The following game is played by 2 players Jack Left Middle Right Adam Up 1,2 3,5 2,1 Middle 0,4 2,1 3,0 Down -1,1 4,3 0,2 a) Determine all dominant strategies for Adam and Jack. b) Solve the equilibrium for this game.
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- Does each individual in a prisoners dilemma benefit more from cooperation or from pursuing self-interest? Explain briefly.Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.Suppose O2 and Vodafone are the only two telecommunicationscompanies in UK. Both companies are considering whether ornot to stop offering unlimited data plans. Each company has twostrategies: stop or don’t stop. The first entry in the brackets is the payoffsof O2 and the second entry is the payoffs of Vodafone, both in $million.What will be the dominant strategies of O2 and Vodafone and what willbe the Nash equilibrium? Explain your answers.
- 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 bargaining problem of splitting a pie of size 1 with utility u(x1) = x1 for player 1 and v(x2) = 2x2 − x22 for player 2, where x1 and x2 denote the share of the pie for player 1 and 2 respectively. a) Consider the bargaining problem of the two players. Find the utility possibility frontier S. b) What is the Nash bargaining solution for this problem (i.e., on which division of the pie (?₁,?₂) will players agree), if the disagreement outcome (the utilities players obtain in case of disagreement) is d1 = d2 = 0? c) What is the Nash bargaining solution if the disagreement outcome is any d1 and d2 in S?Splitting Pizza: You and a friend are in an Italian restaurant, and the owner offers both of you a free eight-slice pizza under the following condition. Each of you must simultaneously announce how many slices you would like; that is, each player i ∈ 1, 2 names his desired amount of pizza, 0 ≤ si ≤ 8. If s1 + s2 ≤ 8 then the players get their demands (and the owner eats any leftover slices). If s1 + s2 > 8, then the players get nothing. Assume that you each care only about how much pizza you individually consume, and the more the better.What outcomes can be supported as pure-strategy Nash equilibria?
- In the attached table determine, for each player, whether any action is strictly or weakly dominated. Justify youranswer. Find the Nash equilibria of the game and determine whether any equilibrium is strict.Suppose two players play a two-period repeated game, where the stage game is the normal-form game shown below. Is there a subgame perfect Nash equilibrium in which the players select (A, X) in the first period? If so, fully describe such equilibrium. If not, explain why not. Player 1 has choice A, B; Player 2 has choice X, Y, Z. Payoff: (A,X)-(5,7), (A,Y)-(2,4), (A,Z)-(3,8), (B,X)-(1,4), (B,Y)-(3,5), (B,Z)-(1,4)Jesse and Janie are playing a game. On the table in front of them, there are 50 coins. They take turns removing some of these coins from the table. At each turn, the person moving can remove either 1,2,3,4,5,6, or 7 coins (they cannot remove 0 or 8, etc.). The person to remove the last coin wins. Suppose Janie is the first player to have a turn (a) In the subgame perfect equilibrium, Janie begins by taking _________coins (b) Suppose Janie makes a mistake in play and removes a number of coins that leaves 27 coins on the table when it is jesses turn. To have a chance at winning, Jesse should remove _________ Coins
- A game involves two players: player A and player B. Player A has three strategies a1, a2 and a3 while player B has three strategies b1, b2 and b3. Player B b1 b2 b3 a1 -40,30 70,20 -10,120 Player A a2 40,60 80,80 60,20 a3 -30,40 -50,110 150, -70 Assuming that this is a one-time game, answer the following questions: Is there any dominant strategy for each player? What is the secure strategy of each player. What is the Nash equilibrium of the game?if Y = 4 (a) If ⟨a,d⟩ is played in the first period and ⟨b,e⟩ is played in the second period, what is the resulting (repeated game) payoff for the row player? (b) What is the highest payoff any player can receive in any subgame perfect Nash equilibrium of the repeated game?3.1 Do either of the two telecommunications firms have a dominant strategy in this interaction?If so, what are these dominant strategies? 3.2 What is the Nash Equilibrium of the game above? Clearly, show the logic you use to reachyour conclusion. What type of game is this? 3.3 Suppose the two firms could incentivize or punish each other, could the two firms find theirway to the socially optimum outcome? How would they do this? After observing the strategic interaction between Globogym and Average Joe’s, the governmentdecides to pass a law that states that the two terms must pre-commit to the quantities of trainingsessions they will supply to the American market.Market demand for training sessions is still modeled as ? = 400 − 0.2?, as before, and the marginalcost of production is constant at R40 per call. Let the number of sessions provided by Globogym berepresented by ?G and the quantity provided by Average Joe’s be represented by ?A. 3.4 Solve the firms’ reaction functions and…