The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

Abstract

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules Pazdera, J.; Schumacher, J.M.; Werker, B.J.M. Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. AbstractWe consider the problem of finding an efficient and fair ex-ante rule for division of an uncertain monetary outcome among a finite number of von Neumann-Morgenstern agents. Efficiency is understood here, as usual, in the sense of Pareto efficiency subject to the feasibility constraint. Fairness is defined as financial fairness with respect to a predetermined pricing functional. We show that efficient and financially fair allocation rules are in one-to-one correspondence with positive eigenvectors of a nonlinear homogeneous and monotone mapping associated to the risk sharing problem. We establish relevant properties of this mapping. On the basis of this, we obtain a proof of existence and uniqueness of solutions via nonlinear Perron-Frobenius theory, as well as a proof of global convergence of the natural iterative algorithm. We argue that this algorithm is computationally attractive, and discuss its rate of convergence.

Multi-Period Risk Sharing Under Financial Fairness

“…We shall show the existence and uniqueness of the PEFF solution, and give a numerical algorithm to find it. This can be seen as a direct generalization of the research by Pazdera et al [33], which explores the Pareto efficient and financially fair risk-sharing rule in a single-period case. Compared to Barrieu and Scandolo [7], we restrict ourselves to the case of expected utility as the preference functional, and risk-neutral valuation is used to determine a unique solution.…”

“…Given the prespecified market values of the payments, the PEFF approach can optimize the risk allocation while keeping the market values equal to the prespecified values. The feasibility of finding a unique PEFF solution has been investigated theoretically by Pazdera et al [33] in a single-period setting and by Bao et al [5] (see Chapter 2) in a multi-period setting.…”

“…It is given exogenously. It can then be shown that it is possible to find such a solution that is PE while the FF constraints are satisfied at the same time; moreover, the PEFF solution is unique; see Bühlman and Jewell [14], Gale [19] and Pazdera et al [33]. The y's are increasing functions of X which can be solved numerically.…”

A Multi-Objective Interpretation of Optimal Transport

“…Convergence of the iterative proportional fitting procedure and of its generalization to infinite dimensions has been studied extensively; see for instance [37,38,39,40,41,42,43]. Motivated by a MOMA problem in risk sharing, Pazdera et al [17] proved convergence of the iterative procedure for a broad class of strictly concave utilities, on the basis of a nonlinear version of the Perron-Frobenius theorem due to [44]. One may write down an analogous iterative procedure for optimal transport problems, or equivalently for their multi-objective counterparts.…”

“…Up to a positive factor, which we can ignore on the basis of Remark 2.5, the corresponding regularization of the MOMA problem is g 1 (x) = 1/x. From (17), it follows that the MOMA regularization in this case can be written as…”