Lots of people wanted game theory, so game theory it is. The logical first question: what is game theory?
Game theory is typical of math. What mathematicians like to do is reduce
things to fundamental abstract structures or systems, and understand them in
terms of the abstraction. So game theory studies an abstraction of games - and
because of the level of abstraction, it turns out be be applicable to a wide
variety of things besides what you might typically think of as games.
Game theory starts with the fundamental idea of a game. What is
A game consists of a system with at least two agents (that is,
entities that can perform actions), who each have their own
interests which they are trying to satisfy. They have a situation,
which consists of the set of things that they have, and the things that
they want. They have a set of rules, which describes what actions they
can take in the situation, and when they can take them.
Game theory looks at that, and tries to understand how the
agents will interact. You can study it from many different points
of view, and with many different objectives.
For example, you can look at it from a very "pure" perspective, and
try to find optimal strategies for a particular kind of game. For example,
if you analyze the game of Nim, you can find that there's a guaranteed
winning strategy for player two. (We'll look at Nim, and why it's always winnable by player two in a later post.)
You can also use game theory for things that seem very non-game like.
Basically, any situation in which you have multiple interacting agents with
distinct, possibly conflicting goals, you can analyze it using game
A classic example of this is the prisoner's dilemma. In the
prisoners dilemma, you have two criminals who've been arrested for a murder.
The two criminals (now prisoners) are the agents. The police know that they
did it; but they don't have enough evidence to convict them of murder, only of
a lesser charge. So the police want to get the prisoners to rat on each other.
You end up with the following situation:
- If neither prisoner rats on the other, they'll both get off with a very light sentence of 6 months in jail.
- If one prisoners rats on the other, that prisoner goes free,
and the other one gets a life sentence.
- If both prisoners rat on each other, they each get 10 years in
If you look at this in the abstract, it's obvious what the prisoners
should do. If they both keep their mouths shut, then they'll both get off
easy. But if you look at it in terms of cost/benefit for one of them,
you get a very different answer: for a prisoner acting only in his own self-interest, the correct choice is to rat on his partner. Game theory
looks at it from that latter point of view: each agent is concerned
only with maximizing the benefit/minimizing the penalty for
To see why that works, just look at it from of one of the prisoners. We'll call him Adam, and we'll call his partner Bert. Adam doesn't know what Bert is going to do. So he looks at the options.
If Bert keeps his mouth shut, Adam has two choices: he can keep quiet, or
he can rat on Bert. If he keeps quiet, he'll go to jail for six months. If he
rats, he'll walk. Clearly, in this case, ratting on Bert is the best choice
If Bert rats, Adam has the same two choices. If he keeps quiet,
he goes to jail for life. If he rats, then he goes to jail for ten years. Once again, clearly, the better choice for Adam is to rat.
So each prisoner maximizes his own benefits by testifying, even though
that means that they'll both wind up spending 10 years in jail, when they
could have gotten off with 1 year each by cooperating. By trying to always
maximize their own benefit, both suffer.
Game theory works out a kind of categorization of games, based on
how many agents, whether they can cooperate or communicate, whether
they move simultaneously or take turns, etc. It also categorizes
based on the kinds of benefit/penalty relationships that define the
possible outcomes of the game. For games of different forms, it
describes strategies, equilibriums, and tipping points in those games.
It turns out to be an incredibly useful framework. It's used in computer
science for things like protocol design; it's used in economics for models of
markets; it's used in negotiation studies; it's used in sociology. The basic
idea of multiple parties acting in their own interest is a fundamental
framework for understanding almost anything involving multiple people or
multiple interacting processes of any kind.