Friday, 22 March 2013

Probability preferences : expectation is secondary

I didn't realise counting was so important to the theory of probability.  First you have the simplified sub-case where all N disjoint outcomes are mutually exclusive, in which case you can use combinatorics to estimate probabilities.  Combinatorics just being counting power tools.  In effect the move is to set all of these $\frac{1}{n}$ probabilities to be mapped to the natural numbers.  Then comparing probability areas becomes a question of counting sample space elementary outcomes.  

Second, even in the case where it is a general (non equi-probable) distribution, you can look at the set of outcomes themselves and map them to a series of numbers on the real (or whole) line.  So say you have a die with six images on them.  You could map those images to six numbers.  In fact, dice normally come with this 1-to-6 mapping additionally etched onto each of the faces.  The move from odds-format to ratio-of-unity format that we see in probability theory is crying out for a second number, representing some kind of value, perhaps a fair value, associated with some game or contract or activity.  In other words, now we've partitioned the sample space into mutually exclusive outcome weights, let's look at finding numerical values associated with the various states.  When it comes to pricing a financial contract which has an element of randomness in it (usually a function of some company's stock price, which serves nicely as such a source), then a careful reading of the prospectus of the derived instrument ought to be able to be cashed out in terms of a future value, given any particular level of the stock.

I've seen Pascal's wager claimed to be the first use of expectation in a founding moment for decision theory.  By the way, that's a poorly constructed wager since it doesn't present value the infinite benefit of God's love. That could make a dramatic difference to the choices made.  Anyway, Huygens himself wrote about expectations in his probability book, but for me, the warm seat problem (the problem of points) represents an attempt to find a mean future value starting from now during a game.  This is an expectation calculation, even though the word may not have been used in this context.