Game Theory is a well known way of analysis used in many fields to determine what people would do, when each of their actions affects the others and neither is aware of what the other is going to do. This dilemma of sorts results in decisions made by all the players that lead to their gaining less than what they could have otherwise.
For example, let's analyze two firms A and B, competing with each other to sell a TV that is similar in every way. Now, before Christmas, they have to decide whether they should offer a 10% discount to customers who buy the TV from them. The profits each of them can make if neither offers a discount is $5 million. If A offers a discount and B doesn't, A makes profits of $4 million and B make profits of $3 million. If B offers a discount and A doesn't the payoff is reversed. In case both offer a discount each of them only makes profits of $3.5 million. Now what would the two do.
As each of the firms wants to maximize its gains, neither of them should offer a discount and make profits of $5 million each. But none is aware of what the other will do which could reduce their profits to $3 million. So, both of them are forced to offer a discount. Though this results in total profits of only $7 million instead of $10 million, this is the decision we find them taking using Game Theory as the tool of analysis.
Game theory is a set of theoretical framework used for understanding and analysing situations in which the decisions by two or more independent groups or persons impact the outcome or results obtained by each other. Example of one such situation is the Prisoner's Dilemma. For additional details of Prisoner's Dilemma please refer to the enotes Q&A, for which link is given below.
Though the theory is named as game theory, it intended to be used as an aid for taking major business decisions rather than in games.
The field of game theory was formally established with the work of the mathematician Lohn Von Neumann, who along with Oskar Morgenstern is the author of the book The Theory of Games and Economic Behavior, published in 1944.
The last of the three enotes pages links provided below gives a list of different games in game theory.