## Introduction to Game Theory

### An Introduction to Game Theory

Game theory analyzes competitive situations to determine possible, probable, and optimal outcomes. Games consist of a set of players and a set of strategies for each player which are given or defined by rules. For each combination of players and possible strategies, there is a payoff.

Game theory is used in a variety of fields such as political science, sociology, etc., not just economics; in fact, its roots are in applied mathematics. It was first developed by John von Neumann in a series of papers in the 1920's and 1930's. In the early 1950's, John Nash greatly expanded on existing game theory and helped it to develop much further.

#### Basic Example: The Prisoners' Dilemma

The prisoners' dilemma is a popular, basic, yet illustrative example of game theory. There are two prisoners, Jack and Tom, who have just been captured for robbing a bank. The police don't have enough evidence to convict them, but know that they committed the crime. They put Jack and Tom in separate interogation rooms and lay out the consequences:

• If both Jack and Tom confess they will each get 10 years in prison.
• If one confesses and the other doesn't, the one who confessed will go free and the other will spend 20 years in prison.
• If neither person confesses, they will both get 5 years for a different crime they were wanted for.
It is easier to see and compare these outcomes if they are put into a matrix:

 Jack Tom C NC C -10,-10 0,-20 NC -20,0 -5,-5

Since Tom's strategies are listed in rows, or the x-axis, his payoffs are listed first. Jack's payoffs are listed second because his strategies are in columns, or on the y-axis. "C" means "confess" and "NC" means "not confess." This matrix is called "Normal Form" in game theory. Moves are simultaneous, which means that neither player knows the other's decision and decisions are made at the same time (in this example, both prisoners are in separate rooms and won't be let out until they have both made their decision).

At first glance, it may seem that both players not confessing is the best choice since each prisoner will only get 5 years, but a more in-depth look will show that this is not the case.

#### Dominant Strategy

By looking at each prisoner individually, we can find the dominant pure strategy. First look at Tom. Let's assume that Jack is going to confess, what's the best strategy for Tom? If Jack confesses and Tom doesn't, Tom will go to prison for 20 years, but if he does confess he'll only go for 10. In this case, it's best for Tom to confess. Let's highlight this payoff in the matrix to keep track:

 Jack Tom C NC C -10,-10 0,-20 NC -20,0 -5,-5

Now let's assume that Jack is not going to confess. If Tom confesses, he will go free, if he doesn't confess, he'll get 10 years. Again, his best option is to confess, no matter what Jack does. Since in both cases his best option was to confess, confessing is his dominant pure strategy (it's also correct to say not confessing is dominated by confessing). A dominant strategy is a strategy that has the best payoff no matter what the other player chooses.

 Jack Tom C NC C -10,-10 0,-20 NC -20,0 -5,-5

Since Tom and Jack both have the same options and payoffs, Jack's dominant pure strategy is also confession. After going through the same steps, you can highlight your matrix as follows:

 Jack Tom C NC C -10,-10 0,-20 NC -20,0 -5,-5

#### Nash Equilibrium

You'll notice that the payoffs for (confess,confess) are both highlighted. Since both prisoners have the same dominant pure strategy, confess, they will both confess and each will get 10 years in prison. This is the Nash Equilibrium, named after John Nash. A Nash Equilibrium is any outcome where there is no unilateral profitable deviation--that is, holding all other players constant, one player cannot choose a strategy that will make him better off. In this case, if either player doesn't confess, he will get 20 years instead of 10. The payoff box (5,5) that results from (NC,NC) is NOT a Nash Equilibrium because either player could confess (holding the other player constant) and go free instead of spend 5 years in prison. In some games there are multiple Nash Equilibria, but that will be discussed later.

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