The Ash Twins Blog

I wrote the following as part of an experimental economics paper, but I decided to scrap it and put it on the blog.

The Prisoners’ Dilemma and Chicken games, one may recall, can both be represented by the following payoff matrix:

PD & Chicken Payoffs

		Player 2
		Cooperate	Defect
Player 1	Cooperate	R, R	S, T
	Defect	T, S	P, P

A PD is distinguished from chicken in the order of inequalities among payoffs.  As a mnemonic device and as trivia, T stands for Temptation, R for Reward, P for Punishment, and S for Sucker’s Payoff.  In PD, the following inequality holds: T>R>P>S; in Chicken, the following inequality holds: T>R>S>P.

The mechanics of both games can be illustrated by evolutionary biology’s model of cooperative hunting.  Two lions hunting prey can give chase (cooperate) or hold back (defect).  Giving chase is costly in expenditures of energy, and the prey will be shared equally regardless, so the highest payoff is to hold back while the other lion gives chase (T).  The second-best payoff, R, represents both lions giving chase, in which the energy costs of hunting are shared.  The P payoff is given to both lions if neither gives chase.  In PD, P is greater than S, the payoff given to a lone cooperator; in Chicken, S is greater than P.  One can conceptualize this difference as differing energy costs of giving chase: In PD, the costs of giving chase are so high that hunting alone is worse than no one hunting at all (S<P); in Chicken, the hunting costs are low enough that it is better to be the only one giving chase than to have no one hunting at all (S>P).

In PD, both players have dominant pure strategies of defect: Mutual defection is a pure-strategy Nash equilibrium.  While cooperation can emerge in an indefinitely repeated game, finitely repeated games unravel into defection by backward induction.[1] In an iterated chicken game, the evolutionarily stable strategy matches the mixed-strategy equilibrium.  That is, the set of strategies (x,y) will be evolutionarily stable only if each (either pure or mixed) strategy x_i and y_i is a best reply to (x,y).[2] Otherwise, a superior mutant strategy will invade and displace it.

The following table uses prevention investments of 25 and 75 to illustrate the decisional dynamics of another game, Crime Deterrence. I use arrows to indicate the rational change in decision at each outcome:

Fig. 3.3 – Snowdrift Payoffs

		Criminal
		Stay	Take
Regulator	Invest 25	75, 25 (→)	-37.5, 87.5 (↓)
	Invest 75	25, 25 (↑)	-12.5, -37.5(←)

The first thing to note is that payoffs are asymmetric.  In terms of the TRPS inequality, T₂>R₁>R₂,T₁,S₂>P1>S₁,P₂.  As with chicken, there is no pure-strategy Nash equilibrium; the optimal strategies for both players would require mixed strategies.  There are two significant differences: 1) Regulator is rewarded for higher investments only when Criminal takes (R₁>T₁); and 2) Stay results in the same payoff for Criminal no matter what Regulator invests (R₂=S₂).

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[1] Robert Axelrod, The Evolution of Cooperation (1984).

[2] Jorgen W. Weibull, Evolutionary Game Theory (1995), ch. 5.