Two players have $1 to divide. Player 1 offers to keep $x. Player 2 either accepts this proposal or rejects it in which case both players get zero.

a) Find the perfect equilibrium.

b) Now suppose that whereas player 1 is self-interested and only cares about his own monetary payoff, player 2 may not be. Player 2's utility function is given by

w2 = u2 - a max (u1 - u2, 0) - b max (u2-u1, 0)

where u1 and u2 are the monetary payoffs for players 1 and 2 respectively and a and b are positive parameters. a measures inferiority aversion and b measures superiority aversion. Player 1 does not know a and b but assumes a=0 with probability 60% and a=4 with probability 40% and b=0 with probability 45% and b=0.6 with probability 55%. How does this change your answer to a)?

c) For the data in b), what is the chance that player 2 rejects player 1's proposal at equilibrium?