6. Battle of the Networks Sport 42;58 12 %share Sitcom Nature Sitcom 48;52 40;60 55:45 58;42 Ans: s; = (0; 0.85; 0.14); V, = 55.43 s; = (0.43; 0.57;0); V½ = 44.57 Sport Nature 56:44 52:48 63:37 60;40 %3D Find Nash Equilibrium.
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- 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:…5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:A writes down either number 1 or number 2, and B must guess which one.If the number that A has written down is i and B has guessed correctly, B receives i units from A.If B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.Suppose B guesses 1 with probability p and 2 with probability 1 −p.Let X1 denote B’s gain (or loss) in a game where A has written down 1.Let X2 denote B’s gain (or loss) in a game where A has written down 2.(a) Find the pmf of X1 and X2(b) Find B’s expected gain for these two cases, E[X1] and E[X2].(c) What value of p maximizes the minimum possible value of B’s expected gain?Now consider the expected loss of player ASuppose that A writes down 1 with probability q and 2 with probability 1 −q.Let Y1 be A’s loss (or gain) if B chooses number 1.Let Y2 be A’s loss (or gain) if B…L 0 4 15.1 X₂ A M R 0 1 4 0 2 B L 4 0 с X3 M R 0 1 3 FIGURE 15.5 Exercise 15.1. Equilibrium Selection: Consider the extensive-form game in Figure 15.5. a. Find all the Bayesian Nash equilibria of this game. b. Which of the Bayesian Nash equilibria are also perfect Bayesian equi- libria? Why?
- In the 3 x 3 matrix below, what is the Nash equilibrium? Р2 N S E U 6,4 7,3 5, 5 P1 H7,3 3,7 4,6 L 8,2 | 6,4 | 3, 7 Select one: a. U,E b. 8,2 О с. 5,5 O d. L,Nt 0886951] Two firms are competing in a market by simultaneously deciding the quality of their manufactured product. The payoff matrix for this competition is depicted in the image. Firm 1 This game can be solved by deleting dominated strategies. Use this tactic to identify all pure-strategy Nash equilibria in this game. Low Medium High 53 54 66 Low A. Firm 1 chooses high and Firm 2 chooses high. 25 34 43 B. Firm 1 chooses low and Firm 2 chooses high. 43 68 77 C. Firm 1 chooses medium and Firm 2 chooses medium. Firm 2 Medium D. Firm 1 chooses medium and Firm 2 chooses low. 12 45 E. Firm 1 chooses low and Firm 2 chooses low. 74 96 89 F. Firm 1 chooses high and Firm 2 chooses low. High G. Firm 1 chooses medium and Firm 2 chooses high. 39 48 H. Firm 1 chooses low and Firm 2 chooses medium. I. Firm 1 chooses high and Firm 2 chooses medium. 50 214. Suppose in 1977 Honda and Toyota each have to decide whether to build an automobile plant in the North American market. The payoff matrix below shows Honda's payoff on the left, and Toyota's on the right. Is there a Nash equilibrium? If so, where, and how do you know? Тoyota Build small Don't build anything plant Build small 16, 16 20, 15 Honda plant Don't build anything 15, 20 18, 18
- Consider the following game: ● ● U P1 M D P2 C R L 1, 24, 36, 5 1,5 2,7 4, 3 2, 43, 25, 2 Identify the undominated strategies of P1 and P2. Which strategies are rationalizable? Find all the Nash equilibrium of this game. Illustrate your answer by a clearly labelled best response graph, with P1's strategy on the x-axis and P2's strategy on the y-axis.rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified 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.on 8.1 Consider the following game: Player 1 A C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1, 1 F 0,0 1,1 4,4 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game.
- Q2. For each of the following extensive form games, identify the subgame perfect Nash equilibrium. Game 5 Game 6 9. 15 10 10 Game 7 {:) 10 11 10ITERATIVE DOMINANCE Consider the following game in strategic form. a₁ 22 a3 bi 35,52 30,30 28,9 (9,11) (45,47) (47,28) (52,16) (28,9) (30,30) (16,45) (11,35) (35,52) b2 11,35 9,11 16,45 b3 52,16 45,47 47,28 Which of the following payoff profiles survive iterative deletion of strictly dominated strategies? (Mark all profiles that survive.)し(5,3) b I(2,2) も(0,0) (4,12) a (12.4) (0,0) i). List all subgame pertect Nash equilibria and name one Nash eqvilibrivm that is not subgame pertect i). How many strategies does playot and player 2 have?