5 players must simultaneously choose a number from the interval [0, 9]. Let xi denote the number chosen by player i; x = (x1, . . . , x5) all the selected numbers; x−i = x\xi all the selected numbers except the one chosen by player i. If player i chooses the number xi ∈ [0, 9], while the others choose numbers x−i, then the utility of player i is given by ui(xi, x−i) = 4(9 − xi) + 2Σ5 j=1 xj. Find the Nash equilibrium of the game and the corresponding utilities. Would all players be better off if they signed a binding contract before the game, obligating them to choose the number 9?
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5 players must simultaneously choose a number from the interval [0, 9]. Let xi denote the number chosen by player i; x = (x1, . . . , x5) all the selected numbers; x−i = x\xi all the selected numbers except the one chosen by player i. If player i chooses the number xi ∈ [0, 9], while the others choose numbers x−i, then the utility of player i is given by ui(xi, x−i) = 4(9 − xi) + 2Σ5 j=1 xj. Find the Nash equilibrium of the game and the corresponding utilities. Would all players be better off if they signed a binding contract before the game, obligating them to choose the number 9?
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- n people guess an integer between 1 and 100, and the winner is the player whose guess is closest to the mean of the guesses + 1 (ties broken randomly). Which of the following is an equilibrium: a) All announce 1. b) All announce 50. c) All announce 75. d) All announce 1005.Each of Player 1 and Player 2 chooses an integer from the set {1, 2, ..., K}. If they choose the same integer, P1 gets +1 and P2 gets -1; if they choose different integers, P1 gets -1 and P2 gets +1. (a) Show that it is a NE for each player to choose every integer in {1, 2, ..., K} with equal probability, K1 . (b) Show that there are no NE besides the one you found in (a).= - Five players must simultaneously choose a number from the interval [0, 9]. Let x; denote the number chosen by player i; x = (x1, ..., x5 ) all selected numbers; x_{ x\x; all selected numbers except the number chosen by player i. If player i chooses the number x; Є [0,9], while the others choose the numbers _, then the utility of player i is u; (xi, x_i) = 4(9 − xi) + 2-1 xj. Find the Nash equilibria of the game and the corresponding utilities. Would all players be better off if they signed a binding contract before the game requiring them to choose the number 9?
- Two players play the following game for infinite times. For the player to continue to cooperate what would be the ranges of their discount factor, δ_1 and δ_2, respectively? cooperate betray cooperate (10,20) (-25,30) betray (15, -22) (-12, -18)An amount of money is accumulating; in period t (= 1, 2, . . . , T) its size is $2t. In each period two people simultaneously decide whether to claim the money. If only one person does so, she gets all the money; if both people do so, they split the money equally; and if neither person does so, both people have the opportunity to do so in the next period. If neither person claims the money in period T, each person obtains $T. Each person cares only about the amount of money she obtains. Formulate this situation as an extensive game with perfect information and simultaneous moves, and find its subgame perfect equilibria). (Start by considering the cases T = 1 and T = 2.)Two oil companies are deciding how much oil to extract from their properties, which lie above the same underground reservoir. The faster that oil is extracted, the less total oil is extracted. Letting x denote the extraction rate for company X and y denote the extraction rate for company Y, we assume that the total amount of oil extracted is 1/(x + y) million gallons of oil. Of the total amount that is extracted, the share going to company X is x/(x + y), and the share to company Y is y/(x + y); that is, a company’s share depends on how fast it extracts compared with the other company. The price of oil is $100 per gallon. Each company chooses its extraction rate from the interval [1,10] in order to maximize the monetary value of the oil that it extracts. Find the Nash equilibrium extraction rates. (Note: You can assume that the payoff function is hill shaped.)
- Kimberly's sister would like to start a business with her brother selling simple T-shirts that are green in color at all stores in the area. Her brother disagrees and thinks that the shirts should have a special logo on them and should be sold only at specific stores. As the deciding vote, what should Kimberly choose and why? Choose one: A selling green T-shirts because prices will be higher as the number of stores increases B. selling green T-shirts because prices will be higher as the shirt becomes more commonplace OC. selling shirts with a special logo because prices will be higher as the shirts becomes more unique OD. selling shirts with a special logo because prices will be higher as the shirts are sold in fewer stores OE. both C and D(ii) A mixed strategy profile (p, q) is one in which p = (p,P2.... P) is the mixed strategy of player 1, and q- (g1, q2,..q4) is the mixed strategy of player 2. Show that if p, >0 in a Nash equilibrium profile (p*, q*), the player 2 must also play i with strictly positive probability q'; > 0. (State clearly any theorem you use to show this. You are not required to justify the theorem.) %3DQUESTION 24 A deck of 52 playing cards consists of 4 suites: clubs, spades, hearts, and diamonds. Each suite consists of 13 cards. Draw one from a deck of cards. Let A=a card "2" is drawn, B=a "diamond" card is drawn. Which is the value of P(A|B), i.e., {probability that the card drawn is "2" given that we have known it is "diamond"} ? 1/52 1/4 1/13 16/52
- Player 2 Y1 Y2 Y3 X1 1,4 8,4 7,4 6,2 4,5 Player 1 X2 3,5 3,5 X3 5,3 2,2Find the perfect Bayesian equilibria in the following two-person game: B. Chance 1/4 U 3/4 2 L \R (:) (?) (3) (6) () (:) (:) (*) 8Arielle is a risk-averse traveler who is planning a trip to Canada. She is planning on carrying $400 in her backpack. Walking the streets of Canada, however, can be dangerous and there is some chance that she will have her backpack stolen. If she is only carrying cash and her backpack is stolen, she will have no money ($0). The probability that her backpack is stolen is 1/5. Finally assume that her preferences over money can be represented by the utility function U(x)=(x)^0.5 Suppose that she has the option to buy traveler’s checks. If her backpack is stolen and she is carrying traveler’s checks then she can have those checks replaced at no cost. National Express charges a fee of $p per $1 traveler’s check. In other words, the price of a $1 traveler’s check is $(1+p). If the purchase of traveler’s checks is a fair bet, then we know that the purchase of traveler checks will not change her expected income. Show that if the purchase is a fair bet, then the price (1+p) = $1.25.