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GAME THEORY The ball has been served and we nd Venus and Serena in the middle of the termis point. At the moment, Venus has

GAME THEORY

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The ball has been served and we nd Venus and Serena in the middle of the termis point. At the moment, Venus has come forward to the net and played a volley to Serene at the baseline. Serena must choose tn hit the ball one of three ways: crocourt (denoted G], downtheline (denoted D}, or lob (denoted L]. As Serena is making her decision, Venus is simultaneously making hers: should she shift her momentum towards the downtheline shot {d}, or towards the crosscourt shot [c]? [f Venus moves towards the incoming shot, this will improve her chances to successfully defend against that shot and to win the point. PayoHs to each player are the probability that they win the point. These payoffs are given in the payoff matrix below: Venus d c 3mm D (cane) (neon) C [e.e,e.1) (ages) L [o.7,e.3) (.,.4] {a} Verify that the game has no pure strategy Nash equilibrium. Is there a MSNE? How can you be sure? {b} Suppose that Venus plays the mixed strategy on; = get + [1 q]c. lGraph Serena's pure strategy payoffs as a. function of q (Hint: you can use a graphical technique similar to what we did in class with the i'alth and Goal" example.) {c} For which g values is D a best response? For which if values is U a best response? For which 9 values is L a best response? For which 9 values could Serena. be indifferent between two or more of her pure strategies while also playing a best response? Please state these :3 values explicitly [i.e. solve them algebraically, rather than estimating them off your graph]. {d} Using the results above, and the observation that high payoffs for Serena mean low payoffs for Venus, which .3 value should Venus play to maximize her expected payoE? {e} Find a mixed strategy Nash equilibrium

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