Optimal risk sharing as a cooperative game

Łukasz Kuciński¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland and Polish Financial Supervision Authority Plac Powstańców Warszawy 1 00-950 Warszawa, Poland

Applicationes Mathematicae, Tome 38 (2011) no. 2, pp. 219-242

Résumé

The problem of choosing an optimal insurance policy for an individual has recently been better understood, particularly due to the papers by Gajek and Zagrodny. In this paper we study its multi-agent version: we assume that insureds cooperate with one another to maximize their utility function. They create coalitions by bringing their risks to the pool and purchasing a common insurance contract. The resulting outcome is divided according to a certain rule called strategy. We address the fundamental questions of profitability of cooperation and existence of strategies not rejected by any of the coalitions. These issues are closely related to the notion of Pareto optimality and the core of a game. We give a characterization of the former and prove the nonemptiness of the latter. Moreover, assuming that the pricing rule used by the insurance company is linear, we formulate necessary and sufficient conditions for the profitability of cooperation.

Détail

DOI : 10.4064/am38-2-7

Keywords: problem choosing optimal insurance policy individual has recently better understood particularly due papers gajek zagrodny paper study its multi agent version assume insureds cooperate another maximize their utility function create coalitions bringing their risks pool purchasing common insurance contract resulting outcome divided according certain rule called strategy address fundamental questions profitability cooperation existence strategies rejected coalitions these issues closely related notion pareto optimality core game characterization former prove nonemptiness latter moreover assuming pricing rule insurance company linear formulate necessary sufficient conditions profitability cooperation