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- 1 True or False : Every finite extensive - form game of imperfect information admits at least one pure - strategy Nash equilibrium . Justify if true or give a counter - example if not1. (Dynamic Game with Perfect Information)33. When neither player has a dominant strategy, A) game theory will not provide information.B) no Nash-Equilibrium exists.C) at least one Nash-Equilibrium exists.D) the game cannot be analyzed.
- 6)Game Theory: The Prisoners’ Dilemma: Assume that the Wilson and Spalding athletic equipment companies are in a one-shot game for market share and profits, but that they have the option of choosing only one of two possible price strategies for basketballs: $20 or $80. Obviously if they choose different strategies, the firm with the lower price will win the entire market. The firms face the following payoff matrix (See Chap. 10). Wilson\ \Spalding Wilson $ 20 Price Wilson $ 80 Price Spalding $ 20 Price $ 400, $ 400 $ 1500, $ 0 Spalding $ 80 Price $ 0, $ 1500 $ 1000, $ 1000 What strategy will each firm choose? Why? Which strategy is dominant? Which strategy is preferred by each firm? What will be the outcome of the one-shot game? Where is the Nash equilibrium? Which outcome would be best for the two firms? What will happen if the game is repeated an infinite number of times?8 Please use the game theory matrix to answer the following question We have 110,000 Homes • The low price is $79.95 and the high price is 10% more • Set up fees of $49.50 + $6.50 per subscriber The company's profit is based on its price and its competitor’s price: What is the contribution margin per house in each scenario? *Contribution margin = Price – total costs3) Indicate whether the following statement is TRUE or FALSE and explain your answer: If one player is the leader in a game it means that s/he has the first move in the game. The key thing is that this move can be observed by the follower before making his/her own decision, otherwise it would be a simultaneous move game that cannot be solved by backwards induction.
- 1. Solve for the mixed strategy Nash equilibrium of the game above. Explain carefully how you solve for it. 2. In the equilibrium you calculated above, what is the probability that both consoles are released in October? In December? What are the expected payoffs of firm A and of firm B in equilibrium?6 Two people will select a policy that affects both of them by applying a "veto" in a sequential and alternate manner, that is: person 1 begins to veto a policy and then person 2 exercises his "veto" with the remaining policies; the process repeats until only one policy remains. Assume that there are 3 policies: X,Y,Z, and that person 1 prefers X to Y to Z and person 2 prefers Z to Y to X. a. Represents the game extensively b. Give the number of subgames C. Indicate the total strategies of the players d. find all subgame perfect nash equilibria e. Find all Nash Equilibriums.H3. Cooperation in a Finite Game We have seen that if we play the Prisoners' dilemma with finite repetition, there will be no cooperation in the subgame-perfect equilibrium. There are other games, however, when cooperation can be enforced for at least some time in a finite repetition as well. Give an example with explanation. (Hint: consider games with more than one Nash equilibria.)
- 12 Please provide explanation and example of -Infinite vs. Finite Horizon Games -ThanksConsider a bankruptcy game with two risk neutral players where V =$800,000, C1= $300,000 and C2=$800,000. a) What is the Nash bargaining solution?John enjoys playing two-player zero-sum games. The matrix below shows the losses to John in a particular two-player zero-sum game. His strategies are denoted by I, II, and III, whereas the strategies for his opponent are denoted by A, B, and C. I II III A 3 2 4 B 0 1 1 C 1 3 0 (a) Explain which of John’s strategies is dominated.The opponent now has the option of a fourth strategy, D, which results in none ofJohn’s strategies being dominated.(b) Suggest possible values for the strategy D.
