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The Board of Supervisors Problem

This is the case involving having to wait a long time for an event to happen. It can be for a haircut where you are told only that it will be a long wait, a place where you select numbers for waiting in line, or a court case or government body case which is in a huge docket of cases. Let's take the last one for an example. Suppose your development is concerned about a developer next door building lower cost homes and your neighborhood is afraid of plummeting market values of its homes. So your development is going to challenge their petition at the Board of Supervisors meeting to change the zoning to apartments, say. It is schedule for some night, but you are afraid that it will take a long time before the case comes up. Let's say it will come up sometime between 7 pm and midnight. So you say, I won't come at 7. I'll come in at 10 and save some time. But you are afraid they may pull a fast one on you and bring it up at 9. Then you'll miss it. What should you do?

It depends on your attitude to the supervisors. You could observe the past history of cases similar to your own and find out what the most likely time is for your case to come up. If it is 10, then you will arrive at 9:45. But suppose you don't have such a history, or you have had bad luck with situations in the past and believe that the system is out to make things the most miserable on you as it can.

In that case it is a 2-person zero-sum game. The theory of such games was worked out about half a century ago. Some of these games have what is known as saddle points, or Nash equilibria (mentioned in the beautiful movie A Beautiful Mind, which came out in 2002). In these equilibria, if you deviate from the solution, you will do worse. Well is there such an equilibrium in this case? You come in at midnight, because you feel that the system will make you wait a long time and delay the thing as much as it can. So then it decides to make it at 8 o'clock, to make you miss it. Seeing this, you want to come in at 8. But it decides to make it 7 instead to make you miss it. So you come in at 7. So then it makes it midnight to make you waste 5 hours of your time. And this thing goes on indefinitely. It is clear that this game has no saddle points.

In such a case, game theory says you should use a random device to determine what you are going to do. That keeps the other side guessing. But with what probabilities should you make the times to be? There is a well-known procedure for solving this, using linear programming. I set up the problem using 33 discrete times on an Excel spreadsheet and solved for it. The solution is this: with some high probability, come in on time. In the above example, arrive at 7; that is, on time. If you draw the other probability, then draw completely at random from all the possible hours, making the later hours more likely. How high a probability should you come on time? It depends on how much you value not missing the case versus wasting your time. Roughly speaking, how long a wait in hours must it be before you are indifferent between waiting that time and missing it? Say it is H hours. Then with probability H/(H+1), arrive on time; else draw completely randomly an arrival time. It turns out this is not a good game to play; missing the event is too likely, and you are apt to wait a long time. But this is the best you can do.

Now suppose everyone in the neighborhood follows this rule. Then a huge bunch, maybe 2/3 of the development, will come in on time to support their cause. While the other cases are going on, other people from the development will dribble in a little at time, with the frequency increasing as the night wears on. A few will miss it, when the case comes up and those who selected later times lose out. The last time I went to such a hearing, I did indeed observed this type of behavior to some extent. So game theory in this case had the power to predict accurately what people will do.