Question: Exercise: (Nash demand game)* Consider the problem of dividing a pie between two players. If we let x and y denote player l's and player 2's payoffs, the vector (x, y) is feasible if and only if x ≥ x0, y ≥ y0, and g(x, y) ≤ 1, where g is a differentiable function with ∂g/∂x > 0 and ∂g/∂y > 0 (for instance, g(x, y) = x + y). Assume that the feasible set is convex. The point (x0, y0) will be called the status quo. Nash (1950a) proposed axioms which implied that the "right" way to divide the pie is the allocation (x*, y*) that maximizes the product of the differences from the status quo (x - x0)(y y0) subject to the feasibility constraint g(x, y) ≤ 1. In his 1953 paper, Nash looked for a game that would give this axiomatic bargaining solution as a Nash equilibrium.
(a) Suppose that both players simultaneously formulate demands x and y. If(x, y) is feasible, each player gets what he demanded. If (x, y) is infeasible, player I gets x0 and player 2 gets y0. Show that there exists a continuum of pure-strategy equilibria, and, more precisely, that any efficient division (x. y) (i.e.. feasible and satisfying g(x, y) = 1) is a pure-strategy-equilibrium Outcome.
(b)** Consider Binmore's (1981) version of the Nash "modified demand game." The feasible set is defined by x ≥ x0, y ≥ y0, and g(x,y) ≤ z, where has cumulative distribution F on (suppose that ∀z, the feasible set is nonempty). The players do not know the realization of z before making demands. The allocation is made as previously, after the demands are made and z is realized. Derive the Nash-equilibrium conditions. Show that when F converges to a mass point at 1, any Nash equilibrium con-verges to the axiomatic bargaining solution.