Sunday, 2 November 2014

John Nash: Game Theory and the Equilibrium

By Calvin Price

In 1950, John Forbes Nash Jr. earned a doctorate for his work on game theory.  His dissertation focused on what players do in non-cooperative strategic games.  Since his original work, game theory has been hailed as one of the greatest intellectual advances in economics and has had applications on a vast range of fields.  It was truly a worthy cause for Nash to win the Nobel Prize in 1994.

Game theory can be summed up in a disarmingly simple question: what will the other guy do?  It focuses on what a person will do when confronted with the move of another person.  This is called game theory because that’s where it can be most obviously applied.  While it’s unnecessary in games such as rock-paper-scissors or tic-tac-toe, game theory becomes much more useful for something like chess.  This has clear implications as a mental construct for thinking about complex games, but it otherwise seems like an exercise in mathematical thinking.  What about real-life situations?

The most famous of game theory applications is called the prisoners’ dilemma:
Two men have been arrested.  The police have them on minor charges, but suspect a larger crime.  If they get the testimony from one of the men, he will be freed and the other will be locked up for 10 years.  If neither man betrays the other, they both get 1 year imprisonment.  If both men talk, then they each get 5 years.

So, what do you do?

Beyond incarceration, game theory applies to all aspects of non-cooperative interactions for n-people. International relations are guided by game theory, so are business strategies in competitive markets and traffic patterns (where can I go to avoid traffic?  Should I take the subway?).  Besides providing a structure to formulate your own strategies, game theory gives an insight into an opponent’s options and creates predictability, one of the most important things in any market.  

Besides contributing to the overall field of knowledge, Nash demonstrated a special case of game theory known as the Nash Equilibrium.  The Nash Equilibrium is a situation in which all parties have perfect information about the situation and do not benefit from changing their strategy.  There is a Nash Equilibrium is X is making the best decision she can, knowing of Y’s position, and Y is making the best decision he can, knowing of X’s position.

For example, in the Prisoners’ Dilemma, a Nash Equilibrium would happen if both prisoners knew they were getting the same deal:

Since each prisoner has perfect information about the situation, they’ll both end up in a situation that is not optimal.  This is what makes the Prisoners’ Dilemma a classic equilibrium problem: both have an incentive to cooperate.  In other words, the outcome of the situation depends on the people involved.

The issue with equilibriums is that perfect knowledge about a situation almost never exists.  In tic-tac-toe, one can be pretty certain about what the other player will do, but what about chess?  Some moves are better than others and not everyone sees the best move or strategy.  The matter becomes even more complicated when businesses, markets, or nations are involved.  The knowledge available to people in those situations is anything but perfect.
Nash was a genius who made many significant contributions to game theory and mathematics.  The Nash equilibrium is an invaluable tool for estimating the outcomes of situations with mostly perfect knowledge.  Traffic conditions, for example are predictable and are a case of a Nash Equilibrium, even though there is never perfect knowledge about the exact conditions.  For all his efforts, human error is beyond modeling.


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