Understanding Nash's Equilibrium (Decision Theory)

The key to the Nash Equilibrium is about maximising benefit for both players (just talking about 2 players here, ourselves and the media, Nash's theory applies to more than just 2, but it's easier to understand this way). The concept is that every action has a certain utility. (utility is expressed as a mathematical value, where higher numbers represent a greater worth) I might attach a 'utility' of 5 to eating a banana, for example, but attach only a 4 to an apple, suggesting that since I prefer a banana, it's of greater worth to me. This is utility. Nash's Theory aims to maximise the product (multiplication) of the utility of all the players - the point at which this occurs, is the equilibrium saddle point.

The easiest demonstration of this theory is when we are dividing up single units, and both players place the same utility on the counter - ie: Dollars (where both people value money equally)
Typically speaking, we refer to money as a 1:1 ratio, 1 dollar = 1 utility point (there are exceptions to this, but I'll go into that later)

Consider a situation as follows: We have 10 dollars to split up between two people, and we need to figure out the most effective way to do so (Nash's theory is built not on fairness, but effectiveness - it's just convenient that they slot together)

If we give 5 dollars to one person, and 5 to another, then we multiply the utility, (5x5) we get a total utility product of 25.
If we give 6 dollars to one person, and 4 to another, then we multiply the utility, (4x6) we get a total utility product of 24.
If we give 7 dollars to one person, and 3 to another, then we multiply the utility, (7x3) we get a total utility product of 21.
If we give 8 dollars to one person, and 2 to another, then we multiply the utility, (8x2) we get a total utility product of 16.

So as we can see - the further we skew the agreement in favour of one party over another, the lower the total utility product (consider this product the group result).
In this instance, we could say that this must mean we have to split things up evenly to be effective. This is approaching the spirit of the position, but it's a little more complicated than that.

Say for example, we have two people once again, only - this time they value money differently. (It's important to understand that risk averse and risk prone tendencies do not result in linear utility transformations, but rather end up in curved graphs, if we were to plot preferences on paper).

We have one rich man, and one poor man - they are being offered 10 thousand dollars to split up between them, if only they can agree on how to share the money. Simply giving 5 thousand each does not work, because of the their utility scales.

10 dollars on its own makes a HUGE difference to the poor man. It means getting a meal, buying a beer, catching a bus, things that he would not have been able to do. 50 dollars is still awesome for the poor man, it means being able to sleep in a motel, buy a nice meal etc. 2000 dollars to the poor man (while he would absolutely be ecstatic to receive it) is much the same to him as 3000 dollars. Or to explain this in a better way - the difference in utility for him between 0 and 100 dollars, is far greater than the difference between 2000 and 2100 dollars.

The rich man of course, has a very different looking scale. If he has 10 million dollars in the bank, the prospect of being offered 5000 (although it's for free) just doesn't have the same impact in his life, and therefore is given a lower utility. (ie: we're probably going to have to give the rich man alot more money for him to feel as enriched as giving money to the poor man in order to maximise the group utility)

You can probably see where this is going. If the Poor man values 100 dollars equally to the way that the rich man values 9900 dollars, then we should actually only give the poor man 100 dollars and the rest to the rich man (if our aim is to maximise utility)

Hence - Nash's theory is awesome for trying to find an equilibrium for various forms of media - but we probably shouldn't be using it to form economic policy. (this is to Phil)

~ Colin