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 4 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 andA) a mixed strategy equilibrium in which one player swerves with probability 0.90 and the other swerves with probability 0.10.B) a mixed strategy in which each player swerves with probability 0.45 and drives straight with probability 0.55.C) a mixed strategy equilibrium in which each player swerves with probability 0.90 and drives straight with probability 0.10.D) two mixed strategies in which players alternate between swerving and driving straight.E) no mixed strategies.

Question

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 4 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 andA) a mixed strategy equilibrium in which one player swerves with probability 0.90 and the other swerves with probability 0.10.B) a mixed strategy in which each player swerves with probability 0.45 and drives straight with probability 0.55.C) a mixed strategy equilibrium in which each player swerves with probability 0.90 and drives straight with probability 0.10.D) two mixed strategies in which players alternate between swerving and driving straight.E) no mixed strategies.

Julieanne Sathoud · Accepted Answer

Final Answer : C, Explanation :  The only mixed strategy that can be an equilibrium is one in which both players are indifferent between swerving and driving straight. This means that both players would have the same expected payoff from each strategy, and therefore would be willing to play either one with equal probability. Since the payoff for swerving is always zero, this means that both players would only want to swerve if the expected payoff for driving straight is lower than zero. Using the payoff formula, we can set up the following equation:4p + ( - 36)(1-p) = 0where p is the probability that each player will choose to drive straight. Solving for p, we get:p = ( - 36)/4This means that both players would choose to drive straight with probability ( - 36)/4 and swerve with probability 1 - ( - 36)/4 = (16- + 36)/4. Since both players would have the same strategy, this is a mixed strategy equilibrium.We can also check that this equilibrium is stable by verifying that neither player can gain by deviating from their strategy. If a player were to change their strategy, they would have to either increase their probability of driving straight or decrease it. If they increase their probability, their expected payoff would also increase, but if they decrease it, their expected payoff would decrease. Therefore, each player's strategy is optimal given the other player's strategy, and this is a Nash equilibrium.

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