# Game Theory

Game Theory is a method for formalizing interactions between players. You have a set of players, a set of strategies that the players can use, and a set of outcomes that result from the cross product of the players and the strategies. If you have a 10 player game and a selection of 5 strategies there are 5*10^10 (50,000,000,000) possible outcomes. That's a lot. Let's look at a 2 player, 2 strategy game instead. The one I like is the ever-distressing question of "should I call her/him or not?"

There's a girl/guy that you like and you don't know if she/he likes you or not. Your two options are to either call her/him or not to. If you call and she/he doesn't like you, you might look dumb, but if you don't call and she/he does like you then you'll miss out. So, what do you do? Just apply game theory! The numbers in the table below indicate how much benefit you gain from your strategy, depending on her feelings for you. (Don't get caught up on exact values, I'm going to pick them arbitrarily. The numbers are very important, however, since they determine what the best strategy is.)

 You \ She Likes you Doesn't like you Call +2 -1 Don't call -1 +1
You want to pick the strategy that maximizes your benefit. If you knew her feelings it would be easy to pick -- call if she likes you, don't if she doesn't. However, since you don't know she feels you'll have to pick the strategy with the highest likely payoff. If you estimate the chances are 50/50 that she likes you then you can estimate your expected gain for each strategy.

If you call, you'd expect a 50% chance of getting +2, and a 50% chance of getting -1.
(0.5)*(+2) + (0.5)*(-1) = 0.5

If you don't call, you'd expect a 50% chance of getting -1, and a 50% chance of getting +1.
(0.5)*(-1) + (0.5)*(+1) = 0

So with those outcome payoffs and those predictions for her behavior you're better off to call than not to call. However, if you think there's only a 25% chance that she likes you then you shouldn't call.

A lot of the trickiness comes from choosing the payoff numbers. Do you really lose something if you call and she doesn't like you? Maybe instead of a "-1" in that position there should be a "0". Similarly, if you really really really super duper like her, then maybe the payoff for calling if she likes you will be much higher than merely "+2". The relative values of these numbers really determine the most beneficial outcome of the game.

That simple example only deals with the benefit to you of your strategy. Let's look at a more complicated game that has benefits for both players: The Prisoners' Dilemma. To set the stage, imagine there are two thieves who have been caught by the police. The cops get the crooks, but they can't convict them unless they find the loot as well, and they don't know where it is. So the police separate the crooks into two rooms so that they can't communicate to each other and they tell each of the crooks: "If you tell us where the loot is and turn on your partner we'll go easy on you, but if your partner turns on you first then we're going to go easy on him and you're going to take all the heat." What do the prisoners choose to do? We can represent the situation with a table similar to the one above, except that this time there will be payoff numbers for both crooks -- the first number in the parenthesis represents the payoff for crook 1, and the second number represents the payoff for crook 2 (the payoffs are negative because they represent how many years the crooks will spend in jail, say).

 Crook 1 \ Crook 2 Doesn't talk Talks Doesn't talk (0, 0) (-10, 0) Talks (0, -10) (-5, -5)
Crook 1 has a decision to make: does he talk and squeal on his partner or doesn't he? If neither talks then both get to go free, but crook 1 knows that if he doesn't talk and crook 2 does then he's going to get screwed. If both talk, then both go to jail, but at least they'll get some time off for cooperating with the cops.

So crook 1 draws the table above and examines his options. If his partner doesn't talk, then crook 1 doesn't go to jail at all no matter what he himself does; however, if his partner does talk then the length of crook 1's sentence will hinge on whether or not he himself cooperated. So, even though crook 1 has no idea what crook 2 will do, he knows that he'll be better off if he talks to the cops, and so he does. If you've ever watched Law & Order then you know that the cops actually do this kind of thing all the time, and it's very effective.

Bah, games! What good are they in real life? Well, they're good for a lot and game theory is very useful in any situation where there is negotiation, such as diplomacy or economics. Saddam Hussein made the determination that he would benefit most by not cooperating with the UN and destroying his WMD, and his mistake was miscalculating the chance that the United States would attack. He knew that the cost of such an attack would be high, but he thought that the likelihood of it actually happening was low. Additionally, there was a cost associated with getting rid of his weapons because it would have weakened his position in the Arab world and within his own country. He may have constructed a table like this:

 America \ Saddam Gets rid of WMD Keeps WMD Invades (-20, -100) (-20*, -100) Doesn't invade (+10*, -20) (-20, +50)
So based on this table, Saddam would determine that no matter what America does he is better off keeping his weapons. If he gets invaded then it won't matter what he does, since he'll be deposed, but if he's not invaded then he will gain a lot of prestige for having faced down the US and will be able to keep his weapons. I put (*) next to two of the values because I wanted to point out that these were Saddam's miscalculations. He believed that America would benefit most by not invading whether he kept his weapons or not, and so he thought that America would use these values and thus not attack. However, these numbers were incorrect.

Once the United States had deployed troops on his border it was inevitable that we would invade and occupy his country, whether he gave up his weapons or not. Why? Because we can't allow countries to manipulate us into spending that kind of money and effort and then escape us just by changing their minds. If Saddam had changed his mind at the last moment and we had withdrawn our troops, then nothing would stop him from simply waiting until they were far away again and then starting his WMD programs. We would begin to deploy, and then at the last moment he could change his mind yet again. America couldn't afford to play this game, and so from the moment we had serious troops on the ground we were committed to an attack. By America's calculation the "+10*" was really a "-20".

Saddam also thought that if he kept his WMD and America did attack, he would be able to inflict substantial losses on our troops. The "-20*" represents this belief -- even if America attacked he thought that it would cost us a lot to do it, and he thought this would dissuade us. Many people around the world didn't think that the US was serious; based on recent history, they believed that as soon as we started taking casualties we would pull out. In fact, we never did start taking heavy casualties, but even if we had we would not have withdrawn our forces. In America's game theory table, this number was more like "+20". We wanted to go in there and shake up the region, undermine support for terrorism, and give the Arabs a bloody nose.

These two miscalculations resulted in Saddam's seemingly irrational behavior. If he had guessed these numbers correctly he would have seen that the strategy that would have given him the most benefit (and least loss) would have been to get rid of his WMD early and hope to avoid a US attack.

Game theory can be applied to almost every area of life. I'll write later about how our entire social fabric is based around game theory and the enforcement of cooperation, even though it is to everyone's individual benefit to cheat and steal.

GameTheory.net
The Prisoner's Dilemma: A Fable
The Prisoners' Dilemma simulation -- a neat Java applet.
Game Theory from Yahoo

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