12. The following table contains the possible actions and payoffs of players 1 and 2. Player 2 Player 1 Cooperate Not Cooperate a. 50 b. 100 Cooperate 450, 450 700, 100 This game is repeated 100 times, and at each period both players must choose their actions simultaneously. The players are deciding whether or not to cooperate. They are considering a simple agreement: Cooperate as long as no firm has deviated, and Not Cooperate forever if some firm deviates. Players discount the future according to the interest rate r=0.60. In this case, how many times will the players cooperate? c. 60 d. 40 e. None of the above Not Cooperate 100, 700 300, 300

Consider the following price game:                   Firm 1     Firm 2                   High         Low         High  20, 20       12, 24          Low  24, 12       14, 14 Remark: In simultaneous move games (games with rows and columns) theconvention is to write the row player’s payoff first and the column player’spayoff second. (a) What is the Nash equilibrium of this game? Recall that for each playeryou should find the best response to each of the opponents’ strategies andunderline the associated payoff. Then look for a cell where both strategiesare best responses to each other. This is a Nash equilibrium. (b) Does either firm have a dominate strategy (a strategy that is always abest response)?

practice

Consider the extensive form game portrayed below. The top number at aterminal node is player 1’s payoff, the middle number is player 2’s payoff,and the bottom number is player 3’s payoff.a. Derive the strategy set for each player. (Note: If you do not want to listall of the strategies, you can provide a general description of a player’sstrategy, give an example, and state how many strategies are in thestrategy set.)b. Derive all subgame perfect Nash equilibria.                                                c. Derive a Nash equilibrium that is not a SPNE, and explain why it isnot a SPNE.

5. Aaron and Betty play the following one-shot game. Aaron Up Middle Down Left 10,0 5,10 1,1 Betty Right 0,5 1,0 2,2 a. Does the game above have any (pure strategy) Nash equilibria? Find any equilibria or explain why an equilibrium doesn't exist. b. Suppose instead that Aaron moves first and Betty moves second. Represent this situation with a game tree and find the subgame perfect Nash equilibrium. Are Aaron and Betty better off playing a simultaneous game or a sequential game in which Aaron moves first? Explain. c.

Use the following one-shot, normal-form game to answer the following questions a. Find each player’s dominant strategy, if it exists. b. Find each player’s secure strategy. c. Find the Nash equilibrium.

2. Consider the following game. Two criminals are thinking about pulling off a bank robbery. The take from the bank would be $20,000 each, but the job requires two people (one to rob the bank and one to drive the getaway car. Each criminal could instead rob a liquor store. The take from robing a liquor store is only $1000 but can be done with one person acting alone. Write the payoff matrix of this game Player A Bank job Liqour store Player B Bank job Liquor store a. What are the Nash equilibria in this game? b. Explain why there can be multiple equilibria in this game. c. How the game will be played if mixed strategies are allowed? Discuss.

K In a game, there are two players: Player 1 and Player 2. They have two strategies to select from, A and B. Where the first payoff and the second payoff in every cell is for Player 1 and Player 2 respectively. Which of the following is an example of a game without a dominant strategy? (Check all that apply.) A. C. Player 1 Player 1 A B A B A 3,-3 1, 1 A 2,1 0,0 Player 2 Player 2 B 3, -3 5, -5 B 0,0 1,2 B. D. Player 1 Player 1 A B A B A 1,-1 -1, 1 Player 2 Player 2 A 3.5, -1.5 2.7, -1 B -1, 1 1,-1 B -5, 3 0,0 If the dominant strategy for Player 1 is selecting strategy A and the dominant strategy for Player 2 is strategy B, then A B is an example of a c From ame C Murat Çok Di id Kushn y My Hand

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5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:ˆA writes down either number 1 or number 2, and B must guess which one.ˆIf the number that A has written down is i and B has guessed correctly, B receives i units from A.ˆIf B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.ˆSuppose B guesses 1 with probability p and 2 with probability 1 −p.ˆLet X1 denote B’s gain (or loss) in a game where A has written down 1.ˆLet 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 AˆSuppose that A writes down 1 with probability q and 2 with probability 1 −q.ˆLet Y1 be A’s loss (or gain) if B chooses number 1.ˆLet Y2 be A’s loss (or gain) if B…