Consider an ultimatum game. P1 offers a share x [0,1] of a dollar to P2; P2 can either accept, in which case the dollar is divided as proposed, or reject, in which case both players get $0. Assume P1 is a standard self- interested rational player, but he is uncertain about P2 and fears he might be playing an envious or spiteful type. Suppose specifically that he believes P2 is a standard rational player with probability = 1 but an envious type with 'envy' parameter > 0 with probability . That is, if P1's share 1 x is larger than P2's share x, then P2 incurs an envy cost (1 2x) if he accepts. Find the equilibrium of this game as a function of . What happens if rises?