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- Consider a game where each player picks a number from 0 to 60. The guess that is closest to half of the average of the chosen numbers wins a prize. If several people are equally close, then they share the prize. The game theory implies that (A) all players have dominant strategies to choose 0 (B) all players have dominant strategies to choose 30 (C) there is a Nash equilibrium where all players pick 0 (D) there is a Nash equilibrium where all players pick positive numbers Behavioral data in such games suggests that (A) most subjects choose 0; (B) most subjects choose 30;(C) common answers include 30, 15, 7.5, and 0; (D) most subjects use randomization.Micro Nash game theory Show that if in a game G, the sets of actions (A i) i in N are compact, and the payment functions (u i) i in N are continuous, then for all i in N, the application of Best response has a closed graph. Conclude that the application of best response has a closed graph.2. Provide the normal-form expression to the extensive-form game below, and find all pure-strategy Nash equilibria there. Find the subgame-perfect Nash equilibrium then. A B X E Y B A B F t G F G BF B D A 0 3. Find the subgame perfect Nash equilibria (may or may not be unique) in the game below. A B 100 30 F 60 9 5 B 3 5 12 10 60 10 80 10 180 60 200 40
- In a game theory payoff matrix. Your company (A) and a major competitor (B) havetwo potential strategies: to advertise or to not advertise during the Super Bowl. The payoffs in each cell represent the change in firm profits from advertising. How would you create payoffs in each cell such that the Nash equilibrium is that both firms advertise despite having a higher profit if neither firm advertised2. Consider a finitely repeated game with a unique Nash equilibrium in each stage game. What is the subgame-perfect outcome for such a game?II. Determine which of the following two-person zero-sum games are strictly determinable and fair. Give the optimum strategies for each player in the case of it being strictly determinable. a- Player A Player B B1 B2 B3 A1 8 2 4 A2 5 -1 3 b. II II V V I -4 -2 -2 1 A II 1 -1 III -6 -5 -2 -4 4 IV 3 1 -6 0 -8 3.
- Solve for the Nash equilibrium (or equilibria) in each of the following games. (a) The following two-by-two game is a little harder to solve since firm 2’spreferred strategy depends of what firm 1 does. But firm 1 has a dominantstrategy so this game has one Nash equilibrium. Firm 2 Launch Don’tFirm 1 Launch 60, -10 100, 0 Don’t 80, 30 120, 0 What is the Nash equilibrium of this simultaneous-move game? (b) What would the outcome of this game be if instead firm 1 moved first and then, after seeing what firm 1 chose, firm 2 chose it strategy? In this case firm 1 doesn’t necessarily need to choose a best response, but firm 2 must choose a best response since it moves second.You are playing a game with a friend. It’s yourmove but you don’t have a dominant strategy.Your payoff depends on what your friend doesafter your move. You consider flipping a coin todecide what to do. You are about to reach for acoin, but then you realize that your friend has adominant strategy. Explain how using backwardinduction (rather than a coin toss) will now determine your next moveSuppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.
- Two farmers have unlimited access to a common plot of land and can let their cows graze on it. The matrix below shows the benefits they get from grazing either 1 or 2+ cows on the land. Farmer 2 Farmer1 1 cow 2+ cows 1 cow 8,8 2,10 2+cows 10,2 4,4 What kind of game is this? What is/are the Nash equilibrium/equilibria? What is/are the Pareto efficient outcome(s) in this game? (Hint: Remember that Pareto efficiency occurs when no one person can be made better off without someone else being made worse off) The government offers a reward or subsidy for communities where farmers only allow 1 cow to graze on the common field, resulting in a new payoff matrix:…3) Consider the following two extensive form games and answer the following questions. How many subgames present in the following games? Find all pure strategies SPNE. Find all pure strategies NE of these games. Game A Game B u (5,3) u (0,2) 2 2 L L (-5,4) d (0,0) 1 1 1 (1,-5) (-10,-10) u R (2,3) A M 2 R 2 r d 1 (8,1) B (2,0) (2,2)1\2 L 1,-1 | 5,0 | 0,0 0,5 4,4 0,0 0,0 3,3 M R C 0,0 В 1. Find all set of rationalizable strategies (Hint: We are allowed to eliminate a pure strategy, if it is strictly dominated by a mixed strategy). 2. Find all set of pure strategy Nash equilibria. 3. Is there a dominant strategy equilibrium for this game?