Consider an application of the Prisoner’s Dilemma game where two countries in a cartel decide whether to Cooperate or Defect. If both choose to follow the cartel agreement and continues cooperating with each other, each receives a payoff of 2. If both decide to defect on each other, they both get -2. If one player cooperates and the other defects, then the one cooperates loses 10 (receives -10) while the other player gets 0 . Denote the probability of cooperating for Player 1 as p and for Player 2 as q.How much does Player 1 earn in the mixed-strategy Nash equilibrium of this game?Write your answer as a decimal with 2 decimal places (e.g. 0.25)

In the mixed-strategy Nash equilibrium of this game, Player 1's earnings can be calculated by considering the probabilities of each player's actions.

Let's denote the probability of Player 1 cooperating as p and the probability of Player 2 cooperating as q. In the mixed-strategy Nash equilibrium, both players choose their actions to maximize their expected payoffs, given the probabilities of the other player's actions.

Player 1's expected payoff can be calculated as follows:

E(Player 1's payoff) = (payoff of cooperation) * (probability of both players cooperating) + (payoff of defection) * (probability of both players defecting) + (payoff of defection) * (probability of Player 1 cooperating and Player 2 defecting)

E(Player 1's payoff) = 2 * p * q + (-2) * (1 - p) * (1 - q) + (-10) * p * (1 - q)

Simplifying the equation:

E(Player 1's payoff) = 2pq - 2(1 - p)(1 - q) - 10p(1 - q)

To find the mixed-strategy Nash equilibrium, we need to find the values of p and q that maximize Player 1's expected payoff. This can be done by taking the partial derivatives of the expected payoff equation with respect to p and q, setting them equal to zero, and solving for p and q.

Differentiating with respect to p:

∂E(Player 1's payoff)/∂p = 2q - 2(1 - q) - 10(1 - q) = 0

Simplifying the equation:

2q - 2 + 2q - 10 + 10q = 0

14q - 12 = 0

q = 12/14 = 6/7

Differentiating with respect to q:

∂E(Player 1's payoff)/∂q = 2p - 2(1 - p) + 10p = 0

Simplifying the equation:

2p - 2 + 2p + 10p = 0

14p - 2 = 0

p = 2/14 = 1/7

Therefore, in the mixed-strategy Nash equilibrium, Player 1 earns:

E(Player 1's payoff) = 2 * (1/7) * (6/7) + (-2) * (1 - 1/7) * (1 - 6/7) + (-10) * (1/7) * (1 - 6/7)

E(Player 1's payoff) = 12/49 - 2/7 + 10/7

E(Player 1's payoff) = 12/49 - 14/49 + 70/49

E(Player 1's payoff) = 68/49

Therefore, Player 1 earns 68/49 in the mixed-strategy Nash equilibrium of this game, which is approximately 1.39 when rounded to 2 decimal places.

In the mixed-strategy Nash equilibrium of this game, Player 1's earnings can be calculated by considering the probabilities of each player's actions.

Let's denote the probability of Player 1 cooperating as p and the probability of Player 2 cooperating as q. In the mixed-strategy Nash equilibrium, both players choose their actions to maximize their expected payoffs, given the probabilities of the other player's actions.

Player 1's expected payoff can be calculated as follows:

E(Player 1's payoff) = (payoff of cooperation) * (probability of both players cooperating) + (payoff of defection) * (probability of both players defecting) + (payoff of cooperation for Player 1) * (probability of Player 1 cooperating and Player 2 defecting) + (payoff of defection for Player 1) * (probability of Player 1 defecting and Player 2 cooperating)

E(Player 1's payoff) = 2 * p * q + (-2) * (1 - p) * (1 - q) + (-10) * p * (1 - q) + 0 * (1 - p) * q

Simplifying the equation:

E(Player 1's payoff) = 2pq - 2(1 - p)(1 - q) - 10p(1 - q)

To find the mixed-strategy Nash equilibrium, we need to find the values of p and q that maximize Player 1's expected payoff. This can be done by taking the partial derivatives of the equation with respect to p and q, setting them equal to zero, and solving for p and q.

Differentiating the equation with respect to p:

∂E(Player 1's payoff)/∂p = 2q - 2(1 - q) - 10(1 - q) = 0

Simplifying the equation:

2q - 2 + 2q - 10q + 10 = 0

12q - 12 = 0

q = 1

Differentiating the equation with respect to q:

∂E(Player 1's payoff)/∂q = 2p - 2(1 - p) + 10p = 0

Simplifying the equation:

2p - 2 + 2p + 10p = 0

14p - 2 = 0

p = 1/7

Therefore, in the mixed-strategy Nash equilibrium, Player 1 earns 2(1/7)(1) - 2(6/7)(0) - 10(1/7)(0) = 2/7.

So, Player 1 earns 0.29 in the mixed-strategy Nash equilibrium of this game.