Tossing away information

Continuing from my initial analysis I would like to model the consequences of sheep bones (just like all other animals' bones) being white.

The crucial human animation in using sheep heel bones as random event generators is the act of tossing.  In the act of tossing, you loose knowledge of the order in which your four bones will lie.  This wouldn't be an issue if all four bones were of a different colour.  Or perhaps if the bones were marked not with 1,3,4,6 on each of the four bones, but with 16 different numbers.  If humans had etched 16 different numbers (pips) on their bones, they'd be using the maximum amount of information possible in that act, namely 6.88 bits.  But that doesn't happen.  Instead we humans make 4 more or less similar sets of markings on the bones.  Then, when we toss, we toss away some information.  But how much?

To answer this question, consider the die.  One die has 2.6 bits per roll.  With two dice, if order is important, then you have 5.16 bits (imagine each of the dice had a different colour).  With three, 7.75 bits (again, imagine each of the three dice a different colour).  You can see how when you run this experiment, information is additive as the sample space size grows multiplicatively.  You can also see that, with the addition of colour, your parallel toss does not lose track of which die had which value.  This parallel toss is the same as if you had tossed one die two (or three) times, and taken note of each result.  It is a kind of 'sampling with replacement' activity in so far as the probability of any single outcome is independent of earlier throws.  (The Markov property).

But bones are white.  And there's a pre-existing tradition of making each bone contain the same number and type of markings as each of the others.  Most likely early dice were crafted out of the astragalus, and they would have inherited this feature of being practically indistinguishable from each other.  That means, when two or three are tossed, information is lost, in comparison to the 'order important' experiment.  Of course, the lost information is precisely that of ordering.  But how much?  For two indistinguishable tossed dice you now only have 4.3 bits of information. When you do the 'order unimportant' analysis on four astragali, the information content drops from 6.88 to 4.3 bits.

Isn't that amazing?  Given what we know about the colour of bones, the number of sheep legs and the casual act of tossing collections of indistinguishable objects, we  built up a randomisation machine which would conveniently deliver for us 4.3 bits.  When we smooth and chip away at a sheep-load of astragalli and make them into 6 sided dice, we manufacture a convenient randomisation machine which delivers 4.3 bits of information to us using only two dice.  That is impressively close to the original.  All those religious-oracular rule books, all those long-forgotten games could still be randomised by a machine of approximately the same degree.  One die would not have been enough, three too much.  But two would have been just right.  Our culture got to keep the innovations built up around the astragali.

My guess as to why each bone (and, later, die) was marked identically is because in the beginning was the single astragalus.  It was a much easier step to make a second instance of that same pip-marked bone.  And a third, and a fourth.  

But why did humans go to the bother of crafting a die if their main practice of four-bone tossing is equivalently replaced with two dice tossing?  Was it a matter of aesthetics?  Did the cubic astragalus with only 4 landable sides present an affront to our sense of symmetry?  Surely it couldn't be to make any kind of mathematical analysis more amenable, though it isn't impossible that some people might have worked out the possibilities of dice.  My money's on our desire to manufacture a symmetric, crafted, more aesthetically pleasing randomiser machine.  The uniform distribution perhaps also made the construction of new equivalence classes and interpretive games based on that uniform element of randomness easier to plan and design.