Regret aversion and opportunity dependence

Transcription

1 Journal of Economic Theory 139 (2008) Regret aversion and opportunity dependence Takashi Hayashi Department of Economics, University of Texas at Austin, Austin, TX 78712, USA Received 21 June 2006; final version received 2 July 2007 Available online 14 July 2007 Abstract This paper provides an axiomatic model of decision making under uncertainty in which the decision maker is driven by anticipated ex post regrets. Our model allows both regret aversion and likelihood judgement over states to coexist. Also, we characterize two special cases, minimax regret with multiple priors that generalizes Savage s minimax regret, and a smooth model of regret aversion Elsevier Inc. All rights reserved. JEL classification: D81 Keywords: Regret; Opportunity dependence 1. Introduction 1.1. Objectives and motivation This paper provides an axiomatic model of decision making under uncertainty in which the decision maker is driven by anticipated ex post regrets. We start from a choice function instead of a binary relation, since the choice behavior of such type may not be rationalized as a preference maximization according to a single binary relation. It is opportunity dependent. For example, Savage s minimax regret choice [24] violates the independence of irrelevant alternatives condition (IIA), which is necessary for preference rationalizability (see [22,2]. See also [6,27] for related properties). The opportunity dependence due to regret aversion is behaviorally of interest. Here we list two points. address: th925@eco.utexas.edu /$ - see front matter 2007 Elsevier Inc. All rights reserved. doi: /j.jet

2 T. Hayashi / Journal of Economic Theory 139 (2008) Example 1 (Non-binding constraints matter). Consider a portfolio choice problem with shortsale constraints. In the standard models of preference maximization, slightly weakening or tightening the non-binding constrains does not change the choice. It does, however, when the decision maker is driven by anticipated regret. This is because at each state it is ex post optimal to short-sale some assets up to the limit. Example 2 (Aggressive behavior). In many practical problems, ambiguity averse choice is observably equivalent to the standard expected utility maximization. The reason is that in many cases low shock is always the worst case scenario. Thus, it is equivalent to having a probabilistic belief that simply assigns high likelihood on low shocks. However, even in such cases, there is a qualitative difference between regret averse choice and the standard expected utility maximization. Consider a seller who is to choose a quantity to supply, under demand uncertainty. Assume the price is fixed. If she is more ambiguity averse, she just lowers the quantity level. This is observably equivalent to the choice made by the standard expected utility maximizer who assigns higher probability on lower demand, or who is more risk averse. Regret averse seller behaves differently. She anticipates regret for two cases, one is where the quantity is too much compared to the actual demand, the other is where it is too little. She may take the second case seriously, and thus she may raise the quantity level. In the standard model, being less risk averse is the only way to appreciate the second case. However, a regret averse decision maker may be either more risk averse or less risk averse. The limitation of the Savage minimax regret model is that it is a model of complete ignorance. In this paper, we provide a general model in which regret aversion and likelihood judgement over states coexist Overview of results First, we identify the minimal necessary set of axioms that characterize the class of choice rules where the decision maker is driven by anticipated ex post regret of any kind, and obtain a general representation. The general class of choice rules is represented by ( ) φ(b) = arg min Φ max u(g( )) u(f ( )), f B g B where Φ is a homothetic function that evaluates ex post regrets. Our main axioms are roughly: (i) a restricted version of Nash/Arrow-type independence axiom for the case where added alternatives are stochastically dominated (in the sense we define below) by the existing ones, since such added alternatives do not change ex post optimal choices. This is a counterpart of Milnor s Special Row Adjunction axiom which is stated in his setting of set-dependent binary relations (see [20]); (ii) an admissibility axiom; and (iii) an independence axiom for mixtures of sets and singleton sets. Also, we obtain that the function Φ is quasi-convex, under an additional axiom of regret aversion which is revealed via hedging acts in the spirit of Gilboa Schmeidler [10]. We emphasize that regret-averse choice is not a reverse version of ambiguity-averse choice by flipping gain to loss. Loss is calculated using a fixed outcome, whereas regret depends on ex post optimal choice which varies as opportunity set varies. For example, choice with maximin

3 244 T. Hayashi / Journal of Economic Theory 139 (2008) expected utility violates one of our axioms. This suggests that regret aversion is different from ambiguity aversion [7,9]. As discussed above, a regret-averse choice may be aggressive. The general representation includes two special classes: 1. Minimax regret with multiple-priors: Given a subjective set of priors P Δ(Ω), the choice is determined by φ(b) = arg min max f B p P ω Ω ( ) max u(g(ω)) u(f (ω)) p(ω). g B This class includes two important subcases. When P is equal to the entire probability simplex it reduces to [ ] φ(b) = arg min max max u(g(ω)) u(f (ω)), f B ω Ω g B which is the standard minimax regret choice by Savage, in which subjective belief is completely indeterminate. When P ={p} it reduces to φ(b) = arg min f B = arg max f B [ max u(g(ω)) u(f (ω)) g B ω Ω u(f (ω))p(ω), ω Ω ] p(ω) which is the standard Bayesian case where subjective belief is probabilistic. 2. Smooth model of regret aversion: Given a probabilistic belief p Δ(Ω) and a coefficient of regret aversion α > 0, the choice is determined by φ(b) = arg min f B ω Ω ( max u(g(ω)) u(f (ω)) g B ) α p(ω) for every B B. Here the second-order risk aversion on regrets can play a different role than the first-order risk aversion described by u does. This type of specification has been used in non-axiomatic models of regret aversion by Krähmer and Stone [13,14]. The case α > 1 corresponds to regret aversion, α < 1 corresponds to regret loving, and α = 1 corresponds to regret neutrality, which is the case of subjective expected utility maximization. The analysis of these special classes involves off-equilibrium comparison of acts, to see the detailed properties of regret aversion. Although a choice function describes equilibrium choices only, the current approach partially allows off-equilibrium comparison of acts through observable choices. This is basically by determining the amount of premium that is necessary for an act to be chosen over a given opportunity set. We call it regret premium. We consider two properties of regret premium: (i) that regret premium is independent of acts with constant regret, where the notion of constant regret is suitably defined; (ii) that regret premium satisfies eventwise separability (sure-thing principle) in the sense of Savage [25]. First one characterizes minimax regret with multiple-priors, and the second one characterizes the smooth model of regret aversion Related literature There are several approaches to opportunity dependence of choice under uncertainty. Here we list two relevant ones.

4 T. Hayashi / Journal of Economic Theory 139 (2008) Milnor [20], and recently Stoye [28], characterize the Savage maximum regret representation basically for a family of binary relations indexed by opportunity sets. Given an opportunity set B, the ranking between acts in B, denoted B, is determined by f B h max ω Ω [ max u(g(ω)) u(f (ω)) g B ] max ω Ω [ max u(g(ω)) u(h(ω)) g B for each f, h B. The main axiom in their approach is a restricted version of independence of irrelevant alternatives type condition that the ranking does not change when added alternatives do not change ex post optimal outcomes. Puppe and Schlag [23] extend those results to the case where outcome spaces are state dependent. In the same framework, Terlizzese [31] obtains a representation in which the value of a choice from a set is given by: (i) dividing its ex post utility by the maximum possible ex post utility of choice from the set, at each state; and (ii) taking the minimum across states. Also he shows that it is equivalent to Savage s minimax regret. Another approach to the opportunity dependence due to regret aversion is to drop the transitivity axiom from the standard axiom set for subjective expected utility. It is taken by, for example, Loomes Sugden [16], Fishburn [8] and Sudgen [30]. To illustrate, consider an urn containing balls that are red, blue and green, where the proportion of each color is one-third. Consider three bets, f = (20, 10, 0), g = (0, 20, 10) and h = (10, 0, 20), where (x,y,z)denotes the bet that delivers x if red is drawn, y if blue is drawn, and z if green is drawn. When the decision maker is to compare f and g, the plausible ranking is f g, since g is very regrettable compared to f when red is drawn, whereas f is not that regrettable for any draw. Similar reasoning leads to g h and h f, and we have a cycle. Bell [3] studies a von-neumann/morgenstern type risk preference over pairs of outcomes, where one is to be received and the other is to be foregone. There he obtains a representation that captures a notion of regret. Our approach differs from the above mainly in three aspects: (i) Choice function is concerned with equilibrium choices only. Apparently our choice representation is telling as if one act in a given opportunity set may have smaller regret evaluation than another in it. However, such off-equilibrium comparison of acts does not imply that one act is chosen over another or that one act is more likely to be chosen than another, since typically neither of them may be chosen from the given set. Off-equilibrium comparison of acts is generally unobservable. In the second half of the paper, however, we deal with off-equilibrium comparison of acts through equilibrium choices. (ii) As for binary choices, our model covers the prediction by the models by Fishburn/Loomes- Sugden/Sudgen. Moreover, our model covers non-binary choices. (iii) Our model is flexible enough to allow that both regret aversion and nontrivial likelihood judgements coexist, whereas Milnor, Stoye and Puppe-Schlag exclude likelihood judgements by assuming symmetry across states and invariance of ranking to eliminating or duplicating states with redundant payoffs. This also allows us to define comparative regret aversion. More recently, another independent paper by Stoye [29] adopts the choice function approach, and gives axiomatizations of several decision rules some of which are popular in statistical decision making, ex post minimax regret, minimax regret with multiple-priors, and interim minimax regret. In contrast, we are interested in characterizing a behavioral class of regret-driven choices. In the axiomatization of minimax regret with multiple-priors, he presents monotonicity, ambiguity aversion and ambiguity neutrality for rectangular menus, that are parallel to but formulated and motivated differently from our axioms of admissibility, regret aversion and constant-regret independence of regret premium, respectively. Some axioms there may be more profoundly or ]

5 246 T. Hayashi / Journal of Economic Theory 139 (2008) elegantly presented than ours, probably because he is concerned with the normative appeals of axioms in statistical decision theory. In contrast, we are concerned with a descriptive method of identifying the observable restrictions for regret-driven choices. Sarver [26] adopts a different approach to a notion of regret. He considers preference over menus of lotteries, and obtains a representation with a subjective set states, in which the ex post value of a choice is discounted according to regret. In contrast, we are rather concerned with the traditional notion of regret in the sense by Savage, which is obtained with regard to objective states of the world. Another difference is that his notion of regret affects choice over menus, but does not affect choice from a menu. There, choices from menus fall in the standard revealed preference argument. There is a vast literature on the minimax regret model in the statistical and econometric decision theory. For example, see Chamberlain [5] for a survey in econometrics, recent papers by Manski [18,19] for application to statistical treatment problems, as well as standard textbooks of statistical decision theory. There are some papers that applied the minimax regret model to describe economic behavior. Linhart and Radner [15] applied it to a model of bilateral bargaining. Recently, Bergemann and Schlag [4] apply it to a monopoly pricing problem in which the monopolist perceives regret about lost profit opportunities. 2. Model The set of possible states of the world, denoted Ω, is assumed to be finite. The set X of pure outcomes is assumed to be compact metric. Let Δ(X) be the set of lottery outcomes (Borel probability measures) over X, endowed with the weak convergence topology. It is again compact metric with respect to the Prokhorov metric. When no confusion arises, an outcome refers to a lottery outcome. Let F ={f : Ω Δ(X)} be the set of acts [1], which is endowed with the product topology. Notice that any lottery is taken to be a constant act, therefore Δ(X) is included in F without loss of generality. The lottery space Δ(X) allows the usual mixture operation. That is, given l,m Δ(X) and λ [0, 1], the mixture λl + (1 λ)m Δ(X) is given by (λl + (1 λ)m)[y ]=λl(y) + (1 λ)m[y ] for every Borel subset Y of X. Also, the act space F allows to define a mixture operation. Given f, g F and λ [0, 1], the mixture λf + (1 λ)g F is given by (λf + (1 λ)g)(ω) = λf(ω) + (1 λ)g(ω) for every ω Ω. In words, mixture of acts λf +(1 λ)g is interpreted as taking randomization of ex post outcomes, in the way that at each state ω the decision maker receives f(ω) with probability λ and g(ω) with probability 1 λ. Denote by B the space of non-empty compact subsets of F, endowed with the Hausdorff metric; then B is compact. We call an element of B, typically denoted by B,anopportunity set. Also, let L denote the space of non-empty compact subsets of Δ(X), which is the space of opportunity sets consisting only of lottery outcomes (constant acts). Obviously we have L B. Given B B, g F and λ [0, 1], the mixture of set and act, λb + (1 λ){g} B, is defined by λb + (1 λ){g} ={λf + (1 λ)g : f B}.

6 T. Hayashi / Journal of Economic Theory 139 (2008) In words, opportunity set λb + (1 λ){g} consists of acts that are made of ex post randomization of each act in B and act g. Also note that we are not concerned with a mixture of set and set here. We model choice behavior by means of choice function φ : B 2 F such that φ(b) B and φ(b) = for all B B Stochastic dominance relations Here we define several notions of stochastic dominance. Notice that the dominance relations defined below need not be transitive at this moment. It is guaranteed by axioms to be introduced in the next section. First we define a weak dominance relation between acts, by looking at choices between lottery outcomes. Given f, g F, say that f weakly dominates g if f(ω) φ({f(ω), g(ω)}) for every ω Ω. When it is the case, we write f g. Also, we define a series of strict dominance relations between acts. Given acts f, g F and an event S Ω, say that f strictly dominates g on S if φ({f(ω), g(ω)}) ={f(ω), g(ω)} for every ω Ω \ S and φ({f(ω), g(ω)}) ={f(ω)} for every ω S. In other words, f and g tie at the complement event of S, whereas f strictly dominates g at all states in S. When this is the case, we write f > S g. When S = Ω, the relation f> Ω g simply says f strictly dominates g at all states. Now we define a stochastic dominance relation between opportunity sets. Given B,C B, B is said to ex post dominate C if for every ω Ω and g C, there is f B such that f(ω) φ({f(ω), g(ω)}). In other words, opportunity set B guarantees a better ex post choice than C does at every state. We denote this dominance relation by B EP C. Notice that when sets are singletons the ex post dominance relation reduces to the weak dominance relation between acts. The word ex post deserves some explanation. Since our model is static (one-shot), there are not really ex post choice problems that the decision maker faces after knowing states. Here f(ω) is formally a lottery, which is a constant act that gives f(ω) regardless of states. Therefore, {f(ω), g(ω)} is formally an unconditional choice problem between lotteries, not a conditional choice problem upon ω. This is without loss of generality, however, since timing does not matter in choosing constant acts Event of ex post optima For later purpose, we define the event where action f is ex post optimal in the opportunity set B. For each B B and f B, define N(f,B) ={ω Ω : f(ω) φ({h(ω) : h B})}, which is the event where f is ex post optimal in B and no regret is anticipated with regard to the choice of f.

7 248 T. Hayashi / Journal of Economic Theory 139 (2008) Main axioms First, we impose a Nash/Arrow-type independence axiom for the restricted cases where opportunity dependence is unchanged. Axiom 1 (Irrelevance of ex post dominated acts). For any B,C B with B EP C, φ(b C) B = φ(b) = φ(b C) B. When we add alternatives which are ex post dominated by the current opportunity set, it does not change how the decision maker perceives regret since it does not change ex post optimal choices. For such cases, it is intuitive to assume that opportunity dependence is absent. Notice that the added alternatives may be chosen. The condition above is excluding the case that the presence of added alternatives changes the choice despite that they are not chosen. One may think of a weaker condition where the addition of alternatives is limited to those that are ex ante dominated by the original set of acts. However, the weaker condition does not have a significant implication when it is combined with the admissibility condition listed below, since the added alternatives are never chosen. Axiom 1 implies the standard axiom of Independence of Irrelevant Alternatives for sets of outcomes, which characterizes that choice over sure outcomes is preference-rationalizabile. 1 In other words, opportunity dependence here is coming only from subjective uncertainty. Lemma 1. Let L s be the family of finite sets of lotteries. Then, Axiom 1 implies IIA for outcome sets: For every B,C L s with B C, φ(c) B = φ(b) = φ(c) B. Moreover, the above implication is true on the entire L when Axiom 4 (Upper Hemi-continuity) is added. Next, we introduce the admissibility condition, which basically says that choice should be undominated. Axiom 2 (Admissibility). For any B B and f φ(b), there is no g B such that g> Ω\N(f,B) f. Consider an act, denoted f. When it is ex post optimal at some states, there is no other act that can dominate f at all states. However, it is natural to exclude f from choice if there is an act that is at least as good as f in every state and dominates f outside the event. This is what the above axiom says. This admissibility condition is stronger than the condition requiring only that a chosen alternative is undominated in the > Ω relation, and weaker than the condition requiring that a chosen 1 I gratefully appreciate one of the referees for pointing this out, where it was assumed as an axiom in the previous version.

8 T. Hayashi / Journal of Economic Theory 139 (2008) alternative is undominated in the > S relation for any non-empty S Ω. Formally, they are stated as Axiom 2 (Weak admissibility). For any B B and f φ(b), there is no g B such that g> Ω f. Axiom 2 (Strong admissibility). For any B B and f φ(b), there is no non-empty event S Ω and no act g B such that g> S f. The weak admissibility condition allows that the decision maker believes some event to be sure, and chooses an act which is ex post optimal at such event, and is unaffected even when there is another act which dominates it at all states outside the sure event. We rule out this type of extremeness. This is vital in our approach, since the absence of null event is the condition required for our off-equilibrium comparison of acts (potential comparison of acts that may not be chosen) through observable choices. The strong admissibility condition rules out the Savage minimax regret choice, which we allow here. Note that Savage s minimax regret allows a weakly dominated strategy (see Milnor [20] for discussion). When an act is not ex post optimal at several states and if the decision maker is extremely averse to regret, improving upon it in only some states may not help to reduce her anticipated regret feeling. Our Admissibility axiom allows that the decision maker may not be moved by such improvement. Axiom 3 (Singleton independence). For any B B, f F and λ [0, 1], φ((1 λ)b + λ{f }) = (1 λ)φ(b) + λ{f }. To illustrate, consider an option that gives an opportunity set B with probability 1 λ and an act f with probability λ. Then it will be intuitive to say that with probability 1 λ the decision maker chooses what she chooses from B and with probability λ she chooses f. Behind this interpretation there is an implicit assumption that the decision maker views such randomization over a set/act pair to be identical to taking the corresponding mixture. The singleton independence axiom implies that choice of an act from a given set depends only on its relative position in the set. Hence it is violated for example by maximin expected utility, which exhibits particular preference for choosing constant acts. When the singleton independence axiom is strengthened into the mixture independence condition between sets, subjective expected utility maximization is the only case. See Corollary 3. The last two axioms are just for regularity, not of conceptual appeals, although Upper Hemicontinuity suggests that one should leave ties as they are. An example of the violation of Upper Hemi-continuity is shown later, where tie-breaking is imposed. Axiom 4 (Upper hemi-continuity). When B ν B and f ν satisfies f ν φ(b ν ) for every ν N and f ν f, then f φ(b). Axiom 5 (Non-degeneracy). There exist l,l Δ(X) such that φ({l,l }) ={l}.

9 250 T. Hayashi / Journal of Economic Theory 139 (2008) General result 4.1. Representation Given x,y R Ω, write x> S y for S Ω if x ω = y ω for all ω Ω \ S, and x ω >y ω for all ω S. Say that Φ : R Ω + R + is weakly monotone if for all N Ω and for all x,y R Ω + with x ω = y ω = 0 for all ω N, x> Ω\N y implies Φ(x) > Φ(y). Say that Φ : R Ω + R + is strongly monotone, if for all non-empty S Ω and x,y R Ω +, x> S y implies Φ(x) > Φ(y). Theorem 1. The choice function φ satisfies Axioms 1 5 if and only if there exists a mixture-linear, continuous and non-constant function u : Δ(X) R, and a homothetic and weakly monotone function Φ : R Ω + R + such that ( ) φ(b) = arg min Φ max u(g( )) u(f ( )) f B g B for every B B. Moreover, u is unique up to positive affine transformations, and Φ is unique up to monotone transformations. Our characterization is tight, in the sense that each of Axioms 1 5 is independent of the others. See appendix for the examples that show the independence. The following corollary shows the consequence of strengthening the admissibility axiom to the stronger one. Corollary 1. The choice function φ satisfies Axioms 1, 2, 3 5 if and only we have the representation result as in Theorem 1 with the additional property that Φ is strongly monotone. Here we show the consequence of imposing IIA on the entire domain. Consider the axiom Axiom 1 (IIA on the full domain). For any B,C B with B C, φ(c) B = φ(b) = φ(c) B. The corollary below shows that maximization of subjective expected utility is the only consequence when we additionally impose full IIA. This is not surprising, since the set of axioms falls into the standard framework of Anscombe Aumann [1] s subjective expected utility. However, our proof in appendix shows this with a different order: given Theorem 1, the full IIA implies that the function Φ is ordinally linear. One limitation is that the obtained probability measure p has to have full-support here. This is due to the Admissibility axiom which excludes null events, while the weak admissibility axiom allows that. Corollary 2. The choice function φ satisfies Axioms 1, 2 5 if and only if there exists a mixturelinear, continuous and non-constant function u : Δ(X) R, and p intδ(ω) such that φ(b) = arg max u(f (ω))p(ω) f B ω Ω for every B B. Moreover, u is unique up to positive affine transformations, and p is unique.

10 T. Hayashi / Journal of Economic Theory 139 (2008) Similar result holds when Singleton Independence is strengthened into an independence axiom for mixtures of sets and sets, which is a well-known property of choice function (see [21]). 5. Regret aversion Axiom 6 (Regret aversion). For any B B and any f, g F with B EP {f, g}, and any λ [0, 1], { } f φ(b {f }) λf + (1 λ)g φ(b {λf + (1 λ)g}). g φ(b {g}) This axiom states that regret aversion is revealed by hedging acts, which is parallel to Gilboa Schmeidler [10]. However, regret aversion coincides with uncertainty (or ambiguity) aversion only when ex post optimal outcomes are equal across all states (up to payoffs). If it is not the case, a regret averse choice may be aggressive rather than conservative. Corollary 3. The choice function φ satisfies Axioms 1 6 if and only if we have the representation as in Theorem 1 with an additional property that the function Φ is quasi-convex. 6. Regret premium and off-equilibrium comparison of acts 6.1. Regret premium Since a choice function deals with equilibrium choices only, it does not seem to allow observing off-equilibrium comparison of acts. 2 However, the current revealed-choice approach partially allows an off-equilibrium comparison of acts through equilibrium choices. Here we define the notion of regret premium, to measure how much a choice is regrettable given an opportunity set. Given B B, denote an ideal act by i(b), which is an act that takes ex post optimal outcome at every states. In other words, it is a dominant strategy that yields no regret at any state, though of course it is not feasible in general. Formally, i(b) is an act such that for every ω Ω, and i(b)(ω) φ({i(b)(ω), g(ω)}) for every g B φ({i(b)(ω), g(ω)}) ={i(b)(ω), g(ω)} for some g B. Given B B and f F with B EP {f }, the regret premium for f with regard to B, denoted by μ(f ; B), is defined by μ(f ; B) = inf μ such that (1 μ)f + μi(b) φ({(1 μ)f + μi(b)} B). In words, μ(f ; B) is the minimum necessary probability to be assigned to the ex post optimal outcome in ex post randomization, so that (1 μ)f + μi(b) be chosen over B. It is easy to see that no premium is needed when f is already an equilibrium choice given B, i.e., μ(f, B) = 0if and only if f φ(b {f }). 2 We use the term equilibrium/off-equilibrium rather than optimal/non-optimal to describe choices and alternatives, since we have departed from the standard binary relation approach.

11 252 T. Hayashi / Journal of Economic Theory 139 (2008) Suppose f and g are off-equilibrium choices given B, and that f requires less premium so as to be chosen over B than g does. This suggests that f is observed to be potentially better in the presence of B than g. Notice that we are still concerned with observable choices only. Determination of premium depends only on whether an act is chosen over a given opportunity set. In particular, positive regret premium is observed only when an act is not chosen over the given opportunity set. The following claim is straightforward. Lemma 2. The regret aversion axiom is equivalent to the following condition: for any B B and any f, g F with B EP {f, g}, and any λ [0, 1], μ(f, B) = μ(g, B) = 0 μ((1 λ)f + λg, B) = Minimax regret with multiple-priors This and next subsections are concerned with off-equilibrium comparison of acts. In this subsection, we consider that acts with constant regrets play a special role in the off-equilibrium comparison. As a consequence, we obtain minimax regret with multiple-priors, which is a counterpart of maximin expected utility by Gilboa Schmeidler [10]. Remember that in the model of maximin expected utility, constant acts have a special role to formalize the axiom of certainty independence. While the notion of constant act is obvious, the notion of constant regret is not. Here we identify the class of opportunity sets where constancy of regrets is associated with constancy of acts. Say that an opportunity set E is uniform if the corresponding ideal act i(e) is a constant act. In words, ex post optimal choices are the same at all states when the opportunity set is uniform. When one chooses a constant act in the presence of a uniform opportunity set, the difference between the ex post optimal outcome and the actual outcome is the same at all states. In this sense, constant acts are understood to generate constant regrets in the presence of a uniform opportunity set. Axiom 7 (Constant-regret independence of regret premium). For any uniform opportunity set E B, f, g F, l Δ(X) with E EP {f, g, l}, and λ [0, 1], if then μ(f, E), μ(g, E), μ((1 λ)f + λl,e), μ((1 λ)g + λl,e) > 0, μ(f, E) μ(g, E) μ((1 λ)f + λl,e) μ((1 λ)g + λl,e) A constant act is viewed as yielding constant regret across states, when it is considered under an additional opportunity set which is uniform. Suppose that f requires less premium so as to be chosen over B than g does. In our term, f is potentially better in the presence of B than g. Consider taking ex post randomization by means of l, which is viewed to generate constant regret across states. Since the only difference between (1 λ)f + λl and (1 λ)g + λl is in f and g, it is appealing to say that (1 λ)f + λl is again potentially better in the presence of B than (1 λ)g + λl. This is parallel to the certainty-independence axiom that is considered in Gilboa Schmeidler [10]. The difference is that it is imposed only on off-equilibrium comparison of acts in the presence

12 T. Hayashi / Journal of Economic Theory 139 (2008) of uniform opportunity set, while certainty-independence is defined for every binary comparison in the absence of opportunity dependence. Theorem 2. The choice function φ satisfies Axioms 1 7 if and only if there exists a mixture-linear, continuous and non-constant function u : Δ(X) R and a closed convex set P Δ(Ω) with P intδ(ω) = such that φ(b) = arg min max [ ] max u(g(ω)) u(f (ω)) p(ω) f B p P g B ω Ω for every B B. Moreover, u is unique up to positive affine transformations and P is unique. Corollary 4. The choice function φ satisfies Axioms 1, 2, 3 7 if and only we have the representation result as in Theorem 1 with the additional property that P intδ(ω) Smooth model of regret aversion In this subsection, we consider that the off-equilibrium comparison of acts satisfies the notion of eventwise separability, which is a counterpart of Savage s eventwise separability in his axiomatization of subjective expected utility (see [25]). Given f, h F and S Ω, f S h denotes an act f(ω) if ω S, (f S h)(ω) = h(ω) if ω / S. Suppose that f S h requires less premium so as to be chosen over B than g S h does. In our term, f S h is potentially better in the presence of B than g S h. Now consider replacing h by h. Since the only difference between f S h and g S h is in f and g, from the eventwise separability viewpoint it is natural to say that the comparison of regret premium depends only on f and g. This idea is formalized below. Axiom 8 (Eventwise separability of regret premium). For all B B, f, g, h, h F with B EP {f, g, h, h }, and S Ω, if then μ(f S h; B), μ(g S h; B), μ(f S h ; B), μ(g S h ; B) > 0, μ(f S h; B) μ(g S h; B) μ(f S h ; B) μ(g S h ; B) Theorem 3. Assume Ω 3. The choice function φ satisfies Axioms 1, 2, 3 5, 8 if and only if there exists a mixture-linear, continuous and non-constant function u : Δ(X) R, a probability measure p intδ(ω) and a number α > 0 such that φ(b) = arg min f B ω Ω ( ) α max u(g(ω)) u(f (ω)) p(ω) g B

13 254 T. Hayashi / Journal of Economic Theory 139 (2008) for every B B. Moreover, u is unique up to positive affine transformations, and α andpare unique. Remark 1. We are not imposing the regret aversion axiom here, which characterizes the case α 1. Otherwise, the decision maker can be regret loving. Also, α = 1 corresponds to the special case of subjective expected utility maximization, and it is easy to see that the choice rule gets closer to Savage s minimax regret choice as α tends to infinity. Remark 2. Here anticipated regrets are not only distorted by a non-linear function that exhibits regret aversion, but also we obtain a constant degree of regret aversion. This is due to the homotheticity of regret evaluation, which is coming from the singleton independence axiom. 7. Comparative regret aversion In this section we define comparative regret aversion. We argue that a decision maker is more regret averse if she is more pessimistic in anticipating regret. However, being more pessimistic here does not necessarily mean that she is more conservative in choice. Thus we restrict attention to a subdomain where pessimism is straightforwardly related to conservative choice. In particular, we consider the subdomain of uniform opportunity sets. When an opportunity set is uniform, it is straightforwardly understood that actions with constant regret are associated with constant acts. Thus we can take an approach parallel to that taken by Ghirardato and Marinacci [9] for comparative ambiguity aversion. Now we say that φ 1 is more regret averse than φ 2 if l φ 2 ({l} E) l φ 1 ({l} E) for every uniform opportunity set E B and l Δ(X). This says that the two choice rules coincide over sets of outcomes, but any constant act chosen by one over a uniform opportunity set is chosen by the more regret averse one. Notice that this comparison is partial in the sense that the degree of regret aversion is comparable only between the rules which exhibit the same choices over risky prospects. Given Φ : R Ω + R + and z R Ω +, define the lower contour set by L(Φ,z)={y R Ω + : Φ(y) Φ(z)}. By homogeneity of Φ, it is enough to write down the relevant condition just for a fixed vector 1 = (1,...,1). Given two functions Φ 1 and Φ 2, the condition L(Φ 1, 1) L(Φ 2, 1) is stating that Φ 1 is more quasi-convex than Φ 2 on the certainty line. In the class of general representations, higher regret aversion is revealed by the function Φ being more quasi-convex on the certainty line. Theorem 4. Consider the class of representations as in Theorem 1. Let (u 1, Φ 1 ) and (u 2, Φ 2 ) represent φ 1 and φ 2, respectively. Then, φ 1 is more regret averse than φ 2 if and only if there exists constants a,b with a>0 such that u 1 = au 2 + b, and L(Φ 1, 1) L(Φ 2, 1). In the class of minimax regret with multiple-priors, higher regret aversion is described by a larger set of priors.

14 T. Hayashi / Journal of Economic Theory 139 (2008) Corollary 5. Consider the class of representations as in Theorem 2. Let (u 1,P 1 ) and (u 2,P 2 ) represent φ 1 and φ 2, respectively. Then, φ 1 is more regret averse than φ 2 if and only if there exists constants a,b with a>0 such that u 1 = au 2 + b, and P 1 P 2. In the class of smooth models of regret aversion, higher regret aversion is described by a larger coefficient of regret aversion. Corollary 6. Consider the class of representations as in Theorem 3. Let (u 1,p 1, α 1 ) and (u 2,p 2, α 2 ) represent φ 1 and φ 2, respectively. Then, φ 1 is more regret averse than φ 2 if and only if there exists constants a,b with a>0 such that u 1 = au 2 + b, and p 1 = p 2, α 1 α Concluding remarks We provide an axiomatic model of choice under uncertainty where the decision maker may be driven by anticipated regrets. One thing to note is that here we limit attention to anticipated regrets, that are directly relevant to choice. What about actual emotion of regret, which the decision maker may feel after making action and seeing resolution of uncertainty? Does it affect her choice in the meantime, in dynamic choice problems? In a companion paper [12], we provide an axiomatic model of dynamic choice with regret, which is an extension of the current model. Still there, the notion of regret is limited to be an anticipated one. We can think of two possibilities: (i) An actual emotion of regret is purely an internal mental state, and it is not revealed by choice, as for example an emotion of well-being is not revealed by choice. It may be detected by a survey method, but will not be detected by the standard revealed choice method; (ii) It does affect choice, and it may be observed by a more sophisticated method of revealed choice analysis. However, it is beyond our current scope. Acknowledgment I gratefully appreciate the associate editor and two anonymous referees for helpful comments and suggestions that improved the paper significantly. I thank the participants at the Fall 2005 Midwest Economic Theory and International Economics Conference at Lawrence, the 2006 North American Summer Meeting of the Econometric Society at Minneapolis, and the seminar participants at Rice, Caltech and Cornell, for helpful comments. Of course, all possible errors are mine. Appendix A. Proofs A.1. Proof of Lemma 1 and related properties Define an ordering over Δ(X),,by l m l φ({l,m}). Lemma 3. Under irrelevance of ex post dominated acts, the relation is complete and transitive. Proof. Completeness follows from non-emptyness of φ. To show transitivity, suppose l m, m n and l / n.

15 256 T. Hayashi / Journal of Economic Theory 139 (2008) Since lotteries are viewed as constant acts and since l φ({l,m}), m φ({m, n}) and l / φ({l,n}) ={n}, wehave{l,m} EP {l, m, n}, {m, n} EP {l, m, n}, and {l,n} EP {l, m, n}. Now suppose that l/ φ({l, m, n}). Then, each of two possible cases leads to a contradiction: Case 1: Suppose m φ({l, m, n}). Then, since φ({l, m, n}) {l,m} =, Irrelevance of ex post dominated acts implies φ({l, m, n}) {l,m} =φ({l,m}), but this contradicts to l φ({l,m}). Case 2: The remaining case is φ({l, m, n}) ={n}. Then, since φ({l, m, n}) {m, n} =, Irrelevance of ex post dominated acts implies φ({l, m, n}) {m, n} =φ({m, n}), but this contradicts to m φ({m, n}). Thus, we have l φ({l, m, n}). However, since φ({l, m, n}) {l,n} =, Irrelevance of ex post dominated acts implies φ({l, m, n}) {l, n} = φ({l, n}), but this contradicts to l/ φ({l, n}). Lemma 4. Under upper hemi-continuity, the relation is continuous. Proof. Let l ν m ν for every ν N, i.e., l ν φ({l ν,m ν }) for every N. Assume l ν l and m ν m. Then the sequence of sets {l ν,m ν } converges to {l,m} in the Hausdorff metric. By upper hemi-continuity we get l φ({l,m}), which implies l m. Lemma 5. Under singleton independence, the relation satisfies mixture-independence in the sense of von-neumann/morgenstern. Proof. Let l m, i.e, l φ({l,m}). By Singleton Independence, φ((1 λ){l,m}+λ{n}) = (1 λ)φ({l,m}) + λ{n}. Therefore l φ({l,m}) is true if and only if (1 λ)l + λn φ((1 λ){l,m} +λ{n}) = φ({(1 λ)l + λn, (1 λ)m + λn}), which is true if and only if (1 λ)l + λn (1 λ)m + λn. Lemma 6. Under irrelevance of ex post dominated acts, singleton independence, upper himicontinuity and non-degeneracy, there exists a mixture-linear, continuous and non-constant function u : Δ(X) R which represents. Moreover, u is unique up to positive linear transformations. Proof. It follows from Grandmont [11]. Now we prove Lemma 1 in text. Let C be a finite set. Transitivity of and finiteness guarantee that there exists l C such that l m for all m C. Takeanyl φ(c) and consider {l,l}, then {l,l} EP C. Since l φ(c) {l,l}, irrelevance of ex post dominated acts implies φ(c) {l,l} = φ({l,l}). Since l φ({l,l}),wehavel φ(c). Thus, φ is rationalized by for finite sets, which immediately implies IIA for outcome sets. Now consider an arbitrary compact set B. Since is continuous, φ (B) ={l B : l r, r B} is non-empty. To show φ (B) φ(b), pick l φ (B) and consider an increasing sequence of finite sets {B n } such that l B n for all n. Since l φ(b n ) for all n, upper hemi-continuity implies l φ(b). To show φ (B) φ(b), pick l φ(b). Also pick any l φ (B). Since we already know φ (B) φ(b), wehavel φ(b). Since {l,l} EP B and φ(b) {l,l} l, irrelevance of ex

16 T. Hayashi / Journal of Economic Theory 139 (2008) post dominated acts implies φ(b) {l,l} =φ({l,l}). Hence l φ({l,l}), which implies l l m for all m B. Thus φ is rationalizable by for all compact sets, and it also implies the second claim of Lemma 1. A.2. Proof of Theorem 1 Necessity of the axioms is straightforward throughout. So we focus on sufficiency. First goal to is to show the following property. Monotonicity. For every B B and f, g B with f g, g φ(b) f φ(b) We show one lemma for this purpose. Lemma 7. For every B B and f F such that B EP f and f > Ω\N(g,B) g for some g φ(b), φ(b {f }) ={f }. Proof. Step 1: Suppose f / φ(b {f }). Then φ(b {f }) B. Hence by the irrelevance of ex post dominated acts, φ(b {f }) B = φ(b). Therefore g φ(b {f }), but this violates admissibility. Step 2: Suppose φ(b {f }) B =, then by the irrelevance of ex post dominated acts φ(b {f }) B = φ(b). This implies g φ(b {f }), which violates admissibility. Now we prove monotonicity. Proof. Suppose for f, g B with f g that g φ(b) and f / φ(b). Case 1: When there is f F such that B EP f and f > Ω\N(g,B) g, we can take a sequence of such acts {f ν } that converges to f. By the above lemma, φ(b {f ν }) ={f ν } for every ν. Since f / φ(b) in the limit, this violates upper hemi-continuity. Case 2: Otherwise, g h for every h B, and the same property is met by f too. Hence one can find a sequence {f ν } converging to f such that f ν > Ω h for every h B, for every ν. By admissibility, φ(b {f ν }) ={f ν } for every ν. Since f / φ(b) in the limit, this violates upper hemi-continuity. The following lemma is immediate from monotonicity. Lemma 8. For any B B and f, g B with u f = u g, f φ(b) g φ(b). Now we show the choice depends only on the sets of induced payoffs. Lemma 9. For any B,C B with u B = u C, f B and g C with u f = u g, f φ(b) g φ(c). Proof. Let f φ(b). By assumption, B EP C.

17 258 T. Hayashi / Journal of Economic Theory 139 (2008) Step 1: Suppose φ(b C) B =, then φ(b C) C.Nowtakeanyg φ(b C), then there is f B such that u f = u g. Here f / φ(b C) since φ(b C) C, but this contradicts to Monotonicity. Step 2: Hence we have φ(b C) B =. Then, irrelevance of ex post dominated acts implies φ(b) = φ(b C) B, hence f φ(b C). By monotonicity, g φ(b C). By assumption, C EP B. Thus irrelevance of ex post dominated acts implies g φ(c). Likewise, g φ(c) implies f φ(b). Given v u F, define u 1 (v) ={f F : u f = v}. Also, define V by V ={V R Ω : V = u B for some B B}. Then the above lemma ensures that there is a mapping ψ : V 2 RΩ such that φ(b) = u 1 (ψ(u B)) B, where ψ satisfies ψ(v ) V for any V V. Lemma 10. For any V V, {v} V and λ [0, 1], ψ((1 λ)v + λ{v}) = (1 λ)ψ(v ) + λ{v}. Proof. When λ = 0 or 1, the claim is trivial. So assume λ (0, 1). Let B B be an opportunity set that gives u B = V and f F be an act that gives u f = v. By Singleton Independence, ψ((1 λ)v + λ{v}) = u φ((1 λ)b + λ{f }) = u ((1 λ)φ(b) + λ{f }) = (1 λ)u φ(b) + λ{u f } = (1 λ)ψ(v ) + λ{v}. Without loss of generality, let us assume that u(l) = 0 for some l Δ(X). Then the above lemma implies ψ(θv) = θψ(v ) for any θ [0, 1]. Denote the space of compact subsets of R Ω by K Ω. Then we can extend ψ to K Ω by ( ) 1 ψ(v ) = θψ θ V for every V K Ω, where θ > 1 is taken so that θ 1 V V. For the extended ψ : K Ω 2 RΩ, we obtain: Lemma 11. For any V K Ω, v R Ω and λ, θ 0, ψ(λv + θ{v}) = λψ(v ) + θ{v}.

18 T. Hayashi / Journal of Economic Theory 139 (2008) To make the mathematical treatment easier, we flip the direction so as to describe regrets by non-negative numbers. Thus, we define ψ such that ψ (Z) = ψ( Z) for every Z K Ω. Notice that all the properties of and ψ are maintained in ψ after flipping the direction. Let Δ denote the unit simplex in R Ω. Let us define a set of vectors Z Δ by Z Δ ={z R Ω + : z ψ ({z} Δ)}. Since Δ R Ω +, ψ (Δ) is non-empty and ψ (Δ) Z Δ by construction, Z Δ is non-empty. Upper Hemi-continuity implies Z Δ is closed. We show several properties of Z Δ below. First, Z Δ has a monotonicity property. Lemma 12. For every z Z Δ, (z R Ω + ) RΩ + Z Δ Proof. Pick any z (z R Ω + ) RΩ +. Suppose z / ψ (Δ {z }). Then it must be that z / ψ (Δ {z, z }), by irrelevance of ex post dominated acts. Case 1: Suppose z ψ (Δ {z, z }). Since z z, monotonicity implies z ψ (Δ {z, z }), which is a contradiction. Case 2: Suppose z/ ψ (Δ {z, z }). Then ψ (Δ {z, z }) Δ, implying that ψ (Δ {z, z }) (Δ {z}) =. Hence by irrelevance of ex post dominated acts, ψ (Δ {z}) = ψ (Δ {z, z }) (Δ {z}), which leads to z/ ψ (Δ {z}), a contradiction to z Z Δ. Second, Z Δ allows strictly positive regret vectors. Lemma 13. There exists λ > 0 such that λ 1 Z Δ, where 1 = (1,...,1) R Ω. Also, combined with the previous lemma, [0, λ 1] Z Δ Proof. Suppose not. Then, there is a sequence of positive numbers λ ν converging to zero such that λ ν 1 / ψ (Δ {λ ν 1}) for every ν. By irrelevance of ex post dominated acts, ψ (Δ {λ ν 1}) = ψ (Δ) for every ν. Takeanyz ψ (Δ). Then z ψ (Δ {λ ν 1}) for every ν. In the limit, ψ(δ {0}) ={0} by Admissibility, which implies z/ ψ(δ {0}). However, this contradicts to upper hemi-continuity. Lastly, we show that regret vectors allowed by Z Δ should be bounded by Δ from above. Let compδ ={z R Ω : z z for some z Δ} denote the comprehensive hull of Δ. Lemma 14. Z Δ comp Δ R Ω +. Proof. Suppose z R Ω + \ comp Δ. Then, there is some z Δ and S = such that z ω >z ω for all ω S and z ω = z ω = 0 for all Ω \ S. By admissibility, z/ ψ(δ {z}), hence z/ Z Δ.

19 260 T. Hayashi / Journal of Economic Theory 139 (2008) Let Z Δ Z Δ denote the upperboundary of Z Δ, Z Δ ={z Z Δ : z Z Δ,z > Ω z}. Lemma 15. ψ (Δ) Z Δ. Proof. Suppose not. Then there exist z ψ (Δ) and z Z Δ such that z > Ω z. Since z ψ (Δ {z }) and z< Ω z, this violates admissibility. Lemma 16. For every z Z Δ, ψ ({z} Δ) = ψ (Δ) {z}. Proof. part: It suffices to show that ψ ({z} Δ) \{z} ψ (Δ). Let z ψ ({z} Δ) \{z}, then it must be that z Δ, hence z ψ ({z} Δ) Δ. By irrelevance of ex post dominated acts, ψ ({z} Δ) Δ = ψ (Δ). Hence z ψ (Δ). part: It suffices to show that ψ ({z} Δ) ψ (Δ). Takeanyz ψ (Δ). Since z Z Δ, there is a sequence z ν in R Ω + \ Z Δ that converges to z. By construction, z ν / ψ ({z ν } Δ), which implies ψ ({z ν } Δ) Δ. Hence by irrelevance of ex post dominated acts, ψ ({z ν } Δ) Δ = ψ ({z ν } Δ) = ψ (Δ) for every ν. Therefore z ψ ({z ν } Δ) for every ν. By upper hemicontinuity, z ψ ({z} Δ). Lemma 17. ψ (Δ Z Δ ) = Z Δ. Proof. part: It suffices to show ψ (Δ Z Δ ) \ Z Δ ψ (Δ), since ψ (Δ) Z Δ. Let z ψ (Δ Z Δ ) \ Z Δ, then z Δ. Hence z ψ (Δ Z Δ ) Δ. By Irrelevance of ex post dominated acts, ψ (Δ Z Δ ) Δ = ψ (Δ), therefore z ψ (Δ). part: This consists of three steps. Step 1: First we show ψ (Δ) ψ (Δ {z 1,z 2 }) for every z 1,z 2 Z Δ. Take any z ψ (Δ). By the previous lemma, z ψ (Δ {z 1 }) and z ψ (Δ {z 2 }). Suppose z/ ψ (Δ {z 1,z 2 }). Case 1: If ψ (Δ {z 1,z 2 }) Δ =, by irrelevance of ex post dominated acts, ψ (Δ {z 1,z 2 }) Δ = ψ(δ). Therefore z/ ψ (Δ), which is a contradiction. Case 2: If ψ (Δ {z 1,z 2 }) Δ =,wehaveψ (Δ {z 1,z 2 }) {z 1,z 2 }. By non-emptyness, let us assume z 1 ψ (Δ {z 1,z 2 }) without loss of generality. Then by irrelevance of ex post dominated acts, we have ψ (Δ {z 1 }) ={z 1 }, which contradicts to z ψ (Δ {z 1 }). Step 2: Next we show {z 1,z 2 } ψ (Δ {z 1,z 2 }). Without loss, it suffices to show z 1 ψ (Δ {z 1,z 2 }). Suppose z 1 / ψ (Δ {z 1,z 2 }). Since we already know = ψ (Δ) ψ (Δ {z 1,z 2 }), ψ (Δ {z 1,z 2 }) (Δ {z 1 }) =. By irrelevance of ex post dominated acts, ψ (Δ {z 1,z 2 }) (Δ {z 1 }) = ψ (Δ {z 1 }). Since we are assuming z 1 / ψ (Δ {z 1,z 2 }),wehavez 1 / ψ (Δ {z 1 }), which is a contradiction. Step 3: Thus we obtain ψ (Δ {z 1,z 2 }) = ψ (Δ) {z 1,z 2 }. Inductively, we obtain ψ (Δ {z 1,...,z ν }) = ψ (Δ) {z 1,...,z ν } for any finite subset of Z Δ, {z 1,...,z ν }. Since Z Δ is compact, there is a sequence of finite sets that is increasing and converges to Z Δ with respect to the Hausdorff metric. By upper hemi-continuity, ψ (Δ Z Δ ) ψ (Δ) Z Δ. Since ψ (Δ) Z Δ, we obtain the desired result.

20 T. Hayashi / Journal of Economic Theory 139 (2008) Given a compact set Z K Ω, denote its upper-envelope set by σ(z) ={z R Ω : z z for some z Z}. Lemma 18. ψ (σ( Z Δ ) [0, θ1]) = Z Δ for every θ 1. Proof. part: Pick any z ψ (σ( Z Δ ) [0, θ1]) and suppose z/ Z Δ. Since σ( Z Δ ) Z Δ = Z Δ,wehavez/ Z Δ. On the other hand, since z Δ {z}, by irrelevance of ex post dominated acts, z ψ (Δ {z}). However this contradicts to z/ Z Δ. part: Pick any z Z Δ and suppose z/ ψ (σ( Z Δ ) [0, θ1]). From the previous step, we already know ψ (σ( Z Δ ) [0, θ1]) Z Δ. Hence ψ (σ( Z Δ ) [0, θ1]) (Δ Z Δ ) =. Therefore ψ (Δ Z Δ ) = ψ (σ( Z Δ ) [0, θ1]) (Δ Z Δ ). By assumption we get z / ψ (Δ Z Δ ), but this contradicts to the conclusion of the previous lemma that ψ (Δ Z Δ ) = Z Δ. Lemma 19. For every z R Ω + \{0}, there exists a unique λ > 0 such that λz Z Δ. Proof. Existence is guaranteed by the fact that [0, λ 1] Z Δ for some λ > 0 and Z Δ is compact. To show uniqueness, let N(z) = {ω Ω : z ω = 0}. Suppose there exist λ > λ > 0 with λz, λ z Z Δ. Then, we have λz > Ω\N(λz) λ z, which contradicts to admissibility since λz Z Δ = ψ (Δ Z Δ ). This immediately implies that Lemma 20. For every z R Ω +, there exists a unique λ 0 such that z λ Z Δ. Now define a function Φ : R Ω + R + by Φ(z) = λ such that z λ Z Δ For any given Z K Ω, define I(Z) R Ω by I ω (Z) = min z Z z ω for each ω Ω. ForΦ given above, define ψ Φ : KΩ 2 RΩ by ψ Φ (Z) = arg min Φ(z I(Z)) z Z Lemma 21. ψ (σ( Z Δ ) [0, θ1]) = ψ Φ (σ( Z Δ) [0, θ1]) for every θ 1. Proof. Immediate from ψ Φ (σ( Z Δ) [0, θ1]) = Z Δ. Lemma 22. ψ (σ(α Z Δ ) [0, θ1]) = ψ Φ (σ(α Z Δ) [0, θ1]) = α Z Δ for every θ α 0. Proof. It is immediate from non-negative-linearity when θ α > 0. Also the case where θ = α = 0 is obvious since σ({0}) [0, 0] =0.