91Ó°ÊÓ

(Nash equilibria of the ultimatum game) Find the values of \(x\) for which there is a Nash equilibrium of the ultimatum game in which person 1 offers \(x\).

(Bargaining over two indivisible objects) Consider a variant of the ultimatum game, with indivisible units. Two people use the following procedure to allocate two desirable identical indivisible objects. One person proposes an allocation (both objects go to person 1 , both go to person 2, one goes to each person), which the other person then either accepts or rejects. In the event of rejection, neither person receives either object. Each person cares only about the number of objects she obtains. Construct an extensive game that models this situation and find its subgame perfect equilibria. Does the game have any Nash equilibrium that is not a subgame perfect equilibrium? Is there any outcome that is generated by a Nash equilibrium but not by any subgame perfect equilibrium?

(Stackelberg's duopoly game with quadratic costs) Find the subgame perfect equilibrium of Stackelberg's duopoly game when \(C_{i}\left(q_{i}\right)=q_{i}^{2}\) for \(i=1,2\), and \(P_{d}(Q)=\alpha-Q\) for all \(Q \leq \alpha\) (with \(P_{d}(Q)=0\) for \(Q>\alpha\) ). Compare the equilibrium outcome with the Nash equilibrium of Cournot's game under the same assumptions (Exercise \(57.2\) ).

(Hungry lions) The members of a hierarchical group of hungry lions face a piece of prey. If lion 1 does not eat the prey, the game ends. If it eats the prey, it becomes fat and slow, and lion 2 can eat it. If lion 2 does not eat lion 1, the game ends; if it eats lion 1 then it may be eaten by lion 3, and so on. Each lion prefers to eat than to be hungry, but prefers to be hungry than to be eaten. Find the subgame perfect equilibrium (equilibria?) of the extensive game that models this situation for any number n of lions.

(Sequential positioning by three political candidates) Consider a further variant of Hotelling's model of electoral competition in which the \(n\) candidates choose their positions sequentially and each candidate has the option of staying out of the race. Assume that each candidate prefers to stay out than to enter and lose, prefers to enter and tie with any number of candidates than to stay out, and prefers to tie with as few other candidates as possible. Model the situation as an extensive game and find the subgame perfect equilibrium outcomes when \(n=2\) (easy) and when \(n=3\) and the voters' favorite positions are distributed uniformly from 0 to 1 (i.e. the fraction of the voters' favorite positions less than \(x\) is \(x\) ) (hard).

(Hungry lions) The members of a hierarchical group of hungry lions face a piece of prey. If lion 1 does not eat the prey, the game ends. If it eats the prey, it becomes fat and slow, and lion 2 can eat it. If lion 2 does not eat lion 1 , the game ends; if it eats lion 1 then it may be eaten by lion 3, and so on. Each lion prefers to eat than to be hungry, but prefers to be hungry than to be eaten. Find the subgame perfect equilibrium (equilibria?) of the extensive game that models this situation for any number \(n\) of lions.