Write down the payoff function upi in the game gpi for

Question

Consider the following two-player game with payoffs in R²:

(a) Write down the payoff function U_π in the game Gπ for every π = (π₁, π₂) ∈ R₂ (for the definition of Uπ see Equation (14.68) on page 588).

b) Draw the graph of the function val(Gπ ). That is, on the two-dimensional plane, where the x axis is identified with π₁, and the y axis is identified with π₂, draw val(Gπ₁,π₂ ) at each point.

(c) For the following sets C, compute the value of min  as a function of π, and determine which of them are approachable by Player 1.

(i) C1 = {(1/2, 1/2)}.

(ii) C2 = [(0,1), (1,0)].

(iii) C3 = [(0,1), (0,0)].

(iv) C4 is the triangle whose vertices are (1,1), (1,0), and (1/2,1/2).

(v) C5 is the parallelogram whose vertices are (1/2, 1), (1/4, 1), (1/2, 0), and (3/4,0).