Q1. Connie and Stephen must decide how to split a pie. Suppose both of them simultaneously formulate demands x and y. These demands are feasible if x ≥ 0 , y ≥ 0 and x + y ≤ 1: If (x, y ) is feasible, Connie and Stephen get exactly what they demanded. If (x, y ) is not feasible, they both get zero.
(a) Show that any efficient allocation (that is, x+y = 1) is a Nash equilibrium in pure strategies.
(b) Can you find a Nash equilibrium in pure strategies that is not efficient?
(c) Suppose now the size of the pie is T. Feasibility requires now that x+y ≤ T. Also, Connie and Stephen don't know the exact value of T but they know that T is a random variable uniformly distributed on [0,1]. Hence, they get their demands if the realized T is greater of equal to x + y, otherwise they get zero. Find all pure strategy Nash equilibria of this game.
Q2. In some legislatures, proposals for modifications of the law are formulated by committees. Under the "closed rule", the legislature may either accept or reject a proposed modification, but may not propose an alternative. In the event of a rejection, the existing law is unchanged. Model an outcome as a number y ∈ [0; 1] , and let y0 represent the status quo. The legislature and the committee do not share the same preferences over policy outcomes. In particular, the legislature has preferences represented by the payoff function ul (y) = - y, while the committee's preferences are represented by the utility function uc (y) = - |y - yc| which yc > 0 .
(a) Model this procedure under the "closed rule" as an extensive game and find the sub game perfect equilibrium as a function of the status quo outcome y0.
(b) Show that for a range of values of y0 , an increase in the value of y0 leads to a decrease in the value of the equilibrium outcome (i.e., in some cases a worse status quo for the legislature may lead to a better policy outcome for the legislature).