Game theroy

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Game Theory

Game theory is defined as the analysis of multiple interacting rational agents. This definition is intentionally vague because game theory has been found to apply to an overwhelming amount of concepts. And before going into some of the more interesting examples and applications of game theory to economic sociology, it should be prudent to provide some explanation.

A game is created when any two or more parties interact, and has four distinct parts.

The players. The players are the interacting parties. One player can be one person, or a committee, a company, or an entire country.

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The rules. The rules of the game are the set feasible strategies that each player can choose, including when to choose them. A normal form game has players making their choices simultaneously, while a successive game has players responding to one another.

The payoffs. The benefit or harm of each possible interaction must be defined (or predicted) in order to make an accurate decision. The payoff is most often given in terms of utility.

Natural events. A natural event is anything that can plausibly happen outside of the control of the players and inside the realm of the game. Natural events are only important in successive games or in normal form games that are repeated over time.

Having defined the component of a game and the purpose of the game, it should also be helpful to define and interpret the different types of equilibriums that may occur. They are presented in the order of plausibility from most to least. It is important to notice that the most plausible form of equilibrium is such because it makes the fewest assumptions.

Dominant strategy equilibrium. This is an equilibrium where all the players have a dominant strategy. That is, each player has a strategy that is their best choice no matter what the other players do. This is the most valid for of equilibrium because it does not assumes nothing, including rationality.

Dominance solvable solution. In this type of solution, dominated strategies are eliminated from the realm of possibility until each player has only one strategy remaining. This is still a highly valid form of equilibrium; however, it assumes rationality and that the players have knowledge of the payoffs.

Nash equilibrium. In Nash equilibrium, each player action is payoff maximizing given their beliefs about the other players. In Nash equilibrium, we make a huge leap of faith and assume that the players' beliefs are always correct.

Mixed strategy equilibrium. If no singular equilibrium exists, it is possible to determine an equilibrium defining the proportion of times each player should choose each of his or her strategies.

Having defined the elements of a game and the different types of equilibriums encountered, I know wish to switch gears away from the dry task of definition to the more interesting application.

The first example is called the "prisoners dilemma," and is the single most popular example in game theory. The story states that two criminals have just been apprehended for committing a crime. They are put in separate rooms and given the following choice, confess or deny. The payoffs are given in the following matrix, in the amount of years each will have to serve in prison.

Player

Confess Deny

Confess 5,5 0,40

Deny 40,0 1,1

Player 1

As we can see, each player has a dominant strategy. No matter what player chooses, player 1 will choose to confess. Notice that if player denies, player 1 will confess and serve 0 time. However, if player confesses, player one must choose confess as well to reduce his sentence from 40 years to 5 years. Since this game is also symmetric, the dominant strategy for both of the players is the same.

The dominant strategy equilibrium is then for both criminals to confess, and the payoff is 5 years in prison each. This is an interesting example to consider especially since the outcome of the game, both criminals confessing, is significantly different from what one would assume. It is a strange thing to imagine that it isn't in the criminal's best interest to deny his crime, however game theory has very nicely shown us that it sometimes is.

The next example is commonly called the "chicken" example. Chicken is a competition wherein two people get in cars and drive directly towards on another. The first person to swerve to avoid the collision is the loser. The payoffs are given in the following matrix, in terms of utility.

Player

Swerve Don't

Swerve , 1,10

Don't 10,1 0,0

Player 1

This example is a bit more complicated. Recalling our definition of Nash equilibrium, we assume that each player has the proper belief about what the other player will do. If player on believes that player two will swerve, then Player 1 will choose to not swerve, and vice versa. Since neither player has a dominant strategy, since each player will change his strategy based upon his belief about the strategy of the other player, and this matrix is not dominance solvable, we check for Nash equilibrium.

What we find is that there are two. Both (don't, swerve) and (swerve, don't) are Nash equilibriums. The thing that makes this example complicated is that it is difficult to determine which equilibrium will be reality, however as long as the two players coordinate properly, and nobody gets hurt in a horrible collision, it doesn't really matter.

Though my examples were quite simple, it is important to note that game theory can be much more deeply applied. Any number of players and any number of strategies may be input to determine what types of actions the players will take. It seems to me that game theory is one of the more promising bridges between economics and sociology.

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