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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)


………………….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.

How to beat the queue in McDonald’s

In popular economics on June 3, 2009 at 4:58 pm

How to select the fastest queue in a fast food restaurant or supermarket?

Let’s take a typical McDonald’s branch. Suppose there are 5 counters. One obvious option is to pick randomly, in which case you have 20% chance of being lucky to be in the fastest line. Can we increase this probability?

Professor Eugene Fama, contender for the Nobel Prize in Economics, will say you cannot beat 20%. That’s because according to the efficient-market hypothesis, all customers are perfectly informed about the quickest line. Perfect information eliminates the quickest line. If every customer knows which line is quickest, then any “get-food-quick” scheme is impossible and all the queues will move as they are supposed to move. (If it’s obvious that the shortest line is the quickest, then the other customers will immediately select it, after which it is no longer the fastest.)

Another option is available when you have brought friends, in which case you can apply the modern portfolio theory, which was pioneered by H.M. Markowitz, a 1990 Nobel Laureate. Scatter (or rather, diversify) yourselves into different queues and you can increase the probability of selecting the quickest. If there are two of you, then your chance of success is doubled to 40%. Nice tactic, but there’s a limit to this.

Supposing you have brought 4 friends so that there are 5 of you. Theoretically, you can apply Markowitz diversification by utilizing all 5, but a cost-benefit analysis will eventually cut your resources to less than that number. Consider this: we already know that the more you increase the friends to fall in line, the sooner the whole group can eat their meal. In effect, you are buying the opportunity to eat as soon as possible or ‘ASAP meal’ like a normal commodity in exchange for a certain number of friends. The only problem is that the ‘ASAP meal’ can be substituted by talking around the table while waiting for the other friends delegated to order the food. Talking is a substitute good for the ‘ASAP meal’, and it is purchased when the ‘ASAP meal’ is too expensive (requires all 5 friends). My hypothesis is that 3 is the maximum number of friends (including you) that a customer can diversify to fall in line, because at least 2 friends are required to purchase talking. Hence, 60% is the maximum probability of success. I say with perfect confidence that this occurs unfailingly in the real world.

Supposing you walked into McDonald’s alone, you cannot use modern portfolio theory (assuming of course that you do not block two adjacent lines, which I do in practice). Your other option is to perform a fundamental analysis. This requires a lot of data, but as a rule-of-thumb, examine the order-takers and avoid the trainees. If you want more sophistication and accuracy, study statistical reports on what type of customers order the most in McDonald’s at what specific time of the day, then it’ll be easy to guess which lines can bring you the ASAP meal.

The behavioral economics school, led most famously by 2002 Nobel Laureate Daniel Kahneman, argues against Fama’s efficient-markets hypothesis, but they both agree that you simply can’t beat the market. So too bad, Kahneman would say, you will never beat the queue in McDonald’s because of too much irrationality and chaos.