Strategy (game theory)

In game theory, a player's strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour. A player's strategy will determine the action the player will take at any stage of the game, for every possible history of play up to that stage.

A strategy profile is a set of strategies for each player which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.

The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for playing the game, implicitly listing all moves and counter-moves for every possible situation throughout the game. The number of "moves" in a Tic Tac Toe game is 4 or 5, depending on whether you start or not, and considering that neither player can skip a turn; while the actual number of "strategies" is over 6 trillion.

Types of strategies
A pure strategy provides a complete definition in which way a player may play a game. In particular, it defines every possible choice a player might have to make, which option the player picks. A player's strategy space is the set of pure strategies available to that player.

A mixed strategy is an assignment of a probability to each pure strategy. It defines a probability over the strategies, and reflects that, rather than choosing a particular pure strategy, the player will randomly select a pure strategy based on the distribution given by their mixed strategy. Of course, every pure strategy is a mixed strategy which selects that particular pure strategy with probability 1 and every other one with probability 0.

Tit for Tat
Strategies in game theory are of essential importance, since the prisoners dilemma was shown never to lead to cooperation unless multiperiod strategies are considered. A highly effective strategy is "Tit for Tat". It was found in a programming contest, with several algorithms competing for the highest utility score.

Roulette
There is a variety of betting strategies and tactics in the Roulette game. The most famous strategy is the doubling strategy:


 * 1) Set 1€
 * 2) If you lose: double your bet
 * 3) Repeat 2. until you have a profit

This was originally called the "Martingale strategy", and was formalized simply to show why it will not create an expected profit. However, it is a common strategy seen in many casinos, especially by beginning players who are sometimes called "system players". The typical casino prefers the "system player" to other types of player because the casino's risk is very low (they stand to lose only the amount of the minimum bet each time the player starts), but their potential reward is extremely high (the entire capital of the gambler). However, most "system players" tend to play only for a short time, and so the casino's edge is not compounded as often against the system player as against the usual roulette players (who vary their bet size less).

Hedging
Hegding is a strategy for financial investments, that searches for the lowest risk or optimal risk to performance ratio. Some kind of hedges are uniquely determined from simple parameters. The Black-Scholes equation demonstrates how a continuous stock buy and selling strategy can replicate an option without risk.