Game of Chicken

The game of chicken is an influential model of conflict for two players in game theory. The principle of the game is that while each player prefers not to yield to the other, the worst possible outcome occurs when both players do not yield. Game which takes its name from 'dare' games said to be played by American teenagers: two teenagers take their cars to opposite ends of Main Street, Middle of Nowhere, USA, at midnight and start to drive toward each other. One must swerves to prevent a collision and the other does not, the one who swerved will be called a 'chicken', meaning a coward (1). The game of chicken is best represented by the following diagram:

Game of Chicken

Buzz

Swerves Keeps going

q 1-q

James Swerves p a, a -b, b

Keeps going 1-p b, -b -c, -c

Where c>b>a>0 and in each box the letter before the comma is what James get and the letter after the comma is what Buzz get. Each player most prefers to win, having the other to be chicken, and each least prefers the crash of the two cars. Form the diagram, it is easy to see that each player has an incentive to try to lock the other into cooperating (swerving), by announcing he or she will defect (keep going) first. If this works, the defector will get b (the best result) and the cooperator will get -b (the third best). However, if both players keep going, they will both get -c (the worst outcome). Moreover, the game faced by each chicken player in deciding whether to commit him to defection before the game start, is a Chicken game. All in all, for any game of chicken, there are four essential features. First, each player has one strategy that is the "tough" strategy and one that is the "weak" strategy. Second, there are two pure strategy Nash equilibriums: James keeps going, Buzz swerves (b, -b) and Buzz keeps going, James swerves (-b, b). Third, each player strictly prefers that equilibrium in which the other player chooses the "weak" strategy. Fourth, if both players choose the "tough" strategy, the worst outcome will appear to all of them.

As mentioned before, the game has two equilibria using just pure strategies, (swerves, keeps going) and (keeps going, swerves). For the mixed strategy:

The expected payoff to James is:

J = a*p*q - b*p*(1-q) + b*(1-p)*q - c*(1-p)(1-Yq)

= [(c - b) - (c - a)*q]*p + (b + c)*q - c

The expected payoff to Buzz is:

B = a*p*q + b*p*(1-q) - b*(1-p)*q - c*(1-p)(1-q)

= [(c - b) - (c - a)*p]*q + (b + c)*p - c

Taking the derivative of J w.r.t. p:

dJ/dp=(c-b)-(c-a)*q=0

Taking the derivative of B w.r.t. q:

dB/dq=(c-b)-(c-a)*p=0

Thus, the equilibrium with mixed strategies is:

(p, 1-p)= [(c-b)/(c-a), (b-a)/(c-a)] = (q, 1-q)

And the expected payoff for each player is:

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