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The Next Page: Everyday uses for game theory (such as, when to wash the dishes)
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Game theory whiz Kevin Zollman
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You're in a traffic jam. You're about to close on an important deal. You need to persuade your spouse to pick up the kids.
When you make a decision, you make it. You don't think about the mathematical models behind it. That's Kevin Zollman's job.
An assistant professor at Carnegie Mellon University, Mr. Zollman lectures on game theory, the mathematical study of human strategy. Spanning economics, politics and science, game theory can tell you the best way to plan your commute, how to negotiate a bargain -- and even when to leave up the toilet seat.
"Game theory is a mathematical theory aimed at understanding strategic situations -- situations where how well one person does depends on what others do, and vice versa," he said. "Chess, poker and football all fit into this category, but so do more weighty decisions like investing, deciding who to marry and deciding what job to take."
Here, he tackles four everyday problems with his mathematical mind and a toolbox of theorems.
Have to make a decision in your life? Maybe game theory can help you, too.
Question: I don't call my mother as often as I should, but she doesn't call me that often, either. When we get on the phone, one of us gives the other a guilt trip about not calling often enough. We're constantly doing this to each other. Is there any way out?
Answer: From a game-theory perspective, the situation you're facing is actually very common. When you call your mother, you have an incentive to guilt-trip her because doing so gives you the moral high ground. If you were never going to interact again, nothing would stop you from giving her a guilt trip. But, thankfully, you will interact again. And that might be an incentive to change things.
PG graphic: Prisoner's Dilemma
(Click image for larger version)
This problem is very similar to the Prisoner's Dilemma. Here's the story:
Two prisoners have been caught red-handed committing a minor crime, but they are suspected of committing a much more serious one. They are arrested and held in separate rooms. They are told that they are suspected of the serious crime, but that there is insufficient evidence to convict them. Despite this, they are offered the opportunity to confess to the serious crime. If neither confesses, they will go to jail only for the minor crime. If one prisoner confesses and the other stays silent, the confessing prisoner will get a free pass on both crimes in exchange for her testimony against the other. If both confess, both go to jail for the more serious crime, but each will get a small amount of time off for confessing.
If the prisoners care only about minimizing their jail time, each has an incentive to snitch. But there is another possibility, known as the "Folk Theorem" in game theory. If there's a high likelihood that the prisoners will interact again, both prisoners can agree to refrain from confessing. To enforce that contract, both can adopt the strategy known as Tit for Tat. If one of the prisoners confesses on this interaction, the other will confess on the next one.
In the right situation, this contract is "self-enforcing." The risk of the other one selling you out tomorrow is sufficient to prevent you from confessing today.
If you and your mother agree that you won't give her a guilt trip so long as she doesn't give you one, either, then you each have a strong incentive to keep the bargain. This solution has been put to use in business contracts and arms treaties. So it will probably work with you and your mother, too.
Question: When should you change lanes in heavy traffic? When the car in front of you changes lanes? When the lane besides you starts speeding up?
Answer: If you're like me, no matter what you do, you'll end up in the slower lane. Just accept your fate.
But perhaps you'd like more than a pessimist's answer. Let's begin by assuming that one lane is moving faster simply because it has fewer cars (there are no accidents or other obstructions).
PG graphic: Changing Lanes
(Click image for larger version)
If you're the impatient type, you might switch lanes. So long as there aren't too many impatient types, this will work out. But if many people switch into the fast lane, it becomes the slow lane. The situation might seem hopeless: Whatever you do only works if other people don't behave like you do. But there is one nice solution that will work out in the long run for everybody: randomly choose when to switch lanes. Every so-many minutes, check the other lane, and if it's moving faster, switch. Otherwise, stay in your lane. Then repeat the process.
Mathematicians have shown that if everyone adopts this strategy in this situation, all will end up in the equilibrium -- the state in which both lanes are moving at the same speed. While being in an equilibrium isn't the best outcome for you --you'd rather be in a lane all by yourself -- it's the best we can do while still being fair to everyone.
The drawback, of course, is that everyone on the road has to have read game theory.
Question: When should you ask your boss for a raise?
Answer: There are more mathematical models of bargaining than an economist can shake a stick at. My favorite is by an Israeli economist, Ariel Rubinstein. Using Mr. Rubinstein's model, economists have found that there are two factors that determine who gets the better end of a bargain.
First, the person who has less to lose gets a better deal. This makes sense. If your boss has 10 other workers who are almost as good as you are, your boss has little incentive to give you very much. On the other hand, if you have another job that is almost as good as your current job, you don't have much to lose by quitting.
The second critical feature is a little less intuitive. Mr. Rubinstein showed that the person who is more patient -- that is, the person who has less to lose from the negotiation dragging out longer and longer -- will end up winning a larger share. If I'm patient, then I can refuse your offer and drag the negotiation out, and if I'm more patient than you, I can use that tactic to get you to give me more upfront.
So, the best time to ask your boss for a raise is when these two factors favor you. Ask your boss for a raise when your boss has more to lose from you quitting than you do (as when you have another offer or when you're difficult to replace). Also, ask when you can be most patient -- don't wait until you really need a raise quickly but instead ask when you can drag the negotiation out.
Question: Should you always clean dishes right after you use them, or is there any justification to let them sit in the sink for a while?
Answer: Well, if you're like me and your significant other hates dishes sitting in the sink, you have a very strong incentive to do them right away. But let's ignore those sorts of considerations for a minute. If you don't have a dishwasher, it's probably better to let them accumulate for a little while before you do them. That's because doing dishes, like many manufacturing processes, features decreasing costs to scale. There is a fixed set-up cost -- putting water in the sink, rolling up your sleeves, etc. So, if you can minimize the number of these fixed costs that you pay in a week, you do better.
If you don't put water on your dishes right away, however, this argument may break down. As food dries, it becomes more difficult to remove. Now you have to consider the impact of the fixed costs against the increased time to clean each dish. Unless you really miss your college calculus tests, calculating this trade-off can be unpleasant. So put water on the dishes to prevent the food from drying.
Lillian Thomas is assistant managing editor/special projects: lthomas@post-gazette.com. Andrew McGill covers Allegheny County government and designs newsroom Web projects: amcgill@post-gazette.com. Kevin Zollman is an assistant professor of philosophy at Carnegie Mellon University. He has a doctorate from the University of California, Irvine, and has published more than 20 peer-reviewed articles on the application of game theory to biology, economics and philosophy: AskAGameTheorist@gmail.com.
The Next Page is different every week: Joe Smydo, editor (jsmydo@post-gazette.com).
First Published February 3, 2013 12:00 am
