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. We can write the payoff matrix for this game as follows: scissors rock paper rock 0 paper 1 0 scissors -1 1 -1 0 (a) Show that this game does not have a pure Nash equilibrium. (b) Show that the pair of mixed strategies x¹ = (₁,4) and y¹ = (₁) together are a Nash equilibrium. 2. Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors 0 -3 1 -1 rock paper 1 0 scissors 3
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. We can write the payoff matrix for this game as follows: scissors rock paper rock 0 paper 1 0 scissors -1 1 -1 0 (a) Show that this game does not have a pure Nash equilibrium. (b) Show that the pair of mixed strategies x¹ = (₁,4) and y¹ = (₁) together are a Nash equilibrium. 2. Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors 0 -3 1 -1 rock paper 1 0 scissors 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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.
We can write the payoff matrix for this game as follows:
rock paper scissors
rock 0 -1
0
1
paper 1
scissors -1
-1
0
(a) Show that this game does not have a pure Nash equilibrium.
(b) Show that the pair of mixed strategies x¹ = (₁,4) and y¹ = (₁) together
are a Nash equilibrium.
2. 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
0
paper 1 0
scissors -1 3
(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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d1b97ca-a014-4a9d-850a-b61b08d119c0%2F00580d9d-b49b-471e-a8d7-39c83ddba5d0%2Fvo8snp2_processed.png&w=3840&q=75)
Transcribed Image Text: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.
We can write the payoff matrix for this game as follows:
rock paper scissors
rock 0 -1
0
1
paper 1
scissors -1
-1
0
(a) Show that this game does not have a pure Nash equilibrium.
(b) Show that the pair of mixed strategies x¹ = (₁,4) and y¹ = (₁) together
are a Nash equilibrium.
2. 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
0
paper 1 0
scissors -1 3
(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.
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