Explain whether the statement is true or false and explain why? Statement 1: If a game has a Pareto efficient outcome, there exists a Nash equilibrium that leads to this outcome. Statement 2: If someone has linear indifference curves between contingent commodity bundles, the she must be risk averse and the risk premium is positive. Statement 3: Suppose we have a game with three pure strategies. In a (fully) mixed strategy Nash equilibrium, a player is indifferent between her three pure strategies
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- 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.For the operating systems game, let us now assume the intrinsic superiorityof Mac is not as great and that network effects are stronger for Windows.These modifications are reflected in different payoffs. Now, the payoff fromadopting Windows is 50 X w and from adopting Mac is 15 + 5 X m;n consumers are simultaneously deciding between Windows and Mac.a. Find all Nash equilibria.b. With these new payoffs, let us now suppose that a third option exists,which is to not buy either operating system; it has a payoff of 1,000.Consumers simultaneously decide among Windows, Mac, and nooperating system. Find all Nash equilibria.Consider Bernard \ Mary Left Center Right Top 0,5 1,0 2,2 Bottom 1,0 0,3 2,2 The first number in a cell denotes the payoff to Bernard and the second number denotes the payoff to MaryForexample: πB(B,L)=1and πM(T,L)=5. a Give all pure strategy Nash equilibria of this one-shot game, if any. Briefly explain.Let Bernard play Top with probability p and Bottom with probability 1 − p; let Mary play Left with probability qL , Center with probability qC and Right with probability qR = 1 − qL − qC . b Give all mixed strategy Nash equilibria of this game.
- Find all NE of the stage game.(b) Consider a two-period game without discounting in which the stage game is played ineach period. Find all pure strategy SPNE.(c) What’s the min-max payoff of each player?(c1) Consider pure strategies only.(c2) Consider all strategies, including the mixed ones.(d) Now suppose the stage game is repeated infinitely many times. Use the Fudenberg-Maskin Folk theorem to find all possible values of payoff that can be supported as aSPNE.We’ll now show how a college degree can get you a better job even if itdoesn’t make you a better worker. Consider a two-player game between aprospective employee, whom we’ll refer to as the applicant, and an employer. The applicant’s type is her intellect, which may be low, moderate,or high, with probability 1/3 , 1/2 , and 1/6 , respectively. After the applicantlearns her type, she decides whether or not to go to college. The personalcost in gaining a college degree is higher when the applicant is less intelligent, because a less smart student has to work harder if she is to graduate. Assume that the cost of gaining a college degree is 2, 4, and 6 for an applicant who is of high, moderate, and low intelligence, respectively.The employer decides whether to offer the applicant a job as a manageror as a clerk. The applicant’s payoff to being hired as a manager is 15,while the payoff to being a clerk is 10. These payoffs are independent ofthe applicant’s type. The employer’s payoff from…Consider the game of Chicken in which each player has the option to “get out of the way” and “hang tough” with payoffs: Get out of the way Hang tough Get out of the way 2,2 1,3 Hang tough 3,1 00 a. Find all pure strategy Nash equilibria, if they exist b. Let k be the probability that player 1 chooses “hang tough” and u be the probability that player two chooses “hang tough.” Find the mixed stragety Nash equilibria, if they exist
- Consider a modified Traveler’s Dilemma. In terms of strategy options that the players have and the dollars they earn, it is like the standard Traveler’s Dilemma, but the players do not have endless appetite for money. Up to 100 dollars, each dollar feels like a dollar. But any moneybeyond 100 is psychologically like 100 dollars. Assuming that players are maximizers of ‘psychological’ dollars instead of real dollars, describe all the Nash equilibria of this modified Traveler’s Dilemma.Consider the following variation to the Rock (R), Paper (P), Scissors (S) game:• Suppose that the Player 1 (row player) has a single type, Normal.• Player 2 (column player) has two types Normal and Simple.• A player of Normal type plays this zero-sum game as we studied in class whereas a player of type Simple always play P.• Player 2 knows whether he is Normal or Simple, but player 1does not.a) Suppose player 2 is of type Normal with probability 1/3 and of type Simple with probability (2/3). Find all pure strategy Bayesian Nash Equilibria.b) Suppose player 2 is of type Normal with probability 2/3 and of type Simple with probability (1/3). Find all pure strategy Bayesian Nash Equilibria.on 8.1 Consider the following game: Player 1 A C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1, 1 F 0,0 1,1 4,4 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game.
- 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 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.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?