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Exercise 5.1. Consider the following game-frame in strategic form, where o,,02,0; and o, are basic outcomes: Player 2 O 02 Player 1 03 OA Both players satisfy the axioms of expected utility. The best outcome for Player 1 is o, ; she is indifferent between outcomes o, and o, and ranks them both as worst; she considers o to be worse than o, and better than of; she is indifferent between o, with certainty and the lottery 9 . The best 0.25 0.75) outcome for Player 2 is of, which he considers to be just as good as of ; he considers o, to be worse than o, and better than o, ; he is indifferent between 0, with certainty and the lottery (0.4 0.6) Find the normalized von Neumann-Morgenstern utility functions for the two players and write the corresponding reduced-form game. 205 GAME THEORY - Giacomo Bonanno Exercise 5.2. Consider the following game-frame, where o,,...,, are basic outcomes. Player 2 C D 01 O4 A N/- O / w N /- Player 1 B 03 w / w Both players have von Neumann-Morgenstern rankings of the basic outcomes. The ranking of Player 1 can be represented by the following von Neumann- Morgenstern utility function outcome: 02 03 04 U. : 12 10 6 16 and the ranking of Player 2 can be represented by the following von Neumann- Morgenstern utility function outcome: 0 02 0; 04 U, : 6 14 8 10 Write the corresponding reduced-form game.Player 1 ranks the outcomes as indicated by A below and Player 2 ranks the outcomes as indicated by B below (if outcome z is above outcome z' then z is strictly preferred to z', and if z and z' are written next to each other then the player is indifferent between the two). N Z 3 , ZA 26 A = 24, 21 B= Z 208 GAME THEORY - Giacomo Bonanno (a) One player has a strategy that is strictly dominated. Identify the player and the strategy. [Hint: in order to answer the following questions, you can make your life a lot easier if you simplify the game on the basis of your answer to part (a).] Player 1 satisfies the axioms of Expected Utility Theory and is indifferent between z, and the lottery and is indifferent between z, and the lottery (b) Suppose that Player 1 believes that Player 2 is going to play d with probability ; and e with probability } . Which strategy should he play? Player 2 satisfies the axioms of Expected Utility Theory and is indifferent between z, and the lottery ( 2 (c) Suppose that Player 2 believes that Player 1 is going to play a with probability + and c with probability 4. Which strategy should she play? (d) Find all the (pure- and mixed-strategy) Nash equilibria of this game. Exercise 5.10. Consider the following game (where the payoffs are von Neumann-Morgenstern payoffs): Player 2 C D X 3 Player 1 6 0 (a) Suppose that x = 2 and y = 2. Find the mixed-strategy Nash equilibrium and calculate the payoffs of both players at the Nash equilibrium. is C )( Nash equilibrium. Exercise 5.11. Find the mixed-strategy Nash equilibria of the game of Exercise 5.3. Calculate the payoffs of both players at every Nash equilibrium that you find. 209Exercise 5.12. In the following game, for each player, find all the rationalizable pure strategies (that is, apply the cardinal IDSDS procedure). Player 2 L M R Player A 3 , 5 2 , 0 2 , 2 5 , 2 1 , 2 2 , 1 9 ,0 1 , 5 3 , 2 Note: The next three exercises are more difficult than the previous ones. Exercise 5.13. Is the following statement true or false? (Either prove that it is true or give a counterexample.) "Consider a two-player strategic-form game with cardinal payoffs. Let A and B be two pure strategies of Player 1. Suppose that both A and B are rationalizable (that is, they survive the cardinal IDSDS procedure). Then any mixed strategy that attaches positive probability to both A and B and zero to every other strategy is a best reply to some mixed strategy of Player 2." Exercise 5.14. Consider the following three-player game, where only the payoffs of Player 1 are shown. Player 2 Player 2 E F E F 3 0 0 0 Player B 0 3 Player B 3 0 1 0 0 0 3 D 2 D 0 2 Player 3: G Player 3: H (a) Show that if Player 1 assigns probability { to the event "Player 2 will play E and Player 3 will play G" and probability { to the event "Player 2 will play F and Player will play H", then playing D is a best reply. 210 GAME THEORY - Giacomo Bonanno Next we want to show that there is no mixed-strategy profile - ((5 ,5) (9 4 of Players 2 and 3 against which D is a best reply for Player 1. We do this in steps. First define the following functions: A(p, q) = II,(A,G_,) (that is, A(p,q) is Player 1's expected payoff if she plays the pure strategy A against o_,), B(p,q) = II, (B,o_,), C(p,q) =II, (C,o_,) and D( p, q) = II, (D, o_,). (b) In the (p,q) plane (with 0 D(p,q). (c) In the (p,q) plane draw the curve corresponding to the equation C(p,q) = D(p, q) and identify the region where C(p, q) > D(p, q). (d) In the (p,q) plane draw the two curves corresponding to the equation B(p,q) = D(p, q) and identify the region where B(p, q) > D(p, q). (e) Infer from parts (b)-(c) that there is no mixed-strategy profile of Players 2 and 3 against which D is a best reply for Player 1.You have to go to a part of town where many people have been mugged recently. You consider whether you should leave your wallet at home or carry it with you. Of the four possible outcomes, your most preferred one is having your wallet with you and not being mugged. Being mugged is a very unpleasant experience, so your second favorite alternative is not carrying your wallet and not being mugged (although not having any money with you can be very inconvenient). If, sadly enough, your destiny is to be mugged, then you prefer to have your wallet with you (possibly with not too much money in it!) because you don't want to have to deal with a frustrated mugger. A typical potential mugger, on the other hand, does not care whether or not you are carrying a wallet, in case he decides not to mug you (that is, he is indifferent between the corresponding two outcomes). Of course his favorite outcome is the one where you have your wallet with you and he mugs you. His least preferred outcome is the one where he attempts to mug you and you don't have your wallet with you (he risks being caught for nothing). Denote the possible outcomes as follows: Potential mugger Not mug Mug You Leave wallet at home Take wallet with you (a) What is the ordinal ranking of the outcomes for each player? (b) Suppose now that both players have von Neumann-Morgenstern utility functions. You are indifferent between the following lotteries: furthermore, you are 1=(678 ) and4-. The potential mugger is indifferent between the two lotteries L - ( 17 ) and 4.- . For each player find the normalized von Neumann-Morgenstern utility function. 241 GAME THEORY - Giacomo Bonanno You have to decide whether or not to leave your wallet at home. Suppose that, if you leave your wallet at home, with probability p (with 0