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Game Theory

#game-theory #mathematics #strategy

Game Theory

A “Game” is any situation in which there are several decision-makers, and each of them wants to optimize their results. The optimizing decision will depend on the decisions of the others.

Game theory attempts to define these situations in mathematical terms, and determine what would happen if every player acts rationally. Maybe an equilibrium can be reached (Which is why we all drive on the same side of the road within a country). Maybe this equilibrium will be worse for all players (Which is why people litter or pollute common resources), or maybe everyone will try to be as unpredictable as possible in their actions (as might happen with troop deployment in war). In essence, it's a way to mathematically model complex human behavior, to try to understand it and predict it.

Every game has players (the decision makers), actions (what the players can do) and payoffs (what motivates them, how they "profit" from each result.) So first you describe the possible universe of results. You take every action player A can take, and put them in columns. Then you take every action player B can take, and put them in rows. The intersections of columns and rows will be the results of each action. After that, you figure out how much each player wins or loses with every result, and write it in your column. Then you can analyze what each player has to do to optimize their payoff. And finally you can figure out what each player is most likely to do, and how this reflects on the system as a whole.

Of course, the whole point of this is that not only can you understand and optimize the game for yourself, you can set out to change the rules of the game in a way that the resulting equilibrium is more favorable for everyone.

I wish I was less tired so I could explain it better. My explanation is a bit simplistic, but honestly, Game Theory is one of the most fascinating and little-explored fields of study today. Its broadness makes it applicable to all kinds of situations, from relationships to job hunting to evolution to urban planning to financial trading algorithms to politics to war. If you combine the power of this tool with the capacity of computers to carry out calculations and the amount of data we have available, game theory can easily become one of the strongest fields in the following decades.

Beginner Resources for Game Theory:

Mind your Decisions, a really amazing blog that writes about Game Theory a lot. If you want an introduction, read this blog (instead of Wikipedia, which can be extremely arid when it comes to maths!)

Free University of Michigan course on Model Thinking a great entry-level course that touches on Game Theory. Fantastic if you want to start thinking of human behavior in more structured ways.

Free Stanford Course on Game Theory, a great mid-level MOOC

The assumption in an intro game theory class is that all players are rational, and purely so, which isn't the case a lot of the time in real life.

For the quintessential example of Prisoner's Dilemma, which was very well played out in the game show Split or Steal, there are SOOOO many other factors into the decision. If I'm in jail for a crime, caught with another person for the same crime, I would consider if the other person is a friend, how well I know them, if they're a moral person, if they're a religious person, etc. It's never as easy as class when you're in the real world.

Fun fact: game theory also explains why we always see gas stations in clumps and why in America political parties nominate candidates that are very moderate (relative to American politics).

One thing that I think can be confusing is when people say things like "game theory explains x".

Game theory doesn't really explain much of anything. That's not the point. The point is to model games.

Occasionally you end up "explaining" something in the sense that you see how something you thought to be irrational is, in fact, rational (like the gas stations), but I think the notion that game theory affords some sort of secret insight is one of the primary things that confuses a lot of people about it. Game theory is just a way of quantizing people's intuitions about what constitutes rational strategy. Once you quantize them, it makes it easier to break down more complex problems in terms of your intuitions about simpler ones.

At no point is any secret math voodoo giving you magical knowledge you couldn't otherwise arrive at.

As an analogy, knowing the equation for the area of a rectangle doesn't mean you've explained why a rectangle that's twice as long has twice as much area. You have to have figured out that fact about the area of rectangles before you write the formula - the formula doesn't reveal it to you.

I've seen a lot of instances of people saying things like "LOOK AT THIS GUY USE THE SECRETS OF GAME THEORY TO WIN". But another way you could phrase that is: look at this guy being clever. Game theory gives you a quantitative framework to reason about strategy, but it doesn't buy you any result you couldn't independently arrive at - it just makes it a little easier to arrive at them. Saying that someone is "using game theory" when they're not actually doing calculations using game theory is just saying that the person is playing a game rationally.