Two players bargain over a pie of size 1, and have a time-based discount factor of δ.  That is, the size of the pie doesn’t change, but the players are impatient (or short-lived) such that the value of all transactions diminishes over time.  Additionally, both players feel a bit of spite towards each other, such that they gain a separate payoff s for rejecting the other player’s offer at the time. For parts (a) through (c), assume s is relatively small (for context, see part d).
a)Describe the outcome of the SPNE for the T=1 game, i.e. the Ultimatum Game, with spite.
b)Re-do part (a) for the infinite horizon game.
c) For the infinite horizon game, what is the critical value s^* such that, for s≥s^*, all SPNE simply result in stubborn rejection of offers ad infinitum?  In other words, at what point do the two sides hate each other too much to come to an agreement?  Hint: It’s less than 1.