2 λ 3 λ 1 p ) ( 3 μ1 2 λ 2 p 1 3 μ 3 λ 3 p 2 3 p 3 ) = p 1 p 2 p 3 p 1 3 p 2 3 p 3 Notce that the par (x 1 x 2 ), (x 3 x 1 3 ),and(x 2 3 x 3 ) have been choen o that the ubjectve probablte μ appear n the nomnator a many tme a n the denomnator, and the ame for λ ; hence, thee term cancel out. Such cancelng out motvate condton () and () n the axom below.

8 SAVAGE IN THE MARKET 1473 Now the concavty of u and the aumpton that x 1 >x 2, x 3 >x 1 3,and x 2 3 >x 3 mply that the product of the prce p 1 p 2 p 3 p 1 3 p 2 3 p 3 cannot exceed 1. Thu, we obtan an mplcaton of SEU on prce, an obervable entty. In general, the aumpton of SEU ratonalty requre that, for any collecton of equence a above, approprately choen o that ubjectve probablte and Lagrange multpler wll cancel out, the product of the rato of prce cannot exceed 1. The formal tatement the followng axom. STRONG AXIOM OF REVEALED SUBJECTIVE UTILITY (SARSEU): Aume any equence of par (x x ) n =1 n whch the followng tatement hold: () We have x >x for all. () Each appear a (on the left of the par) the ame number of tme t appear a (on the rght). () Each appear a (on the left of the par) the ame number of tme t appear a (on the rght). Then the product of prce atfe that n =1 p p 1 THEOREM 1: A data et SEU ratonal f and only f t atfe SARSEU. It worth notng that the yntax of SARSEU mlar to that of the man axomn Kubler, Selden, andwe (2014), wth r-neutral prce playng the role of prce n the model wth objectve probablte. The relaton between the two dcued further n Appendx B. We conclude th ecton wth ome remar on Theorem 1. The proof gven n Appendx A. REMARK 1: In the 2 2 cae n Secton 3.1, requrement (5) and(6) are equvalent to SARSEU. REMARK 2: The proof of Theorem 1 provded n Appendx A.Itreleon ettng up a ytem of nequalte from the frt-order condton of an SEU agent maxmzaton problem. Th mlar to the approach n Afrat (1967) and n many other ubequent tude of revealed preference. The dfference that our ytem nonlnear and mut be lnearzed. A crucal tep n the proof an approxmaton reult, whch complcated by the fact that the unnown ubjectve probablte, Lagrange multpler, and margnal utlte all tae value n noncompact et.

9 1474 F. ECHENIQUE AND K. SAITO REMARK 3: Under the aumpton on the data et, x x f ( ) ( ) SARSEU mple SEU ratonalty ung a mooth ratonalzng u. Th condton on the data et play a mlar role to the aumpton ued by Chappor and Rochet (1987) to obtan a mooth utlty ung the Afrat contructon. REMARK 4: In our framewor we aume choce of monetary act, whch mean that conumpton n each tate one-dmenonal. Our reult are not ealy applcable to the multdmenonal ettng, eentally becaue concavty, n general, equvalent to cyclc monotoncty of upergradent, whch we cannot deal wth n our approach. In the one-dmenonal cae, concavty requre only that upergradent are monotone. The condton that ome unnown functon monotone preerved by a monotonc tranformaton of the functon, but th not true of cyclc monotoncty. If one et up the multdmenonal problem a we have done, then one loe the property of cyclc monotoncty when lnearzng the ytem. Fnally, t not obvou from the yntax of SARSEU that one can verfy whether a partcular data et atfe SARSEU n fntely many tep. We how that not only SARSEU decdable n fntely many tep, but there, n fact, an effcent algorthm that decde whether a data et atfe SARSEU. PROPOSITION 2: There an effcent algorthm that decde whether a data et atfe SARSEU. 2 We provde a drect proof of Propoton 2 n Appendx C. Propoton2 can alo be een a a reult of Lemma 7 together wth the lnearzaton n the proofof Theorem 1. The reultng lnear ytem can be decded by ung lnear programmng. 4. STATE-DEPENDENT UTILITY SEU aert, among other thng, the extence of a (concave) tatedependent utlty (.e., an addtvely eparable utlty acro tate). SEU requre more, of coure, but t nteretng to compare SEU wth the weaer theory of tate-dependent utlty. We hall trace the aumpton of tatedependent utlty to a partcular weaenng on SARSEU. The tate-dependent utlty model ay that an agent maxmze U d (x) = u S (x ), an addtvely eparable utlty, for ome collecton (u ) S of concave and trctly ncreang tate-dependent functon, u : R + R. We hould emphaze that here, a n the ret of the paper, we retrct attenton to concave utlty. 2 Effcent mean that the algorthm run n polynomal tme.

10 SAVAGE IN THE MARKET The 2 2 Cae: K = 2 and S = 2 We argued n Secton 3.1 that requrement (5) and(6) are neceary for SEU ratonalty. It turn out that (5) alone capture tate-dependent utlty. A data et that volate requrement (5) can be vualzed on the left of Fgure 1. The fgure depct choce wth x 1 >x 2 and x 2 >x 2, but where requrement (5) volated. Fgure 1 preent a geometrcal argument for why uch a data et not SEU ratonal. Suppoe, toward a contradcton, that the data et SEU ratonal. Snce the ratonalzng functon u concave, t eay to ee that optmal choce mut be ncreang n the level of ncome (the demand functon of a r-avere SEU agent normal). At the rght of Fgure 1, we nclude a budget wth the ame relatve prce a when x 2 wa choen, but where the ncome larger. The larger ncome uch that the budget lne pae through x 1. Snce her demand normal, the agent choce on the larger budget mut be larger than at x 2 : th ndcated by the dotted lne n the pcture. The choce mut le n the lne egment on the larger budget lne that cont of bundle larger than x 2. But uch a choce would volate the wea axom of revealed preference (WARP). Hence, the (counterfactual) choce mpled by SEU at the larger budget would be ncontent wth utlty maxmzaton, contradctng the aumpton of SEU ratonalty. It ueful to emphaze that requrement (5) a trengthenng of WARP, omethng we hall return to below. So SEU ratonalty mple (5), whch a trengthenng of WARP. Now we argue that tate-dependent utlty mple (5) a well. To ee th, uppoe that the agent maxmze u 1 ( ) + u 2 ( ), whereu concave. A n Secton 3.1, aume that u dfferentable. Then x 1 >x 2 and x 2 >x 1 mply that u (x 1 ) u (x 2 ) and u (x 2 ) u (x 1 ). The frt-order condton are x 2 x 2 x 2 x 2 x 1 x 1 x 1 x 1 FIGURE 1. A volaton of requrement (5).

11 1476 F. ECHENIQUE AND K. SAITO u (x ) = λ p ; hence, p 2 p 1 p 1 p 2 = u ( ) x 2 ( 1 u x 1 Indeed, th requrement (5). ) u ( ) x 1 ( 2 ) 1 u x General K and S Adataet(x p ) K =1 tate-dependent utlty (SDU) ratonal f there an addtvely eparable functon U d uch that, for all, y B ( p p x ) U d (y) U d( x ) In the 2 2 cae, we have een that requrement (5) neceary for ratonalzaton by a tate-dependent utlty. More generally, there a natural weaenng of SARSEU that capture ratonalzaton by a tate-dependent utlty. Th weaenng trong enough to mply WARP. Concretely, f one ubttute condton () n SARSEU wth the tatement =, the reultng axom characterze SDU ratonalty. STRONG AXIOM OF REVEALED STATE-DEPENDENT UTILITY (SARSDU): Aume any equence of par (x x >x () We have x () We have =. for all. ) n =1 n whch the followng tatement hold: () Each appear a (on the left of the par) the ame number of tme t appear a (on the rght). Then the product of prce atfe that n =1 p p 1 It hould be obvou that SARSEU mple SARSDU. SARSDU equvalent to SDU ratonalty. THEOREM 3: A data et SDU ratonal f and only f t atfe SARSDU. The proof that SARSDU neceary for tate-dependent utlty mple and follow along the lne developed n Secton 3. The proof of uffcency mlar to the proof ued for the characterzaton of SEU and omtted.

12 SAVAGE IN THE MARKET 1477 Note that n the 2 2 cae, SARSDU and requrement (5) are equvalent. Hence, (5) characterze tate-dependent utlty n the 2 2cae. By Theorem 3 above, we now that SARSDU mple the wea axom of revealed preference (WARP), but t may be ueful to preent a drect proof of the fact that SARSDU mple WARP. DEFINITION 3: A data et (x p ) K =1 atfe WARP f there no and uch that p x p x and p x >p x. PROPOSITION 4: If a data et atfe SARSDU, then t atfe WARP. PROOF: Suppoe, toward a contradcton, that a data et (x p ) K atfe =1 SARSDU but that t volate WARP. Then there are and uch that p x p x and p x >p x. It cannot be the cae that x x for all, o the et S 1 ={ : x <x } nonempty. Chooe S 1 uch that p p p p for all S 1 Now p x p x mple that ( x ) 1 ( x p p x ) x We alo have that p x >p x,o 0 > ( p x ) ( x + p x ) x ( p x ) p ( x + p p x ) x = / S1 p (1 p p p p ) (x ) x } {{ } A + S 1 \{ } p (1 p p p p ) (x x } {{ } B We hall prove that A 0 and that B 0, whch wll yeld the dered contradcton. )

13 1478 F. ECHENIQUE AND K. SAITO For all / S 1, we have that (x x ) 0. If x >x, then SARSDU mple that p p 1 p p a x <x o that the equence {(x x ) (x x )} atfe (), (), and () n SARSDU. Hence A 0. Now conder B. By defnton of,wehavethat p p 1forall S p p 1. Then (x x )<0 mple that ) (1 p p (x p ) x 0 p for all S 1. Hence B 0. Q.E.D. We mae ue of thee reult n the Supplemental Materal, where we how how SARSDU and SARSEU rule out volaton of Savage axom. The Supplemental Materal alo nclude a condton on the data under whch SEU and SDU are obervatonally equvalent. 5. PROBABILISTIC SOPHISTICATION We have looed at the apect of SARSEU that capture the extence of an addtvely eparable repreentaton. SEU alo affrm the extence of a unque ubjectve probablty meaure gudng the agent choce. We now turn to the behavoral counterpart of the extence of uch a probablty. We do not have a characterzaton of probablty ophtcaton. In th ecton, we mply oberve how SARSEU and requrement (6) are related to the extence of a ubjectve probablty. 3 We alo how how SARSEU related to Epten neceary condton for probablty ophtcaton The 2 2 Cae: K = 2 and S = 2 Conder, a before, the 2 2cae:S = 2andK = 2. We argued n Secton 3.1that requrement (5)and(6) arenecearyforseuratonalty, andn Secton 4 that requrement (5) capture the extence of a tate-dependent repreentaton. We now how that requrement (6) reult from mpong a unque ubjectve probablty gudng the agent choce. 3 In the Supplemental Materal, we alo how that requrement (6) rule out volaton of Savage P4, whch capture the extence of a ubjectve probablty (Machna and Schmedler (1992)).

14 SAVAGE IN THE MARKET 1479 x 2 x 2 x 2 x 2 x 1 x 1 x 1 x 1 FIGURE 2. Volaton of requrement (6). Fgure 2 exhbt a data et that volate requrement (6). We have drawn the ndfference curve of the agent when choong x 2. Recall that the margnal rate of ubttuton (MRS) μ 1 u (x 1 )/μ 2 u (x 2 ). At the pont where the ndfference curve croe the 45 degree lne (dotted), one can read the agent ubjectve probablty off the ndfference curve becaue u (x 1 )/u (x 2 ) = 1 and, therefore, the MRS equal μ 1 /μ 2. So the tangent lne to the ndfference curve at the 45 degree lne decrbe the ubjectve probablty. It then clear from the pcture that th tangent lne mut be flatter than the budget lne at whch x 2 wa choen. On the other hand, the ame reaonng reveal that the ubjectve probablty mut defne a teeper lne than the budget lne at whch x 1 wa choen. Th a contradcton, a the latter budget lne teeper than the former and tangent to an ndfference curve on the 45 degree lne mut be parallel General K and S In the followng dcuon, we focu ntead on the relaton wth probabltc ophtcaton, namely the relaton between SARSEU and the axom n Epten (2000). Epten tude the mplcaton of probabltc ophtcaton for conumpton data et. He conder the ame nd of economc envronment a we do and the ame noton of a data et. He focue on probabltc ophtcaton ntead of SEU and, mportantly, doe not aume r averon. Epten how that a data et ncontent wth probabltc ophtcaton f there ext t S and ˆ K uch that () p p t and p ˆ p ˆ,wthatleat t one trct nequalty, and () x >x t and x ˆ <xˆ. t Of coure, an SEU ratonal agent probabltcally ophtcated. Indeed, our next reult etablhe that a volaton of Epten condton mple a volaton of SARSEU.

15 1480 F. ECHENIQUE AND K. SAITO PROPOSITION 5: If a data et (x p ) K =1 atfe SARSEU, then () and () cannot both hold for ome t S and ˆ K. xˆ t PROOF: Suppoe that t S and ˆ K are uch that () hold. Then {(x x ˆ t ) (x )} atfe the condton n SARSEU. Hence, SARSEU requre that p p t p ˆ t 1, o that p p ˆ p t or p ˆ p ˆ t. Hence, () volated. Q.E.D. Propoton 5 rae the ue of whether SEU and probabltc ophtcaton are dtnguhable. In the followng dcuon, we how that we can ndeed dtnguh the two model: We preent an example of a data et that volate SARSEU, but that content wth a r-avere probabltcally ophtcated agent. Hence the weaenng n gong from SEU to probabltc ophtcaton ha emprcal content. Let S ={ }. We defne a data et a follow. Let x 1 = (2 2), p 1 = (1 2), x 2 = (8 0), andp 2 = (1 1). It clear that the data et volate SARSEU: x 2 >x 1 and x 1 >x 2 whle p 2 p 1 = 2 > 1 p 1 p 2 Oberve, moreover, that the data et pecfcally volate requrement (5), not (6). Fgure 3 on the rght llutrate the data et (budget lne are drawn a thc lne). We hall now argue that th data et ratonalzable by a r-avere probabltcally ophtcated agent. Fx μ Δ ++ wth μ 1 = μ 2 = 1/2, a unform x 2 x 2 p 2 p 1 x 1 x 2 x 1 x 1 FIGURE 3. Probabltcally ophtcated volaton of SARSEU.

16 SAVAGE IN THE MARKET 1481 probablty over S. Any vector x R 2 + nduce the probablty dtrbuton on R + gven by x 1 wth probablty 1/2 andx 2 wth probablty 1/2. Let Π be the et of all unform probablty meaure on R + wth upport havng cardnalty maller than or equal to 2. We hall defne a functon V : Π R that repreent a probabltcally ophtcated preference and for whch the choce n the data et are optmal. 4 We contruct a monotone ncreang and quaconcave h : R 2 R + + and then defne V(π)= h( x π x π ),where x π the mallet pont n the upport of π and x π the larget. A a conequence of the monotoncty of h, V repreent a probabltcally ophtcated preference. The preference alo r-avere. The functon h contructed o that (x 1 x 2 ) h(max{x 1 x 2 } mn{x 1 x 2 }) ha the map of ndfference curve llutrated on the left n Fgure 3.Thereare two mportant feature of the ndfference curve drawn n the fgure. The frt that ndfference curve exhbt a convex preference, whch enure that the agent wll be r-avere. The econd that ndfference curve become le convex a one move up and to the rght n the fgure. A a reult, the lne that normal to p 1 upport the ndfference curve through x 1, whle the lne that normal to p 2 upport the ndfference curve through x 2. On the rght of Fgure 3 are ndfference curve drawn a dahed lne and budget lne drawn a thc lne. It clear that the contructed preference ratonalze the choce n the data et. 6. MAXMIN EXPECTED UTILITY In th ecton, we demontrate one ue of our man reult to tudy the dfference between SEU and maxmn expected utlty. We how that SEU and maxmn are behavorally ndtnguhable n the 2 2 cae, but dtnguhable more generally. The maxmn SEU model, frt axomatzed by Glboa and Schmedler (1989), pot that an agent maxmze mn μ u(x ) μ M S where M a cloed and convex et of probablte. A data et (x p ) K =1 maxmn expected utlty ratonal f there a cloed and convex et M Δ ++, and a concave and trctly ncreang functon u : R + R uch that, for all, y B ( p p x ) mn μ M S μ u(y ) mn μ M μ u ( ) x Note that we retrct attenton to r-avere maxmn expected utlty. S 4 It eay to how that there a probabltcally ophtcated wea order defned on the et of all probablty meaure on R 2 + wth fnte upport, uch that V repreent on Π. The detal of the example are techncal and left to the Supplemental Materal.

17 1482 F. ECHENIQUE AND K. SAITO PROPOSITION 6: Let S = K = 2. Then a data et maxmn expected utlty ratonal f and only f t SEU ratonal. The proofofpropoton 6 gven n the Supplemental Materal. The reult n Propoton 6 doe not, however, extend beyond the cae of two obervaton. In the Supplemental Materal, we provde an example of data from a (r-avere) maxmn expected utlty agent that volate SARSEU. APPENDIX A: PROOF OF THEOREM 1 We frt gve three prelmnary and auxlary reult. Lemma 7 provde nonlnear Afrat nequalte for the problem at hand. A veron of th lemma appear, for example, n Green and Srvatava (1986), Varan (1983),orBayer et al. (2012). Lemma 8 and 9 are veron of the theorem of the alternatve. LEMMA 7: Let (x p ) K =1 be a data et. The followng tatement are equvalent: () We have that (x p ) K =1 SEU ratonal. () There are trctly potve number v, λ, μ for = 1 S and = 1 K uch that (7) μ v = λ p x >x v v PROOF: We hall prove that () mple (). Let (x p ) K =1 be SEU ratonal. Let μ R S and let u : R ++ + R be a n the defnton of SEU ratonal data et. Then (ee, for example, Theorem 28.3 of Rocafellar (1997)) there are number λ 0, = 1 K, uch that f we let v = λ p μ then v u(x) f x > 0, and there w u(x) wth v w f x = 0(here we have u(0) ). In fact, nce u trctly ncreang, t eay to ee that λ > 0 and, therefore, v > 0. By the concavty of u and the conequent monotoncty of u(x ) (ee Theorem 24.8 of Rocafellar (1997)), f x >x > 0, v u(x ),andv u(x ), then v v.ifx >x = 0, then w u(x ) wth v w.sov w v. In econd place, we how that () mple (). Suppoe that the number v, λ,andμ,for = 1 S and = 1 K are a n (). Enumerate the element of X {x K S} n ncreang order: y <y 2 < <y n. Let y = mn{v : x = y } and ȳ = max{v : x = y }. Letz = (y + y +1 )/2, = 1 n 1, z 0 = 0, and z n = y n + 1. Let f be a correpondence defned a [ y ȳ ] f z = y, (8) f(z)= max{ȳ : z<y } f y n >zand (z y ), y n /2 f y n <z.

18 SAVAGE IN THE MARKET 1483 Then, by the aumpton placed on v and by contructon of f, y<y, v f(y) and v f(y ) mply that v v. Then the correpondence f monotone and there ext a concave functon u for whch u = f (ee, e.g., Theorem 24.8 of Rocafellar (1997)). Gven that v > 0, all the element n the range of f are potve and, therefore, u a trctly ncreang functon. Fnally, for all ( ), λ p /μ = v u(v ) and, therefore, the frt-order condton to a maxmum choce of x hold at x.snceu concave, the frt-order condton are uffcent. The data et therefore SEU ratonal. Q.E.D. We hall ue the followng lemma, whch a veron of the theorem of the alternatve. Th Theorem n Stoer and Wtzgall (1970). We hall ue t here n the cae where F ether the real or the ratonal number feld. LEMMA 8: Let A be an m n matrx, let B be an l n matrx, and let E be an r n matrx. Suppoe that the entre of the matrce A, B, and E belong to a commutatve ordered feld F. Exactly one of the followng alternatve true. () There u F n uch that A u = 0, B u 0, and E u 0. () There θ F m, η F l, and π F r uch that θ A + η B + π E = 0, π>0 and η 0. The next lemma a drect conequence of Lemma 8: ee Lemma 12 n Chamber and Echenque (2014) for a proof. LEMMA 9: Let A be an m n matrx, let B be an l n matrx, and let E be an r n matrx. Suppoe that the entre of the matrce A, B, and E are ratonal number. Exactly one of the followng alternatve true. () There u R n uch that A u = 0, B u 0, and E u 0. () There θ Q m, η Q l, and π Q r uch that θ A + η B + π E = 0, π>0 and η 0. A.1. Necety LEMMA 10: If a (x p ) K =1 SEU ratonal, then t atfe SARSEU. PROOF: Let(x p ) K be SEU ratonal, and let μ Δ =1 ++ and u : R + R be a n the defnton of SEU ratonal. By Lemma 7, there ext a trctly potve oluton v, λ, μ to the ytem n tatement () of Lemma 7 wth v u(x) when x > 0andv w u(x) when x = 0. Let (x x ) n =1 be a equence atfyng the three condton n SARSEU. Then x >x. Suppoe that x > 0. Then v u(x ) and v By the concavty of u, t follow that λ μ p λ μ p u(x ). (ee Theorem 24.8

19 1484 F. ECHENIQUE AND K. SAITO of Rocafellar (1997)). Smlarly, f x u(x ).Soλ μ p 1 n =1 λ μ p. Therefore, λ μ p λ μ p = n =1 p p = 0, then v u(x ) and v w a the equence atfe condton () and () of SARSEU, and, hence, the number λ and μ appear the ame number of tme n the denomnator a n the numerator of th product. Q.E.D. A.2. Suffcency We proceed to prove the uffcency drecton. An outlne of the argument a follow. We now from Lemma 7 that t uffce to fnd a oluton to the Afrat nequalte (actually frt-order condton), wrtten a tatement () n the lemma. So we et up the problem to fnd a oluton to a ytem of lnear nequalte obtaned from ung logarthm to lnearze the Afrat nequalte n Lemma 7. Lemma 11 etablhe that SARSEU uffcent for SEU ratonalty when the logarthm of the prce are ratonal number. The role of ratonal logarthm come from our ue of a veron the theorem of the alternatve (ee Lemma 9): when there no oluton to the lnearzed Afrat nequalte, then the extence of a ratonal oluton to the dual ytem of nequalte mple a volaton of SARSEU. The bul of the proof goe nto contructng a volaton of SARSEU from a gven oluton to the dual. The next tep n the proof (Lemma 12) etablhe that we can approxmate any data et atfyng SARSEU wth a data et for whch the logarthm of prce are ratonal and for whch SARSEU atfed. Th tep crucal, and omewhat delcate. One mght have tred to obtan a oluton to the Afrat nequalte for perturbed ytem (wth prce that are ratonal after tang log) and then condered the lmt. Th doe not wor becaue the oluton to our ytem of nequalte are n a noncompact pace. It not clear how to etablh that the lmt ext and are well behaved. Lemma 12 avod the problem. Fnally, Lemma 13 etablhe the reult by ung another veron of the theorem of the alternatve, tated a Lemma 8 above. The tatement of the lemma follow. The ret of the paper devoted to the proof of thee lemma. LEMMA 11: Let data et (x p ) =1 atfy SARSEU. Suppoe that log(p ) Q for all and. Then there are number v, λ, μ for = 1 S and = 1 K atfyng (7) n Lemma 7.

20 SAVAGE IN THE MARKET 1485 LEMMA 12: Let data et (x p ) =1 atfy SARSEU. Then for all potve number ε, there ext q [p ε p ] for all S and K uch that log q Q and the data et (x q ) =1 atfy SARSEU. LEMMA 13: Let data et (x p ) =1 atfy SARSEU. Then there are number v, λ, μ for = 1 S and = 1 K atfyng (7) n Lemma 7. A.2.1. Proof of Lemma 11 We lnearze the equaton n ytem (7) of Lemma 7. The reult (9) log v + log μ log λ log p = 0 (10) x >x log v log v In the ytem compred by (9) and(10), the unnown are the real number log v,logμ,andlogλ for = 1 K and = 1 S. Frt, we are gong to wrte the ytem of nequalte (9) and(10) nmatrx form. We hall defne a matrx A uch that there are potve number v, λ,and μ the log of whch atfy (9) f and only f there a oluton u R K S+K+S+1 to the ytem of equaton A u = 0 and for whch the lat component of u trctly potve. Let A be a matrx wth K S row and K S + S + K + 1 column, defned a follow: We have one row for every par ( ), one column for every par ( ), one column for every, one column for each, and one lat column. In the row correpondng to ( ), the matrx ha zeroe everywhere wth the followng excepton: t ha a 1 n the column for ( ); t ha a 1 n the column for ; thaa 1 n the column for ; tha log p n the very lat column. Matrx A loo le (1 1) ( ) (K S) 1 S 1 K p (1 1) log p 1 1 ( ) log p (K S) log p K S Conder the ytem A u = 0. If there are number olvng (9), then thee defne a oluton u R K S+S+K+1 for whch the lat component. If, on the other hand, there a oluton u R K S+S+K+1 to the ytem A u = 0nwhch the lat component (u K S+S+K+1 ) trctly potve, then by dvdng through by the lat component of u, we obtan number that olve (9).

21 1486 F. ECHENIQUE AND K. SAITO In econd place, we wrte the ytem of nequalte (10) nmatrxform. There one row n B for each par ( ) and ( ) for whch x >x.in the row correpondng to x >x, we have zeroe everywhere wth the excepton of a 1 n the column for ( ) and a 1 n the column for ( ).LetB be the number of row of B. In thrd place, we have a matrx E that capture the requrement that the lat component of a oluton be trctly potve. The matrx E ha a ngle row and K S + S + K + 1 column. It ha zeroe everywhere except for 1 n the lat column. To um up, there a oluton to ytem (9) and(10) f and only f there a vector u R K S+S+K+1 that olve the ytem of equaton and lnear nequalte { A u = 0 S1 : B u 0 E u 0 Note that E u a calar, o the lat nequalty the ame a E u>0. The entre of A, B, ande are ether 0, 1, or 1, wth the excepton of the lat column of A. Under the hypothee of the lemma we are provng, the lat column cont of ratonal number. By Lemma 9, then, there uch a oluton u to S1 f and only f there no vector (θ η π) Q K S+B+1 that olve the ytem of equaton and lnear nequalte { θ A + η B + π E = 0 S2 : η 0 π>0 In the followng dcuon, we hall prove that the nonextence of a oluton u mple that the data et mut volate SARSEU. Suppoe then that there no oluton u and let (θ η π)be a ratonal vector a above, olvng ytem S2. By multplyng (θ η π) by any potve nteger, we obtan new vector that olve S2, o we can tae (θ η π) to be nteger vector. Henceforth, we ue the followng notatonal conventon: For a matrx D wth K S + S + K + 1 column, wrte D 1 for the ubmatrx of D correpondng to the frt K S column; let D 2 be the ubmatrx correpondng to the followng S column; let D 3 correpond to the next K column; and let D 4 correpond to the lat column. Thu, D =[D 1 D 2 D 3 D 4 ]. CLAIM 14: We have () θ A 1 + η B 1 = 0, () θ A 2 = 0, () θ A 3 = 0, and (v) θ A 4 + π E 4 = 0. PROOF: Snceθ A + η B + π E = 0, then θ A + η B + π E = 0for all = 1 4. Moreover, nce B 2, B 3, B 4, E 1, E 2,andE 3 are zero matrce, we obtan the clam. Q.E.D.

22 SAVAGE IN THE MARKET 1487 We tranform the matrce A and B ung θ and η. DefneamatrxA from A by lettng A have the ame number of column a A and ncludng () θ r cope of the rth row when θ r > 0, () omttng row r when θ r = 0, and () θ r cope of the rth row multpled by 1 whenθ r < 0. We refer to row that are cope of ome r wth θ r > 0aorgnal row and refer to thoe that are cope of ome r wth θ r < 0aconverted row. Smlarly, we defne the matrx B from B by ncludng the ame column a B and η r cope of each row (and thu omttng row r when η r = 0; recall that η r 0forallr). CLAIM 15: For any ( ), all the entre n the column for ( ) n A 1 are of the ame gn. PROOF: By defnton of A, the column for ( ) wll have zero n all t entre wth the excepton of the row for ( ). InA,foreach( ), there are three mutually excluve poblte: the row for ( ) n A can () not appear n A, () appear a orgnal, or () appear a converted. Th how the clam. Q.E.D. of par that atfe cond- CLAIM 16: There ext a equence (x x ton () n SARSEU. ) n =1 PROOF: DefneX ={x K S}. We defne uch a equence by nducton. Let B 1 = B. Gven B,defneB +1 a follow. Denote by > the bnary relaton on X defned by z> z f z>z and there at leat one copy of a row correpondng to z>z n B : there at leat one par ( ) and ( ) for whch () x >x, () z = x and z = x, and () the row correpondng to x >x n B had trctly potve weght n η. The bnary relaton > cannot exhbt cycle becaue > >. There, therefore, at leat one equence z 1 z L n X uch that z j > z j+1 for all j = 1 L 1 and wth the property that there no z X wth z> z or 1 z L > z. Oberve that B ha at leat one row correpondng to z j > z j+1 for all j = 1 L 1. Let the matrx B +1 be defned a the matrx obtaned from B by omttng one copy of a row correpondng to z j >z for each j = 1 L j+1 1. The matrx B +1 ha trctly fewer row than B. There, therefore, n for whch B n +1 would have no row. The matrx B n ha row, and the procedure of omttng row from B n wll remove all row of B n. Defne a equence (x x ) n of par by lettng =1 x = z and 1 x = z L.Note that, a a reult, x >x for all. Therefore, the equence (x x ) n =1 of par atfe condton () n SARSEU. Q.E.D.

23 1488 F. ECHENIQUE AND K. SAITO We hall ue the equence (x x ) n =1 of par a our canddate volaton of SARSEU. Conder a equence of matrce A, = 1 n, defned a follow. Let A 1 = A, B 1 = B,andC 1 = [ ] A 1 B. Oberve that the row of 1 C1 add to the null vector by Clam 14. We hall proceed by nducton. Suppoe that A ha been defned and that the row of C = [ ] A B add to the null vector. Recall the defnton of the equence x = z > 1 >z L = x. There no z X wth z> z or 1 z L > z, o for the row of C to add to zero there mut be a 1 na n the column correpondng to 1 ( ) and a 1 n A 1 n the column correpondng to ( ).Letr be a row n A correpondng to ( ) and let r be a row correpondng to ( ). The extence of a 1nA 1 n the column correpondng to ( ) and a 1 n A n the column correpondng to ( 1 ) enure that r and r ext. Note that the row r a converted row whle r orgnal. Let A +1 be defned from A by deletng the two row, r and r. CLAIM 17: The um of r, r, and the row of B that are deleted when formng B +1 (correpondng to the par z j >z j+1, j = 1 L 1) add to the null vector. PROOF: Recall that z j > z for all j = 1 L j+1 1. So when we add row correpondng to z j > z and j+1 z j+1 > z j+2, then the entre n the column for ( ) wth x = z j+1 cancel out and the um zero n that entry. Thu, when we add the row of B that are not n B +1, we obtan a vector that 0 everywhere except the column correpondng to z and 1 z L. Th vector cancel out wth r + r, by defnton of r and r. Q.E.D. CLAIM 18: The matrx A can be parttoned nto par of row (r r ) n whch the row r are converted and the row r are orgnal. PROOF: For each, A +1 dffer from A n that the row r and r are removed from A to form A +1. We hall prove that A compoed of the 2n row r and r. Frt note that nce the row of C add up to the null vector, and A +1 and B +1 are obtaned from A and B by removng a collecton of row that add up to zero, then the row of C +1 mut add up to zero a well. We now how that the proce top after n tep: all the row n C n are removed by the procedure decrbed above. By way of contradcton, uppoe that there ext row left after removng r n and r n. Then, by the argument above, the row of the matrx C n +1 mut add to the null vector. If there are row left, then the matrx C n +1 well defned. By defnton of the equence B, however, B n +1 an empty matrx. Hence, row remanng n A n +1 1 mutadduptozero.byclam15, the entre of a

24 SAVAGE IN THE MARKET 1489 column ( ) of A are alway of the ame gn. Moreover, each row of A ha a nonzero element n the frt K S column. Therefore, no ubet of the column of A 1 can um to the null vector. Q.E.D. CLAIM 19: () For any and, f ( ) = ( ) for ome, then the row correpondng to ( ) appear a orgnal n A. Smlarly, f ( ) = ( ) for ome, then the row correpondng to ( ) appear converted n A. () If the row correpondng to ( ) appear a orgnal n A, then there ome wth ( ) = ( ). Smlarly, f the row correpondng to ( ) appear converted n A, then there wth ( ) = ( ). PROOF: Part () true by defnton of (x x ). Part () mmedate from Clam 18 becaue f the row correpondng to ( ) appear orgnal n A, then t equal r for ome and then ( ) = ( ). The reult mlar when the row appear converted. Q.E.D. CLAIM 20: The equence (x x SARSEU. ) n =1 atfe condton () and () n PROOF: ByClam14(), the row of A 2 add up to zero. Therefore, the number of tme that appear n an orgnal row equal the number of tme that t appear n a converted row. By Clam 19, then, the number of tme appear a equal the number of tme t appear a. Therefore condton () n the axom atfed. Smlarly, by Clam 14(), the row of A 3 add to the null vector. Therefore, the number of tme that appear n an orgnal row equal the number of tme that t appear n a converted row. By Clam 19, then, the number of tme that appear a equal the number of tme t appear a.there- fore, condton () n the axom atfed. Q.E.D. Fnally, n the followng, we how that n of Lemma 11, a the equence (x x SARSEU. CLAIM 21: We have n =1 p p > 1. ) n =1 =1 p p > 1, whch fnhe the proof would then exhbt a volaton of PROOF: ByClam14(v) and the fact that the ubmatrx E 4 calar 1, we obtan ( n ( ) ) 0 = θ A 4 + πe 4 = r + r + π =1 4 equal the

25 1490 F. ECHENIQUE AND K. SAITO where ( n (r =1 + r )) 4 the (calar) um of the entre of A 4. Recall that log p the lat entry of row r and that log p the lat entry of row r, a r converted and r orgnal. Therefore, the um of the row of A are n 4 =1 log(p /p ).Then n =1 log(p /p ) = π<0. Thu, n p > 1. Q.E.D. =1 p A.2.2. Proof of Lemma 12 For each equence σ = (x x ) n =1 that atfe condton (), (), and () n SARSEU, we defne a vector t σ N K2 S 2. For each par (x x ), wehall dentfy the par wth (( ) ( )).Lett σ(( ) ( )) be the number of tme that the par (x x ) appear n the equence σ. One can then decrbe the atfacton of SARSEU by mean of the vector t σ.defne T = { t σ N K2 S 2 σ atfe condton (), (), () n SARSEU } Oberve that the et T depend only on (x ) K n the data et =1 (x p ) K.It =1 doe not depend on prce. For each (( ) ( )) uch that x >x, defne ˆδ(( ) ( )) = log( p ) and defne ˆδ(( ) ( )) = 0whenx p x.then ˆδ a K 2 S 2 - dmenonal real-valued vector. If σ = (x x ) n, then =1 ( ( ˆδ t σ = ˆδ ( ) )) ( ( t σ ( ) )) (( ) ( )) (KS) 2 ( n = log =1 p p ) So the data et atfe SARSEU f and only f ˆδ t 0forallt T. Enumerate the element n X n ncreang order: y 1 <y 2 < <y N and fx an arbtrary (0 1). We hall contruct by nducton a equence {(ε ξ (n))}n, n=1 where ε (n) defned for all ( ) wth x = y n. By the denene of the ratonal number and the contnuty of the exponental functon, for each ( ) uch that x = y, there ext a potve number ε (1) uch that log(p ε(1)) Q and ξ <ε (1) <1. Let ε(1) = mn{ε (1) x = y }. In econd place, for each ( ) uch that x = y, there ext a potve ε (2) uch that log(p ε(2)) Q and ξ<ε(2)<ε(1).letε(2) = mn{ε(2) x = y }.

26 SAVAGE IN THE MARKET 1491 In thrd place, and reaonng by nducton, uppoe that ε(n) ha been defned and that For each ( ) uch that x ξ<ε(n). = y n+1, letε (n + 1)>0 be uch that log(p ε(n + 1)) Q and ξ <ε (n + 1)<ε(n).Letε(n + 1) = mn{ε (n + ) x = y n}. Th defne the equence (ε (n)) by nducton. Note that ε(n+1)/ε(n) < 1 for all n.let ξ<1 be uch that ε (n + 1)/ε(n) < ξ. For each K and S,letq = p ε(n),wheren uch that x = y n.we clam that the data et (x q ) K atfe SARSEU. Let =1 δ be defned from (q ) K n the ame manner a =1 ˆδ wa defned from (p ) K. =1 For each par (( ) ( )) wth x >x,fn and m are uch that x = y n and x = y m, then n>m. By defnton of ε, Hence, ε (n) (m) < ε (n) ε(m) < ξ<1 ε δ ( ( ) ( )) = log p ε(n) p < log (m) p ε = ˆδ ( x x ) p + log ξ<log p p Thu, for all t T, δ t ˆδ t 0, a t 0 and the data et (x p ) K atfe =1 SARSEU. Thu, the data et (x q ) K =1 atfe SARSEU. Fnally, note that ξ<ε (n) < 1forallnand each K, S. So that by choong ξ cloe enough to 1, we can tae the prce (q ) to be a cloe to (p ) a dered. A.2.3. Proof of Lemma 13 Conder the ytem compred by (9) and(10) n the proof of Lemma 11. Let A, B,andE be contructed from the data et a n the proof of Lemma 11. The dfference wth repect to Lemma 11 that now the entre of A 4 may not be ratonal. Note that the entre of E, B,andA, = 1 2 3, are ratonal. Suppoe, toward a contradcton, that there no oluton to the ytem compred by (9)and(10). Then, by the argument n the proof of Lemma 11, there no oluton to ytem S1. Lemma 8 wth F = R mple that there a real vector (θ η π)uch that θ A + η B + π E = 0andη 0, π>0. Recall that B 4 = 0andE 4 = 1, o we obtan that θ A 4 + π = 0. Let (q ) K be vector of prce uch that the data et =1 (x q ) K atfe =1 SARSEU and log q Q for all and. (Such(q ) K =1 ext by Lemma 12.) Contruct matrce A, B,andE from th data et n the ame way a A, B, and E contructed n the proof of Lemma 11. Note that only the prce are

27 1492 F. ECHENIQUE AND K. SAITO dfferent n (x q ) compared to (x p ).SoE = E, B = B, anda = A for = Snce only prce q are dfferent n th data et, only A 4 may be dfferent from A 4. By Lemma 12, we can chooe prce q uch that θ A θ A 4 4 <π/2. We have hown that θ A 4 = π, o the choce of prce q guarantee that θ A < 0. Let 4 π = θ A > 0. 4 Note that θ A + η B + π E = 0for = 1 2 3, a (θ η π) olve ytem S2 for matrce A, B, ande, anda = A, B = B,andE = 0for = Fnally, B 4 = 0, o θ A 4 +η B 4 +π E 4 = θ A 4 +π = 0. We alo have that η 0andπ > 0. Therefore, θ, η, andπ conttute a oluton to S2 formatrce A, B,andE. Lemma 8 then mple that there no oluton to S1 for matrce A, B, and E. So there no oluton to the ytem compred by (9) and(10) n the proof of Lemma 11. However, th contradct Lemma 11 becaue the data et (x q ) atfe SARSEU and log q Q for all = 1 K and = 1 S. APPENDIX B: SUBJECTIVE OBJECTIVE EXPECTEDUTILITY We turn to an envronment n whch a ubet of tate have nown probablte. Let S S be a et of tate and aume gven μ, the probablty of tate for S. We allow for the two extreme cae: S = S when all tate are objectve and we are n the etup of Green and Srvatava (1986), Varan (1983),andKubler, Selden, and We (2014), ors =, whch the tuaton n the body of our paper. The cae when S a ngleton tuded expermentally by Ahnetal. (2014) and Boaert et al. (2010). DEFINITION 4: A data et (x p ) K =1 ubjectve objectve expected utlty ratonal (SOEU ratonal) f there μ Δ ++, η>0, and a concave and trctly ncreang functon u : R + R uch that for all S, μ = ημ and for all K, y B ( p p x ) μ u(y ) μ u ( ) x S S In the defnton above, η a parameter that capture the dfference n how the agent treat objectve and ubjectve probablte. Note that nce η contant, relatve objectve probablte (the rato of the probablty of one tate n S to another) unaffected by η. The preence of η ha the reult, n tude wth a ngle objectve tate (a n Ahn et al. and Boaert et al.), of renderng the objectve tate ubjectve. In tude of objectve expected utlty, a crucal apect of the data et the prce-probablty rato, or r-neutral prce, defned a follow: for K