Game theory was designed as a decision-making tool to be used in situations in which there are factors other than chance and your choice. In game theory related problems, while decision makers are trying to manipulate their environment, their environment is trying to manipulate them.
In general, each player in a game will have a strategy that maximizes his return in the long run, and a goal of many game theory scenarios is to find that strategy. Things aren't always so simple when applying that strategy to real life, because people don't always act in accordance with such a strategy. Consider a game in which you have the choice of the following two possibilities:
Most people would go for option 2, and this option does maximize a player's average return ($5 versus $1). Now consider the following game in which you have the following choices:
Now most people will choose option 1, even though this option leads to a smaller average return ($1,000,000 versus $5,000,000). This is because most people perceive both $1,000,000 and $10,000,000 as "huge sums of money", so the difference is not comprehended as easily as the difference between $1 and $10. Most people want a sure "large sum" rather than a 50% chance of getting nothing.
Another interesting application of game theory is in the game known as Prisoner's Dilemma (which I'll get around to mentioning something about later).
See also Newcomb's Paradox.
Combinatorial game theory is another theory of games. In the examples above, note that the games involving flipping a coin involved an element of chance, while Prisoner's Dilemma involves hidden information (the two prisoners do not know what option the other has chosen). In contrast, combinatorial games (1) have what is known as perfect information (no chance is involved and no player can hide information about the game state from other players) and (2) they are sequential (one player makes a move, then another, and so on). For more information on combinatorial games, see Combinatorial Games.