Saturday, August 4, 2012

How to Win Games: Basic Strategy

As you may know, I just recently was at the Johns Hopkins Center for Talented Youth camp taking Game Theory and Economics. I did a few posts that discussed some stuff we were learning in class, but I never actually talked about how to win a game.

In mathematical terms, games are formatted with a table (known as a matrix), where player one is choosing a row and player two is choosing a column. The rows and columns are their strategies, or choices. Each box in the actual table is the payoff, which is how much they gain from the decision (points, profit, etc). The comma separates player one's payoff from player two's payoff.

Let's look at a quick example. Say Home Depot and Lowe's are selling wood, and they have to choose their price. Home Depot has a choice of:


Lowe's has a choice of:


There are 5000 lumber customers in the area, and if they both had the same price, they would be split equally between the two companies. However, every dollar less one company is than the other, they will steal 500 of the other company's customers. The matrix looks like this with the payoffs as how many thousand dollars the company makes:

$5    10,12    15,12    20,8   
$7   7,16  14,18 21,16

This game is pretty easy to figure out which strategy each player should choose. First off, look at Lowe's choices.

If they choose $8, they are always getting less profit than if they choose $6 or $4, regardless of which price Home Depot chooses. You can phrase this by saying $6 dominates $8, or $4 dominates $8. $8 is then considered a dominated strategy.

Is $4 or $6 a dominated strategy? Well, all of $6's numbers are greater than or equal to all of $4's numbers, so $4 is a dominated strategy as well. This leaves $6, which is the dominant strategy.

Since Home Depot knows that Lowe's will choose $6, they must determine their best price. They would prefer 15k to 14k, so they should choose $5 as the price for their wood.

This point that they led to is called the saddle point. In games with a saddle point, it is very easy to choose your strategy because you just go with the strategy corresponding with the saddle point.

It is pretty simple to determine these saddle points and dominated strategies, and will give you a start in game theory. Next week, we will learn some strategy for games without saddle points.

Answer: A month ago, I gave the problem with the king and the dragon, where the dragon manages to pull through and take over the kingdom even though the king has enormous chances of killing him. Here is what the dragon did.

To stay alive, he drank water from well 1 first, which poisoned him. Then, he received the well 10 water from the king, which cured well one's poison rather than poisoning him.

To kill the king, he knew that the king would drink from well 10 after drinking the dragon's offering. Well 10 would cure any poison, but it still is a poison itself. So, the dragon served the king fresh, unpoisoned water. The king then went and poisoned himself with well 10's water, and died leaving the dragon to rule the kingdom.

1 comment: