Four notions of mean preserving increase in risk, risk attitudes and applications to the Rank-Dependent Expected Utility model

This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel & Lehmann dispersion or (its equal-mean version) Quiggin's monotone mean-preserving increase in risk and Jewitt's location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the Expected Utility setup. These notions of risk are also relevant tothe Quiggin-Yaari Rank-dependent Expected Utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM Expected Utility model by Rothschild & Stiglitz's Mean Preserving Increase in Risk (MPIR). Realizing that in the broader rank-dependent set-up this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose. This paper reviews four notions of mean-preserving increase in risk - MPIR, monotoneMPIR and two versions of location-independent risk (renamed here left and right monotone MPIR) - and shows which choice questions are consistently modeled by each of these four orders.Location-independent risk, monotone increase in risk, rank-dependent expected utility.

Comonotonic Processes

We consider in this paper two Markovian processes X and Y, solutions of a stochastic differential equation with jumps, that are comonotonic, i.e., that are such that for all t, almost surely, X_{t} is greater in one state of the world than in another if and only if the same is true for Y_{t}. This notion of comonotonicity can be of great use for finance, insurance and actuarial issues. We show here that the assumption of comonotonicity imposes strong constraints on the coefficients of the diffusion part of X and Y.Comonotonicity, Comonotonic processes, Jump processes, Risk sharing schemes, Pareto optimal allocations

Responsabilité civile et contrôle des activités représentant des risques mal connus

This note focuses on the design of prevention programmes and the role of tort law regarding the control of risky activities, associated with unknown or imperfectly known risks, such as innovation or (long term) environmental damages. Together with the existence of perception bias on the side of citizens, these risks are specific in that they are not insurable.

Safety and the Allocation of Costs in Large Accidents

We study the characteristics of optimal levels of care and distribution of risk in a extended unilateral accident model, where 1/ parties are Rank Dependant Expected Utility maximizers, which allows us to capture two important behavioral characteristics in risk, both pessimism (probability transformation) and risk aversion; 2/ there exists an aggregate/uninsurable risk in case of accident ; 3/ tortfeasors have the opportunity to invest in damages reduction activities having a monetary cost of effort. Important results show that the optimal care is larger than under the risk neutral/small risks case, it depends on the aggregate wealth of society but does not depend on wealth distribution. We then examine whether ordinary liability rules, with or without insurance, can be used to implement the first-best outcome.K13

Diversification, convex preferences and non-empty core in the Choquet expected utility model

This paper explores risk-sharing and equilibrium in a general equilibrium set-up wherein agents are non-additive expected utility maximizers. We show that when agents have the same convex capacity, the set of Pareto-optima is independent of it and identical to the set of optima of an economy in which agents are expected utility maximizers and have same probability. Hence, optimal allocations are comonotone. This enables us to study the equilibrium set. When agents have different capacities, matters are much more complex (asin the vNM case). We give a general characterization and show how it simplifies when Pareto-optima are comonotone. We use this result to characterize Pareto-optima when agents have capacities that are the convex transform of some probability distribution. comonotonicity of Pareto-optima is also shown to be true in the two-state case if the intersection of the core of agents' capacities is non-empty; Pareto-optima may then be fully characterized in the two-agent, two-state case. This comonotonicity result does not generalize to more than two states as we show with a counter-example. Finally, if there is no-aggregate risk, we show thatnon-empty core intersection is enough to guarantee that optimal allocations are full-insurance allocation. This result does not require convexity of preferences.Choquet expected utility; comonotonicity; risk-sharing; equilibrium

Optimal risk-sharing rules and equilibria with Choquet-expected-utility

This paper explores risk-sharing and equilibrium in a general equilibrium set-up wherein agents are non-additive expected utility maximizers. We show that when agents have the same convex capacity, the set of Pareto-optima is independent of it and identical to the set of optima of an economy in which agents are expected utility maximizers and have same probability. Hence, optimal allocations are comonotone. This enables us to study the equilibrium set. When agents have different capacities, matters are much more complex (as in the vNM case). We give a general characterization and show how it simplifies when Pareto-optima are comonotone. We use this result to characterize Pareto-optima when agents have capacities that are the convex transform of some probability distribution. comonotonicity of Pareto-optima is also shown to be true in the two-state case if the intersection of the core of agents' capacities is non-empty; Pareto-optima may then be fully characterized in the two-agent, two-state case. This comonotonicity result does not generalize to more than two states as we show with a counter-example. Finally, if there is no-aggregate risk, we show that non-empty core intersection is enough to guarantee that optimal allocations are full-insurance allocation. This result does not require convexity of preferences.Choquet expected utility; comonotonicity; risk-sharing; equilibrium

Optimal risk-sharing rules and equilibria with Choquet-expected-utility.

This paper explores risk-sharing and equilibrium in a general equilibrium set-up wherein agents are non-additive expected utility maximizers. We show that when agents have the same convex capacity, the set of Pareto-optima is independent of it and identical to the set of optima of an economy in which agents are expected utility maximizers and have the same probability. Hence, optimal allocations are comonotone. This enables us to study the equilibrium set. When agents have different capacities, the matters are much more complex (as in the vNM case). We give a general characterization and show how it simplifies when Pareto-optima are comonotone. We use this result to characterize Pareto-optima when agents have capacities that are the convex transform of some probability distribution. Comonotonicity of Pareto-optima is also shown to be true in the two-state case if the intersection of the core of agents' capacities is non-empty; Pareto-optima may then be fully characterized in the two-agent, two-state case. This comonotonicity result does not generalize to more than two states as we show with a counter-example. Finally, if there is no-aggregate risk, we show that non-empty core intersection is enough to guarantee that optimal allocations are full-insurance allocation. This result does not require convexity of preferences.Equilibrium; Risk-sharing; Comonotonicity; Choquet expected utility;

Social wealth and optimal care

Many accidents result in losses that cannot be perfectly compensated by a monetary payment. Moreover, often injurers control the magnitude rather than the probability of accidents. We study the characteristics of optimal levels of care and distribution of risk under these circumstances and show that care depends on the aggregate wealth of society but does not depend on wealth distribution. We then examine whether ordinary liability rules, regulation, insurance, taxes and subsidies can be used to implement the first-best outcome. Finally, our results are discussed in the light of fairness considerations (second best).accidents, risk, wealth, care, bodily injury