Non-Expected Utility and the Robustness of the Classical Insurance Paradigm: Discussion

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1 The Geneva Papers on Risk and Insurance Theory, 20:51-56 (1995) The Geneva Association Non-Expected Utility and the Robustness of the Classical Insurance Paradigm: Discussion EDI KARNI Department of Economics, The Johns Hopkins University, Baltimore, MD Abstract This paper discusses some aspects of the robustness of the classical insurance paradigm with respect to departures from the independence axiom of expected utility theory. The discussion focuses on the significance of the distinction between risk aversion and outcome convexity and the role of smoothness of the preferences in nonexpected-utility analysis of insurance. Key words: non-expected utility, insurance economics 1. Introduction The development of non-expected-utility theories has taken two distinct directions: the formulation of axiomatic theories that depart from the axioms of expected-utility theory and the development of local expected-utility analysis. In the preceding paper, Machina applies local expected-utility analysis to insurance. In addition to presenting an excellent review of the main theorems on insurance behavior under the expected-utility hypothesis, he offers an insightful analysis of the robustness of these theorems to the relaxation of various aspects of expected utility theory. To understand Machina's paper it is important to have some notion of the axiomatic approach as well as the meaning of local expected utility analysis. I begin, therefore, with a brief exposition of the axiomatic approach and local expected-utility analysis. 1 L1. Axiomatic theories Although some non-expected-utility theories depart from the axioms of completeness and transitivity, the theoretical developments that attracted most of the attention--and also underlie Machina's analysis of insurance--depart from the separability inherent in expectedutility theory. The formal statement of the separability assumption is the independence axiom in the case of decision making under risk and the sure-thing principle in the case of decision making under uncertainty. Axiomatic models of decision making under risk that depart from the independence axiom may be broadly classified into models with the betweenness property and models of expected utility with rank-dependent probabilities (EURDP). 2 To describe these models we introduce the following notation: Let X be an arbitrary set of consequences and let D(X) denote the set of all probability measures on X. Elements

2 52 EDI KARNI of D(X) are risky prospects. Suppose that decision-maker's preference relation on D(X) are representable by a functional V: D(X) ~ ~, Then a preference relation is said to display betweenness if, for any two risky prospects--say, p and q--such that p is at least as preferred as q the probability mixture up + (1 - o0q is at least as preferred as q and at most as preferred as p for all ~ E [0, 1]. Formally, V(p) >_ V(q) implies V(p) >_ V(ctp + (1 - coq) >- V(q) for all ot E [0, 1]. Examples of non-expected-utility models with the betweenness property are weighted-utility theory and implicit weighted-utility theory. In EURDP theories decision makers' preference relations on the set of risky prospects are representable by the mathematical expectations of a utility function on the set of outcomes with respect to a transformation of the probabilities. The transformed probability of an outcome depends on the relative ranking of this outcome in the set of feasible outcomes. To grasp the meaning of this, let the set of consequences Xbe a set of real numbers and suppose that the set of risky prospects is the set of simple probability measures on X. 3 Consider a simple probability measure p and arrange the elements in the support of p in an ascending order--that is, xl - x2 -< xn. Let Pi denote the probability of xi, then, according to the EURDP model V(p) = Eni=l u(xi) [g(~j=l Pj) - g(~-~ Pj)], where u : X ~ ~ is a monotonic increasing function and the probability transformation function g : [0, 1] ~ [0, 1] is continuous, monotonic increasing, and onto. The probability ~ / ~ i-1 Pi is t r a n s f o r m e d to g( )-]ij=l Pj) 8~, j=l Pj)" Note that expected-utility theory has the betweermess property and is also the special case of the EURDP theory in which the probability transformation function is the identity function Local expected-utility analysis Expected-utility analysis is a set of results describing behavioral implications of the interplay between the decision maker's attitudes toward risk and the properties of the sets of risks from which he must choose. Examples of expected-utility analysis include well-known theorems in insurance theory, which are reviewed in Machina's paper. In a seminal work, Machina [1982] showed that many of these implications are robust to the abandonment of the independence axiom. To see this, let risky prospects be represented by cumulative distribution functions on a closed and bounded interval I in ~. Denote the set of all risky prospects by D(I). Suppose that the decision maker's preferences on D are representable by a functional V : D(I) ~ ~. If the preference relation satisfies the independence axiom, then the representation functional is linear in the probabilities. Machina's key idea is to replace the linearity of the expected-utility functional by a smoothness assumption implying that infinitesimal variations in the value of V may be approximated by local (in the space of distributions, D) expected utility4 (that is, for F and G in D(I), V(F) V(G) = ftu(x; F)d(F(x) - G(x)) + o(llf - GII), where o(.) is a function that is zero at zero and of a higher order than its argument and II. II denotes the L t norm.) {U('; F)IF E D(1)} is the set of local utility functions corresponding to I(. Machina showed that, by imposing on the local utility functions the restrictions that were imposed on the von Neumann-Morgenstern utility function in expected utility analysis, many of the behavioral implications derived from the latter analysis are robust to the relaxation of the form of the representation functional. In the preceding paper, Machina applies local expected-utility analysis to insurance.

3 NON-EXPECTED UTILITY: DISCUSSION The significance of smoothness The preceding paper emphasizes the distinction between risk aversion and outcome convexity, and the robustness of well-known results in insurance theory to the relaxation of the independence axiom. To understand these issues it is important that we grasp (1) the significance of the fact that Machina considers general nonexpected utility functionals that are only restricted by smoothness and (2) the role of the smoothness assumption in the robustness of the implications of expected utility analysis for insurance to the relaxation of the independence axiom. In my remarks I address these two issues. 2.L Risk aversion and outcome convexity The reallocation of risk bearing is attained by restructuring of the random variables that represent risks faced by individuals. Therefore, the analysis of institutions designed to reallocate risk bearing, of which insurance is a prime example, must be analyzed using individual preferences over appropriately defined spaces of random variables. A fortunate aspect of expected-utility theory is that risk aversion is equivalent to quasi-concavity of the preference relation in the space of real random variables representing individual levels of wealth. (This is the property that Machina refers to as outcome convexity.) As emphasized by Machina, outcome convexity yields unique solutions to optimal choice problems and implies that the first-order conditions are both necessary and sufficient. In general, however, outcome convexity implies risk aversion but not vice versa (see Dekel [1989, props. 1 and 2]). Consequently, the optimal design of various insurance policies may not be unique and the analysis of insurance problems is complicated by the need to worry about the secondorder conditions. While the possible divergence of outcome convexity from risk aversion should be kept in mind, it is important to note that in all the axiomatic theories that were discussed in the preceding section, risk aversion is equivalent to outcome convexity. In the case of decision models displaying the betweelmess property, this conclusion is implied by two observations. First, if the representation functional exhibits risk aversion and is quasiconcave (in the space of distributions), then it exhibits outcome convexity (see Dekel [1989, prop. 3]). Second, betweenness implies that the representation functional is quasiconcave. In the case of EURDP theories, risk aversion implies that the preference functional is convex in the space of distributions, yet the equivalence of outcome convexity and risk aversion still holds. We shall illustrate this with an example involving two states. To begin with, note that on EURDP functional displays risk aversion if and only if both the utility function and the probability transformations functions are concave (see Chew, Karni, and Safra [1987]). Let there be two states--say, 1 and 2--with probabilities p and 1 - p, respectively. Denote by Wl and w2 the decision maker's wealth in the two states, and consider the risky prospect, [w 1, p; w2, (1 - p)], that pays wl in state 1 and w2 in state 2. If Wl > WE, then EURDP evaluation of the risky prospect is given by V([Wl, p; w2, (1 - p)]) = u(w2)g(1 - p) + U(Wl)(1,g(1 - p)).

4 54 EDI KARNI The marginal rate of substitution is dw2 = _ u'(wx)(1 - g(1 - p)) v=co,,~t, u ' (w2)g(1 - p) If w2 > Wl then the evaluation of the risky prospect is given by V([Wl, p; WE, (1 -- p)]) = u(wl)g(p ) + u(w2)(1 -- g(p)) and the marginal rate of substitution is dw 2 u'(w1)g(p) aw I I V=const. = - u'(w2)(1 - g(p)) Clearly, if u is concave then, holding V constant, the marginal rate of substitution is increasing in wl whenever wl > w2 or w2 > Wl. Moreover, since g is strictly concave, lira - u'(wl)(1 - g(1 - p)) = (1 - g( - p)) > g(p~) = lira - u'(wl)g(p) Wl~,W2 U' (w2)g(1 -- p) g(1 -- p) (1 -- g(p)) w2j, w I U'(W2)(1 -- g(p)) " Hence, holding V constant, the marginal rate of substitution is monotonic increasing in Wl everywhere. Thus, risk-averse EURDP functionals display outcome convexity Smoothness and the order of risk aversion Throughout his paper, Machina assumes smoothness of the preference functional. This is crucial for some of his results. For instance, the conclusions that under either coinsurance or deductible insurance, "if an individual is risk averse, then full insurance will be demanded if and only if it is actuariallyfair "'7 depend critically on the preference functional being smooth in an appropriate sense. What this sense is, however, is not stated explicitly. In fact, the smoothness assumption required entails that in addition to being L 1 Frechetdifferentiable the represenation functional displays second-order risk aversion. 8 This assumption has bite. To see this, note that, with the exception of expected-utility theory, EURDP models are not L 1 Frechet differentiable. Moreover, if the probability transformation function, g, is strictly concave, then the preference functional displays risk aversion of the first order but not of the second order--namely, for all nondegenerate random variables such that E[~] = 0, the risk premium lrv(t) corresponding to the random variable t~ has the property that 01rV(t)/Ot It=0+ ~ 0. In a two-dimensional space of random variables, the fact that risk-averse EURDP functionals are risk averse of the first but not of the second order implies the existence of a kink in the indifference curves on the certainty line. This kink has important implications for insurance. In particular, contrary to Machina's results CO.2 and DE.2, actuarial fairness of the insurance policies is no longer a necessary condition for full insurance. In other words, risk-averse decision makers may demand full insurance even if the insurance policies are actuarially unfair.

5 NON-EXPECTED UTILITY: DISCUSSION Concluding remarks The objective of Machina's paper is "to examine some of the classic results in individual and market insurance theory from the more general non-expected utility point of view, and determine which of these classic results are robust and which are not." The identification of results that are nonrobust is of particular interest since they may serve to test the alternative theories. An inherent aspect of expected-utility theory that requires special consideration in non-expected-utility theories is dynamic consistency. 9 It will be interesting, therefore, to find out if there are insurance-market phenomena, possibly involving sequential insurance decisions, that may shed some light on the issue of dynamic consistency. Besides dynamic consistency, insurance behavior may be used to test alternative nonexpected-utility models. For instance, according to Borch [1974], the result that decision makers will refrain from taking out full insurance whenever the insurance policy has positive proportional loading contradicts observed insurance behavior. In view of the discussion in the preceding section, if Borch is right, EURDP theory may explicate insurance behavior better than expected-utility theory and non-expected-utility theories that imply risk aversion of the second order. Finally, what Machina's paper and this brief discussion shows is that insurance results are robust to the relaxation of the expected-utility hypothesis, somewhat less so with respect to quasiconcavity, and significantly less when it comes to risk aversion and smoothness. No~s 1. The scope of these comments does not permit mentioning, let alone discussing, the contributions to non-expectedutility analysis. The reader will find a more detailed survey and references in Karni and Schmeidler [1991]. 2. For more details see Karni and Schmeidler [1991]. 3. A probability measure is simple if it has finite support. 4. Formally, Machina assumed that the preference functional Vis continuous in the topology of weak convergence and is Frechet differentiable with respect to the L 1 norm. 5. In fact, betweenness implies that the indifference surfaces in the space of distributions are convex, and the preference relation is both quasiconcave and quasiconvex. 6. This analysis may be generalized to any number of states. 7. See results CO.2 and DE.2 in Machina's paper. 8. A decision maker's risk aversion is of second order if, for all nondegenerate random variables ~" such that Eft] = 0, the risk premium lrv(t) corresponding to the random variable t~ has the property that 07rV(t)/Ot It=0 = O, and O2rv(t)/Ot2lt= o ~ 0 +. For more details see Segal and Spivak [1990]. 9. See Karni and Schmeidler [1991] for discussion and references. References BORCH, K. [1974]: The Mathematical Theory of Insurance, Lexington Books, Lexington (Mass.). CHEW, S.H., KARNI, E., and SAFRA, Z. [1987]. "Risk Aversion in the Theory of Expected Utility with Rank- Dependent Probabilities" Journal of Economic Theory, 42, DEKEL, E. [1989]. "Asset Demands Without the Independence Axiom" Econometrica, 57,

6 56 EDI KARNI KARNI, E., and SCHMEIDLER, D. [1991]. "Utility Theory with Uncertainty;' in Handbook of Mathematical Economics, (Vol. 4), W. Hildenbrand and H. Sormenschein (eds.), North-Holland, Amsterdam. MACHINA, M.J. [1982]: "Expected Utility Analysis Without the Independence Axiom;' Econometrica, SEGAL, U., and SPIVAK, A. [1990]. "First Order Versus Second Order Risk Aversion" Journal of Economic Theory, 51,