Construct the normal representation and the fully reduced normal representation of the game shown in Figure 1. 0 (0.5) (0.5) 1.r 1.s W1 X1 yı Z1 0,0 2.ti 2.t 0,4 a2 b2 a2 b2 Figure 1: Extensive-form game 2,-2 -2,2 -2,2 2,-2
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- P1 P1 P2 P2 F F Y 3, 4 0,3 0,3 2,5 3,6 5, 2 Which of the following is a subgame perfect equilibrium of this game? (RDJ,FQY) (RMU,ZTH) (LDU,ZQY) (RDU,ZTY) O (LMJ,ZTY) (RMU,FTH)Exercise 6.1Suppose that two airlines decide to collude. Analyse the game between these two companies. Suppose that each of them can charge for tickets a high price or a low price. If one of them charges 100 euros, it gets few profits if the other also charges 100 euros and high profits if the other charges 200 euros. On the other hand, if the company charges 200 euros, it obtains very little profit if the other charges 100 euros and an average profit if the other also charges 200 euros. a) Represent the matrix of results of this game. b) What is the Nash equilibrium in this game? Explain your answer. c) Is there an outcome that would be better than the Nash equilibrium for the two airlines? How could it be achieved? Who would lose out if it were reached?6. Battle of the Networks 1\2 %share Sitcom Sitcom 48;52 Sport 42;58 Nature 55:45 58;42 Ans: s; = (0; 0.85; 0.14); V, = 55.43 s; = (0.43; 0.57;0); V½ = 44.57 40;60 63;37 60;40 Sport Nature 56:44 52;48 Find Nash Equilibrium.
- Consider the following extensive form game between player 1 and player 2. T B (2, 2) L R R (3, 1) (0, 0) (5, 0) (0, 1) (a). Find the normal form representation of this game. (show the bimatrix) (b). Find all pure strategy NE. (c). Which of these equilibria are subgame perfect?Suppose 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.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?
- Question 1 Consider the following game. Find all Nash equilibria, subgame perfect Nash equilibria, and weak perfect Bayesian equilibria in pure and mixed strategies. a P2 (12, 12) b (12,-8) (7.-3) P1 m d (-3,2) r P2 C (2,7) d (22,17)Consider the Normal Form Game characterized in the following figure: P1 \ P2 b1 b2 b3 a1 (1,2) (0,1) (0,3) a2 (3,-1) (1,1) (-1,0) a3 (2,2) (1,-1) (3,0) a4 (-5,1) (-1,0) (-3,-2) a5 (-3,3) (3,0) (2,4) What is the set of actions of player 1? O la 1, a2, a3, a4, a5) O (b1,b2, b3} O {a2, a3, a4) O (bl, b2}Consider the Normal Form Game characterized in the following figure: P1 \ P2 A1 A2 A3 A4 B1 B2 B3 B4 (-1,1) (0,0) (1,-1) (2,0) (-2,-2) (2,7) (-1,-1) (-1,1) (5,7) (3,5) (0,0) (0,8) (0,3) (-1,-1) (10,2) (0,0) Which is the set of rationalizable actions? O (A2, A4)x(B2, B3) O (A1, A2)x(B1, B2, B4) O (A1, A3)x(B1, B4) O (A1, A3, A4}x{B1, B3)
- 1. Nash Equilibrium (a) Find all pure Nash Equilibria P1/P2 W X Y Z 9,9 0,7 5,5 1,1 7,0 0,0 4,2 7,7 5,5 2,4 0,0 1,1 1,1 7,7 1,1 0,0 A B C D (b) Consider the following picnic game. There are N players in the game each who can chose to bring food to the picnic or not. Let b;= {0, 1} be player i's choice of bringing food with 0 as not and 1 as yes. If they decide to bring food it comes at a cost c;(1) = 1 and no cost to bring food. Payoffs are sum of food brought minus individual cost. i. Write out the normal form matrix if N = 2 ii. If N = 2 find all pure Nash Equilibria iii. Find all pure Nash Equilibria in the general gamerock 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.Game Theory Question A non-profit firm is on a local community online donation platform for a community event it wants to hold (only community members can donate via the website). The event will be held only if the non-profit firm collects $20,000 total from members of the community. Each member values the event at $500. Suppose that there are 100 community members. Community members can only donate by purchasing a lottery ticket from the firm. Each ticket costs $200 and only one ticket can be purchased per member. The proceeds will be collected by the firm. The lottery winner gets a premier meal at a local restaurant that's worth $100. Remember, the firm keeps all the donated money. If the amount of donations is less than $20,000, then the firm returns the donated money to the community members (since there'll be no event held but the lottery winner still gets to eat that fancy meal). If the donations sum up to $20,000, the community event will take place. What are the Nash…