Monday, 11 April 2011

Americans play craps because sheep walk on four legs

The sheep's astragalus can have its anatomically asymmetric shape scrubbed, chipped, smoothed so that it becomes a modern, more-or-less symmetrical, more or less fair die.  The first evidence for this happening, according to F.N. David is around 3000 B.C.  in Iran and India.  What does this allow?  Well, first, you now get a manufactured randomisation machine with an information content of 2.6 bits per die instead of 1.7 per bone.  This is achieved by there being six instead of 4 possible outcomes, in addition to the fact that they are now all, more or less, equally likely.
However, as David points out in her book, for thousands of years popular games and religious divination was performed with four bones.  You get a lot more combinations with four.  And the particular shape of the astragalus facilitates walking and running.  Bipeds, for example, would only have two such bones.  So for many millennia humans have been de-boning their sheep, four running astragali at a time, and inventing games and religious-oracular practices based on the resulting information revealed when the bones are thrown.  By my calculations, four bones rolled in parallel will result in a 6.88 bits of information being revealed.

Now, turning to the die.  If you roll one die, you get 2.6 bits of information.  Two dice deliver 5.16 bits of information, three, 7.75 bits of information.  If your culture had already invested several millennia worth of games of chance and religious divination based on the randomisation machine of choice delivering no more than 6.88 bits of information, you may not need the excess 0.87 bits of information implied in deciding to use three dice and may be content with the reduction in information implied in the pair of dice.

Take craps, for example, which is a slightly simplified version of the game Hazard.  These rules are a gambling cloak around a core randomisation/information generating  machine consisting of two tossed dice.  One of the steps in the game involves the active player making a choice between a target summed score of 5, 6, 7, 8 or 9.  There are close probabilities of winning associated with each of the five choices the player could make, but one had a clear edge - the choice of 7.  So clearly at one time, this knowledge was not widespread, and probably some regular players who worked it out would clear up at Hazard gatherings.  Eventually this kind of secret could not be kept private for too long.  Once everybody knows that you are best advised to pick 7 when faced with the choice, there's no real point in having that choice in your game rules.  The game rules morphed as a result in the 19th Century.  Why then?  Pascal, Fermat, Huygens were all long in their graves by then.   Perhaps it was as a result of the game-busting brilliance of Pierre Remond de Montmort, arguably the worlds first quant, insofar as he took a gambling practice and, through a clear-sighted analysis of the games, blew apart their inner logic and flaws.  One of his analyses was on the game of two dice Hazard.

He also worked on finding the sum of the first n terms in a finite difference calculation, which is of course a supremely quant-like activity.

Another way of looking at de Montmort's effect on Hazard is to say he made it more efficient.  By revealing seeming choices which, through analysis, were no choice at all, he invented the concept of a rational gambler.  And the effect of the rational gambler on games of chance was to put the weaker ones (both players and games played) out of business - or rather - to require them to become more efficient.  This is not so different from the way that the branch of relative value trading known as convertible arbitrage has had real effects on the terms and conditions of real convertible bond issues.