Suppose two players are asked to split $100 in a way that is agreeable to both.
A: The structure for the game is as follows: Player 1moves first—and he is asked to simply state some number between zero and 100. This number represents his “offer” to player 2 — the amount player 1 offers for player 2 to keep, with player 1 keeping the rest. For instance, if player 1 says “30”, he is offering player 2 a split of the $100 that gives $70 to player 1 and $30 to player 2. After an offer has been made by player 1, player 2 simply chooses from two possible actions: either “Accept” the offer or “Reject” it. If player 2 accepts, the $100 is split in the way proposed by player 1; if player 2 rejects, neither player gets anything. (A game like this is often referred to as an ultimatum game.)
(a)What are the sub game perfect equlibria in this game assuming that player 1 is restricted to making his “offer” in integer terms—i.e. assuming that player 1 has to state a whole number.
(b) Now suppose that offers can be made to the penny — i.e. offers like $31.24 are acceptable. How does that change the sub game perfect equilibria? What if we assumed dollars could be divided into arbitrarily small quantities (i.e. fractions of pennies)? (c) It turns out that there are at most two sub game perfect equilibria to this game (and only 1 if dollars are assumed to be fully divisible)—but there is a very large number of Nash equilibria regardless of exactly how player 1 can phrase his offer (and an infinite number when dollars are assumed fully divisible). Can you, for instance, derive Nash equilibrium strategies that result in player 2 walking away with $80? Why is this not sub game perfect?
(d) This game has been played in experimental settings in many cultures — and, while the average amount that is “offered” differs somewhat between cultures, it usually falls between
$25 and $50, with players often rejecting offers below that. One possible explanation for this is that individuals across different cultures have somewhat different notions of “fairness”— and that they get utility from “standing up for what’s fair”. Suppose player 2 is willing to pay $30 to stand up to “injustice” of any kind, and anything other than a 50-50 split is considered by player 2 to be unjust. What is now the sub game perfect equilibrium if dollars are viewed as infinitely divisible? What additional sub game perfect equilibrium arises if offers can only be made in integer amounts?
(e) Suppose instead that player 2 is outraged at “unfair” outcomes in direct proportion to how far the outcome is removed from the “fair” outcome, with the utility player 2 gets from rejecting an unfair offer equal to the difference between the amount offered and the “fair” amount. Suppose player 2 believes the “fair” outcome is splitting the $100 equally. Thus, if the player faces an offer x < 50, the utility she gets from rejecting the offer is (50−x). What are the sub game perfect equilibria of this game now under the assumption of infinitely divisible dollars and under the assumption of offers having to be made in integer terms?
B: Consider the same game as that outlined in A and suppose you are the one that splits the $100 and I am the one who decides to accept or reject. You think there is a pretty good chance that I am the epitome of a rational human being who cares only about walking away with the most I can from the game. But you don’t know me that well—you think there is some chance ρ that I am a self-righteous moralist who will reject any offer that is worse for me than a 50-50 split. (Assume throughout that dollars can be split into infinitesimal parts.)
(a) Structure this game as an incomplete information game.
(b) There are two types of pure strategy equilibria to this game (depending on what value ρ takes). What are they?
(c) How would your answer change if I, as a self-righteous moralist (which I am with probability ρ) reject all offers that leave me with less than $10?
(d) What if it’s only less than $1 that is rejected by self-righteous moralists?
(e) What have we implicitly assumed about risk aversion?