Problem of Points - The Solution

The solution to the problem of points tells you how you would divide up the stakes in a fair game (fair in the sense of each step outcome being equally likely to favour any player) between two players A and B if A needed $n_A$ more wins and B needs $n_B$.  Pascal and Fermat both end up counting the set of all possibilities and comparing the respective counts to each other and come up with the ratio 

$\sum_{k=0}^{n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}$ to $\sum_{k=n_B}^{n_A+n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}$.  

This rather ugly looking formulation is something I'll be looking at over the next couple of posts, in a way mathematicians usually don't.  Enamoured of Euclid, they think interesting maths involves proof and concise statement.  That does not work for me.  I want to unpack it and see some examples with real numbers.  Get a feel for it in use.  And after those posts, I'll be doing the same for gambler's ruin, which I personally think has caused me to think a lot more generally than this solution to the problem of points.

Before I finish on this short post, I'd like to say that this solution to the problem of the division of stakes, if you think about it, is the price of the seat if somebody wanted to buy you out of the game.  This is the fair price of your seat at that moment, or the fair price of your position in the same.  And given that the moment in question can analysed at any point in the game, including the moment before the game starts, it also represents an algorithm for working out the fair price of the game for both players, at all points.  That is, it tells you fully at all moments in the game the expected value of each hand.

If the stakes are a value of S then the expected value of one player is 

$S\frac{\sum_{k=0}^{n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}}{\sum_{k=0}^{n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!} + \sum_{k=n_B}^{n_A+n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}}$

and for the other just has the other sum on the numerator.

Warm Seat

I am really rather pleased with my reading of the history of the theory of probability.  Four points struck me about it, firstly that Cardano has a much stronger claim than the authors of histories of probability give him credit for.  Second that Pascal was wrong in criticising Fermat's combinatorial approach in the case of more than two players in the problem of points and that his mistake was an equivalence class / ordering misunderstanding about the reading of three thrown dice.  Third, that Pascal's solution is a bit like using dynamic hedging for an exotic option (one which doesn't exist yet, but which I'll call a one-touch upswing option).  And fourth, that Huygens's gambler's ruin can be made into a problem of points by using participant stakes and separately some tokens which are transferred from the loser to the winner after each throw.  On the last three of these points Todhunter and the authors Shafer and Vovk agree with me, variously.

A better name for the problem of points is the warm seat price.  And the original first-to-six game, and also Gambler's ruin with plastic tokens and stakes can both be seen as specific games for which there's a warm seat price - the fair value of the game for a participant if he wanted to get out of the game immediately.  Gambler's ruin doesn't have a definite time in the future at which point it will with certainty be known who the winner is.

It is also amusingly my warm seat moment since I didn't discover anything myself, but followed in other peoples' footsteps, and have experienced the warm seat experience of discovery others had made before me.

One gambler wiped out, the other withdraws his interest

In so far as odds are products of a book maker, they reflect not true chances but bookie-hedged or risk-neutral odds.  So right at the birth of probability theory you had a move from risk-neutral odds to risk neutral slices, in the sense of dividing up a pie.  The odds, remember, reflect the betting action, not directly the likelihood of respective outcomes.  If there's heavy betting in one direction, then the odds (and the corresponding probability distribution) will reflect it, regardless of any participant's own opinion on the real probabilities.  Those subjective assessments of the real likelihood start, at their most general, as a set of prior subjective probability models in each interested party's head.  Ongoing revelation of information may adjust that probability distribution.  If the event being betted on is purely random (that is, with no strategic element, a distinction Cardano made), then one or more participants might correctly model the situation in a way which is as good as they'll want, that is immune to new information.  For example, the rolling of two dice and the relative occurrence of pips summing to 10 versus the relative occurrence of pips summing to 9 is the basis of a game where an interested party may well hit upon the theoretical outcomes implied by Cardano and others, and would stick with that model.  

Another way of putting this is to say that probability theory only co-incidentally cares about correspondence to reality.  This extra property of a probability distribution over a sample space is not in any way essential.  In other words, the fair value of these games, or the various actual likelihoods are just one probability distribution of infinitely many for the game.  

Yet another way of putting this is to say that the core of the theory of probability didn't need to require the analysis of the fair odds of a game.  The discoverers ought to have been familiar with bookies odds and how they may differ from likely outcome odds.  Their move was in switching from hedge odds of "a to b" to hedge probabilities of $\frac{b}{a+b}$.  That it did bind this up with a search for fair odds is no doubt partly due to the history of the idea of a fair price, dating back in the Christian tradition as far back as Saint Thomas Aquinas.

Imagine two players, Pascal and Fermat, playing a coin tossing game.  They both arrive with equal bags of coins which represent their two wagers.  They hand these wagers to the organisers, who take care of the pair of wagers.  Imagine they each come with 6,000,000 USD.  The organisers hand out six tokens each , made of plastic and otherwise identical looking.  Then the coin is brought out.  Everyone knows that the coin will be very slightly biassed, but only the organisers know precisely to what degree, or whether towards heads or tails.  The game is simple.  Player 1 is the heads player, player 2 tails.  Player 1 starts.  He tosses a coin.  If it is heads, he takes one of his opponent's plastic coins and puts it in his pile.  If that happened, he'd have 7 to his opponent's 6.  If he's wrong, then he surrenders one of his tokens to his opponent.  Then the opponent takes his turn collecting on tails and paying out on heads.  The game ends when the winner gets to have all 12 tokens and the loser has 0 tokens.  The winner keeps the 12,000,000 USD, a tidy 100% profit for an afternoon's work.  The loser just lost 6,000,000 USD.  Each player can quit the game at any point.

Meanwhile this game is televised and on the internet.  There are 15 major independent betting cartels around the world taking bets on the game.  In each of these geographic regions, the betting is radically different, leading to 15 sets of odds on a Pascal or a Fermat victory.

Totally independent to those 15 cartels of betting, there are a further 15 betting cartels which have an inside bet on, which pays out if you guessed who would see 6 victories first, not necessarily in a row.

Now this second have is inside the first, since you can't finish the first game unless you collected 6 points too.  Pascal and Fermat don't know or care about the inner game.  They're battling it out for total ownership of the tokens, at which point their game ends.  The second betting cartel are guaranteed to finish in at most 11 tosses every time, and possibly as few as 6 tosses.

Just by coincidence, Fermat, player 1, gets 4 heads in a row, to bring him to 10 points of total ownership of all the tokens.  He only needs 2 more heads to win.  At this point Pascal decides to quit the game.  To betters in cartel 1 it looks like Pascal and Fermat are playing gambler's ruin, to cartel 1 it looks like they're playing 'first to get six wins', which is the game the real Pascal and Fermat analyse in their famous letters.

Soon after, Pascal's religious conversion wipes out his gambling dalliance, and Fermat, only partly engaged with this problem, withdraws his interest.  Both men metaphorically enacting gambler's ruin and the problem of points.