Game theory is the mathematical study of the strategic interactions of two "players" in a game situation. Although this could be the case of two people playing an actual game, game theory is more commonly applied to the behaviour of firms, particularly in an oligopolistic market (a market made up of a small number of large firms), the interactions of countries in a political negotiation, or even how states will respond to nuclear attack. Game theory is a very wide ranging and adaptable subject, and can also get very complicated. Here though, are the basics.
One of the most famous, and first, game theorists was John Nash, who formulated his fundamental concept of game theory in 1951; the Nash equilibrium. He is also the subject of the academy award winning film, "A Beautiful Mind", and won the Nobel Prize for his work in 1994. A Nash equilibrium is defined thusly: it is a set of choices such that, when both players (games are usually only played with two people for simplicity) reveal their choices, neither wishes to change his or her strategy.
Perhaps the best-known and most interesting example of this is that of the Prisoner's Dilemma. The situation is as follows. Two criminals have been caught on suspicion of a crime, and are being held in separate cells. Each prisoner is given the option of grassing his partner up, or keeping silent. Should they both keep silent, they will each receive a year in prison on a technicality, and then go free (total = 2 years). If they both tell, then they both receive five years in prison (total = 10 years). Finally if one tells but the other keeps silent, then the silent one gets ten years, but the other gets off with no years (total = 10 years). Clearly the strategy most in their favour is the first one, where they both keep silent.
This is not a Nash equilibrium however; if they both reveal their preferences like this, then they will each want to change their mind and tell on the other one, because that way they will get off with nothing. But of course they both want to do this, so they will both end up with five years - as such this is the Nash equilibrium, and is also the dominant strategy equilibrium, because it is the only Nash equilibrium in the game so the players will always end up there - a shame because if they were able to coordinate, then they would achieve the best outcome.
This game may seem somewhat divorced from reality, but in actual fact it has a number of real world applications. Take the example of firms that collude to maximise their profits - this is done by restricting output and charging a higher price, effectively the two firms act as a monopolist. But if one firm defects, increasing their output, then they will make more profit than their partner for the brief period until their partner realises and punishes them, meaning that defection and the breakdown of the cartel is the dominant strategy (and Nash equilibrium) and thus cartels are inherently unstable.
Another interesting and important game with applications in the field of economics is a sequential game. Here, rather than revealing their choices together, one player acts first and then the second acts based on their choices. This is normally modelled using a tree diagram sort of affair, but that won't work here, so I will just describe an application.
A common real-world example of such a game is a game of entry deterrence. Say we have a market with a single firm, a monopoly. Monopolies enjoy their position because they are able to charge a higher price and make more profit than a firm in a competitive market, and as such they don't want any new firms entering the market and spoiling their fun. To ensure that this does not happen, they can imagine a sequential game.
The potential entrant goes first, and decides whether or not to enter the market. Although he acts first, he is able to work out what the response of the monopolist will be, because the monopolist, it is assumed, will choose the course of action that will maximise its profits (and indeed this tends to be a reasonable assumption). So, it looks at what options the monopolist has available to it, in this case either starting a price war, or not starting a price war.
Now the monopolist may be prepared for a price war, or it may not be - preparation takes the form of maintaining excess production capacity so that it can rapidly produce more goods without increasing its costs astronomically. If it is not prepared then a price war will be mutually assured destruction - costs for both firms will spiral out of control and won't be covered by sales. Both firms go bust. So if the monopolist is not ready for a price war, the entrant knows that it is safe - it is not in the monopolist's interest to try and drive it from the market, and the potential entrant will decide to enter.
If the monopolist is ready for a price war however, then the entrant knows that it will lose - the monopolist will be able to win the war, losing some profits temporarily, but then when the entrant goes bust it can go back to its previous privileged position. In this case, the entrant will not choose to enter.
Looking at this situation tells us interesting things about monopolists - namely that they are likely to keep excess capacity in order to maintain their position, which is a waste of scarce resources; something that economists get very upset about. Similar interesting results are often found by looking at the strategic interactions of players in simple games and applying them to the real world, and game theorists have come up with countless other examples, even running competitions to see whose strategy at playing games such as the Prisoner's Dilemma works best. Game theory is a new and exciting subject, and one that people who are mathematically minded can use very effectively. For the rest of us, we just have to look at their results and go "ooh, that's interesting!"
See here for a list of interesting games. The princess/monster game and the pirates game are particularly good fun, and both have surprising results!