Consider the following variant of chicken in whihc James col

Consider the following variant of chicken, in whihc Jame\'s (column player) payoff from being \"tough\" when Dean (row player) is \"chicken\" is 2, rather than 1.

(a) Find the mixed-strategy equilibrium in this game, including the expected payoffs for the players.

(b) Compare the results with those of the original game in Section 4.B of this chapter. Is Dean\'s probability of playing Straight (being tough) higher now than before? What about James\'s probability of playing Straight?

(c) What has happened to the two players\' expected payoffs?Are these differences in the equilibrium outcomes paradoxical in light of the new payoff structure? Explain how your findings can be understood in light of the opponent\'s-indifference principle.

Solution

(a) the mixed-strategy equilibrium       

0p – 1(1 – p) = 2p – 2(1 – p)

0q + 1(1 – q) = -1q – 2(1 – q)

p=1/3, q=2/3

In the mixed-strategy Nash equilibrium James plays

1/3(Swerve) + 2/3(Straight),

and Dean plays

2/3(Swerve) + 1/3(Straight).

James’ expected payoff = 2/3 – 2(1/3) = 0

Dean’s expected payoff = -2/3 – 1/3 = – 1

(c)     

These expected payoffs are much worse than the collusion example or the mixed-strategy equilibrium. In this case both players are mixing with the wrong (that is, not the equilibrium) mixture. Neither is player is best responding to the other’s strategy, and in this situation—with the very real possibility of reaching the – 10 payoff—the expected consequences are dire.