Like most sciences, Game theory is an attempt to model the world in a useful and objective way. The simplest model, the strategic game, is often a suitable analog for an analysis. Strategic games have three components: a set of players, for each player a set of strategies, and additionally a payoff function indicating how desirable each game outcome is for the player. The players in the game are the same as the players in the strategic model. Each player plays the game based on the strategic preference of that particular player. By giving each player a strategy, we get the outcome of the game. In mathematical notation:
- Strategy sets: where is the set of possible strategies for player i and we could denote a specific strategy in the set by
- Payoffs: where each function is defined over the whole outcome of the game (i.e. the chosen strategies for each player)
- If a player has "rational" preferences as defined earlier and there are a finite number of possible game outcomes then we can always represent the preferences using a function.