Probability preferences : the source of randomness is not the game

Fermat's version of the solution to the problem of points was to create a grid of possibilities reaching fully out to the point beyond which no doubt could exist as to who the winner would be.  This grid of possibilities included parts of the tree which, on one view, would be utterly irrelevant to the game in hand, and on another view, incorrectly modelled the set of possibilities embedded in the game.

Pascal's solution, by way of contrast, was a ragged tree of possibilities stretching out along each branch only as far as was needed to resolve the state of the game in question, and no further.

Pascal additionally made the mistake, in interpreting Fermat's solution, of ignoring order when tossing three dice/coins and in this mis-interpretation came up with an answer in the case of three players which diverged from his own reverse recursive solution based on the principle of fair treatment at each node of his ragged tree.

Because Pascal's wrong-headed idea of Fermat's solution did not match his own, he jumped to the conclusion that what must be wrong in Fermat's method was the extension of the tree of possibilities beyond those parts which the game in hand required.  Pascal consulted Roberval on the likely legitimacy of this fully rolled out tree of possibilities and Roberval seems to have told Pascal that this is where Fermat is going wrong, namely that this 'false assumption' of theoretical play of zombie-games leads to bad results.  It doesn't.

The evolution in time of a source of randomness was seen clearly by Fermat as separate from the rule, game or activity sitting on top of it.  In this case the game was the 'first to get N wins'   Modern derivatives when tree based methods are used all apply this same move.  First the random process's set of possibilities are evolved on a lower, supporting layer, then the payoff of the contract is worked out at the terminal time horizon.  Both in De Mere's game and with an option, there's a clearly defined termination point.  With De Mere's game, the point happens when the first player reaches N wins.  With options, the termination point is the expiry of the option.  Gambler's ruin, as I'll discuss later, doesn't have such a straightforward termination point.  So step 1 is to lay out all the possible states from now to the termination point, the tree of possibilities for the stochastic process.  Then you work out the terminal value of the contract or game and use Pascal's fairness criterion to crawl back up the second tree, until you reach the 'now' point, which gives you the fair value of the contract.  This is the essence of the finite difference solution set, and it works for path dependent and path independent pricings.  The implications of the game is that the tree is re-combinant, which means the binomial coefficients become relevant when working out the probability that each path is traversed.

Fermat has a clearer and earlier conception of this separation.  But Roberval and Pascal were right to flag this move up - what grounds did Fermat give for the move?  In modern parlance, we can see that the stochastic process, often a stock price or a spot FX or a tradeable rate, is independently observable in the market.  But back then, Pascal was struggling to separate the game from the source of randomness.  F. N. David suggests that Pascal sets Roberval up as the disbeliever as a distancing mechanism for his own failure to grasp this point.  Likewise, David suggests perhaps Pascal only solved his side of the problem after initial prompting from Fermat, in a letter which starts off the correspondence but which unfortunately no longer exists.

Of course, this isn't a solution of an unfinished game, but the fair value of the game at any point during its life. Each author I read seems clear in his mind that one other other of the great mathematicians' solution is preferred.  Is this just ignorance, aesthetic preference masquerading as informed opinion?  Yes, largely.  But my own opinion is that the both solutions share many similarities - both need to evolve a tree of possibilities, a binary tree, for which the binomial coefficients come in handy as the number of steps increases.  Both then involve evaluating the state of the game at the fixed and known horizon point.  Fermat's tree is a set of possibilities of a stochastic process.  His solution takes place exclusively at that final set of terminal nodes, but working out the ratio of the set of nodes in which player A is the winner over the total set of terminal nodes.  Pascal's tree is the tree of game states.  He reasons in a reverse iterative way until he reaches the start point, and the start point gives him his final answer.  The arithmetic triangle could help both these men build their trees as the number of steps increases.

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.

Probability preferences: the irrelevance of parallel/sequential distinction

In a sense, whether you throw one die sequentially n times to get a $6^n$ event space, or whether you simultaneously toss n distinguishable dice at one time, it doesn't matter.  As long as you read your die results in a way which preserves the identity of the die the number appears.  I'll leave off talking about what implication this has for the famous Pascal-Fermat problem of points until a later posting.  For now, consider what this means for the classic repeated experiment in probability theory.  If the events are genuinely independent, then it doesn't matter what relative time it is when you toss each one.  The law of large numbers could equally well be satisfied with a single massively parallel experiment in, say, tossing a coin than it is in tossing a coin sequentially n times.

Likewise in set theory, there's a curious atemporality to Venn diagrams.   And when discussing the joint probability of $A \cap B$, which is of course not the same as A then B.  Even with Bayes' theorem it is important to realise that the 'given' meaning in A|B is with respect to our knowledge of the occurrence of B, not that B happened first and then A subsequently happened.