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.