We study game-theoretic models of human evolution to analyze fundamentals of human nature. Rival-claimants games represent common situations in which animals can avoid conflict over valuable resources by mutually recognizing asymmetric claiming rights. Unlike social-dilemma games, rival-claimants games have multiple equilibria which create a rational role for communication, and so they may be good models for the role of language in human evolution. Many social animals avoid conflict by dominance rankings, but intelligence and language allow mutual recognition of more complex norms for determining political rank or economic ownership. Sophisticated forms of economic ownership could become more advantageous when bipedalism allowed adaptation of hands for manufacturing useful objects. Cultural norms for claiming rights could develop and persist across generations in communities where the young have an innate interest in learning from their elders about when one can appropriately claim desirable objects. Then competition across communities would favor cultures where claiming rights are earned by prosocial behavior, such as contributions to public goods. With the development of larger societies in which many local communities share a common culture, individuals would prefer to interact with strangers who identifiably share this culture, because shared cultural principles reduce risks of conflict in rival-claimants games.