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Nash Equilibrium & Game Theory

Nash Equilibrium & Game Theory
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  • 0:06 Review of Game Theory
  • 2:12 Nash Equlibrium
  • 4:07 Lesson Summary
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Instructor: Dr. Douglas Hawks

Douglas has two master's degrees (MPA & MBA) and is currently working on his PhD in Higher Education Administration.

The 2001 movie, 'A Beautiful Mind,' told the story of John Nash, a Nobel Prize-winning economist who theorized the Nash equilibrium. In this lesson, we'll learn about Nash equilibrium by studying game theory.

Review of Game Theory

Before we can talk about Nash equilibrium, we need to review the idea of game theory. Game theory is the idea that the results of one, or more, players' decisions are based on the behaviors or choices of other players.

The classic example typically used to teach game theory is the prisoner's dilemma. In the prisoner's dilemma, two crooks are caught in a burglary attempt. The police separate the two men and offer each of them the following deal: rat out your buddy and you'll be off the hook; refuse to cooperate and you'll be punished. The catch is your punishment depends on the decision your partner made.

There are four combinations of outcomes: you rat out your partner or you don't, and he has the same two choices. That gives us the following possibilities:

  1. You both turn each other in
  2. You both stay quiet
  3. You turn him in but he stays quiet
  4. He turns you in but you stay quiet

Each of these outcomes has a different punishment. In option one, when you both turn each other in, you each have to serve three years in prison. In option two, when you both stay quiet, you both only have to serve one year. That sounds good, but in option three, if your partner stays quiet and you turn him in, you don't have to serve any time, but your partner serves five years. The opposite is true in option four - you serve five years because you stayed quiet and your partner ratted you out.

What's the best option? The answer to that depends on what your partner does, just like his best answer depends on what you say. Since you can't coordinate with him, you have to anticipate - as best you can - what he might do and use that information to determine your action. If you are certain he'll be quiet, you should be quiet, too - since doing that will get you both a year. But, then you think 'I'm sure he'll stay quiet, and he'll have that same confidence in me, so if I turn him in and he stays quiet, he gets 5 years and I get nothing.' Of course, he's thinking the same thing. And if you both act on that, you both end up serving three years.

What to do? The process of finding the best answer to that question is the focus of game theory.

Nash Equilibrium

That review of game theory was important because the prisoner's dilemma is the simplest scenario in which to describe Nash equilibrium. The definition of Nash equilibrium is the outcome where both (or all) players are assumed to know the decisions of the other players and make the best decision for themselves, given the other players' decisions.

Let's try and figure out the best option and the Nash equilibrium of our prisoner's dilemma situation we just described. What's the best decision - not for each individual, but for both of them? If they could coordinate their answers, and they knew each other would stick to it, what would they do? The best option for them is to both stay quiet, thereby each taking one year in prison. Of course, for that to work, they both need to do what they agree to do - stay quiet.

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