Claim games for estate division problems

The estate division problem considers the issue of dividing an estate when the sum of entitlements is larger than the estate. This paper studies the estate division problem from a noncooperative perspective. The integer claim game introduced by O''Neill (1982) and extended by Atlamaz et al. (2011) is generalized by specifying a sharing rule to divide every interval among the claimants. We show that for all problems for which the sum of entitlements is at most twice the estate the existence of a Nash equilibrium is guaranteed for a general class of sharing rules. Moreover, the corresponding set of equilibrium payoffs is independent of which sharing rule in the class is used. Well-known division rules that always assign a payoff vector in this set of equilibrium payoffs are the adjusted proportional rule, the random arrival rule and the Talmud rule.