4. Countries A and B are engaged in a high stakes game of chicken. Both have two

Question

4. Countries A and B are engaged in a high stakes game of chicken. Both have two pure strategies: act tough or back down. They will make their decisions simultaneously. The payoffs are

B
Back Down Act Tough
A Back Down 0,0 -1, k
Act Tough k, -1 -10, -10

where k > 0 is the payoff from acting tough when your opponent backs down.

(a) Show that this game has two pure strategy Nash equilibria.

(b) The game also has a mixed strategy Nash equilibrium. Find the equilibrium probability that each country will act tough as a function of k.

(c) Does the equilibrium probability that a country will act tough rise, fall, or remain constant as k rises?

Explanation / Answer

(a) The two pure strategy Nash equilibria are (tough, back) and (back, tough). We know that because k>0 and -10 0 Analogously: Dq = 9/(9+k)^2 > 0 So, as k increases, p and q each increase. As the value of k increases, the probability that each player will act tough increases.

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