(Symmetric mixed strategy Nash equilibrium) A profile α∗ of mixed strategies in a strategic game with vNM preferences in which each player has the same set of actions is a symmetric mixed strategy Nash equilibrium if it is a mixed strategy Nash equilibrium and α∗ i is the same for every player i.Solve this problem:At a large round table sit n ≥ 2 players, each holding 3 cards: one white, one black, and one red. Each player must secretly choose one of their cards and then, when the bell rings, simultaneously reveal it publicly with all the others. If all players choose the white card, each of them receives 6 points. If player i chooses the white card, and at least one of the other players chooses a card of a different color, player i receives 1 point. If player i chooses the black card, they receive 3 points, regardless of the decisions of the other players. If player i chooses the red card, they receive 0 points, regardless of the decisions of the other players. Find all symmetric Nash equilibria of the given game in strategic form.Solution is on the picture. Why we have E(W) = 6p + q ; E(B) = 3q; E(R) = 0 To find the symmetric Nash equilibria of the game, let's denote the probability of aplayer choosing the white card as p, the probability of choosing the black card as q,and the probability of choosing the red card as r. Since the game is symmetric, allplayers will have the same strategy, so p₁ = p, q₁ = q, and r; = r for all players i.Now, let's analyze the payoffs for each player for different strategies:1. If all players choose the white card:• Each player gets 6 points.2. If a player chooses the white card and at least one other player chooses a differentcolor card:• The player gets 1 point.3. If a player chooses the black card:• The player gets 3 points.4. If a player chooses the red card:⚫ The player gets O points.Now, we can write down the expected payoffs for each player for each strategy:• Expected payoff for choosing the white card:E(W)=6p+qExpected payoff for choosing the black card:E(B)=3qExpected payoff for choosing the red card:E(R) 0For a symmetric mixed strategy Nash equilibrium, no player should have an incentiveto unilaterally deviate from their strategy given the strategies of the other players.First, let's consider the case where p > 0, meaning at least one player chooses thewhite card with positive probability. In this case, players choosing the white card willwant to make sure that switching to another card will not improve their payoff.• If q> 0, players choosing the white card will earn 1 point instead of 6, so they will betempted to switch to the black card to earn 3 points instead of 1. Thus, q = 0.• Similarly, if > 0, players choosing the white card will earn O points instead of 6, sothey will be tempted to switch to the black card to earn 3 points instead of O. Thus,r = 0.Now, we have simplified the game to only consider the case where players onlychoose the white or black card. In this simplified game, if p > 0, players choosing thewhite card will earn 6 points, while players choosing the black card will earn 3 points.Therefore, there is no incentive for players to deviate from their chosen strategy.Since p > 0 (at least one player chooses the white card with positive probability), andq = r = 0, the symmetric Nash equilibrium strategy is for all players to choose thewhite card with probability 1.So, the symmetric Nash equilibrium of the game is for all players to choose the whitecard with probability 1.

Mixed strategy Nash equilibrium is calculated by observing the payoffs for each player. When a…

Answered: (Symmetric mixed strategy Nash…

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(Symmetric mixed strategy Nash equilibrium) A profile α∗ of mixed strategies in a strategic game with vNM preferences in which each player has the same set of actions is a symmetric mixed strategy Nash equilibrium if it is a mixed strategy Nash equilibrium and α∗ i is the same for every player i.

Solve this problem:

At a large round table sit n ≥ 2 players, each holding 3 cards: one white, one black, and one red. Each player must secretly choose one of their cards and then, when the bell rings, simultaneously reveal it publicly with all the others. If all players choose the white card, each of them receives 6 points. If player i chooses the white card, and at least one of the other players chooses a card of a different color, player i receives 1 point. If player i chooses the black card, they receive 3 points, regardless of the decisions of the other players. If player i chooses the red card, they receive 0 points, regardless of the decisions of the other players. Find all symmetric Nash equilibria of the given game in strategic form.

Solution is on the picture. Why we have E(W) = 6p + q ; E(B) = 3q; E(R) = 0

To find the symmetric Nash equilibria of the game, let's denote the probability of a
player choosing the white card as p, the probability of choosing the black card as q,
and the probability of choosing the red card as r. Since the game is symmetric, all
players will have the same strategy, so p₁ = p, q₁ = q, and r; = r for all players i.
Now, let's analyze the payoffs for each player for different strategies:
1. If all players choose the white card:
• Each player gets 6 points.
2. If a player chooses the white card and at least one other player chooses a different
color card:
• The player gets 1 point.
3. If a player chooses the black card:
• The player gets 3 points.
4. If a player chooses the red card:
⚫ The player gets O points.
Now, we can write down the expected payoffs for each player for each strategy:
• Expected payoff for choosing the white card:
E(W)=6p+q
Expected payoff for choosing the black card:
E(B)=3q
Expected payoff for choosing the red card:
E(R) 0
For a symmetric mixed strategy Nash equilibrium, no player should have an incentive
to unilaterally deviate from their strategy given the strategies of the other players.
First, let's consider the case where p > 0, meaning at least one player chooses the
white card with positive probability. In this case, players choosing the white card will
want to make sure that switching to another card will not improve their payoff.
• If q> 0, players choosing the white card will earn 1 point instead of 6, so they will be
tempted to switch to the black card to earn 3 points instead of 1. Thus, q = 0.
• Similarly, if > 0, players choosing the white card will earn O points instead of 6, so
they will be tempted to switch to the black card to earn 3 points instead of O. Thus,
r = 0.
Now, we have simplified the game to only consider the case where players only
choose the white or black card. In this simplified game, if p > 0, players choosing the
white card will earn 6 points, while players choosing the black card will earn 3 points.
Therefore, there is no incentive for players to deviate from their chosen strategy.
Since p > 0 (at least one player chooses the white card with positive probability), and
q = r = 0, the symmetric Nash equilibrium strategy is for all players to choose the
white card with probability 1.
So, the symmetric Nash equilibrium of the game is for all players to choose the white
card with probability 1.

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