In a discussion about decision theory recently, I realized that my friend was confused about the difference between game theory and decision theory. In fact, he seemed to equate them. The difference seems obvious to me, but perhaps this mistake is made more often than I thought.
In pure decision theory, a rational person (I know that might be an oxymoron) is faced with a physical situation that might have many variables, some of which can be random. But there is an implicit assumption that the Universe is not cheating, trying to hinder, or helping. It just is. We who live in the real world might feel this is a bad assumption since we know that things are going against us on purpose. Murphy’s Law was coined to express that feeling. The lesser known “first law of perversity of small inanimate objects” is another attempt to catch the feeling that it is “them” against “us” where “them” includes anything that stands in our way of doing something we really want to do. In truth, we know that “they” do not really exist just to bother us, and the Universe continues to exist without caring too much about what we want to do. When attempting to make the best decision about a physical situation, decision theory was developed to help bias the outcome to the desired one. Note that due to the presence of random variables, the computed best decision in only a probable best decision. Betting on a lottery is not a good financial decision on the average, but if you win the mega-dollars drawing, then for you it was a good decision.
Game theory, on the other hand, explicitly assumes other rational beings are involved in what you are doing — or trying to do. They might be helping; they might be hindering; or they might ignore you. Either way, the world defined by your problem includes other rational people who must be dealt with intelligently to get the job done.
There is a murky border where the person trying to make the best decision is working in a situation with insane (or, at least, not always rational) adversaries. We can simplify what follows by ignoring that possibility or by considering it to be a subset of pure decision theory. That is a cheat, but in one post, we cannot consider all cases.
Here’s an example. You need a new computer. How do you decide which one is best and where to buy it? That is a decision problem. Now suppose you located the computer you want and it is up for auction in a public place. You and another person both want it, but if you bid against each other, you will run the price up. No one else is bidding. What do you do? That is a game problem. Remember that the auctioneer is part of the game.
Here’s another example. Artificial constructs — like the puzzles based on trying to obtain valid information from a person who either always lies or always tells the truth — do not use game theory. This class of puzzles can be transformed into game theoretical puzzles by giving the antagonist rationality and a goal. For instance, consider the classic case of a person shipwrecked on an island known to contain two tribes: one always lies, and the other always tells the truth, but either will only answer one question. A fork in the road can either lead to a village and help or to a dismal swamp and certain death. A native stands at the fork. What do you ask? This is a logic problem, not a game problem.
However, assume the same situation, only now the native belongs to a tribe that does not like foreigners (except as food). The village is populated by a tribe that has good relations with everyone, including the nasty native. If he helps you to the village, he aids an enemy and loses a potential dinner. If he directs you to the swamp, he can collect your clothes and gain respect in his village. You do not know if he is nasty or nice. Neither of you are armed. How do you approach this dilemma?
One answer that I rather like is to walk straight up to the native and ask if he knows they are giving out free beer in the village. Then follow him. If the nasty native is truly dedicated, he will run the wrong way, but at a terrible cost. Most likely he will lead you to the village.
Maybe you can think of a better approach, but whatever you come up with, it must consider the fact that you are facing another rational human who has different needs than yours.
This difference between decision theory and game theory seems to be particularly important at this time of year in which we try to be more than usually concerned with the needs of other people. It is also important when you consider that the USA just pulled out of a war. Think of the various viewpoints that must be addressed to come to the conclusion that pulling out now was the best decision. This includes voters, politicians, insurgents, terrorists, and just about everyone. That makes a very complex situation. We can only hope that the analysis was done correctly.
When people learn that I am a physicist, they are sometimes impressed, but I am quick to point out to them that the genius of the physical sciences is that they attack problems that can be solved. I can, and have, designed instruments to fly in space and used them to study natural plasmas. That is really simple compared to trying to compute what is the best action to take in Iraq. So you will always see more puzzles from me based on decision theory than based on game theory. Game theory involves other human beings and that makes such puzzles much more difficult.