(a) For the following extensive-form game: 1. Identify the pure and mixed strategy Nash Equilibria ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in part (a)? Explain (a couple of sentences should suffice). (3,1) Player 1 1 a1 az D (-2,-2) B Player 2 bi b₂ 3,0 0,1 2,1 2,1 9 (2,5) D (b) Consider the simultaneous-move game below with two players, 1 and 2 Each player has two pure strategies. If a player plays both strategies with strictly positive probability, we call it a strictly mixed strategy for tha player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. (0,7) ||
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- FOOP 6. (a) For the following extensive-form game: i. Identify the pure and mixed strategy Nash Equilibria. ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in part (a)? Explain (a couple of sentences should suffice). (3,1) A D B C (-2,-2) (2,5) D (0,7)NE 2). Consider the following extensive form game between two players. (1,10) u D 1 B X (a) List all pure strategies of player 2. (b) Represent this game in normal form. (c) Find all pure-strategy Nash equilibria of this game. (d) Find all SPNE (in pure strategies) of this game. (6,3) (4,2) (5,1)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.
- . Consider the 2-player, zero-sum game "Rock, Paper, Scissors". Each player chooses one of 3 strategies: rock, paper, or scissors. Then, both players reveal their choices. The outcome is determined as follows. If both players choose the same strategy, neither player wins or loses anything. Otherwise: • "paper covers rock": if one player chooses paper and the other chooses rock, the player who chose paper wins and is paid 1 by the other player. • "scissors cut paper": if one player chooses scissors and the other chooses paper, the player who chose scissors wins and is paid 1 by the other player. • "rock breaks scissors": if one player chooses rock and the other player chooses scissors, the player who chose rock wins and is paid 1 by the other player.6. (a) For the following extensive-form game: 1. Identify the pure and mixed strategy Nash Equilibria. ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in parf (a)? Explain (a couple of sentences should suffice). с (3,1) Player 1 a a2 A D (-2,-2) B Player 2 bi b₂ 3,0 0,1 2,1 2,1 C (2,5) D (b) Consider the simultaneous-move game below with two players, 1 and 2. Each player has two pure strategies. If a player plays both strategies with strictly positive probability, we call it a strictly mixed strategy for that player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. (0,7)3. (a) Give an example of a 3 x 3 zero-sum game where all possible pairs of pure strategies give a pure Nash Equilibrium. (b) Describe all possible 3 x 3 non-zero-sum games where all possible pairs of pure strategies give a pure Nash Equilibrium. Give reasons for your answer.
- 1. Consider the following normal-form game: Player 2 Left Right 30,20 10,10 10,30 20,20 Player 1 How many pure-strategy Nash Equilibria are there in this game? (a) 0 (b) 1 Up Down (c) 2 (d) 3 (e) 4 (f) None of the above Answer: 1b. The only Nash Equilirbium is (Up,Left).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.3. Determine the Nash Equilibrium for the following normal form of the game. A) MARY B) UP DOWN PLAYER 1 HARLEY UP (10, 10) (4,8) LEFT (3,0) UP MIDDLE (1, 1) DOWN (0,1) DOWN (8,4) (5,5) PLAYER 2 CENTER RIGHT (2,1) (1,1) (4,2) (0,0) (5,0) (0,1)
- Player 2 E F H A 6, 5 6, 7 9, 6 7,6 В Player 1 C 6, 7 6, 9 8, 5 9, 7 5, 8 5, 6 7,5 7,5 7,9 8, 7 11, 6 5, 6 (1) In the Unique Nash equilibrium of this game, which strategy does Player1 play? And why? (2) In the Unique Nash equilibrium of this game, which strategy does Player2 play? And why? (3) Is this game dominance solvable? And Why? (4) Does this game have at least one inadmissible Nash equilibrium? And Why?1. Consider the two-player game described in figure 1. Player 1 has three possible actions to choose from and player 2 has two. The payoffs to each player are shown in the normal-form representation of the game below. (a) Player 2 LR U 5,1 4,0 Player 1 M 6,0 3,1 D 7,3 5,4 Figure 1 Do either players have any dominated strategies? (b) What is the Nash Equilibrium of this game?Jack and Jill are the only two suppliers of water in their village. They go up the hill once a day and come back carrying one bucket of water each (each has one small and one large bucket). They can each choose the big bucket that brings 40 gallons of water, or the small bucket that brings 15 gallons of water. If both use the big buckets (40-gallon buckets): they bring down a total of 80 gallons of water (total production) that they sell to the villagers; they equally split the market. They each make $1,400 in profit. If both use the small buckets (15-gallons buckets): they produce a total of 30 gallons of water that they sell to the villagers, once again they equally split the market. They each make $1,650 profit. If one uses the big bucket and the other uses the small bucket (for a total production of 55 gallons), then: the one with small bucket gets $1,200 profit the one with the big bucket gets $2,000 profit Jack is the row 1. Draw the payoff matrix (1 player and Jill is the column…