Introduction to Game theory

Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose. The essential feature is that it provides a formal modelling approach to social situations in which decision makers interact with other agents. Game theory extends the simpler optimisation approach developed in neoclassical economics.
In game theory, normal form is a way of describing a game. Unlike extensive form, normal form representations are not graphical per se, but rather represent the game with a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive form representations. The normal form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, of each player.
In static games of complete, imperfect information, a normal form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, where a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility - often cardinal in the normal form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output.
Economists have long used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, fair division, oligopolies, social network formation, and voting systems. This research usually focuses on particular sets of strategies known as equilibria in games. These "solution concepts" are usually based on what is required by norms of rationality. The most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. So, if all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.
The payoffs of the game are generally taken to represent the utility of individual players. Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's utility. This assumption, however, can be faulty.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use should this information be put. Economists and business professors suggest two primary uses.
In game theory, dominance (also called strategic dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.
For example:
When a player tries to choose the "best" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better. The result of the comparison is one of:
B dominates A: choosing B always gives at least as good an outcome as choosing A. There are 2 possibilities:
B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do.
B weakly dominates A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give B at least the same payoff as A.
B and A are intransitive: B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing B is better in other cases, depending on exactly how the opponent chooses to play. For example, B is "throw rock" while A is "throw scissors" in Rock, Paper, Scissors.
B is dominated by A: choosing B never gives a better outcome than choosing A, no matter what the other player(s) do. There are 2 possibilities:
B is weakly dominated by A: There is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give A at least the same payoff as B. (Strategy A weakly dominates B).
B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B).
This notion can be generalized beyond the comparison of two strategies.
Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy.
Strategy B is weakly dominant if strategy B dominates all other strategies, but some are only weakly dominated.
Strategy B is strictly dominated if some other strategy exists that strictly dominates B.
Strategy B is weakly dominated if some other strategy exists that weakly dominates B.

If a strictly dominant strategy exists for one player in a game, that player will play that strategy in each of the game's Nash equilibrium. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium -- however, that Nash equilibrium is not necessarily Pareto optimal, meaning that there may be non-equilibrium outcomes of the game that would be better for both players. The classic game used to illustrate this is the Prisoner's Dilemma.
Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such it is irrational for any player to play them. On the other hand, weakly dominated strategies may be part of Nash equilibria. For instance consider the payoff matrix pictured to the right.
Strategy C weakly dominates strategy D. Consider playing C: if one's opponent plays C one gets 1; if one's opponent plays D one gets 0. Compare this to D where one gets 0 regardless. Since in one case one does better by playing C instead of D and never does worse, C weakly dominates D. Despite this, (D, D) is a Nash equilibrium. Suppose both players choose D. Neither player will do any better by unilaterally deviating -- if a player switches to playing C, they will still get 0. This satisfies the requirements of a Nash equilibrium.