Scale-Invariant Uncertainty-Averse Preferences and Source-Dependent Constant Relative Risk Aversion

Scale-Invariant Uncertainty-Averse Preferences and Source-Dependent Constant Relative Risk Aversion Costis Skiadas October 2010; this revision: May 7, 2011 Abstract Preferences are defined over consumption that is contingent on a finite number of states representing a horse race Knightian uncertainty) and a roulette objective risk). The class of scale-invariant SI) ambiguity-averse preferences, in a broad sense, is uniquely characterized by a multiple-prior utility representation. Adding a weak certainty independence axiom is shown to imply either unit CRRA toward roulette risk or SI maxmin preferences. Removing the weak independence axiom but adding a separability assumption on preferences over pure horse-race bets leads to source-dependent constant-relative-risk-aversion expected utility with a higher CRRA assigned to horserace uncertainty than to roulette risk. The multiple-prior representation in this case is shown to generalize entropic variational preferences. Small-risk simplifying approximations for certainty equivalents with source-dependent CRRA are derived, thus relating them to continuous-time formulations in the literature. An appendix characterizes the functional forms associated with SI ambiguity-averse preferences in terms of suitable weak independence axioms in place of scale invariance. Kellogg School of Management, Department of Finance, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208. I thank Nabil Al-Najjar, Sandeep Baliga, Snehal Banerjee, Luciano De Castro, Peter Klibanoff, Camelia Kuhnen, Dimitri Papanikolaou, Jacob Sagi, Todd Sarver and Tomasz Strzalecki for helpful discussions or feedback. I am responsible for any errors. 1

1 Introduction Assuming we agree to use the definition of Gilboa and Schmeidler 1989) for ambiguity aversion without their certainty-independence axiom), this paper characterizes all preferences within a broad class that are risk/ambiguity averse and scale invariant or homothetic). Ambiguity aversion means aversion to Knightian uncertainty, as is commonly motivated by the experiments of Ellsberg 1961). Scale invariance of preferences means that the ranking of any two choices is not reversed if all contingent consumption amounts are scaled by the same constant. Scale invariance is a common assumption in models of macroeconomics and finance for well-known reasons: It provides the simplest reasonably realistic way of capturing wealth effects, it justifies Gorman aggregation and the associated representativeagent arguments, it is an essential component of balanced growth models, and generally lends numerical tractability by reducing a model s dimensionality. For example, increasingly common in economic modeling is the use of Epstein-Zin-Weil utility, 1 whose certainty equivalent CE) corresponds to expected utility with a constant coeffi cient of relative risk aversion CRRA), the only possible type of homothetic expected utility. Suppose we are interested in relaxing the assumption of an expected-utility CE, while requiring risk aversion and ambiguity aversion, without sacrificing scale invariance. What CE parameterizations should we consider? This paper gives a parsimonious answer to this question based on a simple axiomatic foundation. Although the paper s main results do not include probabilities among their primitives, let us temporarily focus on the Anscombe and Aumann 1963) type setup of a preference order on state-contingent objective lotteries. The states can be thought of as outcomes of a horse race and the lotteries as roulette bets, the idea being that it is easier to assign probabilities to roulette outcomes than to horse-race outcomes. Ambiguity aversion implies that the agent is less averse to roulette uncertainty than to horse-race uncertainty. A seminal contribution that quantifies this idea is Gilboa and Schmeidler 1989), henceforth GS, whose Uncertainty Aversion axiom A.5 forms the basis for the definition of uncertainty 1 Epstein-Zin-Weil utility is a parameterization of homothetic Kreps and Porteus 1978) utility, and is named in recognition of the contributions of Epstein and Zin 1989) and Weil 1990). Expected discounted power or logarithmic utility is a special case. Another widely used tractable preference class, exemplified by expected discounted exponential utility, is characterized by translation invariance relative to a constant consumption stream. See Chapter 6 of Skiadas 2009) for the corresponding recursive-utility formulation.) The analysis of translation-invariant preferences also reduces to the scale-invariant case by passing to logconsumption. 2

aversion in this paper, too albeit without reference to objective probabilities). GS also assume what they call Certainty Independence, which is key in generating their familiar multiple-prior representation. If the assumptions of this paper s first theorem were interpreted in the GS setting, they would essentially 2 amount to replacing certainty independence with scale invariance. The resulting multiple-prior utility representation associates a unique coeffi cient of relative risk aversion CRRA) γ with roulette risk. For γ = 1, the utility form is within a class studied by Maccheroni, Marinacci, and Rustichini 2006), henceforth MMR. For γ 1, the utility form is similar to but not the same as) representations appearing in Chateauneuf and Faro 2006), henceforth CF, and Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio 2008), henceforth CMMM. Whereas GS, MMR, CF and CMMM 3 derive functional forms for aggregating horse-race uncertainty using some type of weak independence axiom, in this paper s first theorem similar utility structures are derived as a consequence of scale invariance, without any weak independence axiom. The paper s first theorem is specialized in two different directions. One specialization essentially fleshes out the scale-invariant case of MMR preferences. The main conclusion is that the introduction of a weak certainty independence axiom, analogous to the one adopted by MMR, implies either an MMR-type utility form with unit CRRA toward roulette risk, or a utility form within the maxmin class of GS. Moreover, the latter case is given an alternative multiple-prior representation in which the set of priors is replaced with a finite-valued multiplicative cost function for deviating from a reference prior. The other specialization of the first theorem leaves out weak certainty independence, and instead imposes separability of preferences over pure horse-race contingent payoffs. The result is source-dependent constant relative risk aversion, with the CRRA associated with horserace uncertainty being potentially higher than the CRRA associated with roulette risk. The utility form in this case is within a class of second-order expected utility appearing in Nau 2006) and Ergin and Gul 2009) see also Chew and Sagi 2008)). It also corresponds to the formulation of Schroder and Skiadas 2003) in the context of continuous-time recursive utility, a connection that is made in the final section through small-risk approximations. The dual, multiple-prior representation of source-dependent constant relative risk aversion 2 Increasing preferences and strictly positive consumption are also assumed. 3 The reference here is to Theorem 26 of CMMM, which is not the paper s main focus. The main contribution of CMMM is the application of a general form of quasiconcave duality to establish a unified multiple-prior representation of ambiguity-averse preferences in the GS setting, without any weak independence axiom. As explained in Section B.3, the duality results in the present paper build on the CMMM duality, based on the implications of scale invariance. 3

is shown to include a new parametric multiplicative variational form, which converges to the entropic variational preferences of Hansen and Sargent 2001) as the CRRA toward roulette uncertainty converges to one. The unit-crra case overlaps with the analysis of Strzalecki 2010). Scale invariance makes possible a simplified axiomatic foundation that requires no objective probabilities or a full-blown subjective-expected utility theory. The underlying reason is a representation theorem for separable scale-invariant preferences that is formally a special case of this paper s first main result but is also a stepping stone in its proof, as discussed in Appendix B along with related literature. Also discussed in an appendix is the characterization of the considered functional forms for aggregating horse-race uncertainty in terms of weak independence axioms in place of scale invariance. For γ = 1, the functional form implied by scale invariance corresponds to the weak certainty independence axiom of MMR. For γ 1, the relevant weak independence axioms which are different for γ < 1 and γ > 1) are shown to be closely related to but different than Axiom 5 of CF and Axiom 10 of CMMM. Thus scale invariance substitutes for weak independence axioms in shaping the utility functional form, and also restricts the type of weak independence that can be assumed in a way that depends on the CRRA toward objective roulette) uncertainty. The remainder of the paper is organized in four sections and three appendices. Section 2 introduces the preference restrictions that are adopted throughout the main part of the paper, and presents the first main representation theorem. Section 3 specializes the first theorem by imposing a weak certainty independence axiom. Section 4 presents the third main representation theorem, relating to source-dependent CRRA. Section 5 concludes the main part of the paper with an account of small-risk approximations corresponding to source-dependent CRRA. Appendix A relates the implications of scale invariance to weak independence axioms, as discussed in the last paragraph. Appendix B proves the main results and develops their underlying structure in a way that enhances understanding relative to the theorem statements of the main part of the paper. Finally, Appendix C collects proofs omitted up to that point in the paper. 4

2 Scale-Invariant Ambiguity-Averse Preferences There are two sources of uncertainty, represented by the two factors of the state space {1,..., R} {1,..., S}. Informally, we think of 1,..., S as states representing Knightian uncertainty, for example, the possible outcomes of a horse race. We think of 1,..., R as states representing better understood uncertainty, for example, the possible outcomes of a roulette spin. A generic element of the state space, or just state, is denoted r, s). We refer to elements subsets) of {1,..., R} as roulette states events) and to elements subsets) of {1,..., S} as horse-race states events). A payoff is any mapping of the form x : {1,..., R} {1,..., S} 0, ), with x r, s) or x s r denoting the value of x at state r, s). We write X for the set of all payoffs, which we identify with 0, ) R S. A roulette payoff is any payoff x whose value is a function of the roulette outcome only, that is, x r, s) = x r, s ) for all r {1,..., R} and s, s {1,..., S}. If x is a roulette payoff, we write x r instead of x s r. Analogously, a payoff x is a horse-race payoff if x r, s) = x r, s) for all r, r {1,..., R} and s {1,..., S}, in which case we write x s instead of x s r. The set of all roulette resp. horse-race) payoffs is denoted X R resp. X S ) and is identified with 0, ) R resp. 0, ) S ). So while X R and X S are subsets of X, we also think of a payoff x as an R-by-S matrix, whose columns, denoted x 1,..., x S, are roulette payoffs. For any x, y X and roulette event B, x B y denotes the payoff x B y) r, s) = Note that x, y X R implies x B y X R. { x s r if r B, y s r if r / B. The central object of study is a binary relation on the set of payoffs X, representing an agent s preferences: x y means that the agent prefers to switch from y to x for a suffi ciently low but positive cost. As usual, the corresponding relations and on X are defined by [x y not y x] and [x y x y and y x]. The restriction of on X R is denoted R : x R y x, y X R and x y. The following definition lists properties of that will be imposed in each of the main representation theorems in this paper. 5

Definition 1 The relation is increasing if for all x, y X, x y x implies y x. continuous if for all x X, the sets {y : y x} and {y : x y} are open. a preference order if is complete 4 and transitive. 5 scale invariant if x y implies αx αy for all α 0, ). R -monotone if for all x, y X, x s R y s for all s {1,..., S} = x y. ambiguity averse if for all x, y X S, x y = x B y x for every roulette event B. The first three conditions are commonplace, while the fourth condition is the familiar homotheticity condition; it formalizes the notion of scale invariance that along with ambiguity aversion is this paper s focal point. The last two conditions of Definition 1 are analogous to assumptions A.4 and A.5 of Gilboa and Schmeidler 1989). R -monotonicity requires that the agent prefers payoff x to payoff y if for every horse-race outcome s, the roulette payoff x s is preferred to roulette payoff y s. Ambiguity aversion requires that if the agent is indifferent between horse-race payoffs x and y, then the agent weakly) prefers to spin the roulette and select x if the balls settles in B and y otherwise. A commonly used illustration is as follows. Example 2 There are only two horses S = 2), about which the agent has no information. Suppose x = 1, 0) is a unit bet on the first horse and y = 0, 1) is a unit bet on the second horse. The agent s indifference between x and y reflects the symmetry of the situation but conceals the agent s discomfort with the fact that the probability π of horse one winning is unknown. Suppose also that B is a roulette event that the agent knows to have probability one half. Then x B y pays one with probability one half for any given value of π 0, 1). For this reason, x B y is preferred to either x or y. 4 The relation is complete if for all x, y 0, ) n, either x y or y x. 5 The relation is transitive if x y and y z implies x z. 6

The properties defined below will be additionally imposed on R. Definition 3 The relation R is separable if for all x, y, z, z X R and B {1,..., R}, x B z R y B z x B z R y B z. somewhere risk averse if there exists an open set O 0, ) R such that for every x O, the set {y : y R x} O is convex. In the representation theorems to follow, R has a power-or-logarithmic expected utility representation relative to a unique probability over roulette outcomes. It is a rather remarkable fact that such a representation of R follows entirely from the assumptions that R is an increasing, continuous and scale-invariant preference order with the properties of Definition 3, and R > 2. The argument can be found as Theorem 14 in Appendix B. A certainty equivalent CE) is any increasing 6 and continuous function of the form ν : X 0, ) satisfying 7 ν α1) = α for all α 0, ). The CE ν is said to represent if ν x) > ν y) is equivalent to x y. Theorem 4 below describes the functional form of every CE representing a preference order on X satisfying the restrictions listed in Definitions 1 and 3. The following notation is used in its statement as well as throughout the rest of this paper. For any positive integer n, n = {p 0, 1) n : n } p i = 1. 1) i=1 Given any γ [0, ), u γ denotes the real-valued function on 0, ) defined by { z 1 γ / 1 γ) if γ 1 u γ z) = 2) log z) if γ = 1 and C γ denotes the set of every function C : S R + { } with the properties: If γ = 1, then min C = 0 and C is convex and lower semicontinuous. If γ 0, 1), then C is valued in 0, ), min C = 1, and 1/C is concave. If γ 1, ), then C is valued in 0, ), max C = 1 and 1/C is convex. 6 Throughout this paper, we use the term increasing in the strict sense: x y x implies ν x) > ν y). 7 We use the notation 1 = 1, 1,..., 1), the dimensionality being implied by the context. 7

This paper s first main result follows. Theorem 4 Assuming R > 2, the following two conditions are equivalent: 1. is a continuous, increasing, scale-invariant, R -monotone and ambiguity-averse preference order, and R is separable and somewhere risk averse. 2. There exist p R, γ [0, ) and C : S R + { } such that S s=1 u γ ν x) = min q R s r=1 p ru γ xr)) s + C q) if γ = 1 S q S s=1 q R s r=1 p ru γ xr)) s C q) if γ 1 3) defines a CE ν : X 0, ) that represents. Suppose the two conditions are satisfied. Then 3) holds for a unique pair p, γ) in R [0, ) and a function C : S R + { } that can be uniquely selected to be minimal if γ [0, 1] and maximal if γ 1, ). The C so selected is necessarily a member of C γ. Proof. See Appendix B. By a minimal resp. maximal) C we mean that if 3) holds with C : S R + { } in place of C, then C q) C q) resp. C q) C q)) for all q S. We henceforth refer to the unique C in representation 3) that is minimal if γ [0, 1] and maximal if γ 1, ) as the unique extremal C. For γ = 1, representation 3) is within the utility class characterized by Maccheroni, Marinacci, and Rustichini 2006), as further discussed in the following section. For γ 1, the unique extremal C, which is finite-valued, has the property that 1/C is either concave or convex, and therefore C is necessarily continuous on the open) set S. The corresponding representation 3) is closely related to but different than the representation of Chateauneuf and Faro 2006) and its extension by Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio 2008), as further explained in Appendix A. 8

3 Scale Invariance with Weak Certainty Independence The class of scale-invariant ambiguity-averse preferences characterized in Theorem 4 is specialized in this section by imposing what is essentially the weak certainty independence condition of Maccheroni, Marinacci, and Rustichini 2006). A main conclusion is that given weak certainty independence, scale invariance of preferences implies that either the agent has unit CRRA toward roulette risk, or that preferences are within the maxmin class characterized by Gilboa and Schmeidler 1989). As in the Introduction, we will refer to these two references by their acronyms, MMR and GS. Conditions that are analogous 8 to MMR s weak certainty independence and GS s certainty independence are stated below. We write α for the constant payoff that takes the value α 0, ) at every state. Weak Certainty Independence WCI): event B, and α 0, ), For any horse-race payoff s x, y, roulette x B 1 y B 1 x B α y B α. Certainty Independence CI): Given any horse-race payoffs x, y, roulette events A, B, and α 0, ), x A 1 y A 1 x B α y B α. The motivation behind these conditions is that roulette mixing with a constant payoff cannot provide the type of hedging with respect to lack of knowledge of the prior suggested by the ambiguity aversion condition and illustrated in Example 2. Note, however, that the likelihood of the roulette mixing event A or B does affect the overall proportion of the bet that is exposed to ambiguity, making it potentially plausible that WCI but not CI is satisfied. Clearly, CI implies WCI, corresponding to the fact that the GS preference class is a subset of the MMR preference class. The MMR representation is of the form 4) below, but with an arbitrary von Neumann-Morgenstern index u in place of u γ. The GS case corresponds to the function C being {0, }-valued, meaning that C q) is either zero 8 MMR and GS consider a different formal setting of preferences over acts that are horse-race contingent objective probability distributions. Moreover, their conditions are postulated relative to any acts x, y, not necessarily pure horse-race payoffs, as assumed in conditions WCI and CI here. In the presence of our other assumptions, the latter difference is immaterial. 9

or infinity for every q S. If C is {0, }-valued then the minimization in 4) can be equivalently stated as minimization over the convex set of priors {q : C q) = 0}, without the additive C term. This paper s second main result is stated below using notation introduced prior to Theorem 4. In particular, recall that C 1 denotes the set of every function of the form C : S R + { } that is convex, lower semicontinuous and satisfies min C = 0. Theorem 5 Assuming R > 2, the following two conditions are equivalent: 1. In addition to condition 1 of Theorem 4, satisfies Weak Certainty Independence. 2. There exist p R, γ [0, ) and C : S R + { } such that u ν x) = min q S S R q s s=1 r=1 p r u γ x s r) defines a CE ν : X 0, ) that represents, and ) + C q) 4) either γ = 1 or C is {0, } -valued. 5) Suppose the two conditions are satisfied. Then representation 4) holds for a unique pair p, γ) in R [0, ) and a function C : S R + { } that can be uniquely selected to be minimal, in which case C C 1 and 5) is true. The minimal C is {0, }-valued if and only if satisfies Certainty Independence. In particular, if γ 1, then Certainty Independence must be satisfied. Proof. See Appendix B. The Theorem s last sentence clarifies the sense in which MMR-type scale-invariant preferences must either exhibit unit CRRA toward roulette uncertainty or they must be within the maxmin class of GS. Clearly, condition 1 of Theorem 5 also implies the representation of Theorem 4. For γ = 1, the two representations are identical. For γ 1, we have a unique minimal representation of the form 4) in which C is {0, }-valued and a member of C 1 ), as well as a unique extremal representation of the form 3) in which C is finite-valued and a member of C γ ). The circumstances of the overlap of the two representations within the scope of Theorem 4 can be summarized as follows. 10

1/C 1.5 1.0 0.5 0.0 0.0 0.5 1.0 probability Figure 1: Graph 1/C π, 1 π) as a function of π in Example 7. The bottom thick line corresponds to γ [0, 1), and the top thin line corresponds to γ 1, ). Corollary 6 Suppose satisfies condition 1 of Theorem 4 and admits the CE representation ν of equation 4) for some p R, γ [0, ) and minimal C : S R + { }. Then C C 1, and either γ = 1 or C is {0, }-valued. Proof. The assumed representation of ν implies Weak Certainty Independence. Here is a concrete illustration of the overlap of the two representations. Example 7 Suppose S = 2 and x y if and only if U x) > U y), where { R ) R [ ) U x) = min π p r u γ x 1 r + 1 π) p r u γ x 2 1 r : π 3, 2 3] }, r=1 r=1 for some p R and CRRA γ 0. Clearly, representation 4) is valid with { 0 if π [1/3, 2/3], C π, 1 π) = otherwise, 6) which defines an element of C 1. Example 22 of Appendix B.2 has the following implications. For any value of γ, equation 6) defines the unique minimal C consistent with representation 4). For γ 1, the multiplicative representation in 3) is also valid for a unique extremal finite-valued) C, which is an element of C γ and is given concretely by { 1 C π, 1 π) = min {3π, 1, 3 1 π)} if γ 0, 1), max {3/2) 1 π), 1, 3/2) π} if γ 1, ). The above function is graphed in Figure 1, with the case γ [0, 1) corresponding to the bottom thick line and the case γ 1, ) corresponding to the top thin line. 11

4 Source-Dependent CRRA In the last section, the scale-invariant and ambiguity-averse preference class of Theorem 4 was specialized by imposing Weak Certainty Independence. In this section, the same class of preferences is specialized in another direction by imposing separability on preferences over pure) horse-race payoffs, without any weak independence axiom. The result is a utility representation in which both preferences on horse-race payoffs and preferences on roulette payoffs admit a constant-relative-risk-aversion expected-utility representation with a unique prior. The key feature of the representation is that the CRRA assigned to horserace payoffs can be higher than the CRRA assigned to roulette-payoffs, as a reflection of ambiguity aversion. The dual, multiple-prior version of this representation is also given and is shown to generalize a well-known entropic specification. For any x, y X S and A {1,..., S}, x A y denotes the horse-race payoff defined by x A y ) s = { x s if s A, y s if s / A. Let S denote the restriction of to the set of horse-race payoffs: x S y x, y X S and x y. We say that S is separable if for every x, y, z, z X S and A {1,..., S}, Here is this paper s third main result. x A z S y A z x A z S y A z. Theorem 8 Assuming R, S > 2, the following three conditions are equivalent: 1. Condition 1 of Theorem 4 is satisfied and S is separable. 2. There exist unique p, γ) R [0, ) and ˆp, ˆγ) S [γ, ) such that the CE ν representing is given by uˆγ ν x) = S s=1 ˆp s uˆγ u 1 γ R ) p r u γ x s r), x X. 7) 3. There exist unique p, γ) R [0, ), θ 0, ] and η [, 1) 1, ] such that the CE ν representing is given by 3), with the function C is specified as follows: r=1 12

If γ = 1, then If γ 1, then C q) = θ S q s log s=1 qs ˆp s ), q S with 0 = 0). 8) S 1 C q) = ) ) 1/η η qs ˆp s, 9) ˆp s=1 s where the cases η {0, ± } are computed by taking a corresponding limit: S ) 1 exp s=1 ˆp s log q s /ˆp s ) if η = 0, C q) = max s {q s /ˆp s } if η =, 10) min s {q s /ˆp s } if η =. Finally, the parameters in conditions 2 and 3 are related by θ = 1 ˆγ 1 and η = ˆγ 1 ˆγ γ, 11) and the function C specified in condition 3 is the unique extremal C : S R + { } consistent with representation 3). Proof. See Appendix B. Representation 7) is the scale-invariant case of a class of second-order expected utilities characterized by Ergin and Gul 2009) and Nau 2006). Here scale invariance simplifies the axiomatic foundations. As explained in the following section, the continuous-time version of representation 7) already appears in Schroder and Skiadas 2003), albeit without a decision-theoretic foundation. The dual representation for γ = 1 is an instance of the entropic variational preferences of Hansen and Sargent 2001), and the preceding characterization is consistent with Strzalecki 2010). For γ 1, the characterization is new, and extends the case γ = 1, which can be obtained as a limiting version by letting γ 1, as outlined in the following remark. Remark 9 Let ν be the CE of the second condition of Theorem 8. The restriction of ν to the set of horse-race payoff s is the expected-utility CE S ) ν S x) = u 1 ˆγ ˆp s uˆγ x s ), x 0, ) S. s=1 13

For γ = 1, the fact that 7) is equivalent to 3) with C given in 8) is equivalent to the well-known identity log ν S x) = min q S { S q s log x s ) + θ s=1 S q s log s=1 qs ˆp s ) }, x 0, ) S, 12) with the convention 0 = 0. For γ 1, the fact that 7) is equivalent to 3) with C given in 9) is equivalent to the new identity S s=1 u γ ν S x) = min su γ x s ) q S S ) 1/η, s=1 ˆp1 η s qs η x 0, ) S, 13) with the limit conventions in 10). Given 11), identity 12) is the limiting version of identity 13) as γ 1. To verify this claim, multiply 13) by 1 γ), take logs, divide by 1 γ) and use 11) to find { log ν S x) = min q S 1 1 γ log S s=1 q sx 1 γ s θ 1 η log S s=1 ) )} η qs ˆp s. ˆp s As γ 1, and therefore η 1, the first term inside the curly brackets converges to S s=1 q s log x s ) and the second term converges to S s=1 q s log q s /ˆp s ), thus reducing the above expression to 12). 5 Small-Risk Approximations As noted earlier, the source-dependent CRRA representation of Theorem 8 already appears in Schroder and Skiadas 2003) in a continuous-time Brownian formulation, where the tractability of the utility is demonstrated in a context of optimal consumption/portfolio choice. To clarify this connection, as well as for its general interest in applications, this section provides small-risk approximations of the CE 7) in the tradition of Arrow 1965, 1970) and Pratt 1964). The discussion goes beyond the Brownian case to include Poisson-type arrivals, which arise naturally in modeling rare unpredictable events. A poorly understood rare event is arguably the epitome of Knightian uncertainty. We are interested in simplifying approximations of the CE 7) applied to small risks. The size of a risk is parameterized by h 0, 1), representing the length of a short time period at the end of which uncertainty resolves and a payoff x h) is realized. The CE is defined in terms of roulette probabilities p R and horse-race probabilities ˆp h) S. 14

As the notation indicates, ˆp h) but not p can depend on the parameter h. The reason for this will become clear in Proposition 11. The CE ν h is defined to take the form of Theorem 8: uˆγ ν h x) = S s=1 ˆp s h) uˆγ u 1 γ R ) p r u γ x s r), x X, 14) for a CRRA γ associated with roulette risk, and a CRRA ˆγ associated with horse-race uncertainty. Note that the function ν h potentially depends on h only through the probabilities r=1 ˆp h) the CRRA parameters γ and ˆγ do not vary with h. We will approximate ν h x h)) for small h, assuming the payoff structure 9 x h) = 1 + µh + σb h) + ˆσ ˆB h) + o h), 15) for constant µ, σ, ˆσ R and random variables B h) and ˆB h), representing roulette risk and horse-race uncertainty, respectively. Accordingly, we assume that B h) depends only on the roulette state and ˆB h) depends only on the horse-race state. We summarize this assumption by writing B r h) = B h) r, s) and ˆBs h) = B h) r, s), for every state r, s). 16) The factors B and ˆB are normalized so that R p r B r h) = 0, r=1 R p r [B r h)] 2 = h + o h), 17) r=1 S ˆp s h) ˆB S s h) = 0, ˆp s h) [ ˆB s h)] 2 = h + o h). 18) s=1 s=1 In a dynamic setting in which ν h represents the CE of a recursive utility specification with discrete intervals of length h, the pair B h), ˆB h)) should be thought of as a copy of i.i.d. increments of a martingale that generates the underlying information filtration. The first proposition below corresponds to the case in which this martingale becomes twodimensional standard Brownian motion in the limit as the frequency goes to infinity and therefore h goes to zero. The proposition refers to a single period that can be thought of as a single node in the usual binomial approximation of each Brownian motion. 9 We use the standard little-oh notation. Every occurence of o h) stands for some function ε h), not the necssarily the same function every time, such that lim h 0 ε h) /h = 0. 15

Proposition 10 Suppose ˆp = ˆp h) does not vary with h, and therefore neither does the CE ν = ν h defined in 14). Suppose further that the payoffs x h) are defined by 15) for constant µ, σ, ˆσ R and random variables B h) and ˆB h) satisfying 16), 17) and 18). Then Proof. See Appendix C. ) ν x h)) = 1 + µ γ σ2 ˆσ2 ˆγ h + o h). 19) 2 2 Another type of approximation arises if the limiting martingale involves unpredictable jumps. We illustrate with a simple case in which a martingale whose increments are i.i.d. copies of the horse-race factor ˆB h) converges to a compensated Poisson process with arrival rate λ as the frequency goes to infinity. Roulette uncertainty is again assumed to be of the Brownian type although it could alternatively be assumed to be another Poisson-type factor, with the obvious modifications to the approximation). Proposition 11 Suppose that S = 2 and for some constant λ > 0, ˆp 1 h) = 1 ˆp 2 h) = λh + o h) and ˆB1 h) = 1 ˆp 1 h), ˆB2 h) = ˆp 1 h). 20) Suppose further that the payoffs x h) are defined by 15) for constant µ, σ, ˆσ R and random variables Bh) and ˆBh) satisfying 16) and 17). Note that 18) follows from 20).) Then the CE ν h defined in 14) satisfies ) ) ν h x h)) = 1 + µ γ σ2 2 ˆσ 1 + ˆσ)1 ˆγ 1 1 ˆγ λ h + o h). Proof. See Appendix C. Skiadas 2008, 2010) explains how CE approximations such as those of the preceding two propositions correspond to continuous-time recursive utility formulations, extending the Duffi e and Epstein 1992) continuous-time formulation of Kreps and Porteus 1978) utility. The resulting continuous-time utility is within a class of utilities introduced by Schroder and Skiadas 2003, 2008) in the context of optimal consumption and portfolio choice. 16

A Appendix: Weak Independence Axioms Theorem 4 derives a functional structure 3) for evaluating horse-race uncertainty. For γ = 1, this functional structure is recognized in Theorem 5 to correspond to the specification of MMR, which is characterized by Weak Certainty Independence WCI). Moreover, Theorem 5 implies that if γ 1, the multiplicative functional structure in 3) is consistent with WCI if and only if preferences are within the GS class, meaning that Certainty Independence CI) is satisfied. In the case in which CI is not satisfied and γ 1, the functional structure in 3) is related to formulations by CF and CMMM acronyms defined in the Introduction), but lacks an exact foundation based on weak independence conditions, rather than scale invariance. The purpose of this appendix is to close this gap, formulating weak independence axioms that characterize all functional forms for aggregating horse-race uncertainty in 3), without assuming scale invariance. The relevant weak independence conditions, in addition to WCI see Section 3), are listed below. Recall that for any α 0, ), α denotes the payoff that takes the value α at every state. Low-Constant Independence LCI) exists ε > 0 such that Given any x, y X S and roulette event B, there x y for all α 0, ε), x B α y B α. High-Constant Independence HCI) Given any x, y X S and roulette event B, there exists M > 0 such that x y for all α M, ), x B α y B α. Suppose the CE representation ν of admits the representation 3) for some p S, γ [0, ) and C C γ. Then the following implications are easily seen to be true γ = 1 = WCI, γ 0, 1) = LCI, γ 1, ) = HCI. LCI and HCI are variants of Axiom 5 of CF and Axiom A.10 of CMMM. The latter formulates weak independence relative to a fixed reference outcome, while the former further assumes that the outcome is the worst possible. In our setting, there is no worst or best) outcome LCI is weak independence relative to all suffi ciently bad constant payoffs, and HCI is a weak independence relative to all suffi ciently good constant payoffs. 17

We will show that in the absence of scale invariance, WCI, LCI and HCI entirely characterize the functional structure 3) toward horse-race uncertainty. For technical reasons, we will do so in a modified model in which roulette outcomes are uniformly distributed on [0, 1], essentially embedding our earlier treatment in a model with objective roulette probabilities, just as in the related literature of GS, MMR, CF and CMMM. For the remainder of this Appendix, the roulette state space {1,..., R} is replaced with the unit interval [0, 1]. An objective distribution over roulette outcomes is given as Lebesgue measure λ on [0, 1]. A roulette event is now any Borel subset of [0, 1]. A roulette payoff is any Borel-measurable simple random variable of the form z : [0, 1] 0, ), meaning that there exist finitely many disjoint roulette events B 1,..., B n and corresponding z 1,..., z n 0, ) such that z = n i=1 z i1 Bi. The corresponding expectation is Ez = n i=1 z iλ B i ). A payoff is any mapping of the form x : [0, 1] {1,..., S} R such that for every horse-race state s, the section x s : [0, 1] R defined by x s r) = x r, s)) is a roulette payoff. As before, we identify a roulette payoff with a payoff that does not depend on the horse-race state, while a horse-race payoff can be viewed as either a payoff that does not depend on the roulette state or an element of 0, ) S. As in Section 2, we take as given a relation on the set of payoffs X, whose restriction on the set of roulette payoffs X R resp. horse-race payoffs X S ) is denoted R resp. S ). We further assume that R has a von Neumann-Morgenstern vnm) representation. For the purpose of this discussion, a vnm index is any increasing continuous function of the form u : 0, ) R and is said to represent R if x R y is equivalent to Eu x) > Eu y) for all x, y X R. We focus on the case in which R has an unbounded vnm representation u. Since we are free to choose any positive affi ne transformation of u, we assume, without loss of generality, that the range of u is R or ± 0, ) meaning 0, ) or, 0)), choices that correspond to the possible ranges of u γ. The representation theorem that follows essentially modifies Theorem 4 by replacing scale invariance with a weak independence axiom, which one depending on the range of u. Recall the notation u γ and C γ introduced prior to Theorem 4. Note that the definition of C γ depends on γ only through its range u γ 0, ). Extending this notation, we define the set C D, where D is R or ± 0, ), by letting R if γ = 1, C D = C γ if D = u γ 0, ) = 0, ) if γ 0, 1),, 0) if γ 1, ). 21) 18

Theorem 12 Suppose the vnm index u is such that u 0, ) = R or u 0, ) = ± 0, ). Then the following conditions are equivalent: 1. The relation is an increasing, R -monotone and ambiguity-averse preference order, and S is continuous. Moreover, R has the vnm representation u, and satisfies WCI if u 0, ) = R, LCI if u 0, ) = 0, ), and HCI if u 0, ) = 0, ). 2. There exists C : S R + { } such that u ν x) = min q S S s=1 q seu x s )) + C q) if u 0, ) = R, S s=1 q seu x s )) C q) if u 0, ) = ± 0, ) 22) defines a CE ν : X 0, ) representing. Suppose the two conditions are satisfied. The function C in 22) can be uniquely selected to be minimal if 0, ) u 0, ) and maximal if u 0, ) =, 0). If C is so selected, it is necessarily a member of C u0, ). Finally, for the same C and assuming S > 2, the preference order S on X S is separable if and only if the following are true: If u 0, ) = R, then C is given by 8) for some unique θ 0, ] and ˆp S. If u 0, ) = ± 0, ), then C is given by 9) and 10) for some unique ˆp S and η [, 1) if u 0, ) = 0, ) and η 1, ] if u 0, ) =, 0). Remark 13 Representation 22) formulates a dual representation of a function that maps the vector [Eu x s )] s=1,...,s to u ν x). The Theorem s proof explains the primal form of this function, which in the case of a separable S is given in Example 23 if u 0, ) = R and Example 24 if u 0, ) = ± 0, ). In the case u 0, ) = R, the utility representation 22) and the characterization of the entropic form 8) are familiar thanks to MMR and Strzalecki 2010), respectively. The case u 0, ) = ± 0, ) is related to CF and Theorem 26 of CMMM, but is different in terms of the restrictions placed on C, reflecting the difference between LCI or HCI and the corresponding weak-independence assumptions of CF and CMMM. The characterization of a separable S in terms of 9) in the case u 0, ) = ± 0, ) is new. 19

B Appendix: Proof of Representation Theorems The purpose of this appendix is to prove Theorems 4, 5, 8 and 12, as well as explain their underlying structure. The first section presents a key representation theorem for scale-invariant separable preferences that is of interest in its own right. The second section relates preference properties to primal CE representations. The third section develops convex duality results that in conjunction with the primal representations lead to the CE forms of the main results, whose proof is concluded in the last four sections. Omitted lemma proofs can be found in Appendix C. B.1 Scale-Invariant Separable Preferences In preparation for the main analysis, this section states and proves Theorem 14, providing a characterization of scale-invariant separable preferences The result, which is of interest in its own right, is a variant of Theorem 3.37 of Skiadas 2009); whereas in the latter utility is assumed to be continuously differentiable in some arbitrarily small neighborhood, here it is assumed to be somewhere risk averse, a purely ordinal property. The remarkable aspect of Theorem 14 is that separability together with scale invariance substitutes for a subjective expected-utility foundation on a finite state space, delivering the power-or-logarithmic expected utility structure under a unique probability. Related insights were provided by Hens 1992) and Werner 2005). Hens noted that if a continuously differentiable additive utility has a constant marginal rate of substitution along the certainty line, then it must take the form of expected utility. Werner showed that an additive utility that is more risk averse than risk risk-neutral relative to an exogenously given probability must be expected utility relative to this probability. These arguments are not special to scale invariant preferences, but rely on non-ordinal assumptions. Theorem 14 on the other hand makes only ordinal assumptions utility smoothness and the existence of the unique probability p are consequences of these ordinal assumptions. Theorem 14 is stated in terms of a binary relation on 0, ) n, for some positive integer n. This is not the same as in the main part of the paper the result will be applied to R with n = R.) We refer to Definition 1 for the meaning of the terms increasing, continuous, scale-invariant, and preference order. We also use Definition 3. In particular, writing x A y for the element of 0, ) n defined by { x i if i A, x A y) i = y i if i / A, 20

we define to be separable if x A z y A z implies x A z y A z, for all x, y, z, z 0, ) n and any A {1,..., n}, and we say that is somewhere risk averse if there exists an open set O 0, ) n such that for every x O, the set {y : y x} O is convex. We refer to 1) and 2) for the definition of the notation n and u γ. Theorem 14 Suppose is a binary relation on 0, ) n for an integer n > 2. Then the following two conditions are equivalent: 1. is an increasing, continuous, separable and scale-invariant preference that is somewhere risk averse. 2. There exist unique p n and γ [0, ) such that x y n p iu γ x i ) > n p iu γ y i ), for all x, y 0, ) n. 23) i=1 i=1 Proof. That the second condition implies the first one is immediate. Conversely, suppose that satisfies the first condition. By Debreu s additive representation theorem see Debreu 1983), Krantz, Luce, Suppes, and Tversky 1971) and Wakker 1988)), there exist increasing and continuous functions U i : 0, ) R such that x y n U i x i ) > n U i y i ), for all x, y 0, ) n. 24) i=1 i=1 Moreover, if 24) holds for functions Ũi : 0, ) R in place of the U i, there exist a 0, ) and b R n such that Ũi = au i + b i for all i. Given any s 0, ), scale invariance states that x y sx sy, and therefore the functions Ũi z) = U i sz) define another additive representation of. There exist, therefore, functions a : 0, ) 0, ) and b : 0, ) R n such that U i sz) = a s) U i z) + b i s), s, z 0, ), i = 1,..., n. 25) Let us also define the functions F i, α : R R by F i x) = U i e x ) and e fx) = a e x ). We can then restate restriction 25) as F i x + y) = e fx) F i y) + b i e x ), x, y R, i = 1,..., n. 26) 21

Subtracting from this equation its special case obtained by setting y = 0, we obtain the identity F i x + y) F i x) y = e fx) F i y) F i 0), y 0, x R, i = 1,..., n. 27) y Since F i is increasing, it is almost everywhere differentiable see, for example, Theorem 7.2.7 in Dudley 2002)). Choosing any x where F i is differentiable, and letting y converge to zero in 27), it follows that F i is also differentiable at zero and F i x) = efx) F i 0). Given now that F i is differentiable at zero, letting y go to zero in 27) shows that F i must also be differentiable at every x R, and the last equation holds for every x R. We restate this conclusion as f x) = log F i x), x R, i = 1,..., n. 28) 0) F i Differentiating 26) with respect to y and taking logarithms, we also have f x) = log F i x + y) F i y) = log F i x + y) F i 0) log F i y) F i x, y R, i = 1,..., n. 29) 0), Recall that f x) does not depend on i. Equations 28) and 29) imply that f must solve what is known as the Cauchy functional equation: f x + y) = f x) + f y), x, y R; f 0) = 0. It is well-known see, for example, Aczél 2006)) that if f solves the Cauchy equation and is continuous at some point, then f x) = 1 γ) x, x R, 30) for some constant scalar γ. A well-known theorem, proved in Yaari 1977), states that if d i=1 U i is quasiconcave on some set d i=1 I i, where each I i is an open interval, then at most one of the U i : I i R is not concave. The assumption that is somewhere risk averse allows the application of this theorem to conclude that for some i, U i is concave on some open interval. Since U i is differentiable, it must be continuously differentiable on an open interval where it is concave. This proves that F i is somewhere continuous. By equation 28), f is also somewhere continuous and is therefore given by 30) for some γ R. Integrating 28), it follows that there exists some p n such that U i is a positive affi ne transformation of pu γ, and we can therefore set U i = pu γ in 24). Uniqueness of the additive representation up to a positive affi ne transformation implies there is a unique choice of p n and γ R that is consistent with representation 23). The assumption that is somewhere risk averse implies that γ 0. 22

B.2 Primal CE Representation Up to Section B.7, where we prove Theorem 12, we assume the finite state-space setting of Section 2. In this section we derive primal CE representations for the preference classes considered in the paper s main results. Multiple-prior representations are derived in the next section by convex duality arguments. In addition to the terms of Section 2, we use the following terminology. Definition 15 Suppose D R is an interval. A certainty equivalent CE) on D S is any increasing and continuous function f : D S D with the property f α1) = α for all α D. The CE f is defined to be scale invariant SI) if f αz) = αf z) for all α 0, ) and z D S such that αz D S. translation invariant TI) if f z + α1) = f z) + α for all α R and z D S such that z + α1 D S. From Section 2, recall that is a relation on the set of payoffs X and R is its restriction on the set of roulette payoffs X R. Lemma 16 The following two conditions are equivalent: 1. is a continuous, increasing and R -monotone preference order, and R is separable, scale invariant and somewhere risk averse. 2. The CE ν : X 0, ) representing exists and takes the form R u γ ν x) = f ) ) ru γ x 1 R r=1 r,..., p ) ru γ x S r=1 r, x X, 31) for unique p R, γ [0, ) and CE f on u γ 0, ) S. Proof. That 2 = 1) is immediate. Conversely, suppose condition 1 is satisfied. Since is a continuous, increasing preference order, ν x) = inf {α 0, ) : α1 x} 32) defines a CE ν that represents. Applying Theorem 14 to R, it follows that there exist unique p R and γ [0, ) such that x R y is equivalent to R r=1 p ru γ x r ) > 23

R r=1 p ru γ y r ) for all x, y X R. The fact that is increasing and R -monotone implies that the function f : D S D is well-defined by 31). Indeed, suppose x, y X are such that R r=1 p ru γ x s r) = R r=1 p ru γ yr) s, and therefore x s R y, s for every s. For any ε > 0, x + ε1) s ys, and therefore ν x + ε1) > ν y) by R -monotonicity. Letting ε 0, we have ν x) ν y). By symmetry, ν x) = ν y), and therefore f takes the same value whether it is defined in terms of x or y in 31). The proof that f is a unique CE is also straightforward and is left to the reader. Given the representation of the last Lemma, the following lemma maps properties of to corresponding properties of the CE f : u γ 0, ) S u γ 0, ). Lemma 17 Assume that R > 1 and the two equivalent conditions of Lemma 16 are satisfied. Then the following are true: a) is ambiguity-averse if and only if f is quasiconcave. b) is scale invariant if and only if { SI if γ 1, f is TI if γ = 1. c) Weak Certainty Independence see Section 3) is satisfied if and only if f is TI. Proof. a) That quasiconcavity of f implies ambiguity aversion of is immediate. Conversely, suppose is ambiguity-averse. Fix any B {1,..., R} such that π r B p r 0, 1). Then for any horse-race payoffs x, y X S, we can write R r=1 p ru γ x B y) s r ) = πu γ x s ) + 1 π) u γ y s ). Given representation 31), ambiguity aversion requires that for all x, y X S, L f u γ x 1 ),..., u γ x S )) = f u γ y 1 ),..., u γ y S )) implies f πu γ x 1 ) + 1 π) u γ y 1 ),..., πu γ x S ) + 1 π) u γ y S )) L. The same condition can be stated more simply as f x) = f y) = f πx + 1 π) y) f y), for all x, y u γ 0, ) S. 33) Because f is increasing and continuous, condition 33) is equivalent to its apparently stronger version f x) f y) = f πx + 1 π) y) f y), for all x, y u γ 0, ) S. 34) 24

To see why, suppose x, y u γ 0, ) S satisfy f x) > f y). Pick any z u γ 0, ) S such that z x, y and let δ = x z 0. Since f x) > f y) f z), δ is nonzero. The decreasing continuous function h : [0, 1] R defined by h α) = f x αδ) satisfies h 0) > f y) > h 1). Let α 0, 1) be such that h α) = f y). By monotonicity and 33), we conclude that f πx + 1 π) y) f π x αδ) + 1 π) y) f y). This proves 34). Applying the same conclusion with the complement of B in place of B, and the notation for x and y interchanged, we also have f y) f x) = f πx + 1 π) y) f x), for all x, y u γ 0, ) S. 35) Using 34) and 35) together, we show that f is quasiconcave. For any given z u γ 0, ) S, we are to prove that the set C {x : f x) f z)} is convex. Suppose x, y C. Using condition 34) if f x) f y), and condition 35) if f y) f x), it follows that x, y C implies πx + 1 π) y C. 36) This is not quite the definition of convexity of C, since π is fixed, but it implies convexity of C given the continuity of f. To show this claim, let J 0 = {0, 1} and J n+1 = {πα + 1 π) β : α, β J n }, n = 1, 2,... The set J = n=1 J n is dense in [0, 1]. Fix any x and y in C and consider the set K = {φ [0, 1] : φx + 1 φ) y C}. An induction using 36) shows that J K. Since K is closed and contains a dense subset of [0, 1], it contains all of [0, 1]. Therefore C is convex. b) Expression 32) implies that is scale invariant if and only if ν is SI that is, homogeneous of degree one). definitions. Given this observation, the claim is immediate from the c) We prove the only if part, the converse being straightforward. Suppose WCI is satisfied and fix any roulette event B such that π r B p r 0, 1). Suppose we are given any a, b u γ 0, ) S and t R is such that a + t1, b + t1 u γ 0, ) S. Suppose further that t is restricted so that t 1 γ) > 0. It is then not hard to show that there exist x, y X S and α, β 0, ) such that a s = πu γ x s ) + 1 π) u γ α), b s = πu γ y s ) + 1 π) u γ α), t = 1 π) u γ β) u γ α)). The idea is to pick α so that u γ α) is close to zero: if γ = 1 choose α = 1, if γ [0, 1) choose α very small, and if γ 1, ) choose α very large. Given suffi ciently small u γ α), 25