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- Consider the following normal form game Player 2 L C R Player 1 5,4 4, 5 3,8 8,3 8,0 8,8 5,5 6,3 0,8 UMD5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:A writes down either number 1 or number 2, and B must guess which one.If the number that A has written down is i and B has guessed correctly, B receives i units from A.If B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.Suppose B guesses 1 with probability p and 2 with probability 1 −p.Let X1 denote B’s gain (or loss) in a game where A has written down 1.Let X2 denote B’s gain (or loss) in a game where A has written down 2.(a) Find the pmf of X1 and X2(b) Find B’s expected gain for these two cases, E[X1] and E[X2].(c) What value of p maximizes the minimum possible value of B’s expected gain?Now consider the expected loss of player ASuppose that A writes down 1 with probability q and 2 with probability 1 −q.Let Y1 be A’s loss (or gain) if B chooses number 1.Let Y2 be A’s loss (or gain) if B…Example 1: Zero Sum Game Game Table 2 Game Table 1 Player 2 Player 2 S1 S2 Si S2 RI (3,-3) (-5,5) Player 1 R1 3 -5 RI (0,0) (7,-7) R1 7 Player 1 All players payoff shown =) Fnd Nasn ewwliLtlun> zero sum shorthand
- Please help awnser 7. Game TheoryAlice and Bob playing the following 2x3 game. In the payoff matrix below, Alice’s payoffsare the first number and Bob’s are the second number.you and a friend decide to run a three mile race. If you agree to run together, you keep up with himfor the first mile, but you overexert yourself and run the last two miles at slower paces on your own. Tomake up for lost time, your friend runs the last two miles at a faster pace. Your mile times are 6:30, 7:00,and 7:30. Your friend’s times are 6:30, 6:00, and 6:00. If you both agree to run on your own, you run aconstant pace of 7:05 while your friend runs at a constant pace of 6:05. If you want to run together butyour friend wants to run solo, he runs his constant pace of 6:05. You, on the other hand, want to showhim that you can run faster, but you end up overexerting yourself after the first mile. You run times of6:20, 7:05, and 7:30. If he wants to run together but you do not, you both run at your pace of 7:05. Thissituation can be turned into an economic game, with the payoffs the overall race times. You each wantto run the fastest time you possibly can.(a) Who are the players in…Two friends, Khalid and Mahmood, are going to a watch a world cup football match. They play a simple game in which they hold out one or two fingers to decide who will pay for the other's ticket. Khalid wins if the fingers held out add up to an even number; Mahmood wins if the fingers held out add up to an odd number. The price of the ticket is 25 OMR. Construct a payoff matrix for the game. Is there a unique Nash equilibrium in this game? Which strategy should a player use to maximize her chances of winning the game?
- Consider “Providing a Public Good under Incomplete Information". (refer to pages 70-74, lecture notes.) If c and cz have the following distribution. 2. 0.5 1.2 C2 0.5 1.2 Prob 1/2 1/2 Prob 1/3 2/3 Find all Bayesian Nash Equilibria of this game. Select all cOTrect amswers. Don't copy Chegg give new answer ASAP4 Problem: Ticket Exchange Each of two players draw a lottery ticket with a random prize written on it. You can assume that prizes a uniformly drawn from an interval between 0 and 100 (can you solve it without uniform assumption?). Both player simultaneously decide whether to keep the prize or exchange the ticket with another player. If both decide to exchange, then the tickets are swapped and each player collects a prize from the newly acquired ticket. Find the BNE of the game. Hint: consider cut-off strategies, i.e. that prizes above some x are kept, and below z are offered for exchange.rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.
- 3 Use the following payoff matrix for a one-shot game to answer the accompanying questions. 20 oints eBook Player 1 Strategy A B Player 2 X Y 12, 12 -30, 6 6, -30 30, 30 a. Determine the Nash equilibrium outcomes that arise if the players make decisions independently, simultaneously, and without any communication. Instructions: In order to receive full credit, you must make a selection for each option. For correct answer(s), click the box once to place a check mark. For incorrect answer(s), click twice to empty the box. ? (-30, 6) (12.12) ? (6.-30) ? (30, 30) Which of these outcomes would you consider most likely? O (30,30) O(-30, 6) O (12, 12) O(6,-30) b. Suppose player 1 is permitted to "communicate" by uttering one syllable before the players simultaneously and independently make their decisions. What should player 1 utter? OA OB What outcome do you think would occur as a result? O (12, 12) O (-30, 6) O (30, 30) 0 (6,-30) c. Suppose player 2 can choose its strategy before player 1,…QUESTION 9 y a 3, 5 4, 2 4, 1 2, 3 b 2, 4 6, 6 5, 1 1, 2 1,3 7,3 4, 2 3, 7 d 1, 2 5, 2 3, 4 4, 1 How many rationalizable outcomes are there in this game? Give your answer as a whole number. C.05. Solve the following game. Player A 1 0 5 732 Player B 276 4 8 10