Consider the stage game below. Can we get (A,Y) played each date in an infinitely repeated game? Explain. You may assume our players have the same discount factor d. 6. The minimum value of the discount factor 8 needed to support (A,Y) being played each date in an infinitely repeated game is ? Assume a grim trigger strategy. 112 Y Z A 12, 6 2,8 14, 2 4, 4
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- Consider the game with the payoffs below. Which of the possible outcomes are MORE efficient than the Nash Equilibrium (NE)? Note, they do NOT need to be Nash equilibria themselves, they just need to be more efficient than the NE. Multiple answers are possible, but not necessary. You need to check ALL correct answers for full credit. JILL High Medium LowMAGGIE Left 3,4 2,3 2,2Center 4,8 9,7 8,7Right 7,6 8,5 9,4Group of answer choices (Left, Low) There is no strategy combination that is more efficient than the Nash equilibrium for this game. (Right, Medium) (Left, High) (Center, Medium) (Center, High) (Center, Low) (Left, Medium) (Right, Low) (Right, High)Consider the following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.64. (This problem assumes knowledge of the basic rulesof baseball.) George Lindsey (1959) looked at boxscores of more than 1000 baseball games and foundthe expected number of runs scored in an inning foreach on-base and out situation to be as listed in the fileP09_64.xlsx. For example, if a team has a man on firstbase with one out, it scores 0.5 run on average untilthe end of the inning. You can assume throughout thisproblem that the team batting wants to maximize theexpected number of runs scored in the inning.a. Use this data to explain why, in most cases,bunting with a man on first base and no outs isa bad decision. In what situation might buntingwith a man on first base and no outs be a gooddecision?b. Assume there is a man on first base with one out.What probability of stealing second makes an attempted steal a good idea?
- 5. 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…A) Focus on the strategic game at the lower-right side of the gametree. Find all the Nash equilibria for this subgame, including the mixed-strategyones. (b) Find all the subgame perfect equilibria for the entire game, allowingfor both pure and mixed strategies1) What are the Nash equilibria? Which one is unreasonable/non-credible threat? 2) What are the subgame perfect Nash equilibria? Does SPNE concept eliminate the unreasonable Nash equilibrium?
- Consider the location game we covered in Lecture 3. Now assume there arethree players (vendors). As we assumed in the lecture, consumers in each area choosethe closest vendor and if there are multiple closest vendors then these vendors receiveequal share of consumers in the area. Notice Si = {1, 2, 3, ...., 9} for i = 1, 2, 3. Here aresome examples of payoffs: u1(1, 1, 1) = 3, u1(1, 1, 9) = u2(1, 1, 9) = 2.25, u3(1, 1, 9) =4.5, u1(1, 5, 9) = u3(1, 5, 9) = 2.5 and u2(1, 5, 9) = 4. (a) Is s′1 = 1 strictly dominated by s′′1 = 2 for player 1?(b) Is s′1 = 1 weakly dominated by s′′1 = 2 for player 1?(c) Can you find a Nash equilibrium in pure strategies?w= 7 To which strategy t2 is the strategy t1 = 4 a best response? (If your answer is afraction, report it in lowest terms.)1. Consider the pricing game between firm 1 (row) and firm 2 (column) belowwhere each number represents the profits made by each firm.:Table 1: Pricing gameLow HighLow 288,288 360,216High 216,360 324,324 a) Point out the Nash equilibrium. Is this a unique Nash equilibrium?c) What type of commitment can firm 1 make to improve its pay-off?Can it be classified as a threat or promise? (Assume that the commitmentis credible.)d) How would you represent the game as a game tree with the option tocommit?2. Consider the signalling / production game with the order of play given below.Assuming that the signal is informative (i.e. education is sufficiently costly),how would you change the order of the game to make sure that poolingequilibriums are ruled out? Verbally motivate your answer.• Nature chooses ability of the workers a ∈ (H, L), ability is observed byworkers but not employers.• The worker choose an education level s ∈ (0, 1).• The employer offer contracts w(s).• The worker accepts or…
- What is Ann's maximin strategy? Game Bob L RAnn U 10,-1 4, 4 D 4, 1 8, -1 Select one: a.none of the other answers b.4/7 U + 3/7 D c.2/5 U+ 3/5 D d.2/7 U + 5/7 D e.3/5 U + 2/5 DConsider the strategic form game shown. a. Assume that both players are rational. What happens?b. Assume that both players are rational and that each believes that theother is rational. What happens?c. Find the strategies that survive the ISDS.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.
