Nash Equilibrium

John Nash

John Nash grew up in Bluefield, West Virginia. His father was an engineer, and his mother a teacher, instructing him to read and learn Latin by the age of four. Like so many brilliant minds, Nash was often shunned by his teachers - in fact his fourth grade teacher claimed he simply was incapable of doing math. In high school, Nash would solve his teachers long ugly proofs, with a simple, tidy, and elegant solution. His brilliance in school landed him the George Westinghouse Award, granting him a full scholarship to Carnegie Institute of Technology where he studied mathematics.

Nash attended grade school at Princeton, where he created the game of Hex and discovered his love and fascination for game theory. In 1950, at the age of 22, Nash earned his PhD. His 28 page dissertation explored non-cooperative games and outlined what would later be revered as the Nash Equilibrium.

Although admired by his colleagues for his incredible mind, Nash did not have the best reputation as a person. In 1951 he began working as faculty at MIT where his colleagues described him as arrogant, haughty, selfish...but brilliant. He had an affair with a nurse, ended up impregnating her, and refused both to put his name on their child's birth certificate and support their child financially. A few years later Nash met Alicia Larde, a physics major at MIT, the married in 1957.

Around the age of 30 Nash was diagnosed with schizophrenia. For over twenty five years Nash struggled with this disease. Twice he was involuntary admitted into hospitals, one time reciving insulin-coma therapy. He was often found wandering around the Princeton campus talking about himself in third person. Twice he ran away to Europe, and threatened to not return. He resigned from MIT and struggled to hold a job. His behavior was bizarre, and he was often aggressive. Alicia divorced him in 1962.

After years of struggling with Schizophrenia, Nash received a job at Brandeis University. He began to see a psychiatrist and started taking medication. He and Alicia remarried. By the 1980's, Nash had slowly begun his recovery from schizophrenia. His brilliance and contributions to the fields of economics and mathematics were recognized in 1994, where, at the age of 66, Nash was awarded the Nobel Prize in Economics. Nash died on May 23, 2015 in a car crash with Alicia.

Figure 1: John Nash

Necessary Background Information

 Up to this point in the semester we have used game theory to strictly analyze games. However, game theory may also be used as a tool to analyze human interactions. We will see how human interactions can be modeled as games - in war, in politics, in law, in divorce, in sports, in agriculture, in economics, in EVERYTHING - but to do so we must first understand some key terminology.

Cooperative Game: Games in which players are working together. Players work together to make a decision (think coalitions).

Non-Cooperative Game: Games in which players are looking after their own individual interest. Players make decisions independently.

All games must contain the following elements:
        i. Set of Players
        ii. Set of actions/strategies for each player
        iii. Payoff/utility for each player based on their action

Figure 2: Game Theory Everywhere

Nash Equilibrium

A Nash Equilibrium is the optimal outcome of a game, such that neither player has an incentive to change their strategy based on what the other player does. So even if the other player changed their strategy, it would not be in your best interest to change your strategy.

Le us think about this in a real life example. Imagine you live in a country that has no form of law enforcement, no police, nothing. A Nash Equilibrium could be symbolized as a law passed, that everyone followed, even though they knew they would receive no punishment for not following this law. 

Let us think about what side of the road you are driving on. Imagine one car is driving in one direction, and the second car in the opposite direction, presumably if both are in the same lane they will crash. I claim, it is both car's best interest to follow the rules of driving in their country, and drive on the specified side of the road - in this example suppose they are driving in America, in which the right side of the road is the specified driving side. Take a look at the payoff matrix in Figure 3.

Figure 3: Driving Road Side

The utility of Driver 1's decision is represented in green, and the utility of Driver 2's decision is represented in blue. If Driver 1 decides to drive on the left hand of the road, Driver 2 would gain the highest utility from driving on the left side of the road (because $1>-1$). If Driver 1 decided to drive on the right side of the road, it would be in Driver 2's best interest to drive on the right side of the road (again $2>-1$). Repeating the logic, with Driver 2, if Driver 2 decides to drive on the left hand side, Driver 1 would also want to drive on the left hand side ($1>-1$). And if Driver 2 decides to drive on the right hand side, Driver 1 would choose to drive on the right hand side ($2>-1$). Hence we have two Nash Equilibriums in this example (Left, Left) and (Right, Right) as seen in Figure 4 below.

Because (Right, Right) yields both players a higher utility ($2>1$), it would be in both drivers best interest to drive on the right side of the road.

Now let us look at Figure 5.

Figure 5: General Theorem for Nash Equilibrium 

In general, (T,L) is a Nash Equilibrium if the following is true:
        i. $a\geq{e}$
        ii. $b\geq{d}$


Real World Application

So far we have used very trivial examples of the Nash Equilibrium. However, the Nash Equilibrium is used everywhere, and with very important decisions. Let us take a look at the decision of the U.S.A and the U.S.S.R to initiate a conflict or to disarm in the Cold War - can the countries reach a Nash Equilibrium?

Figure 6: Nuke?  

Nike and Adidas both come out with a new pair of running shoes, They are face with the decision to spend money on advertising, or to not advertise - what should they do?

Figure 7: Advertise?

Try out these example on your own, and we will go over them collectively in class tomorrow.








Work Cited
http://www.usaheadlinenews.com/wp-content/uploads/2015/05/john-nash.jpg
http://www.pbs.org/wgbh/amex/nash/index.html
https://www.youtube.com/watch?v=0i7p9DNvtjk
http://economics.fundamentalfinance.com/game-theory/nash-equilibrium.php