Altruism and the ultimatum game Imagine the ultimatum game from Figure 11.1 played by two utilitarians with u (x, y) = √x + √y. Find the unique subgame-perfect equilibrium in this game. The literature on social preferences underscores a point from the previous chapter, namely that the game people are playing depends on their utility functions. The following exercise illustrates how agents with different utility functions end up playing very different games, even when their interactions superficially might look identical.