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

Jaroslav Pazdera, Hans Schumacher, Bas Werker

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We 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.
Original languageEnglish
Pages (from-to)122-133
JournalJournal of Mathematical Economics
Volume72
DOIs
Publication statusPublished - Oct 2017

Fingerprint

Risk Sharing
Fairness
Pareto Efficiency
Composite
Iteration
Monotone Mapping
Composite materials
One to one correspondence
Existence and Uniqueness of Solutions
Global Convergence
Eigenvalues and eigenfunctions
Iterative Algorithm
Eigenvector
Pricing
Division
Rate of Convergence
Costs
Sharing rule
Risk sharing
Pareto efficiency

Keywords

  • risk sharing
  • fair division
  • Perron-Frobenius theory
  • eigenvector computation
  • collectives

Cite this

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abstract = "We 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.",
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The composite iteration algorithm for finding efficient and financially fair risk-sharing rules. / Pazdera, Jaroslav; Schumacher, Hans; Werker, Bas.

In: Journal of Mathematical Economics, Vol. 72, 10.2017, p. 122-133.

Research output: Contribution to journalArticleScientificpeer-review

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AB - We 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.

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