Abstract: We systematically study pairwise counter-monotonicity, an extremal notion of negative dependence. A stochastic representation and an invariance property are established for this dependence structure. We show that pairwise counter-monotonicity implies negative association, and it is equivalent to joint mix dependence if both are possible for the same marginal distributions. We find an intimate connection between pairwise counter-monotonicity and risk sharing problems for quantile agents. This result highlights the importance of this extremal negative dependence structure in optimal allocations for agents who are not risk averse in the classic sense.
Abstract: We design the insurance contract when the insurer faces arson-type risks. We show that the optimal contract must be manipulation-proof. Therefore, it is continuous, has a bounded slope, and satisfies the no-sabotage condition when arson-type actions are free. Any contract that mixes a deductible, coinsurance, and an upper limit is manipulation-proof. A key feature of our models is that we provide a simple, general, and entirely elementary proof of manipulation-proofness that is easily adapted to different settings. We also show that the ability to perform arson-type actions reduces the insured’s welfare as less coverage is offered in equilibrium.
Abstract: We address the problem of sharing risk among agents with preferences modelled by a general class of comonotonic additive and law-based functionals that need not be either monotone or convex. Such functionals are called distortion riskmetrics, which include many statistical measures of risk and variability used in portfolio optimization and insurance. The set of Pareto-optimal allocations is characterized under various settings of general or comonotonic risk sharing problems. We solve explicitly Pareto-optimal allocations among agents using the Gini deviation, the mean-median deviation, or the inter-quantile difference as the relevant variability measures. The latter is of particular interest, as optimal allocations are not comonotonic in the presence of inter-quantile difference agents; instead, the optimal allocation features a mixture of pairwise counter-monotonic structures, showing some patterns of extremal negative dependence.
Abstract: This paper examines optimal risk sharing for empirically realistic risk attitudes, providing results on Pareto optimality, competitive equilibria, utility frontiers, and the first and second theorems of welfare. Empirical studies suggest, contrary to classical assumptions, that risk seeking is prevalent in particular subdomains, and is even the majority finding for losses, underlying for instance the disposition effect. We first analyze cases of expected utility agents, some of whom may be risk seeking. Yet more empirical realism is obtained by allowing agents to be risk averse in some subdomains but risk seeking in others, which requires generalizing expected utility. Here we provide first results, pleading for future research. Our main new tool for analyzing generalized risk attitudes is a counter-monotonic improvement theorem.
Keywords: Gambling behaviour, counter-monotonicity, competitive equilibrium, convex utility functions, rank-dependent utility
* Previously circulated as Negatively dependent optimal risk sharing
Abstract: We examine the trade-off between the provision of incentives to exert costly effort (ex-ante moral hazard) and the incentives needed to prevent the agent from manipulating the profit observed by the principal (ex-post moral hazard). Formally, we build a model of two-stage hidden actions where the agent can both influence the expected revenue of a business and manipulate its observed profit. We show that manipulation-proofness is sensitive to the interaction between the manipulation technology and the probability distribution of the stochastic output. The optimal contract is manipulation-proof whenever the manipulation technology is linear. However, a convex manipulation technology sometimes leads to contracts with manipulations in equilibrium. We identify a regularity condition guaranteeing that we can always find a manipulation technology for which the optimal contract is not manipulation-proof.
Abstract: We define a class of optimization problems for which the value function is always almost everywhere differentiable, even when the objective function is discontinuous in the choice sets. We call this class of optimization problem positioning choice problems as they have a straightforward interpretation of a choice of position. We show that the Dini superdifferential is always well-defined for maxima of positioning choice problems. This property allows stating first-order necessary conditions in terms of Dini superdifferential. We then prove our main result, an ``ad-hoc" envelope theorem for positioning choice problems. Lastly, we apply our results to the producer's problem of choosing an optimal level of capital when output is indivisible (Tobin's $q$) and to the design of mechanisms.