My “Day I Left Pennsylvania” led me to some archived website posts (before blogs were invented) I had written many years ago. I’m re-posting them now. Bear in mind that most of the content in this series is over 5 years old. I have left the content more or less intact. I have removed some links and added some others — but that’s it. Enjoy!
Perhaps best known for the in-depth foray into the fundamentals and theory of zero-sum games is John Nash. A zero-sum game is where the winnings and losses of all players involved is always balanced. Poker is a zero-sum game, Blackjack is not.
John Nash said that during any given game, there is an equilibrium point, where no player has any advantage over another. Comparing this to poker, this would be like if all players had an equal number of chips. While this isn’t required to happen, it must be possible for it to happen.
Game theory is applied to phenomena well beyond those of board games and card games. It has military implications, economic and commercial strategy applications as well.
A common example used in game theory is a matrix like this:
|
A
|
B
|
C
|
|
| X |
-10
|
8
|
2
|
| Y |
-5
|
5
|
5
|
| Z |
10
|
-8
|
-5
|
Now for this example, we’ll say the rules work like this: Player 1 picks a column, and Player 2 picks a row. Then they switch. Whomever picks the row gets (or loses) that many points. For example, say Player 1 (P1) picks column C, secretly. Player 2 (P2) does not know which column P1 picked, but he can look at the rows and decide which is most likely to give him the advantageous outcome. For the sake of example, let’s say he feels lucky and picks row Z. Both Players now reveal their choices, “C” and “Z”, and you find the corresponding score change, which is “-5″. So P2 loses five points.
Now notice something: both players must consider what the other player is doing, or the “metagame” of the strategy. The best situation for the column-choosing player is column “A” because it has two losses and 1 (big) winner. The least advantageous is column “B”, because it has two reasonable winners and one loser. Column “C” is therefore the best ‘average’ pick for the Column-picker.
For the row-picker, the person who’s score will be affected, row “Z” may be tempting, as in our example, because of the possibility of scoring 10 points. However, there is also a 2 in 3 “chance” that the column picker picked B or C, meaning he would lose 5 points, as in our example. Row X, while offering two wins, also has the huge downfall of the -10 in column A. Therefore, Row Y is the best choice for the row-picker.
The best choices for each are Column “C” and Row “Y”. The value at this position is “5″. “5″ is said to be the “value” of this particular game, because it is the the value that is most likely to come up if both players play with perfect knowledge, that is, if they both hypothetically knew what the other was thinking.
These types of game matricies are common in game theory, and illustrate a very valuable point about games: the human interaction. The idea of being able to “bluff”, and the fact that humans are prone to irrational decision-making makes games playable and fun. Tic Tac Toe is a perfect example of game theory. There are a finite number of different combinations, and both players are working directly against each other to both win, or cause a draw. Ultimately, with perfect knowledge, all TicTacToe games would end in a draw.
