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This paper considers the estate division problem from a non-cooperative perspective. The integer claim game initiated by O'Neill (1982) and extended by Atlamaz et al. (2011) is generalized by considering different sharing rules to divide every interval among the claimants. For problems with an estate larger than half of the total entitlements, we show that every sharing rule satisfying four fairly general axioms yields the same set of Nash equilibrium profiles and corresponding payoffs. Every rule that always results in such equilibrium payoff vector is characterized by the properties minimal rights first and lower bound of degree half. Well-known examples are the Talmud rule, the adjusted proportional rule and the random arrival rule. Then our focus turns to more specific claim games, i.e. games that use the constrained equal awards rule, the Talmud rule, or the constrained equal losses rule as a sharing rule. Also a variation on the claim game is considered by allowing for arbitrary instead of integer claims.