TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES

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1 New title: Teting theorie of financial deciion making Publihed in the Proceeding of the National Academy of Science, Volume 113, Number 15; April 12, TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES CHRISTOPHER P. CHAMBERS, FEDERICO ECHENIQUE, AND KOTA SAITO Abtract. We provide revealed preference axiom that characterize model of tranlation invariant preference. In particular, we characterize the model of variational, maxmin, CARA and CRRA utilitie. In each cae we preent a revealed preference axiom that i atified by a dataet if and only if the dataet i conitent from the correponding utility repreentation. Our reult complement traditional exercie in deciion theory that take preference a primitive. 1. Introduction We work out the tetable implication of model with tranlation invariant preference. Given a finite dataet on purchae of tatecontingent aet, we give a revealed preference axiom that decribe the dataet that are conitent with different model of tranlation invariant preference. Thee model include rik neutral variational preference (Maccheroni et al., 2006), rik neutral maxmin preference (Gilboa and Schmeidler, 1989), and ubjective expected utility preference with contant abolute rik averion: o-called CARA preference. Analogouly to the Chamber i affiliated with the Department of Economic, Univerity of California, San Diego. Echenique and Saito are affiliated with the Diviion of the Humanitie and Social Science, California Intitute of Technology, Paadena CA cpchamber@ucd.edu (Chamber), fede@caltech.edu (Echenique), kaito@h.caltech.edu (Saito). We are very grateful to Simon Grant and Maimo Marinacci, who poed quetion to u that led to ome of the reult in thi paper. 1

2 2 CHAMBERS, ECHENIQUE, AND SAITO CARA cae, we alo work out the tetable implication of ubjective expected utility preference with contant relative rik averion, the CRRA preference (thee form the homothetic cla alluded to in the title). The model have been ued by economit for different purpoe. Variational and maxmin preference are the mot commonly-ued model of ambiguity averion. They are alo ued to capture model robutne (Hanen and Sargent, 2008). CARA and CRRA preference are very common in applied work in macroeconomic and finance, among other field. Our contribution i to tart from finite data on tate-contingent conumption purchae, uch a one would oberve from a market experiment on choice under uncertainty (Hey and Pace, 2014; Ahn et al., 2014; Bayer et al., 2012; Boaert et al., 2010). We decribe the dataet that are rationalizable a conitent with a preference relation that atifie tranlation invariance. When we ay that we decribe the dataet that are rationalizable, we mean that we provide a property, a revealed preference axiom, that the data atifie if and only if it i conitent with the theory in quetion. The model we tudy have well known axiomatization when one take preference a primitive, but not when one take conumption data a given. The axiomatization of variational preference i due to Maccheroni et al. (2006) (ee alo Grant and Polak (2013) for a variation on their argument). The axiomatization of maxmin i due to Gilboa and Schmeidler (1989). Thee paper are often thought to provide the behavioral counterpart of certain theorie of choice: the preference relation capture an agent behavior, and the theorem in thee paper decribe the behavior that are conitent with the theory. Our focu i on behavior in the market, not on preference. The primitive i a finite lit of purchae of tate-contingent payment, each one made at a different price vector. In contrat with mot paper on ambiguity, we do not work in the Ancombe-Aumann framework. For thi reaon, we mut retrict attention to rik-neutral variational and maxmin preference. The Ancombe- Aumann framework ha the advantage that it (eentially) allow the

3 TRANSLATION INVARIANCE AND HOMOTHETICITY 3 utility over outcome to be obervable. In a imilar vein, our reult extend beyond the rik neutral cae by adding a utility function a an obervation to our dataet. It would of coure be deirable to obtain reult without the aumption of rik neutrality; but thee are likely difficult to come by. One exception i the cae of maxmin utility with two tate: we give a characterization of the data et that are rationalizable with rik neutral (concave utility over money) maxmin in Section 6. The two-tate cae i of coure retrictive, but probably of interet for experiment on ambiguity: ome of the mot baic experiment illutrating ambiguity averion involve two tate. The cloet paper to our are Varian (1988), Bayer et al. (2012) and Polion and Quah (2013). Our reult on CARA and CRRA are cloe to Varian (1988). The main difference i that Varian conider the cae of objective probabilitie, not ubjective. Bayer et al. (2012) and Polion and Quah (2013) look at the tetable implication of model of ambiguity averion for the ame kind of data that we aume in thi paper. They give a characterization in term of the olution of a ytem of inequalitie. Our contribution i different becaue we give a revealed preference axiom that ha to be atified for the data to be rationalizable. The paper by (Kubler et al., 2014) and (Echenique and Saito, 2013) are alo related. Kubler et al. olve the ame problem a we do here, but for the cae of expected utility theory with known (objective) probabilitie over tate. Echenique and Saito olve the problem for ubjective expected utility. 2. Definition. Let S be a finite et of tate of the world. An act i a function from S into R. So R S i the et of act. An act can be interpreted a a tate-contingent monetary payment. Define x 1 = x. A preference relation on R S i a binary relation that i complete and tranitive. Given a preference relation, we denote by the trict part of. A function u : R S R define a preference relation by

4 4 CHAMBERS, ECHENIQUE, AND SAITO x y if and only if u(x) u(y). We ay that u repreent, or that it i a utility function for. A preference relation on R S i locally nonatiated if for every x and every ε > 0 there i y uch that x y < ε and y x. 3. Preference, utilitie, and data. A data et D i a finite collection {(p k, x k )} K k=1, where each pk R S ++ i a vector of trictly poitive (Arrow-Debreu) price, and each x k R S i an act. The interpretation of a dataet i that each pair (p k, x k ) conit of an act x k choen from the budget {x R S : p k x p k x k } of affordable act. 1 A data et {(p k, x k )} K k=1 i rationalizable by a preference relation if x k x whenever p k x k p k x. So a data et i rationalizable by a preference relation when the choice in the dataet would have been optimal for that preference relation. A data et {(p k, x k )} K k=1 i rationalizable by a utility function u if it i rationalizable by the preference relation repreented by u. So a data et i rationalizable by a utility function when the choice in the dataet would have maximized that utility function in the relevant budget et. A preference relation i tranlation invariant if for all x, y R S and all c R, we have x y if and only if x+(c,..., c) y+(c,..., c). A preference relation i homothetic if for all x, y R S α > 0, we have x y if and only if αx αy. and all A preference i a (rik-neutral) variational preference if there i a convex and continuou function c uch that the utility function inf π x + c(π) π (S) repreent. If a data et i rationalizable by a variational preference relation, we will ay that the dataet et i (rik-neutral) variationalrationalizable. 1 Arrow-Debreu price make ene in a etting of complete market and abence of arbitrage. Arrow-Debreu price can then be recovered from aet price. We alo imagine experimental data from market in which Arrow-Debreu ecuritie are traded (Hey and Pace, 2014; Ahn et al., 2014; Bayer et al., 2012).

5 TRANSLATION INVARIANCE AND HOMOTHETICITY 5 A pecial cae of variational preference i maxmin: A preference relation i (rik-neutral) maxmin if there i a cloed and convex et Π (S) uch that the utility function inf π x π Π repreent. If a data et i rationalizable by a rik neutral maxmin preference relation, we will ay that the dataet et i (rik-neutral) maxmin-rationalizable. A utility u : R S R i contant abolute rik averion (CARA) if there i a > 0 and π (S) for which u(x) = S π ( exp( ax)). Note that CARA i a pecial cae of ubjective expected utility. A utility u : R S R i contant relative rik averion (CRRA) if there i a (0, 1) and π (S) for which u(x) = ( ) x 1 a π. 1 a S If a data et i rationalizable by a CARA (CRRA) utility, we will ay that the dataet et i CARA (CRRA) rationalizable. 4. Variational preference We preent the reult on variational and maxmin rationalizability a Theorem 1 and 3. In each cae, the model in quetion aume a linear utility index: o the model capture ambiguity averion but rik neutrality. Thee reult beg the quetion of the empirical content of rik averion together with ambiguity averion. In Section 6 we preent a reult on maxmin utility with rik averion. It i retricted to environment with two tate. 1. Theorem. The following tatement are equivalent: (1) Dataet D i rationalizable by a locally nonatiated, tranlation invariant preference.

6 6 CHAMBERS, ECHENIQUE, AND SAITO (2) Dataet D i rationalizable by a continuou, trictly increaing, concave utility function atifying the property u(x+(c,..., c)) = u(x) + c. (3) Dataet D i variational-rationalizable. (4) For every l = 1,..., M, and every equence {k l } {1,..., K}, we have M p k l l=1 p k l 1 (x k l+1 x k l) 0 (here addition i modulo M, a uual). 2. Remark. The preceding reult can be generalized. Suppoe we were intereted in the tetable implication of preference which are β-tranlation invariant, for ome β 0, β 0. That i, we want to know whether for all x, y, we have x y if and only if x + β y + β. Define the eminorm x β 1 = i β ix i. Then it i an eay exercie to verify that the tetable implication of β-tranlation invariance are given by equation (4), replacing 1 with β 1. Hence, the tet given here hould be compared with the one given by Brown and Calamiglia (2007), and other tet for rik preference. We now turn out attention to maxmin preference. Note that the equivalence between (2) and (3) in Theorem 3 i well known, but here we prove it through an application of Theorem 1. We ay that a function u : R S R i linearly homogeneou if for all x R S and all α > 0, we have u(αx) = αu(x). 3. Theorem. The following tatement are equivalent: (1) Dataet D i rationalizable by a locally nonatiated, homothetic and tranlation invariant preference. (2) Dataet D i rationalizable by a continuou, trictly increaing, linearly homogeneou and concave utility function atifying the property that u(x + (c,..., c)) = u(x) + c. (3) Dataet D i maxmin-rationalizable. (4) For every k and l, p k x k pl p l 1 x k.

7 TRANSLATION INVARIANCE AND HOMOTHETICITY 7 5. CARA and CRRA The previou ection conider tranlation invariance and homotheticity a general propertie of preference in choice under uncertainty. Here we focu on the cae of ubjective expected utility. So we conider model in which the agent ha a ingle prior over tate, and maximize expected utility. The prior i unknown though, and mut be inferred from her choice. In the ubjective expected utility cae, tranlation invariance give rie to CARA preference, and homotheticity to CRRA. 4. Theorem. A dataet D i CARA rationalizable if and only if there i α > 0 uch that (1) hold; and CRRA rationalizable if and only if there i α (0, 1) uch that (2) hold. (1) (2) α (x k t x k + x k x k t ) = ln( pk p k t α ln( xk t x k x k x k t ) = ln( pk p k t p k t p k p k t p k ) ) The condition in Theorem 4 may look like exitential condition: eentially Afriat inequalitie. Afriat inequalitie are indeed the ource of equation (1) and (2), a evidenced by the proof of Theorem 4, but note that the tatement are equivalent to non-exitential tatement. Equation (1) ay that when (x k t x k + x k x k t ) 0, ln( pk p k t p k t p k (x k t x k + x k x k i independent of k, t, k and ; and that when (x k t x k + x k x k t ) = 0 then ln( pk p k p k t ) = 0. Similarly for Equation (2). t p k It i worth pointing out that, except in the cae when for all obervation, all price are equal, and conumption of all good are equal, equation (1) can have only one olution. Hence, rik preference are uniquely identified. The next corollary alo how that belief are identified. When π (S) and a > 0, let U a = S π ( exp( ax)) denote the aociated ubjective expected CARA utility. ) t )

8 8 CHAMBERS, ECHENIQUE, AND SAITO 5. Corollary. α > 0 olve (1) if and only if there i π (S) uch that π and U α CARA rationalize D. Further, for any uch α > 0, there i a unique π (S) uch that if π and U α CARA rationalize D, then π = π. Similarly for (2) and CRRA rationalizability. 6. Rik avere max-min with two tate The prior reult i about rik neutral maxmin. Here we turn to maxmin with rik averion. A preference relation i maxmin if there i a cloed and convex et Π (S) and a concave utility u : R S R uch that the utility function inf π u(x ) π Π =1,2 repreent. If a data et i rationalizable by a maxmin preference relation, we will ay that the dataet et i maxmin-rationalizable. Aume a dataet {(p k, x k )} K k=1 in which xk x k (k, ). when (k, ) Let K 1 be the et of all k uch that x k 1 < x k 2, and K 2 be the et of all k uch that x k 1 > x k 2. Suppoe that K = K 1 K 2. Given a equence of pair (x k i i, x k i i) n i=1, conider the following notation: Let I l, = {i : k i K l and i = } I l, = {i : k i K l and i = }, for l = 1, 2 and = 1, 2. Strong Axiom of Revealed Maxmin Utility (SARMU): For any equence of pair (x k i i, x k i i) n i=1 in which (1) x k i i x k i for all i; i (2) each k appear a k i (on the left of the pair) the ame number of time it appear a k i (on the right); (3) I 1,1 I 1,1 = I 2,1 I2,1 0 The product of price atifie that n i=1 p k i i p k i i 1. One can alternatively define the axiom with I 2,2 I 2,2 = I 1,2 I 1,2 0 in condition (3).

9 TRANSLATION INVARIANCE AND HOMOTHETICITY 9 6. Theorem. A dataet i maxmin rationalizable if and only if it atifie SARMU Dicuion. Echenique and Saito (2013) how that the following axiom characterize rationalizability by ubjective expected utility. Strong Axiom of Revealed Subjective Expected Utility (SARSEU): For any equence of pair (x k i i, x k i i) n i=1 in which (1) x k i i x k i for all i; i (2) each k appear a k i (on the left of the pair) the ame number of time it appear a k i (on the right); (3) I 1,1 + I 2,1 = I 1,1 + I 2,1 The product of price atifie that n i=1 p k i i p k i i 1. Note that condition (3) of SARSEU i equivalent to I 2,2 + I 1,2 = I 2,2 + I 1,2 becaue I 1,1 + I 2,1 + I 2,2 + I 1,2 = n = I 1,1 + I 2,1 + I 2,2 + I 1,2. Inpection of SARSEU and SARMU yield the following 7. Propoition. If a dataet atifie SARSEU then it atifie SARMU. For a dataet to be maxmin rationalizable, but inconitent with ubjective expected utility, it need to contain a equence in the condition of SARSEU in which I 1,1 + I 2,1 = I 1,1 + I 2,1, but where I 1,1 I 1,1 > 0. A we have emphaized, the reult in Theorem 6 i for two tate. There are two implification afforded by the aumption of two tate, and the two are crucial in obtaining the theorem. The firt i that with two tate there are only two extreme prior to any et of prior. With the aumption that u i monotonic, one can know which of the two extreme i relevant to evaluate any given act. The econd implification i a bit harder to ee, but it come from the fact that one can normalize the probability of one tate to be one and only keep

10 10 CHAMBERS, ECHENIQUE, AND SAITO track of the probability of the other tate. Then the property of beign an extreme prior carrie over to the probability of the tate that i left free Proof 7.1. Proof of Theorem 1. That (3) = (1) i obviou. We hall firt prove that (1) = (4) Suppoe, toward a contradiction, D i a dataet atifying (1) but not (4). Then we have a cycle M p k l l=1 p k l 1 (x k l+1 x k l) < 0. Let u without lo aume the equence i x 1,..., x M o a to avoid cumberome notation. Let Z = M p l l=1 p l 1 (x l+1 x l ) < 0. Define a new equence (y 1,..., y M ) inductively. Let y 1 = x 1, and let y k = x k + (c k,..., c k ) where c k p i choen o that k (y k+1 y k ) = Z. M Specifically, c 1 = 0 and c k+1 = c k + Z M for k = 1,..., M 1. Let q k = k = 1,... M. Oberve that M 1 k=1 q k (y k+1 y k ) + q M (y 1 y M ) = pk (x k+1 x k ) pk and conider the dataet (q k, y k ), M k=1 p k (x k+1 x k ) = Z, and that q k (y k+1 y k ) = Z/M for k = 1,..., M 1. q M (y 1 y M ) = Z/M Therefore, The original dataet i rationalizable by ome locally non-atiated and tranlation invariant preference. It i eay to ee that the ame preference rationalize the dataet (q k, y k ). Indeed, if q k y k q k y then p k x k p k (y (c k,..., c k )), by definition of y k and q k. So x k (y (c k,..., c k )), and thu y k y by tranlation invariance of. M 1 k=1 Oberve that q k (y k+1 y k ) + q M (y 1 y M ) = M 1 k=1 p k (x k+1 x k ) + c M = Z, 2 Thi can be een in the proof of Lemma 8 when we go from π π to µ 1 µ 1.

11 TRANSLATION INVARIANCE AND HOMOTHETICITY 11 and that q k (y k+1 y k ) = Z/M for k = 1,..., M 1. Therefore, q M (y 1 y M ) = Z/M. In particular, q k (y k+1 y k ) = Z/M < 0 for k = 1,..., M (mod M). Thu y k y k+1 a (q k, y k ) i rationalizable by and i locally nonatiated. Thi contradict the tranitivity of. Now we how that (4) = (2). Let x R S. Let Σ x be the et of all ubequence {k l } M l=1 {1,..., K} for which k 1 = 1 and define x k M+1 = x. By (4), if {k l } M l=1 Σ x ha a cycle (meaning that k l = k l for l, l {1,..., M} with l l ), then there i a horter equence {k j } M j=1 Σ x with M j=1 p k j p k j 1 (x k j+1 x k j ) M l=1 p k l p k l 1 (x k l+1 x k l ). Therefore, u(x) = inf{ M p k l l=1 p k l 1 (x k l+1 x k l) : {kl } M l=1 Σ x} i well defined, a the infimum can be taken over a finite et. That u : R S R defined in thi fahion i concave, trictly increaing and continuou i immediate. To ee that it rationalize the data, uppoe that p k x l p k x k. Then pk x l pk x k. It i clear then by definition that u(x l ) u(x k ) + pk (x l x k ) u(x k ). Finally, to how that u(x) + (c,..., c)) = u(x) + c, note that for any p k p k, we have (x + (c,..., c)) = c + pk x. The reult then follow by contruction. We end the proof by howing that (2) = (3) Let u : R S R be a in the tatement of (2). Define the concave conjugate of u by f(π) = inf{π x u(x) : x R S } = inf{π x + cπ 1 u(x) c : x R S, c R} = inf{π x c(1 π 1) u(x) : x R S, c R}, where the econd inequality ue that u(x+(c,..., c)) = u(x)+c. Now note that f(π) = if (1 π 1) 0. Note alo that the monotonicity of u implie that f(π) = if there i uch that π S < 0. Hence the domain of f i a ubet of (S).

12 12 CHAMBERS, ECHENIQUE, AND SAITO Now ince u i continuou, we have that u(x) = inf π (S) π x f(p). Since u rationalize the dataet, the dataet i variational rationalizable Proof of Theorem 3. It i obviou that (3) = (2) and that (2) = (1). Hence, to how the theorem, it uffice to how that (4) implie (3) and that (1) implie (4). For a dataet D, let π k = pk. It i eay to ee that (4) = (3). Let Π be the convex hull of {π k : k = 1,..., K}. Then it i immediate that U(x) = min π Π π x rationalize D. We prove that (1) = (4). Suppoe that D atifie (1) but not (4). Then there i k and l for which π l x k < π k x k. Let be a preference relation a tated in (1). Homotheticity implie that rationalize the data {(x j, π j ) : j = 1,..., K} {(θx l, π l )} for any calar θ > 0. Now, π l x k < π k x k implie that x k (π l π k ) + θx l (π k π l ) < 0 for θ > 0 mall enough. But thi i a violation of (4) in Theorem 1. A contradiction becaue i tranlation invariant and locally nonatiated Proof of Theorem 4. The idea in the proof i to olve the firtorder condition for the unknown term. Conider firt the cae of CARA. Let π (S) and α > 0 rationalize D. Then we know that x k maximize π exp( αx ) ubject to p k x p k x k. By conidering the Lagrangean and the firt order condition, we may conclude that for every, t S and every k {1,..., K}, we have π exp( αx k ) p k = π t exp( αx k t ). p k t Conclude that pk πt p k t π become: = exp( α(xk x k t )). By taking log, the ytem (3) ln(π ) ln(π t ) + α(x k t x k ) = ln(p ) ln(p t ).

13 TRANSLATION INVARIANCE AND HOMOTHETICITY 13 In the cae of CRAA, the exitence of a rationalizing π and parameter α imply a firt-order condition of the form (4) ln(π ) ln(π t ) + α ln(x k t /x k ) = ln(p ) ln(p t ). We can denote ln(π ) by z in Equation (3) and (4). Thu we obtain that D i rationalizable if and only if there exit z R and α > 0 uch that the following equation i olved for all, t, k with t: z z t + α(y k t y k ) = ln(p k ) ln(p k t ), where y k t = x k t for CARA rationalizability, and y k t = ln x k t for CRRA rationalizability. Now the neceity of the axiom i obviou. Let k k, then α(y k t y k ) ln(p k /p k t ) = z z t = α(y k t y k ) ln(p k /p k t ) for any and t. Thu α(y k t y k y k t + y k ) = ln( pk p k t So (1) i atified for the cae of CARA rationalizability, and (2) i atified for the cae of CRRA rationalizability. To prove ufficiency, let d p (, t, k) = log(p k /p k t ) d x (, t, k) = y k y k t. Let α be uch that for all k, k,, and t, α (y k t y k y k t + y k ) = ln( pk p k t Then in particular, for all k, k,, and t, (5) d p (, t, k) + α d x (, t, k) + d p (t,, k ) + α d x (t,, k ) = 0. Note alo that (6) d p (, t, k) + d p (t,, k) + d p (,, k) p k t p k p k t p k +α (d x (, t, k) + d x (t,, k) + d x (,, k)) = 0. ). ).

14 14 CHAMBERS, ECHENIQUE, AND SAITO Fix 0 S and let z 0 R be arbitrary. For any S, define z by z = z 0 + α d x ( 0,, k) + d p (, 0, k), for ome k. In fact, by Equation (5) thi definition i independent of k becaue d p (, 0, k) + α d x (, 0, k) = d p (, 0, k ) + α d x (, 0, k ). Given thi definition, note that z z t = α (d x ( 0,, k) d x ( 0, t, k)) + d p (, 0, k) d p (t, 0, k) = α (d x ( 0,, k) d x ( 0, t, k)) + d p (, 0, k) d p (t, 0, k) + d p (, t, k) + d p (t, 0, k) + d p ( 0,, k) + α (d x (, t, k) + d x (t, 0, k) + d x ( 0,, k)) = d p (, t, k) + α d x (, t, k). Where the econd equality ue Equation (6). Hence, with the contructed (z t ) t S we have z z t + α (yt k y k ) = log(p k /p k t ), for all, t, and k. The firt-order condition for rationalizability are therefore atified Proof of Theorem Lemma. A dataet D i rationalizable if and only if there are v k, λ k, = 1, 2, k = 1,..., K, and π, π 0 with π π, uch that: πv k 1 = λ k p k 1 v k 2 = λ k p k 2, for all k = 1,..., K, where π = π when x k 1 < x k 2 and π = π when x k 1 > x k 2. The number alo atify that v k v k when xk > x k. Proof. To prove ufficiency, let v k, λ k, = 1, 2, k = 1,..., K, and π, π 0 with π π be a in the tatement of the lemma. Define µ, µ (S) a follow. Let µ 1 = π/(1 + π) µ 2 = 1/(1 + π) and µ 1 = π/(1 + π) µ 2 = 1/(1 + π) Note that µ 1 µ 1 and µ 2 µ 2, a π π. Define θ k = λ k /(1 + π) if x k 1 < x k 2 and θ k = λ k /(1 + π) if

15 TRANSLATION INVARIANCE AND HOMOTHETICITY 15 x k 1 > x k 2. Then we have that µ v k 1 = θ k p k 2, with µ = µ when x k 1 < x k 2; and µ = µ when x k 1 > x k 2. Given the number v k it i now routine to define a correpondence ρ uch that if x x, y ρ(x) and y ρ(x ) then y y > 0, and with ρ(x k ) v k. Thi give a concave and increaing function u with u(c) = ρ(x). So θk p k µ u(x k ) for all (x, ), and hence the firt order condition are atified for maxmin rationalization. We omit the proof of neceity. Let A be a matrix with 2K K + 1 column, and 2K row. The firt 2K column are labeled with a different pair (k, ). The next 2 column are labeled π and π. The next K column are labeled with a k {1,..., K}. Finally the lat column i labeled p. For each (k, ) with k K 1, A ha a row with all zero entrie with the following exception. It ha a 1 in the column labeled (k, ), among the firt group of 2K column. It ha a 1 in the column labeled k. In the column labeled p it ha log(p k ). Finally, if = 1 then it ha a 1 in the column labeled π. For each (k, ) with k K 2, A ha a row defined a above. The only difference i that when = 1 then it ha a 1 in the column labeled π intead of having one in π. Let B be a matrix with the ame number of column a A, and one row for each pair (x k, x k ) with xk > x k. The column of B are labeled like thoe of A. The row for x k > x k ha all zeroe except for a 1 in column (k, ) and a 1 in column (k, ). Finally, B ha one more row. Thi row a a 1 in the column for π and a 1 in the column for π, and it i labeled = 1 for future reference. Let E be a matrix with the ame number of column a A, labeled a above, and a ingle row. The row ha all zeroe except for a 1 in column p. By Lemma 8, there i no rationalizing maxmin preference if and only if there i no olution to the ytem of inequalitie A x = 0, B x 0 and E x > 0.

16 16 CHAMBERS, ECHENIQUE, AND SAITO Suppoe that all log(p k ) are rational number. We hall ue the following verion of the Theorem of the Alternative, which can be found a Theorem in (Stoer and Witzgall, 1970). 9. Lemma. Let A be an m n matrix, B be an l n matrix, and E be an r n matrix. Suppoe that the entrie of the matrice A, B, and E belong to a commutative ordered field F. Exactly one of the following alternative i true. (1) There i u F n uch that A u = 0, B u 0, E u 0. (2) There i θ F r, η F l, and π F m uch that θ A + η B + π E = 0; π > 0 and η 0. Then the non-exitence of a olution to the ytem A x = 0, B x 0 and E x > 0 i equivalent to the exitence of integer vector η, θ and γ uch that θ 0, γ > 0, and η A + θ B + γe = 0. For a matrix D with 2K K + 1 column, let D 1 denote the ubmatrix correponding to the firt 2K column, D 2 correpond to the next 2, D 3 to the next K, and D 4 to the lat column. Note that, by contruction of A, B and E, η A + θ B + γe = 0 implie that η A 1 + θ B 1 = 0, η A 2 + θ B 2 = 0, η A 3 = 0, η A 4 + γ = 0. In fact, we can without lo aume that η, θ and γ take value of 1, 0 or 1. (Thi aumption i without lo becaue we can replace each row of matrice A, B and E with a many copie a indicated by the correponding vector η, θ or γ.) From the exitence of uch vector it follow that we can obtain a equence (x k i i, x k i ) n i=1 with x k i i i > x k i. The ource of each pair (x k i i i, x k i i) i that the column (k i, i ) of A i multiplied by η (ki, i ) > 0 and the column (k i, i) of A i multiplied by η (k i, i ) < 0. The vector η mut then have η (ki, i ) > 0 and η (k i, i ) > 0, with a 1 in the firt column and a 1 in the econd. We hall prove that the equence (x k i i, x k i i) n i=1 atifie the propertie tated in the axiom. Firtly, η A 3 = 0 mean that for each k, the number of i for which k = k i equal the number of i for which k = k i.

17 TRANSLATION INVARIANCE AND HOMOTHETICITY 17 Secondly, η A 2 + θ B 2 = 0 implie that: k K 1 η (k,1) + θ =1 = 0 k K 2 η (k,1) θ =1 = 0. Note that k K 1 η (k,1) = {i : k i K 1, = 1} {i : k i K 1, = 1}, and imilarly for k K 2 η (k,1). Hence, a {i : k i K 1 and i = 1} {i : k i K 1 and i = 1} = {i : k i K 2 and i = 1} {i : k i K 2 and i = 1} 0, η (k,1) = η (k,1) = θ =1 0. k K 1 k K 2 Therefore the equence (x k i i, x k i i) n i=1 atifie the econd property tated in the axiom. Finally, n i=1 a η A + γ = 0. Hence log(p k i /p k i i i ) = η (k,) = γ < 0, (k,)) n i=1 p k i i p k i i > 1. The above proof aume that the log of price i rational. The proof of the theorem follow along the ame line a Echenique and Saito (2013). Specifically, we have hown the following 10. Lemma. If {(x k, p k )} i a dataet atifying SARMU, in which log p k Q for all k, then the dataet i maxmin rationalizable. One can then prove the following 11. Lemma. If {(x k, p k )} i a dataet that atifie SARMU, and ε > 0 then there i a collection of price {q k )} uch that log q k Q +, p k q k < ε, and the dataet {(x k, q k )} atifie SARMU.

18 18 CHAMBERS, ECHENIQUE, AND SAITO The proof of Lemma 11 i exactly a in (Echenique and Saito, 2013). Lemma 10 etablihe the reult in dataet in which the log of price i rational. Conider an arbitrary data et {(x k, p k )}, with price that may not be rational. Suppoe toward a contradiction that the dataet atifie SARMU, but that it i not maxmin rational. Specifically then, by Lemma 8, uppoe that there i no olution to the ytem A x = 0, B x 0 and E x > 0. Then by Lemma 9 there are real vector η, θ and γ uch that θ 0, γ > 0, and η A + θ B + γe = 0. Let (q k ) K k=1 be vector of price uch that the dataet (xk, q k ) K k=1 atifie SARMU and log q k Q for all k and. (Such (q k ) K k=1 exit by Lemma 11.) Furthermore, the price q k can be choen arbitrarily cloe to p k. Contruct matrice A, B, and E from thi dataet in the ame way a A, B, and E above. Note that only the price are different in {(x k, q k )} compared to {(x k, p k )}. So E = E, B = B and A i = A i for i = 1, 2, 3. Since only price q k are different in thi dataet, only A 4 may be different from A 4. By Lemma 11, we can chooe price q k uch that θ A 4 θ A 4 < γ/2. We have hown that θ A 4 = γ, o the choice of price q k guarantee that θ A 4 < 0. Let γ = θ A 4 > 0. Note that θ A i + η B i + γ E i = 0 for i = 1, 2, 3. And B 4 = 0 o θ A 4 + η B 4 + γ E 4 = θ A 4 + γ = 0. We alo have that η 0 and γ > 0. Therefore θ, η, and γ exhibit a olution to the dual ytem for dataet {(x k, q k )}, a contradiction with Lemma 10. Reference Ahn, D., S. Choi, D. Gale, and S. Kariv (2014): Etimating ambiguity averion in a portfolio choice experiment, Forthcoming, Quantitative Economic. Bayer, R., S. Boe, M. Polion, and L. Renou (2012): Ambiguity Revealed, Mimeo, Univerity of Eex.

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