Undeterred by their experiences with chicken so far (see Section 4.B), James and Dean decide to increase the excitement (and the stakes) by starting their cars farther apart. This way they can keep the crowd in suspense longer, and they’ll be able to accelerate to even higher speeds before they may or may not be involved in a much more serious collision. The new game table thus has a higher penalty for collision.
(a) What is the mixed-strategy Nash equilibrium for this more dangerous version of chicken? Do James and Dean play Straight more or less often than in the game shown in Figure 7.4?
(b) What is the expected payoff to each player in the mixed-strategy equilibrium found in part (a)?
(c) James and Dean decide to play the chicken game repeatedly (say, in front of different crowds of reckless youths). Moreover, because they don’t want to collide, they collude and alternate between the two pure-strategy equilibria. Assuming they play an even number of games, what is the average payoff to each of them when they collude in this way? Is this better or worse than they can expect from playing the mixed-strategy equilibrium? Why?
(d) After several weeks of not playing chicken as in part (c), James and Dean agree to play again. However, each of them has entirely forgotten which pure-strategy Nash equilibrium they played last time and neither realizes this until they’re revving their engines moments before starting the game. Instead of playing the mixed-strategy Nash equilibrium, each of them tosses a separate coin to decide which strategy to play. What is the expected payoff to James and Dean when each mixes 50–50 in this way? How does this compare with their expected payoffs when they play their equilibrium mixtures? Explain why these payoffs are the same or different from those found in part (c).