B (4,1) D E E F (1,0) (5,2) (2,2) (1,1) (a) Find the pure-strategy Nash equilibria of the game. (b) Find the pure-strategy subgame-perfect equilibria of the game. (c) Find the mixed-strategy subgame-perfect equilibrium of the game. (Hint: Find the mixed-strategy Nash equilibrium of the subgame. Given this is played in the subgame, which action is chosen initially? Hence construct the mixed-strategy subgame-perfect equilibrium.)

Game theory problems For the game: row a b X Column у -3 2 4 -1 1. Does this game have a saddle point? 2. Find the optimal mixed strategy for each player. 3. What is the value of the game? 4. Interpret the solution. Please be as clear and legible as possible. Show and explain all the steps in detail. Thank you a lot.

.  The table below states the payoffs in political points (measured in billions of rubles) to two nations that are rivals in world politics, Russia and Ukraine. Each country can take one of two courses: peace; or war.  In each cell, the first payoff is for Russia, and the second payoff is for Ukraine.             (a)  Assume that neither country observes the military strategy of its rival, and solve the game (if it can be solved).  Explain your solution step-by-step.  Does this outcome maximize total political points?             (b)  In general, what is a Nash equilibrium?  Is the solution to this game a Nash equilibrium?             (c)  Suppose that each country deposits a fund of two billion rubles with the United Nations.  Either country would forfeit this fund if it wages war. What is the solution now to the game?  Is this a Nash equilibrium?        Russia, Ukraine Peace War Peace 4,4 1, 5 War 5, 1 2,2

4

Internet Audience At the end of a certain year, the four companies with the largest number of home Internet users in the United States were Microsoft, Time Warner, Yahoo, and Google, with a combine audience of 284 million users. Taking x to be the Microsoft audience in millions, y the Time Warner audience in millions, z the Yahoo audience in millions, and u the Google audience in millions, it was observed that z u 3(x- y) +26 = x+y=46+z+u and x y + zu 54. How large was the audience of each of the four companies at the end of that year? Microsoft Time Warner Yahoo Google

(4) Three firms, F₁, F2, F3 share the market for a certain commodity. F₁ has 20%, F2 has 60% and F3 has 20%. In the following year the changes accour: F₁ keeps 85% of its consumers, loses 5% to F2 and 10% to F3 F2 keeps 55% of its consumers, loses 10% to F₁ and 35% to F3 F3 keeps 85% of its consumers, loses 10% to F₁ and 5% to F2. Write the matrix S = the initial share and the matrix T = tij, the transformation matrix, where tij = what F; gets from Fj. Compute T - S,T· (TS), T · (T · (T· S)). What do they represent? (5) Calculate the following determinants: 3 -1 2 0 −1 -1 6 1 2 2-0 3 -1 0 4 0 0 0 1 -1 2 3 2 5 -3 30 1 1 0 -1 20 5

Q1

Ant Co. has developed a new product, the A-Warren. It is now time to bring the A-Warren product to market. There are two alternatives for Ant-Co either market the product only in the local area, or market the product nationally. If Ant Co. rolls out the A-Warren product locally and it is successful, then the company will receive $1.4M from product sales. However, if the local rollout is unsuccessful, then the company will lose $100,000 ($0.1M) due to the costs of advertising. If Ant Co. rolls out the A-Warren product nationally and it is successful, then the company will receive $3M from product sales. However, if the national rollout is unsuccessful, then the company will lose $1M due to the costs of advertising. Historically, 40% of Ant Co.'s product rollouts have been successful. Which of the following statements are true? A decision alternative is to rollout the A-Warren product successfully. O An event outcome is a successful national A-Warren rollout with a payoff of $1.4M. An…

Ant Co. has developed a new product, the A-Warren. It is now time to bring the A-Warren product to market. There are two alternatives for Ant-Co either market the product only in the local area, or market the product nationally. If Ant Co. rolls out the A-Warren product locally and it is successful, then the company will receive $1.4M from product sales. However, if the local rollout is unsuccessful, then the company will lose $100,000 ($0.1M) due to the costs of advertising. If Ant Co. rolls out the A-Warren product nationally and it is successful, then the company will receive $3M from product sales. However, if the national rollout is unsuccessful, then the company will lose $1M due to the costs of advertising. Historically, 40% of Ant Co.'s product rollouts have been successful. What is the most that Ant Co should pay for any information regarding the success of the A-Warren product? $0.00M $0.54M None of the answers are correct. $1.14M $0.60M

Ant Co. has developed a new product, the A-Warren. It is now time to bring the A-Warren product to market. There are two alternatives for Ant-Co either market the product only in the local area, or market the product nationally. If Ant Co. rolls out the A-Warren product locally and it is successful, then the company will receive $1.4M from product sales. However, if the local rollout is unsuccessful, then the company will lose $100,000 ($0.1M) due to the costs of advertising. If Ant Co. rolls out the A-Warren product nationally and it is successful, then the company will receive $3M from product sales. However, if the national rollout is unsuccessful, then the company will lose $1M due to the costs of advertising. Historically, 40% of Ant Co.'s product rollouts have been successful. Ant Co. could choose to use a focus group to provide feedback on the A-Warren. In the past when Ant Co.'s products were successful (locally or nationally), the focus group predicted this 80% of the time.…