2. In game theory, the Nash equilibrium, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, thus no one has anything to gain by changing only one's own strategy. In this question we consider the system of two players playing against each other. Let R be the set of available strategies. If the first player choose strategy & E R and the second player choose strategy y E R then f(x, y) = 4at - jyx3 - 4x2 + yx - by2 + 2y represents the loss of the first player (that is also the gain of the second player). In order for (To, yo) to be a Nash equilibrium (a stable state such that no participant can gain by changing his strategy if the strategies of the others remain unchanged), the point r = To must be a global minimum of go(@) := f(x, yo) and the point y = yo must be a global maximum of holy) := f(To, y) (a) Find Vf(x, y) and the Hessian matrix I(x, y). [3] (b) Find all stationary points of f. [4] Hint: there are three stationary points. (c) Classify the stationary points of f as local maxima/minima or saddle points. [4] (d) Explain why any Nash equilibrium for the system of two players in the question description [3] is one of the saddle points found in (c). (More generally, we have that in a two-player zero sum game defined on a continuous space, the equilibrium point is a saddle point). (e) Which of the saddle points found in (c) are Nash equilibria? Justify your answer. [4]