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

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Handbook of Insurance

Abstract

This chapter uses the technique of “generalized expected utility analysis” to explore the robustness of some of the basic results in classical insurance theory to departures from the expected utility hypothesis on agents’ risk preferences. The topics include individual demand for coinsurance and deductible insurance, the structure of Pareto-efficient bilateral insurance contracts, the structure of Pareto-efficient multilateral risk sharing agreements, self-insurance vs. self-protection, and insurance decisions under ambiguity. Most, though not all, of the basic results in these areas are found to be quite robust to dropping the expected utility hypothesis.

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Notes

  1. 1.

    Depending upon the context, the probabilities in these distributions can either be actuarially determined chances, or a decision-maker’s personal or “subjective probabilities” over states of nature or events.

  2. 2.

    An interpretive note: The rectangle property is essentially the condition that (smooth) expected utility preferences are separable across mutually exclusive states of nature. Given the rectangle property, the MRS at certainty property is equivalent to “state-independent” preferences, a property we shall assume throughout this chapter. For important analyses of state-dependent preferences under both expected utility and non-expected utility, see Karni (1985,1987). For a specific application to insurance theory, see Cook and Graham (1977).

  3. 3.

    For example, Rothschild and Stiglitz (1970,1971).

  4. 4.

    An alternative term for property (3.3) is quasiconvexity in the outcomes.

  5. 5.

    As before, they satisfy risk aversion since they are steeper/flatter than the iso-expected value lines in the region above/below the 45∘ line, so mean-preserving increases in risk make them worse off.

  6. 6.

    For an explicit example, based on the proof of Dekel’s Proposition 1, let \(\mathcal{V}(\mathbf{P}) \equiv {[\sum \sqrt{x_{i}}\cdot p_{i} - 5]}^{3}+\) \(8\cdot {[\sum x_{i}\cdot p_{i} - 49]}^{3}\). Since the cube function is strictly increasing over all positive and negative arguments, this preference function is strictly increasing in each x i and satisfies strict first-order stochastic dominance preference. Since any mean-preserving spread lowers the first bracketed term yet preserves the second, \(\mathcal{V}(\cdot )\) is also strictly risk averse. Calculation reveals that \(\mathcal{V}(\$100, \frac{1} {2}; \$0, \frac{1} {2}) = \mathcal{V}(\$49, \frac{1} {2}; \$49, \frac{1} {2}) = 8\) but \(\mathcal{V}(\$74.5, \frac{1} {2}; \$24.5, \frac{1} {2}) \approx 6.74\). But since the latter probability distribution is a 50:50 outcome mixture of the first two, \(\mathcal{V}(\cdot )\) is not outcome convex.

  7. 7.

    For example, Machina (1982,1983).

  8. 8.

    Algebraically, {U(x 1), …, U(x n )} forms a concave sequence if and only if its point-to-point slopes (U(x 2) − U(x 1))∕(x 2 − x 1), (U(x 3) − U(x 2))∕(x 3 − x 2), etc. are successively nonincreasing.

  9. 9.

    {U 1(x 1), …, U 1(x n )} is at least as concave than {U 2(x 1), …, U 2(x n )} if and only if each ratio of adjacent point-to-point slopes \([(U(x_{i+1}) - U(x_{i}))/(x_{i+1} - x_{i})/[(U(x_{i}) - U(x_{i-1}))/(x_{i} - x_{i-1})\)] is no greater for {U 1(⋅ ))} than for {U 2(⋅ ))}.

  10. 10.

    For the appropriate definition of “at least as risk averse as” under non-expected utility, see Machina (1982,1984).

  11. 11.

    This follows from applying Machina (1982, eq. 8) to the path \(F(\cdot;\alpha ) \equiv (x_{1},p_{1}; \ldots; x_{i-1},p_{i-1};\alpha,p_{i};\) \(x_{i+1},p_{i+1}; \ldots; x_{n},p_{n})\).

  12. 12.

    In some of our more formal analysis below (including the formal theorems), we use the natural extension of these ideas to the case of a preference function \(\mathcal{V}(F)\) over cumulative distribution functions F(⋅ ) with local utility function U(⋅ ; F), including the smoothness notion of “Fréchet differentiability” (see Machina 1982).

  13. 13.

    The reader wishing self-contained treatments of the vast body of insurance results can do no better than the excellent survey by Dionne and Harrington (1992, pp.1–48) and volume by Eeckhoudt and Gollier (1995). For more extensive treatments of specific topics, see the rest of the chapters in Dionne and Harrington (1992), as well as Schlesinger (2013) and the other chapters in the present volume.

  14. 14.

    We consider non-differentiabilities (“kinks”) in the outcomes and probabilities in Sects. 3.7 and 3.8.

  15. 15.

    This point is nicely made by Karni (1992).

  16. 16.

    The case when the individual faces additional “background risk” is considered in Sect. 3.9.

  17. 17.

    As demonstrated in Pratt (1964), further results which link increasing/decreasing absolute and/or relative risk aversion to changes in as an individual’s wealth changes can be derived as corollaries of result CO.3.

  18. 18.

    So can result CO.1, if one calculates the slope of the budget lines in Fig. 3.7a and b.

  19. 19.

    This close correspondence of expected utility and non-expected utility first-order conditions will come as no surprise to those who have read Chew, Epstein and Zilcha (1988), and will appear again.

  20. 20.

    We consider the nondifferentiable case in Section 8 below.

  21. 21.

    A NOTE ON BELIEFS: Although CO.2 accordingly survives dropping the assumption of expected utility risk preferences, it does not survive dropping the assumption that the individual’s subjective probabilities exactly match those of the “market,” that is, the probabilities by which an insurance policy is judged to be actuarially fair or unfair. If—for reasons of moral hazard, adverse selection, or simply personal history—the individual assigns a higher probability to state 2 than does the market, then the indifference curves in Fig. 3.7a will be flatter than and cut the dashed lines at all certainty points, and an individual with a smooth (differentiable) U(⋅ ) may well select point C on an actuarially unfair budget line like A–C. How far must beliefs diverge for this to happen? Consider earthquake insurance priced on the basis of an actuarial probability of.0008 and a loading factor of 25%. Every smooth risk averter with a subjective probability greater than.001 will buy full insurance.

  22. 22.

    Readers will recognize this argument (and its formalization in the proofs of the theorems) as an application of the well-known “single-crossing property” argument from incentive theory, as in Mirrlees (1971), Spence (1974), and Guesnerie and Laffont (1984), and generalized and extended by Milgrom and Shannon (1994).

  23. 23.

    Thus, sgn(max\(\{\tilde{l}-\alpha, 0\})\), equals 1 when l > and equals 0 when l ≤ α.

  24. 24.

    This was shown by Schlesinger (1981) and Karni (1983).

  25. 25.

    See also Arrow (1974), Blazenko (1985), Gollier (1987), and Marshall (1992), and the survey by Gollier (1992, Sect. 2).

  26. 26.

    Note, however, that derivative I ′(l) in PE.2 or PE.3 need not be constant, but as Raviv (1979, pp. 90,91) has shown, depends upon each party’s levels of risk aversion, as well as marginal indemnity cost C ′(I).

  27. 27.

    Thus, (I ∗(⋅ ), π ∗) is a solution to problem (3.40) for some given w 1 and w 2, though it needn’t be a unique solution.

  28. 28.

    By way of clarification, note that d π is a differential change in the scalar π, while dI(⋅ ) is a differential change in the entire function I(⋅ ), in the sense being some differential change dI(l) in I(l) for every value of l.

  29. 29.

    Readers intrigued by this type of argument are referred to Chew, Epstein, and Zilcha (1988) who, under slightly different assumptions (namely, uniqueness of maxima), demonstrate its surprising generality.

  30. 30.

    See also Gerber (1978), Moffet (1979), Bühlman and Jewell (1979), and Eliashberg and Winkler (1981) for important subsequent contributions, and Lemaire (1990) and Gollier (1992, Sect. 1) for insightful surveys.

  31. 31.

    We consider what happens when agents may not have subjective probabilities at all in Sect. 3.10.

  32. 32.

    We say risk tolerance since i (x) is the reciprocal of the standard Arrow–Pratt measure of absolute risk aversion.

  33. 33.

    Like the 2-state formula (3.16), its n-state equivalent follows immediately from Eq. (3.15).

  34. 34.

    For example, Jullien, Salanié, and Salanié (1999, Sect. 3), Eeckhoudt and Gollier (2005).

  35. 35.

    For clarity, the iso-expected values lines are not shown in Figs. 3.10b or 3.11b, but do appear in Fig. 3.11a.

  36. 36.

    Can Fig. 3.11a and b also be used to illustrate the demand for conditional insurance in states 1 and 2 when states 3,…,nare uninsured? Only when the insurance contract refunds the premium in every uninsured state. If the premium is retained in every state, then moving along the coinsurance budget line in the figure also changes the outcomes in states 3, …, n, so the x 1, x 2 indifference curves in the figure will shift.

  37. 37.

    Since the kinks generated here are convex kinks, this may occur even without full outcome-convexity.

  38. 38.

    See, for example, Quiggin (1982,1993), Ritzenberger (1996), Röell (1987), and Bleichrodt and Quiggin (1997).

  39. 39.

    For the following equation, define \(\hat{x}_{0}\) (resp. \(\hat{x}_{n+1})\) as any value lower (resp. higher) than all of the outcomes in P.

  40. 40.

    That is, \(U(\cdot; \mathbf{P}) \equiv a_{k} \cdot \upsilon (\cdot ) + b_{k}\) over \([\hat{x}_{k},\hat{x}_{k+1})\), where \(a_{k} = {G}^{{\prime}}(\sum \nolimits _{i=1}^{k}\hat{p}_{j})\) and \(b_{k} =\sum \nolimits _{ i=k+1}^{n}\upsilon (\hat{x}_{i}) \cdot [{G}^{{\prime}}(\sum \nolimits _{j=1}^{i}\hat{p}_{j})\) \(-{G}^{{\prime}}(\sum \nolimits _{j=1}^{i-1}\hat{p}_{j})]\) are constant over each interval \([\hat{x}_{k},\hat{x}_{k+1})\).

  41. 41.

    From Note 39, υ(⋅ ) concave is necessary and sufficient for U(⋅ ; P) to be concave within each interval \([\hat{x}_{k},\hat{x}_{k+1})\), in which case G(⋅ ) concave (hence G ′(⋅ ) decreasing) is necessary and sufficient for U(⋅ ; P) to be concave across these intervals.

  42. 42.

    Again from Note 39, comparative concavity of υ ∗(⋅ ) and υ(⋅ ) is necessary and sufficient for comparative concavity of U ∗(⋅ ; P) and U(⋅ ; P) within each interval [x i , x i+1), in which case comparative concavity of G ∗(⋅ ) and G(⋅ ) (G ∗‘(⋅ ) decreasing proportionately faster than G ′(⋅ )) is necessary and sufficient for comparative concavity of U ∗(⋅ ; P) and U(⋅ ; P) across these intervals.

  43. 43.

    So called because the linearity/nonlinearity properties of payoff and probability are reversed relative to the expected utility form.

  44. 44.

    See Wang (1995), Wang and Young (1998), and van der Hoek and Sherris (2001) for the development of some measures of risk along the lines of the Dual model and their application to insurance, and Sung, Yam, Yung, and Zhou (2011) for an analysis of optimal insurance policies under the general rank-dependent form.

  45. 45.

    The following is an example of the general observation of Markowitz (1959, Ch.11), Mossin (1969), Spence and Zeckhauser (1972), and others that induced risk preferences are generally not expected utility maximizing.

  46. 46.

    For example, Alarie, Dionne, and Eeckhoudt (1992), Eeckhoudt and Kimball (1992), Gollier and Eeckhoudt (2013), Gollier and Pratt (1996), Mayers and Smith (1983), Pratt (1988), and Schlesinger and Doherty (1985).

  47. 47.

    See Pratt (1964,Thm.5), Kreps and Porteus (1979), and Nachman (1982) for analyses of how various properties of the underlying utility function U(⋅ ) do or do not carry over to the derived utility function U Q (⋅ ).

  48. 48.

    Thus, λ P + (1 −λ) ⋅ P ∗ is the single-stage equivalent of a coin flip that yields probability of winning the distribution P and probability (1 −λ) of winning P ∗.

  49. 49.

    For example, Whitmore (1970).

  50. 50.

    See, for example, the surveys of Camerer and Weber (1992), Siniscalchi (2008), Hey, Lotito, and Maffioletti (2010), Etner, Jeleva, and Tallon (2011), and Gilboa and Marinacci (2012).

  51. 51.

    For example, Savage (1954), Machina and Schmeidler (1992).

  52. 52.

    The implications of such findings for insurance against environmental risks are discussed in Kunreuther (1989).

  53. 53.

    In the Choquet model, risks that are positively correlated across events are termed comonotonic.

  54. 54.

    See, however, the predominantly negative findings of Ryan and Vaithianathan (2003).

  55. 55.

    See also Winter (2013), Dionne, Doherty, and Fombaron (2013), and the other related chapters in this volume.

  56. 56.

    See Jeleva and Villeneuve (2004), however, for some expected utility results that do not carry over.

  57. 57.

    The expected utility property only enters the Rothschild–Stiglitz analysis in their Eq. (3.4) (p.645), which gives conditions for an optimal insurance contract. As in the above analyses, these first-order conditions will continue to hold for general (risk averse, outcome convex) non-expected utility preferences, with individuals’ von Neumann–Morgenstern utility functions replaced by their local utility functions.

  58. 58.

    From here until the end of the paragraph following (3.91), all equations and discussion refer to this point (ρ, c).

  59. 59.

    Since \(c +\rho \cdot \tilde{z} \geq c_{0} +\rho \cdot \tilde{z} = w_{0} - (1+\lambda ) \cdot E[\tilde{\ell}] +\rho \cdot ((1+\lambda ) \cdot E[\tilde{\ell}]-\tilde{\ell}) =\rho \cdot (w_{0}-\tilde{\ell}) + (1-\rho ) \cdot (w_{0} - (1+\lambda ) \cdot E[\tilde{\ell}])\), nonnegativity of \(c +\rho \cdot \tilde{z}\) on the set {(ρ, c) | ρ ∈ [0, 1], c ≥ c 0} follows from nonnegativity of \(w_{0}-\tilde{\ell}\) and \(w_{0} - (1+\lambda ) \cdot E[\tilde{\ell}]\). Note that since c ≥ c 0 > 0, the condition c −ρ ⋅ k < 0 also implies that ρ must be nonzero, and hence positive.

  60. 60.

    From here until (3.101), all equations and discussion refer to this point (α, w).

References

  • Akerlof G (1970) The market for ‘lemons’: quality uncertainty and the market mechanism. Q J Econ 84:488–500. Reprinted in Akerlof (1984) and in Diamond and Rothschild (1989)

    Google Scholar 

  • Akerlof G (1984) An economic theorist’s book of tales. Cambridge University Press, Cambridge

    Google Scholar 

  • Alarie Y, Dionne G, Eeckhoudt L (1992) Increases in risk and the demand for insurance, in Dionne G (ed) Contributions to insurance economics. Kluwer Academic, Boston

    Google Scholar 

  • Alary D, Gollier C, Treich N (2012) The effect of ambiguity aversion on insurance and self-protection. Econ J forthcoming

    Google Scholar 

  • Allen B (1987) Smooth preferences and the local expected utility hypothesis. J Econ Theory 41:340–355

    Google Scholar 

  • Arrow K (1963) Uncertainty and the welfare economics of medical care. Am Econ Rev 53:941–969. Reprinted in Arrow (1971) and in part in Diamond and Rothschild (1989)

    Google Scholar 

  • Arrow K (1965a) Aspects of the theory of risk bearing. Yrjö Jahnsson Säätiö, Helsinki

    Google Scholar 

  • Arrow K (1965b) Theory of risk aversion, in Arrow (1965a). Reprinted in Arrow (1971)

    Google Scholar 

  • Arrow K (1968) The economics of moral hazard: further comment. Am Econ Rev 58:537–539. Reprinted in Arrow (1971).

    Google Scholar 

  • Arrow K (1971) Essays in the theory of risk-bearing. North-Holland, Amsterdam

    Google Scholar 

  • Arrow K (1974) Optimal insurance and generalized deductibles. Scand Actuarial J 1:1–42

    Google Scholar 

  • Billot A, Chateauneuf A, Gilboa I, Tallon JM (2002) Sharing beliefs and the absence of betting in the choquet expected utility model. Stat Paper 43:127–136

    Google Scholar 

  • Blazenko G (1985) The design of an optimal insurance policy: note. Am Econ Rev 75:253–255

    Google Scholar 

  • Bleichrodt H, Quiggin J (1997) Characterizing QALYs under a general rank-dependent utility model. J Risk Uncertainty 15:151–165

    Google Scholar 

  • Borch K (1960) The safety loading of reinsurance premiums. Skandinavisk Aktuarietidskrift 163–184. Reprinted in Borch (1990)

    Google Scholar 

  • Borch K (1961) The utility concept applied to the theory of insurance. Astin Bull 1:245–255. Reprinted in Borch (1990)

    Google Scholar 

  • Borch K (1962) Equilibrium in a reinsurance market. Econometrica 30:424–444. Reprinted in Borch (1990) and in Dionne and Harrington (1992)

    Google Scholar 

  • Borch K (1990) Economics of insurance. North Holland, Amsterdam (Completed by K. Aase and A. Sandmo)

    Google Scholar 

  • Boyer M, Dionne G (1983) Variations in the probability and magnitude of loss: their impact on risk. Can J Econ 16:411–419

    Google Scholar 

  • Boyer M, Dionne G (1989) More on insurance, protection and risk. Can J Econ 22:202–205

    Google Scholar 

  • Briys E, Schlesinger H (1990) Risk aversion and propensities for self-insurance and self-protection. South Econ J 57:458–467

    Google Scholar 

  • Briys E, Schlesinger H, Schulenburg J-M (1991) Reliability of risk management: market insurance, self-insurance, and self-protection reconsidered. Geneva Papers Risk Insur Theory 16:45–58

    Google Scholar 

  • Brunette M, Cabantous L, Couture S, Stenger A (2012) The impact of governmental assistance on insurance demand under ambiguity: a theoretical model and experimental test. Theor Decis 13:1–22

    Google Scholar 

  • Bryan G (2010a) Ambiguity and insurance. manuscript, Yale University

    Google Scholar 

  • Bryan G (2010b) Ambiguity and the demand for index insurance: theory and evidence from Africa. manuscript, Yale University

    Google Scholar 

  • Bühlman E, Jewell H (1979) Optimal risk exchanges. Astin Bull 10:243–262

    Google Scholar 

  • Cabantous L (2007) Ambiguity aversion in the field of insurance: insurers’ attitude to imprecise and conflicting probability estimates. Theor Decis 62:219–240

    Google Scholar 

  • Camerer C, Kunreuther H (1989) Experimental markets for insurance. J Risk Uncertainty 2:265–299

    Google Scholar 

  • Camerer C, Weber M (1992) Recent developments in modeling preferences: uncertainty and ambiguity. J Risk Uncertainty 5:325–370

    Google Scholar 

  • Chang YM, Ehrlich I (1985) Insurance, protection from risk and risk bearing. Can J Econ 18:574–587

    Google Scholar 

  • Chateauneuf A, Dana RA, Tallon JM (2000) Optimal risk-sharing rules and equilibria with choquet-expected-utility. J Math Econ 34:191–214

    Google Scholar 

  • Chew S (1983) A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the allais paradox. Econometrica 51

    Google Scholar 

  • Chew S, Epstein L, Segal U (1991) Mixture symmetry and quadratic utility. Econometrica 59:139–163

    Google Scholar 

  • Chew S, Epstein L, Zilcha I (1988) A correspondence theorem between expected utility and smooth utility. J Econ Theory 46:186–193

    Google Scholar 

  • Chew S, Karni E, Safra Z (1987) Risk aversion in the theory of expected utility with rank-dependent probabilities. J Econ Theory 42:370–381

    Google Scholar 

  • Clarke D (2007) “Ambiguity aversion and insurance. manuscript, Oxford University, Balliol College

    Google Scholar 

  • Cohen M (1995) Risk aversion concepts in expected- and non-expected utility models. Geneva Paper Risk Insur Theory 20:73–91. Reprinted in Gollier and Machina (1995)

    Google Scholar 

  • Cook P, Graham D (1977) The demand for insurance and protection: the case of irreplaceable commodities. Q J Econ 91:143–156. Reprinted in Dionne and Harrington (1992)

    Google Scholar 

  • Courbage C (2001) Self-insurance, self-protection and market insurance within the dual theory of choice. Geneva Paper Risk Insur Theory 26:43–56

    Google Scholar 

  • Debreu G (1959) Theory of value: an axiomatic analysis of general equilibrium. Yale University Press, New Haven

    Google Scholar 

  • Dekel E (1989) Asset demands without the independence axiom. Econometrica 57:163–169

    Google Scholar 

  • di Mauro C, Maffioletti A (1996) An experimental investigation of the impact of ambiguity on the valuation of self-insurance and self-protection. J Risk Uncertainty 13:53–71

    Google Scholar 

  • Diamond P, Rothschild M (eds.) (1989) Uncertainty in economics: readings and exercises, 2nd edn. Academic Press, New York

    Google Scholar 

  • Dionne G (ed) (1992) Contributions to insurance economics. Kluwer Academic, Boston

    Google Scholar 

  • Dionne G (ed) (2013) Handbook of insurance, 2nd edn. Springer, New York; forthcoming in 2013

    Google Scholar 

  • Dionne G, Doherty N, Fombaron N (2013) Adverse selection in insurance contracting, in Dionne G (ed) Handbook of insurance, 2nd edn. Springer, New York

    Google Scholar 

  • Dionne G, Eeckhoudt L (1985) Self-insurance, self-protection and increased risk aversion. Econ Lett 17:39–42

    Google Scholar 

  • Dionne G, Harrington S (eds) (1992) Foundations of insurance economics: readings in economics and finance. Kluwer Academic, Boston

    Google Scholar 

  • Doherty N, Eeckhoudt L (1995) Optimal insurance without expected utility: the dual theory and the linearity of insurance contracts. J Risk Uncertainty 10:157–179

    Google Scholar 

  • Drèze J (1981) Inferring risk tolerance from deductibles in insurance contracts. Geneva Paper Risk Insur 20:48–52

    Google Scholar 

  • Drèze J (1986) Moral expectation with moral hazard, in Hildenbrand and Mas-Colell (1986)

    Google Scholar 

  • Dupuis A, Langlais E (1997) The basic analytics of insurance demand and the rank-dependent expected utility model. Finance 18:47–745

    Google Scholar 

  • Eeckhoudt L, Gollier C (1995) Risk: evaluation, management and sharing. Harvester Wheatleaf, New York

    Google Scholar 

  • Eeckhoudt L, Gollier C (2005) The impact of prudence on optimal prevention. Econ Theory 26:989–994

    Google Scholar 

  • Eeckhoudt L, Gollier C, Schlesinger H (1991) Increases in risk and deductible insurance. J Econ Theory 55:435–440

    Google Scholar 

  • Eeckhoudt L, Kimball M (1992) Background risk, prudence, and the demand for insurance, in Dionne G (ed) Contributions to insurance economics. Kluwer Academic, Boston

    Google Scholar 

  • Ehrlich I, Becker G (1972) Market insurance, self-insurance, and self-protection. J Polit Econ 80:623–648. Reprinted in Dionne and Harrington (1992)

    Google Scholar 

  • Eliashberg J, Winkler R (1981) Risk sharing and group decision making. Manag Sci 27:1221–1235

    Google Scholar 

  • Ellsberg D (1961) Risk, ambiguity, and the savage axioms. Q J Econ 75:643–669

    Google Scholar 

  • Etner J, Jeleva M, Tallon J-M (2011) Decision theory under ambiguity. J Econ Surv 26:234–270

    Google Scholar 

  • Gerber H (1978) Pareto-optimal risk exchanges and related decision problems. Astin Bull 10:25–33

    Google Scholar 

  • Gilboa I, Marinacci M (2012) Ambiguity and the Bayesian paradigm, in Acemoglu D, Arellano M, Dekel E (eds) Advances in economics and econometrics: theory and applications, tenth world congress of the econometric society. Cambridge University Press, Cambridge

    Google Scholar 

  • Gollier C (1987) Pareto-optimal risk sharing with fixed costs per claim. Scand Actuarial J 13:62–73

    Google Scholar 

  • Gollier C (1992) Economic theory of risk exchanges: a review, in Dionne G (ed) Contributions to insurance economics. Kluwer Academic, Boston

    Google Scholar 

  • Gollier C (2013a) Optimal insurance design of ambiguous risks, manuscript, University of Toulouse

    Google Scholar 

  • Gollier C (2013b) The economics of optimal insurance design, in Dionne G (ed) Handbook of insurance, 2nd edn. Springer, New York

    Google Scholar 

  • Gollier C, Eeckhoudt L (2013) The effect of changes in risk on risk taking: a survey, in Dionne G (ed) Handbook of insurance, 2nd edn. Springer, New York

    Google Scholar 

  • Gollier C, Machina M (1995) Non-expected utility and risk management. Kluwer Academic, Dordrecht

    Google Scholar 

  • Gollier C, Pratt J (1996) Risk, vulnerability and the tempering effect of background risk. Econometrica 64:1109–1124

    Google Scholar 

  • Gollier C, Schlesinger H (1996) Arrow’s theorem on the optimality of deductibles: a stochastic dominance approach. Econ Theory 7:359–363

    Google Scholar 

  • Gould J (1969) The expected utility hypothesis and the selection of optimal deductibles for a given insurance policy. J Bus 42:143–151

    Google Scholar 

  • Guesnerie R, Laffont J-J (1984) A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm. J Public Econ 25:329–369

    Google Scholar 

  • Hagen O (1979) Towards a positive theory of preferences under risk, in Allais M, Hagen O (eds) Expected utility hypotheses and the allais paradox. D. Reidel Publishing, Dordrecht

    Google Scholar 

  • Heilpern S (2003) A rank-dependent generalization of zero utility principle. Insur Math Econ 33:67–73

    Google Scholar 

  • Hey J, Lotito G, Maffioletti A (2010) The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. J Risk Uncertainty 41:81–111

    Google Scholar 

  • Hibert L (1989) Optimal loss reduction and risk aversion. J Risk Insur 56:300–306

    Google Scholar 

  • Hildenbrand W, Mas-Colell A (eds) (1986) Contributions to mathematical economics. North-Holland, Amsterdam

    Google Scholar 

  • Hirshleifer J (1965) Investment decision under uncertainty: choice-theoretic approaches. Q J Econ 79:509–536. Reprinted in Hirshleifer (1989)

    Google Scholar 

  • Hirshleifer J (1966) Investment decision under uncertainty: applications of the state-preference approach. Q J Econ 80:252–277. Reprinted in Hirshleifer (1989)

    Google Scholar 

  • Hirshleifer J (1989) Time, uncertainty, and information. Basil Blackwell, Oxford

    Google Scholar 

  • Hirshleifer J, Riley J (1979) The analytics of uncertainty and information – an expository survey. J Econ Lit 17:1375–1421. Reprinted in Hirshleifer (1989)

    Google Scholar 

  • Hirshleifer J, Riley J (1992) The analytics of uncertainty and information. Cambridge University Press, Cambridge

    Google Scholar 

  • Hogarth R, Kunreuther H (1985) Ambiguity and insurance decisions. Am Econ Rev 75:386–390

    Google Scholar 

  • Hogarth R, Kunreuther H (1989) Risk, ambiguity and insurance. J Risk Uncertainty 2:5–35

    Google Scholar 

  • Hogarth R, Kunreuther H (1992a) How does ambiguity affect insurance decisions?, in Dionne G (ed) Contributions to insurance economics. Kluwer Academic, Boston

    Google Scholar 

  • Hogarth R, Kunreuther H (1992b) Pricing insurance and warranties: ambiguities and correlated risks. Geneva Papers Risk Insur Theory 17:35–60

    Google Scholar 

  • Jeleva M (2000) Background risk, demand for insurance, and choquet expected utility preferences. Geneva Papers Risk Insur Theory 25:7–28

    Google Scholar 

  • Jeleva M, Villeneuve B (2004) Insurance contracts with imprecise probabilities and adverse selection. Econ Theory 23:777–794

    Google Scholar 

  • Jullien B, Salanié B, Salanié F (1999) Should more risk-averse agents exert more effort? Geneva Papers Risk Insur Theory 24:19–28

    Google Scholar 

  • Kaluszka M, Krzeszowiec M (2012) Pricing insurance contracts under cumulative prospect theory. Insur Math Econ 50:159–166

    Google Scholar 

  • Karni E (1983) Karni E (1983) Risk aversion in the theory of health insurance, in Helpman E, Razin A, Sadka E (eds) Social policy evaluation: an economic perspective. Academic Press, New York

    Google Scholar 

  • Karni E (1985) Decision making under uncertainty: the case of state dependent preferences. Harvard University Press, Cambridge, MA

    Google Scholar 

  • Karni E (1987) Generalized expected utility analysis of risk aversion with state-dependent preferences. Int Econ Rev 28:229–240

    Google Scholar 

  • Karni E (1989) Generalized expected utility analysis of multivariate risk aversion. Int Econ Rev 30:297–305

    Google Scholar 

  • Karni E (1992) Optimal insurance: a nonexpected utility analysis, in Dionne G (ed) Contributions to insurance economics. Kluwer Academic, Boston

    Google Scholar 

  • Karni E (1995) Non-expected utility and the robustness of the classical insurance paradigm: discussion. Geneva Papers Risk Insur Theory 20:51–56. Reprinted in Gollier and Machina (1995)

    Google Scholar 

  • Kelsey D, Milne F (1999) Induced preferences, non additive probabilities and multiple priors. Int Econ Rev 2:455–477

    Google Scholar 

  • Knight FH (1921) Risk, uncertainty, and profit. Houghton Mifflin, Boston/New York

    Google Scholar 

  • Konrad K, Skaperdas S (1993) Self-insurance and self-protection: a non-expected utility analysis. Geneva Papers Risk Insur Theory 18:131–146

    Google Scholar 

  • Kreps D, Porteus E (1979) Temporal von Neumann-Morgenstern and induced preferences. J Econ Theory 20:81–109

    Google Scholar 

  • Kunreuther H (1989) The role of actuaries and underwriters in insuring ambiguous risks. Risk Anal 9:319–328

    Google Scholar 

  • Kunreuther H, Hogarth R, Meszaros J (1993) Insurer ambiguity and market failure. J Risk Uncertainty 7:71–87

    Google Scholar 

  • Kunreuther H, Meszaros J, Hogarth R, Spranca M (1995) Ambiguity and underwriter decision processes. J Econ Behav Organ 26:337–352

    Google Scholar 

  • Lemaire J (1990) Borch’s theorem: a historical survey of applications, Loubergé H (ed.) Risk, information and insurance. Kluwer Academic, Boston

    Google Scholar 

  • Machina M (1982) ‘Expected Utility’ analysis without the independence axiom. Econometrica 50:277–323

    Google Scholar 

  • Machina M (1983) Generalized expected utility analysis and the nature of observed violations of the independence axiom, in Stigum B, Wenstøp F (eds.) Foundations of utility and risk theory with applications. D. Reidel Publishing, Dordrecht, Holland. Reprinted in Dionne and Harrington (1992)

    Google Scholar 

  • Machina M (1984) Temporal risk and the nature of induced preferences. J Econ Theory 33:199–231

    Google Scholar 

  • Machina M (1989) Comparative statics and non-expected utility preferences. J Econ Theory 47:393–405

    Google Scholar 

  • Machina M (1995) Non-expected utility and the robustness of the classical insurance paradigm. Geneva Papers Risk Insur Theory 20:9–50. Reprinted in Gollier and Machina (1995)

    Google Scholar 

  • Machina M (2001) Payoff kinks in preferences over lotteries. J Risk Uncertainty 23:207–260

    Google Scholar 

  • Machina M (2005) ‘Expected Utility/Subjective Probability’ analysis without the sure-thing principle or probabilistic sophistication. Econ Theory 26:1–62

    Google Scholar 

  • Machina M, Schmeidler D (1992) A more robust definition of subjective probability. Econometrica 60:745–780

    Google Scholar 

  • Markowitz H (1959) Portfolio selection: efficient diversification of investments. Yale University Press, New Haven

    Google Scholar 

  • Marshall J (1992) Optimum insurance with deviant beliefs, in Dionne G (ed) Contributions to insurance economics. Kluwer Academic, Boston

    Google Scholar 

  • Martinez-Correa J (2012) Risk Management and Insurance Decisions under Ambiguity, manuscript, Georgia State University

    Google Scholar 

  • Mashayekhi M (2013) A note on optimal insurance under ambiguity, Insur Markets Companies Anal Actuarial Comput 3:58–62

    Google Scholar 

  • Mayers D, Smith C (1983) The interdependence of individual portfolio decisions and the demand for insurance. J Polit Econ 91:304–311. Reprinted in Dionne G, Harrington S (eds) (1992) Foundations of insurance economics: readings in economics and finance. Kluwer Academic, Boston

    Google Scholar 

  • Milgrom P, Shannon C (1994) Monotone comparative statics. Econometrica 62:157–180

    Google Scholar 

  • Mirrlees J (1971) An exploration in the theory of optimal income taxation. Rev Econ Stud 38:175–208

    Google Scholar 

  • Moffet D (1977) Optimal deductible and consumption theory. J Risk Insur 44:669–682

    Google Scholar 

  • Moffet D (1979) The risk sharing problem. Geneva Papers Risk Insur Theory 11:5–13

    Google Scholar 

  • Mossin J (1968) Aspects of rational insurance purchasing. J Polit Econ 79:553–568. Reprinted in Dionne and Harrington (1992)

    Google Scholar 

  • Mossin J (1969) A note on uncertainty and preferences in a temporal context. Am Econ Rev 59:172–174

    Google Scholar 

  • Nachman D (1982) Preservation of ‘More Risk Averse’ under expectations. J Econ Theory 28:361–368

    Google Scholar 

  • Ozdemir O (2007) Valuation of self-insurance and self-protection under ambiguity: experimental evidence. Jena Econ Res Paper 2007–034

    Google Scholar 

  • Pashigian B, Schkade L, Menefee G (1966) The selection of an optimal deductible for a given insurance policy. J Bus 39:35–44

    Google Scholar 

  • Pauly M (1968) The economics of Moral Hazard. Am Econ Rev 58:531–537

    Google Scholar 

  • Pauly M (1974) Overinsurance and public provision of insurance: the role of Moral Hazard and adverse selection. Q J Econ 88:44–62. Reprinted in part in Diamond and Rothschild (1989)

    Google Scholar 

  • Pratt J (1964) Risk aversion in the small and in the large. Econometrica 32:122–136. Reprinted in Diamond and Rothschild (1989) and in Dionne and Harrington (1992)

    Google Scholar 

  • Pratt J (1988) Aversion to one risk in the presence of others. J Risk Uncertainty 1:395–413

    Google Scholar 

  • Quiggin J (1982) A theory of anticipated utility. J Econ Behav Organ 3:323–343

    Google Scholar 

  • Quiggin J (1993) Generalized expected utility theory: the rank-dependent model. Kluwer Academic, Boston, MA

    Google Scholar 

  • Quiggin J (2002) Risk and self-protection: a state-contingent view. J Risk Uncertainty 25:133–146

    Google Scholar 

  • Raviv A (1979) The design of an optimal insurance policy. Am Econ Rev 69:84–96. Reprinted in Dionne and Harrington (1992)

    Google Scholar 

  • Rigotti L, Shannon C (2012) Sharing risk and ambiguity. J Econ Theory 147:2028–2039

    Google Scholar 

  • Ritzenberger K (1996) On games under expected utility with rank-dependent probabilities. Theor Decis 40:1 27

    Google Scholar 

  • Röell A (1987) Risk aversion in Quiggin’s and Yaari’s rank-order model of choice under uncertainty. Econ J 97(Supplement):143–159

    Google Scholar 

  • Rothschild M, Stiglitz J (1970) Increasing risk: I. A definition. J Econ Theory 2:225–243. Reprinted in Diamond and Rothschild (1989) and in Dionne and Harrington (1992)

    Google Scholar 

  • Rothschild M, Stiglitz J (1971) Increasing risk: II. Its economic consequences. J Econ Theory 3:66–84

    Google Scholar 

  • Rothschild M, Stiglitz J (1976) Equilibrium in competitive insurance markets: the economics of markets with imperfect information. Q J Econ 90:629–650. Reprinted in Diamond and Rothschild (1989) and in Dionne and Harrington (1992)

    Google Scholar 

  • Ryan M, Vaithianathan R (2003) Adverse selection and insurance contracting: a rank-dependent utility analysis. Contrib Theor Econ 3(1):chapter 4

  • Samuelson P (1960) The St. Petersburg paradox as a divergent double limit. Int Econ Rev 1:31–37

    Google Scholar 

  • Savage L (1954) The foundations of statistics. Wiley, New York. Revised and Enlarged Edition, Dover Publications, New York, 1972

    Google Scholar 

  • Schlee E (1995) The comparative statics of deductible insurance in expected- and non-expected utility theories. Geneva Papers Risk Insur Theory 20:57–72. Reprinted in Gollier and Machina (1995)

    Google Scholar 

  • Schlesinger H (1981) The optimal level of deductibility in insurance contracts. J Risk Insur 48:465–481

    Google Scholar 

  • Schlesinger H (1997) Insurance demand without the expected utility paradigm. J Risk Insur 64:19–39

    Google Scholar 

  • Schlesinger H (2013) The theory of insurance demand, in Dionne G (ed) Handbook of insurance, 2nd edn. Springer, New York

    Google Scholar 

  • Schlesinger H, Doherty N (1985) Incomplete markets for insurance: an overview. J Risk Insur 52:402–423. Reprinted in Dionne and Harrington (1992)

    Google Scholar 

  • Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57:571–587

    Google Scholar 

  • Schmidt U (1996) Demand for coinsurance and bilateral risk-sharing with rank-dependent utility. Risk Decis Pol 1:217–228

    Google Scholar 

  • Segal U, Spivak A (1990) First order versus second order risk aversion. J Econ Theory 51:111–125

    Google Scholar 

  • Segal U, Spivak A (1997) First order risk aversion and non-differentiability. Econ Theory 9:179–183

    Google Scholar 

  • Shavell S (1979) On Moral Hazard and insurance. Q J Econ 93:541–562. Reprinted in Dionne and Harrington (1992)

    Google Scholar 

  • Siniscalchi M (2008) Ambiguity and ambiguity aversion. In: Durlauf SN, Blume LE (eds) The new palgrave dictionary of economics, 2nd edn. Palgrave Macmillan

    Google Scholar 

  • Smith V (1968) Optimal insurance coverage. J Polit Econ 76:68–77

    Google Scholar 

  • Snow A (2011) Ambiguity aversion and the propensities for self-insurance and self-protection. J Risk Uncertainty 42:27–44

    Google Scholar 

  • Spence M (1974) Competitive and optimal responses to signals: an analysis of efficiency and distribution. J Econ Theory 7:296–332

    Google Scholar 

  • Spence M, Zeckhauser R (1972) The effect of the timing of consumption decisions and the resolution of lotteries on the choice of lotteries. Econometrica 40:401–403

    Google Scholar 

  • Sung K, Yam S, Yung S, Zhou J (2011) Behavioral optimal insurance. Insur Math Econ 49:418–428

    Google Scholar 

  • Sweeney G, Beard T (1992) The comparative statics of self-protection. J Risk Insur 59:301–309

    Google Scholar 

  • Tobin J (1958) Liquidity preference as behavior toward risk. Rev Econ Stud 25:65–86

    Google Scholar 

  • van der Hoek J, Sherris M (2001) A class of non-expected utility risk measures and implications for asset allocations. Insur Math Econ 28:69–82

    Google Scholar 

  • Vergnaud J-C (1997) Analysis of risk in a non-expected-utility framework and application to the optimality of the deductible. Finance 18:156–167

    Google Scholar 

  • Viscusi K (1995) Government action, biases in risk perception, and insurance decisions. Geneva Papers Risk Insur Theory 20:93–110. Reprinted in Gollier and Machina (1995)

    Google Scholar 

  • Wang T (1993) Lp-Fréchet differentiable preference and ‘Local Utility’ analysis. J Econ Theory 61:139–159

    Google Scholar 

  • Wang S (1995) Insurance pricing and increased limits ratemaking by proportional Hazard transforms. Insur Math Econ 17:43–54

    Google Scholar 

  • Wang S, Young V (1998) Ordering risks: expected utility theory versus Yaari’s dual theory of risk. Insur Math Econ 22:145–161

    Google Scholar 

  • Whitmore G (1970) Third-degree stochastic dominance. Am Econ Rev 60:457–459

    Google Scholar 

  • Winter R (2013) Optimal insurance contracts under Moral Hazard, in Dionne G (ed) Handbook of insurance, 2nd edn. Springer, New York

    Google Scholar 

  • Wilson R (1968) The theory of syndicates. Econometrica 36:119–132

    Google Scholar 

  • Yaari M (1965) Convexity in the theory of choice under risk. Q J Econ 79:278–290

    Google Scholar 

  • Yaari M (1969) Some remarks on measures of risk aversion and on their uses. J Econ Theory 1:315–329. Reprinted in Diamond and Rothschild (1989)

    Google Scholar 

  • Yaari M (1987) The dual theory of choice under risk. Econometrica 55:95–115

    Google Scholar 

  • Young V, Browne M (2000) Equilibrium in competitive insurance markets under adverse selection and Yaari’s dual theory of risk. Geneva Papers Risk Insur Theory 25:141–157

    Google Scholar 

  • .

    Google Scholar 

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Acknowledgements

This chapter is derived from Machina (1995), which was presented as the Geneva Risk Lecture at the 21st Seminar of the European Group of Risk and Insurance Economists (“Geneva Association”), Toulouse, France, 1994. I have benefited from the comments of Michael Carter, Georges Dionne, Christian Gollier, Peter Hammond, Edi Karni, Mike McCosker, Garey Ramey, Suzanne Scotchmer, Joel Sobel, Alan Woodfield, and anonymous reviewers.

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  1. Department of Economics, University of California, San Diego, La Jolla, California, 92093, USA

    Mark J. Machina

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    Georges Dionne

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Machina, M.J. (2013). Non-Expected Utility and the Robustness of the Classical Insurance Paradigm. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_3

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