03.30.06

A Beautiful Mind

Posted in MBA, Marketing at 3:56 pm by Jason Permalink

Ok… So I just said I wouldn’t talk about other classes. Now I’m about to talk about something that I learned about first by watching a movie then in more detail in economics. What is so striking about this topic, though, is that I then heard about it in finance, marketing, statistics, and decision theory, but it was invented by a mathematician… So when that old dude from the Nobel Committee tells John Nash that his theory has had an incredible impact on a vast array of subjects, he wasn’t kidding

Unless you were hibernating in a cave for all of 2001, you’ve proly heard of the movie ‘A Beautiful Mind’ and it’s main character John Nash (if you haven’t I highly recommend it). So you may have a flavor for what I’m about to lay out. In short John Nash figured out that Adam Smith’s ‘Invisible Hand’ was only partially optimal. Smith said that in a capitalist free market economy, the ‘Invisible Hand’ would guide the market to an optimal equilibrium. He called it the invisible hand because by the mere fact that all parties are acting for the sole purpose of profit maximization prices could be set and quantities determined. Inefficient firms would leave the market and overpriced markets would attract competition…

What Nash realized is that this is in-fact not always the best way… A firm should also take into account the actions of their competition. This is simple game theory… I will illustrate with a very simple example.

- First, we assume that all parties act rationally (I know this is a BIG assumption But mathematicians and economists are optimistic folks)

- Second, for this example, we will assume a simultaneous move one shot game. This means that both players move at the same time and there is only one move to the game. We can extrapolate this out to multi move games; then it gets even more interesting. (I’ll write that up in another post, I promise )

OK… Here we go…

We have two players Sav-On and Walgreen’s. They are located on the same corner of some street USA and each is trying to decide how to set prices on a new product. They can choose to set price either HIGH or LOW. The decisions are interdependent…

We represent the game in a matrix normal form:

From the matrix we see the various outcomes of total profits to each firm depending on the combination of moves by each player.

Now we’ll analyze the game from the perspective of Sav-On:

First we ask what is Sav-On’s best response if we think Walgreen’s will price HIGH? From the above diagram we see it is to price LOW.

Next we ask what is Sav-On’s best response if we believe Walgreen’s will price LOW? Again, we see that Sav-On should price LOW.

We see that no matter what Walgreen’s does, Sav-On’s best response is to price LOW. This represents a DOMINANT strategy for Sav-On.

Now we analyze the game from the perspective of Walgreen’s and we again see that there exists a dominant strategy, which is to price LOW.

A Nash Equilibrium exists that takes into account each player’s BEST response, which is for both firms to price LOW. This maximizes the expected profits assuming that neither player is allowed to collude.

We see that there is a better equilibrium that can only exist in the presence of some sort of collusion… These equilibriums tend to be unstable, but stabilization strategies do exist. Perhaps I will post on that sometime

OK… This was a very very simple example of a Nash equilibrium. A Nash equilibrium doesn’t necessarily have to exist, neither does a Dominant strategy. In these cases there are certain decision strategies that can be employed, but optimality is no longer guaranteed.

Here we have used game theory to set the optimal price for a new item. We can also use the same approach to determine if market entry would be profitable…

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