Consider a game that game theory people refer to as the “ultimatum game.” We will refer to our two players as the “offerer” and the “decider”. How the game works is that the offerer proposes a way to split $1000 between the two players. While this could be done in a variety of ways, we will assume that the offerers only has two possible proposals: Either a 50-50 split, or she offers the decider $50 and keeps the rest. The decider can either accept or reject the offer. If the offer is accepted, the money is split as proposed. If the offer is rejected, the money spontaneously combusts and nobody gets anything.
a) Assume that the offerer acts first, then the decider. List the strategies for each player and write an extensive form version of the game with payouts.
b) List all the Nash equilibria of this game.
c) Explain which, if any of the Nash equilibrium are not sub-game perfect.
d) Write the game as a normal form game assuming decisions must be made simultaneously. Find the Nash equilibrium of this game and compare with your answers to (b) and (c).