Game Theory

Game theory is an attempt to model the way decisions are made in competitive situations. It has obvious applications in economics. But it has also been applied to a huge range of other areas, including politics, philosophy, biology, and computer science.

A basic starting point is the “two person game”. Two players must decide how they will move, and are rewarded or punished accordingly. There are many variations, such as one player moving first and the other responding, or both moving simultaneously, or taking turns for a sequence of moves, etc. The goal of the theory is to analyze how they should move so as to optimize the outcome. It hopes to provide a “rational basis” for playing the game. Naturally, the players do not have to be “persons”. They can be corporations or other entities, and, for applications to biology, they can be species or even Nature.

Of course, “games” in the real world tend to be very complicated, too much so to be readily subjected to analysis. In a way, it is surprising how much we can learn about real situations from simplified formalized versions. They do not tell us exactly how to proceed in various real situations, but they do provide insight into what is going on. One of the most difficult situations to address, either in theory or in practice, is an opponent who behaves irrationally. Most theory about games assumes that all players behave rationally. It is difficult even to see how to incorporate irrational behaviour into the analysis of a game. But even in this situation, we can learn something from game theory.

A simple kind of game is the two-person “matrix game”. The idea is that each player can choose from a list of strategies, and the outcome of the game is determined by the strategies chosen by both of them. For each possible combination of strategies, one for each player, a payoff is specified for each player. For example, suppose one player can choose either of two strategies, a and b, and the other can choose from a list of three: A, B, C. Then the game is completely specified by a table of the payoffs to each player.

Here is such a table: The idea is that the two numbers in each cell represent the payoffs to the two players for each combination of strategies. ABC a2, -13, 1-1, 3 b0, 3-1, 23, -1 Player #1 Player #2 For example, if Player #1 plays strategy a, and Player #2 plays C, then Player #1 loses 1 point and Player #2 wins 3 points. A table like this is called a “matrix”, and that is why this type of game is called a “matrix game”. -1, 3

The horizontal lines in the matrix are called the “rows”, while the vertical lines are the “columns”. ABC a2, -13, 1-1, 3 b0, 3-1, 23, -1 Player #1 Player #2 With this terminology in mind, Player #1 is sometimes called “Rose”, and Player #2 is “Colin”. Rose Colin

How should the players play? Rose would like to get 3 points, which is possible if she plays a or b. But in either case, it is also possible that she will lose a point. ABC a2, -13, 1-1, 3 b0, 3-1, 23, -1 Player #1 Player #2 Colin could get 3 from either A or C, but in either case might lose 1. Rose Colin A “safe” strategy for Colin is B: he will get either 1 or 2.

If Rose observes or predicts that Colin will play it safe with B, then she should play a. Unfortunately for Colin, this limits his winnings. ABC a2, -13, 1-1, 3 b0, 3-1, 23, -1 Player #1 Player #2 If he realizes that she is going to play a, then he should play C. Rose Colin And so it goes. If either player’s intentions are predictable, then the opponent can take advantage of this knowledge.

Example: Stag Hunt This game is used to explore questions of cooperation among players. The idea is that two wolves working together can catch a stag, whereas each on his own can only catch a hare. staghare stag3, 30, 2 hare2, 02, 2 If they work together, they can catch a stag, and each has the highest possible payoff. But if either refuses to hunt a stag and goes after a hare, the other is better off also going after a hare. Wolf #2 Wolf #1

This is a simple version of something that really happens. If people work together and compromise to reach a consensus, they can often achieve the best results for everybody. staghare stag3, 30, 2 hare2, 02, 2 But as soon as one participant chooses to put his self interest first, then there is an incentive for the others to do the same, for fear of losing out. 3, 3 2, 0 2, 2

You could argue that this is what happens to global efforts to control the environmental crisis. A multilateral agreement like the Kyoto Accord offers a good hope for improving things for everybody. staghare stag3, 30, 2 hare2, 02, 2 But if a few countries refuse to sign, it makes others reluctant to sign. Or, if they have signed, it makes them reluctant to act on their commitments. They fear that the cost of emission reductions will put them at a competitive disadvantage. The end result is that everybody may continue polluting, which is bad for everybody.

This offers an interesting insight into the way we often seem to end up doing something that is not really good for anybody. It is especially likely to happen when there are multiple “players”. staghare stag3, 30, 2 hare2, 02, 2 But even two players can easily find themselves in a stalemate in which neither is prepared to budge, even though both could do better if they changed their behaviour. A classic example is the division of property after a divorce.

A special kind of game is one in which the payoffs are each other’s negatives: In this situation, it is not really necessary to write both payoffs. It is enough to write the payoffs for Rose only. ABC a2, -2-1, 1-3, 3 b0, 0-2, 23, -3 Player #1 Player #2 Rose Colin A game like this is called a “zero-sum game”, because the payoffs in any situation add up to zero. ABC a2-3 b0-23 Rose Colin

Many real-world games are zero-sum games. A typical example involves the division of fixed resources, financial or otherwise. An analysis of how best to achieve this can be complicated and difficult, even for simple examples. ABC a2-3 b0-23 Rose Colin