Example of exam

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GAME THEORY 5 cfu 2022 Surname: Name: Matricola: The exam, or the single exercise, will not be evaluated if: • The Last name and name are not clearly written in the right place; • The explanations are not clearly written and contain cancellations. Exercise 1 Given the following bimatrix game, depending on the parameters a, b ∈ R,   (12, 6) (4 , 4) (3 , 5) (3, 6) (8 , 5) (2 , 7) (a, 3) (6 , 7) (3 , b)   , 1. find the best reaction of the first player to ( 1 3 , 1 3 , 1 3 ) for every a, b; 2. find a, b such that there is a Nash equilibrium with second player playing ( 1 3 , 1 3 , 1 3 ), and write the NEp; 3. show that there exist a, b such that there is a Nash equilibrium with the first player playing a pure strategy and the second one a mixed strategy where exactly two pure strategies are played with strictly positive probability, and specify the strategies. Answer 1 Answer 2 Answer 3 Short explanations 1 • In the boxes only answers; • If the answer is correct, explanations are revised and the score is assigned according to the accuracy of the explanations, if the answer is wrong 0 points are assigned. 2 Exercise 2 Consider the following glove game, with 4 players: • Player 1 has two right gloves: • Player 2 has one right glove; • Player 3 and 4 have a left glove each. 1. Is the game superadditive, convex?; 2. Find the core of the game; 3. Find the Shapley value of Player 1. Answer 1 Answer 2 Answer 3 Short explanations 3