Games

约翰·冯·诺依曼和奥斯卡·Morgenstern introduced game theory to the world in 1943 with "Theory of Games and Economic Behavior." They hoped to find mathematical answers to economic problems.

According to economic theory, producers could make a greater profit by reacting to conditions such as supply and demand. But these theories fail to account for the strategies of other producers, and how the anticipation of those strategies affects each producer's moves. Game theory attempted to account for all of these strategic interactions. It didn't take long for military strategists to see the value in this.

When we discuss game theory, we assume a few things:

Agameis considered any scenario in which two players are able to strategically compete against one another, and the strategy chosen by one player will affect the actions of the other player. Games of pure chance don't count, because there's no freedom of choice, and thus no strategy involved. And one-player games, such as solitaire, aren't considered by game theorists to be games, because they don't require strategic interaction between two players.
Players in a game know every possible action that any player can make. We also know all possibleoutcomes. All players havepreferencesregarding these possible outcomes, and, as players, we know not only our own preferences but also those of the other players.
Outcomes can be measured by the amount ofutility, or value, a player derives from them. If you prefer reaching point A to reaching point B, then point A has higher utility. By knowing that you value A over B, and B over C, a player can anticipate your actions, and plan strategies that account for them.
All players behave rationally. Even seemingly irrational actions are rational in some way. For instance, if you were to play two games ofpool, you wouldn't intentionally lose yourmoneyon the first game unless you believed that doing so would bolster your opponent's confidence when he or she was deciding how much to bet on game 2 -- a game you anticipate winning. This is an essential difference betweenone-shotandrepeatinggames. In a one-shot game, you play once; in a repeating game, you play multiple times. (A little later, we'll look at how rational thinking varies between one-shot and repeating games.)
If no player can reach a better outcome by switching strategies, the game reaches an impasse called theNash Equilibrium. Essentially, this boils down to players keeping their current strategies (even if they don't have the highest preference) because switching won't accomplish anything.

In the next section, we'll put this information to use and see what we can learn about strategy by plotting it on a game tree.

Game theory in the bathroom?

Mathematician John Nash -- winner of the 1994Nobel Prizein economics and namesake of the Nash Equilibrium -- invented a game in 1948 that became known as "Nash" or "John." It was often played on the hexagonal floor tiles in the bathrooms of Fine Hall, home to Princeton's mathematics department.

Nash was a two-person game in which players took turns placing pieces on open spaces of adiamond-shaped board formed of hexagons. The winner was the first to form an unbroken chain connecting his two sides of the board. This game was independently invented by a mathematician in Denmark and was known as "Polygon." In 1952, Parker Brothers commercialized the game and marketed it as "Hex" [source:Weisstein].