What is the (pure-strategy) Nash equilibrium? O There is no (pure-strategy) Nash equilibrium. O (P1, P2) = (A, B) is the only (pure-strategy) Nash equilibrium. O (P1, P2) = (B. A) is the only (pure-strategy) Nash equilibrium. O There are more than one (pure-strategy) Nash equilibria. (P1, P2) = (A, A) is the only (pure-strategy) Nash equilibrium.
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- GAME 5 Player B B2 B1 Player A A1 7, 3 5, 10 A2 3, 8 9, 6 In Game 5 above, O there are no Nash equilibria in pure strategies. Player A choosing A1 and Player B choosing B2 is a Nash equilibrium. O Player A choosing A1 and Player B choosing B1 is a Nash equilibrium. O Player A choosing A2 and Player B choosing B1 is a Nash equilibrium.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.5. Consider a simultaneous game in which player A chooses one of two actions (Up or Down), and B chooses one of two actions (Left or Right). The game has the following payoff matrix, where the first payoff in each entry is for A and the second for B.(8 points) B Right Left 3,3 5,1 Down 2,2 4,4 a. Find the Nash equilibrium or equilibria. b. Which player, if any, has a dominant strategy? A Up
- GAME UUU B1 Player B B2 A1 7,13 5, 10 A2 3,8 9,16 Player A A3 5,8 4,7 In Game UUU (see table above), assuming players move simultaneously, Player A choosing A1 and Player B choosing B3 is a Nash equilibrium. Player A choosing A3 and Player B choosing B2 is a Nash equilibrium. Both Player A choosing A1 and Player B choosing B1 and Player A choosing A2 and player B choosing B2 are Nash equilibria in pure strategies Player A choosing A1 and Player B choosing B2 is a Nash equilibrium.Consider the following game: Player 2 In Out Player 1 In -2,-2 2, 0 Out 0, 2 0, 0 (a) What is the Nash equilibrium of this game, or what are the Nash equilibriaof this game? (b) Does either firm have a dominate strategy (a strategy that is always abest response)? Which? (c) Suppose Player 1 could move before Player 2 and Player 2 could observe Player 1’s move. What do you think would happen?5 Suppose two players play one of the two normal-form games shown in Figure 1. L U 0,-1 D 2,4 R 2,0 6,0 L U | 4,-1 D 2,-2 R 2,0Now suppose that Player 2 knows which game is being played, but Player 1 does not. Find the pure strategy Bayesian Nash equilibrium of this game.
- Refer to the accompanying figure. What is the Nash equilibrium of this game? Player A O Multiple Choice O Up O Down Player B Left 5 for A 30 for B -2 for A 10 for B A chooses Up, B chooses Right A chooses Up, B chooses Left Right 10 for A 12 for B 8 for A 15 for B A chooses Down, B chooses Right A chooses Down, B chooses LeftThe following table contains the possible actions and payoffs of players 1 and 2. Player 1 U M D L 2,10 2,5 8,7 Player 2 C 5,2 11,8 8,4 R 5,4 2,9 8,3 This is a simultaneous move game. In a pure strategy Nash equilibrium of this game, Player 1 receives a payoff of ✓and player 2 receives a payoff of If, instead of playing simultaneously, player 2 moves first, then in the Nash equilibrium player 1 receives a payoff of ✓and player 2 receives a payoff ofO Macmillan Learning In this game table, the Nash equilibrium is Mario's Fancy Mario's Casual Isabella's Fancy Isabella's Casual 3,6 4,8 O Casual, Fancy = (4,8) O Fancy, Casual = (7, 2) O Casual, Casual = (9, 7) O Fancy, Fancy = (3,6) 7,2 9,7
- Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.Consider the following classic game, called Hawk and Dove. Player 2 Hawk Dove Hewk 0,0 4,1 Player 1 Dove 1,4 33 Select all pure strategy Nash equilibria and check the corresponding option if you think there exists a mixed strategy Nash equilibrium too. O There exists also a mixed strategy Nash equilibrium O Hawk, Dove O Dove, Hawk O Hawk, Hawk O Hawk, Hawk1. 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