1. Consider the following game: not A (2,6) L R' L R' (0,1) (3,2) (-1,3)' (1,5) a) ^Write down the normal form representation of the game and find its Nash Equilibria. b) and beliefs). Find a sequential equilibrium of the game (a profile of strategies
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- 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 game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?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.
- Question 5 Consider the following Extended Form Game: A' (1,1) A P2 B' O [(B, B')} (2,3) O ((B.(B',A')). (B.(B¹, B'))) P1 A' What is the set of Nash Equilibria? O {(B.(A', B)), (B,(B', B'))) (4,5) B O ((B.(A'A')). (B.(A, B')). (B,(B,A')), (B,(B', B'))} P2 B' (6,7)rock paper scissors rock 0. -3 0. раper scissors -1 -1 3. 0. (a) Show that xT= (,) and yT= (,) together are not a Nash equilibrium 3' 3 3131 for this modified game.Which is the correct mixed strategy Nash equilibrium for the below game? A B Confess Confess (3,2) Doesn't (0,0) confess a. (2/5); (3/5) b. (1/4); (3/4) c. (1/2); (1/2) d. (1/3); (2/3) Doesn't confess (0,0) (2,3)
- 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.Aedri Quick Lesson in Game Theory A Nash Equilibrium is an outcome in which neither player is better off by changing their strategy. 2.7 Is a Dominant Strategy equilibrium also a Nash equilibrium? a) Yes b) No esc The Ice Cream Guys $3.99 $4.99 + Chucky's Chunky CCT: $20,000 $3.99 ICG: $20,000 CCT: $60,000 ICG: $10,000 tab Treats CCT: $10,000 $4.99 ICG: $60,000 CCT: $40,000 ICG: $40,000 The table above is the payoff matrix for the annual profit of the only two ice-cream-truck firms operating in Beach City. They are deciding the price of an ice cream cone. %3D caps lo 2.8 What is Chucky's dominant strategy? a) $3.99 b) $4.99 c) Not enough information hift 2.9 What is the dominant strategy equilibrium in this situation? a) Both charge $3.99 b) Both charge $4.99 c) CCT charges $3.99 and ICG charges $4.99 d) CCT charges $4.99 and ICG charges $3.99 2.10 Suppose these two firms engaged in collusion (which, of course, totally doesn't happen because it is against the law). Which outcome would…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?
- 1. Consider the following normal form game. X Y (a) What is the set of rationalizable strategies for this game? Player 1 Player 2 W Q 1,7 1,5 2,3 Z 3,4 0,4 0,6 R= (b) The game has only one Nash equilibrium and it is a mixed strategy Nash equilibrium. Compute and report this equilibrium. Equil. strategy profile:a. Consider the following sequential game. Player 1 A B Player 2 Player 1 a a B Player 2 (4) () y i. Determine the subgame perfect Nash equilibria and their outcomes. ii. Determine the Nash equilibria of the game. For each of the Nash equilibria, argue why no player has a profitable deviation. ii. Determine which parts of the Nash equilibrium strategies involve uncredible threats.4. Correlated EquilibriaConstruct an example (not one from class or the reading) of a Normal form game with a correlated equilibrium that is not a Nash equilibrium.
