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What is Game Theory?

Game Theory is an exciting set of ideas based on the combination of pure mathematics and psychology aiming to develop simplified models for 'games'. A game is a situation in which two people (agents) have each one two or more mutually exclusive options (generally called 'strategies'). The matrix built upon the two sets of strategies is called a 'payoff matrix' and shows the different exclusive scenarios that may be reached by the agents. A game is said to be solved when the analyst finds the most likely scenario as well as the probability distribution of the final outcome.

Game Theory aims to forecast the final outcome of a two-person conflict. It was originally proposed by John Von Neumann and Oskar Morgenstern in 1939, who were researching techniques to analyze bargaining problems. Von Neumann and Morgenstern wrote The Theory of Games and Economic Behaviour (1944).

Albert W. Tucker created the model known as the Prisonner's Dilemma. John Forbes Nash created the concept of a Nash Equilibrium (1950). A Nash Equilibrium is a specific scenario which both competitive agents seek to defend, because shifting the strategies will generate only disadvantages for any agent.

Pay-Off Matrix

Pay-off matrix is dual-entry table shpwing the names of the strategies that each agent may choose. Generally, the choices of the first agent are shown in the left side, while the choices of the second agent are shown on top of the table. The cells inside the table are the different possible final scenarios for the game. The information of each cell is the pay-off for the first and for the second agent. Let's see the following example:

B1 B2
A1 10, -10 -8, 8
A2 -12, 12 11, -11

The first value in a cell is the payoff for the first agent, and the second value is the payoff for the second agent.

Reaction Curves

Reaction curves are loci in the Cartesian system that show combinations of decision and payoffs.One example is the supply and demand curves. Suppose that these curves are built by tatonnement, so that the supplier and tye buyer propose each one of them a price. The supplier is therefore proposing one price and willing to sell a large quantity of goods. Regarding that price, the buyer is willing tro buy a much smaller quantity of goods. Each price will draw two points, one belonging to the supply curve and the other one belonging to the demand curve. It's easy to see that there exist infinity pairs of price and the response in quantity. These pairs will define the two loci, one locus being the supply curve and the other one being the demand curve. These two curves will cross at one (maybe more and maybe not even one) point, which is the market equilibrium. The point here is: you can trade goods with another agent, and then each one of the agents will react to the proposal of one price with a desired quantity to offer or demand. Reaction curves show how one agent will react (his reaction function) to any specific value of a variable.

Sequential moves' trees

Tree diagrams are useful when analyzing sequential games. Starting from one point A two extensions grow, in order to show the moves available for the first player. At the open end of these extensions one will specify the options available for the second player, which may also be two. So one will have now four branches. If we are intended to make a new move, then we will draw new branches starting form the new open endpoints. The resulting diagram has the shape of a tree, and the terminal endpoints will show the different scenarios for the conflict.

Zero Sum Games

Zero Sum games are those games in which payoffs to player A come from player B's pocket, and, vice versa, payoffs to player B should come from player A's pocket. So, if one player wins a prize, the other is having a loss of the very same amount. See the following zero sum game matrix:

B1 B2
A1 10, -10 -8, 8
A2 -12, 12 11, -11

Some games, though not all, may be noted with a matrix. Game Theory's main models are noted by matrices.