What Is Game Theory? A Basic Introduction and Example
Game Theory is one of the most fascinating branches of mathematics with tons of applications to fields ranging from the social sciences to the biological sciences. Game Theory has even found its way into mainstream media through movies such as A Beautiful Mind, with Russell Crowe.
This article will explain some of the fundamentals of game theory and work through a simple example.
Definition of a "Game"
Game Theory is the study of "games." Games, in the mathematical sense, are defined as strategic situations in which there are multiple participants. Furthermore, the outcome of the decision any individual makes is dependent on the decision that individual's decision and the decisions made by all of the other participants.
Is Sudoku a "game?"
No, not the way we defined "game." Sudoku is not a "game" because what you do when solving the game is independent of what anyone else does.
Is Chess a "game?"
Yes! Imagine that you are playing a game of chess with a friend. Whether you win or not will be dependent on the moves you make and the moves your friend makes. At the same time, whether or not they win will be dependent on the moves they make and the moves you make.
NOTE: The most important thing to realize in the chess example is that at least 2 "participant's" decisions were affected by the decisions of other participants. Solving a Sudoku puzzle is not a game since how you solve the puzzle is not affected by anyone else's decisions.
Ok, I get what a "game" is, but what is Game Theory?
Game Theory is the study of "games." Game theorists try to model "games" in a way that makes them easy to understand and analyze. A lot of "games" end up having similar properties or reoccurring patterns, but sometimes it is hard to understand a complicated game.
Let's work through an example of a game and how a game theorist might model it.
Example: The Game of Chicken
Consider the "game" of chicken. In the game of chicken we have 2 people, Bluebert and Redbert, who drive their cars at full speed towards each other. They each have to make the decision just before crashing to either drive straight ahead or to swerve at the last minute. The possible results are as follows:
Bluebert is happy he wins, Redbert is sad he loses
Bluebert is sad he loses, Redbert is happy he wins
They stare at each other shocked at what they've done
Now that we know the general results, this isn't the easiest way of understanding the game. Let's reorganize the possible results into a matrix.
This is called a payoff matrix. The rows represent the possible actions of Bluebert. The columns represent the possible actions of Redbert. Each box represents the result from each combination of decisions. By using this matrix, it is easy to see what the result of different combinations of actions is.
A quick example: If Bluebert swerves, then we know the result will be one of the top 2 boxes, depending on what Redbert decides to do. On the other hand, if Blubert goes straight, then we know the result will be one of the bottom two boxes, depending on what Redbert decides to do.
Let's replace the illustrations of the results with some numbers to make things easier to analyze.
- Both swerving and staring at each other = 0 for both
- Both going straight and crashing = -5 for both
- One swerving and one going straight = 1 for the winner (straight) and -1 for the loser (swerve)
Some Simple Analysis:
Now that we have organized this game theoretic "game" into an easily readable payoff matrix, let's see what we can learn about how the game will be played out.
The first thing we will look at is something called a best response. Essentially, lets imagine that we are Bluebert and we KNOW what Redbert will do. How do we react?
If we KNOW Redbert will swerve, we need only look at the left column. We see that if we swerve we get 0 and if we go straight, we get 1. So the best response is to go straight.
On the other hand, if we KNOW Redbert will go straight, we need only look at the right column. We see that if we swerve we get -1 and if we go straight, we get -5. So the best response is to go straight.
In this game, Redbert has similar best responses.
If you have seen the Ron Howard movie, A Beautiful Mind, with Russell Crowe, you may remember that it was about the Mathematician John Nash. Nash Equilibriums are named after this very Nash!
A Nash Equilibrium is when all players play a best response. In the game of chicken above, both players going straight is not a Nash Equilibrium because at least one player would have preferred to swerve. In the game of chicken, both players swerving is not a Nash Equilibrium because at least one player would have preferred to go straight.
However, when one player swerves, and one player goes straight, this is a Nash Equilibrium because neither player can improve their outcome by changing their action. Another way of saying this is that both players are playing a best response.
If you've made it this far congrats! You've learned the basics of game theory. It wasn't the most fun we can have with game theory, but it did lay a solid foundation to understand this amazing branch of mathematics, and you can see how applicable it is to many different disciplines.
If you have questions, comments, or suggestions, please let me know. In particular, if something was unclear above, let me know so I can try to explain it better. Thanks!