Two players are engaged in a game of "chicken." There are two possible strategies, Swerve and Drive Straight.A player who chooses to Swerve is called "chicken" and gets a payoff of zero, regardless of what the other player does.A player who chooses to Drive Straight gets a payoff of 15.43 if the other player Swerves and a payoff of -36 if the other player also chooses to Drive Straight.This game has two pure strategy equilibria and
A)two mixed strategies in which players alternate between swerving and driving straight.
B)a mixed strategy in which each player swerves with probability 0.35 and drives straight with probability 0.65.
C)a mixed strategy equilibrium in which one player swerves with probability 0.70 and the other swerves with probability 0.30.
D)a mixed strategy equilibrium in which each player swerves with probability 0.70 and drives straight with probability 0.30.
E)no mixed strategies.

Question

Quizplus · Accepted Answer

The payoff matrix for the game is:
\begin{array}{c|cc} & Swerve & Drive Straight \ \hline Swerve & 0, 0 & 15.43, F1F1F1S1-F1F1F1036 \ Drive Straight &F1F1F1S1-F1F1F1036 , 15.43 & 15.43, 15.43 \end{array}
For there to be a mixed strategy equilibrium, the players must be indifferent between Swerve and Drive Straight. This means that the expected payoff for each player must be the same for both strategies. Let p be the probability that Player 1 chooses Drive Straight and q be the probability that Player 2 chooses Drive Straight. Then the expected payoffs are:

For Player 1:
\begin{aligned} 0p+(F1F1F1S1-F1F1F1036)(1-p)&=15.43p+15.43(1-p)\ (F1F1F1S1-F1F1F1036)+1036p-1036p&=15.43+15.43p-15.43p\ F1F1F1S1&=15.43 \end{aligned}
This equation has no solution, so there is no mixed strategy equilibrium in which both players choose Drive Straight with positive probability.

For Player 2:
\begin{aligned} 0q+(F1F1F1S1-F1F1F1036)(1-q)&=15.43q+15.43(1-q)\ (F1F1F1S1-F1F1F1036)+1036q-1036q&=15.43+15.43q-15.43q\ F1F1F1S1&=15.43 \end{aligned}
This equation also has no solution, so there is no mixed strategy equilibrium in which both players choose Drive Straight with positive probability.

Therefore, the only possibility is that one player chooses Swerve and the other player chooses Drive Straight with positive probability. Let p be the probability that Player 1 chooses Swerve and q be the probability that Player 2 chooses Swerve. Then the expected payoffs are:

For Player 1:
\begin{aligned} 0p+(F1F1F1S1-F1F1F1036)(1-p)&=15.43q+0(1-q)\ (F1F1F1S1-F1F1F1036)+1036p-1036p&=15.43q\ F1F1F1S1-F1F1F1036&=15.43q\ q&=\frac{F1F1F1S1-F1F1F1036}{15.43} \end{aligned}
For Player 2:
\begin{aligned} 0q+(F1F1F1S1-F1F1F1036)(1-q)&=15.43p+0(1-p)\ (F1F1F1S1-F1F1F1036)+1036q-1036q&=15.43p\ F1F1F1S1-F1F1F1036&=15.43p\ p&=\frac{F1F1F1S1-F1F1F1036}{15.43} \end{aligned}
Since F1F1F1S1>F1F1F1036, both probabilities are greater than zero. Therefore, a mixed strategy equilibrium is:
Answer: D
 Both players swerve with probability 0.70 and drive straight with probability 0.30.

Two Players Are Engaged in a Game of "Chicken