We consider estate division problems and show that for any claim game based on a (estate division) rule satisfying efficiency, equal treatment of equals, and order preservation of awards, all (pure strategy) Nash equilibria induce equal division. Next, we consider (estate division) rules satisfying efficiency, equal treatment of equals, and claims monotonicity. Then, for claim games with at most three agents, again all Nash equilibria induce equal division. Surprisingly, this result does not extend to claim games with more than three agents. However, if nonbossiness is added, then equal division is restored.