# The recalcitrant James and Dean are playing their more dangerous variant of chicken again (see…

The recalcitrant James and Dean are playing their more
dangerous variant of chicken again (see Exercise S6). They’ve noticed that
their payoff for being perceived as “tough” varies with the size of the crowd.
The larger the crowd, the more glory and praise each receives from driving
straight when his opponent swerves. Smaller crowds, of course, have the
opposite effect. Let k . 0 be the payoff for appearing “tough.” The game may
now be represented as:

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(a) Expressed in terms of k, with what probability does
each driver play Swerve in the mixed-strategy Nash equilibrium? Do James and
Dean play Swerve more or less often as k increases?

(b) In terms of k, what is the expected value of the game
to each player in the mixed-strategy Nash equilibrium found in part (a)?

(c) At what value of k do both James and Dean mix 50–50
in the mixed-strategy equilibrium?

(d) How large must k be for the average payoff to be
positive under the alternating scheme discussed in part (c) of Exercise S6?