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Game Theory of Washing Dishes

In academia, popular economics on June 3, 2009 at 5:00 pm

Ben and Jack are roommates in a condo unit. Someone cooked dinner and did not wash the dishes. Each person has two possible choices: to wash the dishes or not. Let ‘Y’ be to wash the dishes and ‘N’ to do nothing.

Since each outcome will produce a pair of strategies, we will signify the pairs as (Y,Y), (Y,N), (N,Y), and (N,N). The first item in each pair is Ben’s choice, while the second item in each pair is Jack’s. The four possible outcomes are:

1. Ben will wash the dishes, Jack will do nothing (Y,N)
2. Ben will do nothing, Jack will wash the dishes (N,Y)
3. Both will wash the dishes together (Y,Y)
4. Nobody will wash the dishes (N,N)

Now we need to know which outcomes are preferred and which are not. Let’s assign a “payoff” for each outcome to signify what the roommates want and care about. The following figures are units of pleasure or utility (the higher, the happier) from the point of view of Ben:

3 if Jack will wash the dishes alone (N,Y)
2 if both will wash the dishes together (Y,Y)
1 if nobody will wash the dishes (N,N)
0 if Ben will wash the dishes alone (Y,N)

But Ben’s payoffs apply to the other roommate in reverse, so that Jack’s payoffs are:

3 if Ben will wash the dishes alone (Y,N)
Same payoffs for (Y,Y) and (N,N)
0 if Jack will wash the dishes alone (N,Y)

A roommate’s best-case scenario is to get the other to wash the dishes. The worst-case scenario is that a roommate will wash the dishes alone while the other slacks off, because we assume that resentment is costlier and more disadvantageous than having a dirty condominium where nobody washes the dishes (N,N).

Now let us set up the matrix, where each pair has the form (Ben’s payoff, Jack’s payoff).

Outcome and Payoff Matrix (please ignore the dots)

…………………Jack

………………….N       Y

Ben………. N  (1,1)  (3,0)

…………….Y  (0,3)  (2,2)
If you’re Ben, should you wash the dishes or not? Pause for a moment and think about it.

Assuming Ben is rational, the best answer is N, he will not wash the dishes. Why? Because whatever Jack chooses to do, Ben’s payoff in choosing N is always greater than his payoff in choosing Y. If Jack chooses Y, Ben’s choosing N will have a payoff of 3 while his choosing Y will only have a payoff of 2. If Jack chooses N, Ben’s choosing N will have a payoff of 1 while his choosing Y will only have a payoff of 0.

Hence, N strictly dominates strategy Y. But assuming Jack is also rational and utility-maximizing, he will also choose N, so that the outcome is (N,N) and they both get 1, which is Pareto inefficient.

Lesson: rational choices can lead to bad outcomes.

This example constitutes what is called the prisoner’s dilemma in the Game Theory of applied mathematics. Game Theory is a study of strategic situations. Strategic situations are situations where a person’s success in making choices depends on the choices of others.

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