An entry today in Bruce Schneier’s blog alerted me to the existence of a paper by Diaconis, Holmes, and Montgomery called “Dynamical Bias in the Coin Toss” and published in* SIAM Review* **49**, 211 (2007) (a preprint is available from Schneier’s entry). It reminded me of a little blurb I wrote around 2005 or so about a similar sort of thing, and I figured I’d put it up here with minor edits and a few links added:

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Consider the familiar six-sided die. It is common practice to simply state that the probability of any of the six outcomes for a toss is 1/6. Let’s look at this in some more detail. The probabilistic constructions are mostly straightforward, even in detail: the probability space is {1, 2, 3, 4, 5, 6} and the -algebra of events is generated by all the singletons. *Justifying* the uniform measure, however, is a task for physics—not mathematics or probability. The die is a physical object and subject to physical laws, and we require a connection from the physics to obtain the common idealization.

A qualitative sketch of the justification is not too hard: a gambler’s tosses of a die (or dice) at a table are represented by subtly different initial conditions collectively generating the phase space average. The dynamics of the die-table system are well understood: there are gravitational and hard-core potentials and a dissipative frictional contribution. Over short time scales after the initial impact of the die on the table, dissipation can be ignored. One could argue by analogy with ergodic results for billiards in an attempt to justify the ergodic hypothesis, and by extension the uniform probability measure.

The lines of this sketch become less clear when upon closer examination, though: it is plausible that there might be correlations in the tosses that result in a “non-metrically indecomposable” system, for instance. Such an objection might be overcome by formally stipulating that the initial conditions of the die are given by suitably nice distributions on initial position, velocity, and orientation. But there still might be some correlations hidden between these distributions and the symmetries of the table, or in the dissipative dynamics. Upon further reflection people generally admit that the problem is so difficult to really break down that it appears operationally intractable, [1] and *this intractability is the actual* *justification for selecting the uniform 1/6 measure*. [2]

Despite this apparent intractability, there might be deviations from uniformity arising from determinate causes, as the following argument suggests (but does not establish or even claim). The moment of inertia of a cube of mass *M* and edge length *a* about its center of mass is Now consider a typical die as indicated:

Each of the 21 indentations or “pips” affects the moment of inertia about one of the principal axes, which is taken to (approximately) coincide with the coordinate axes, which in turn are taken to have their origin at the center of the convex hull of the die (i.e., the cube without pips). Assume each pip corresponds to a negative volume . Now and the corresponding mass deficit for a pip is

Assume for convenience that the pips are small cylindrical indentations of negligible depth. If, for instance, a pip’s small dimension is along the *z*-axis, then the parallel-axis theorem shows that its moment of inertia about that axis is of the form where *r* is the pip’s radius and *d* the distance from the pip’s center to the *z*-axis. Its moments of inertia about the *x*- and *y*-axes are of the form

The geometry of the faces is essentially captured by the following figure:

Note that Regardless of the number of pips on a face, each pip is either in a position as indicated in the figure above, or in the center of the face.

The die’s moment of inertia about the *z*-axis is given by

where and This simplifies to

Similarly,

Moreover, substituting for shows that in fact This is not an accident: it depends on not only the assignment of outcomes to particular faces, but also the orientations of faces relative to each other. It also explains why the particular configuration of pips and faces depicted above is almost always chosen for casino dice.

However, the center of mass of the die is not at the center of the corresponding cube, but rather is slightly displaced in the direction . Thus the moments of inertia about the principal axes most nearly parallel to the *z*, *x*, and *y* axes have, respectively the greatest, intermediate, and least values. Consequently, rotation about the *x*-principal axis is unstable, and the other two principal rotations are stable. [3] The instability is of course weak because of the small contribution from the pips. (An essentially identical derivation shows that the same stability result holds for painted or even filled pips, so long as the densities of the die body and filling are not identical: however, the instability will be correspondingly weaker in these cases.)

Nevertheless, there are the beginnings of a possibly viable strategy for craps, based on the (questionable) supposition that a dedicated gambler might be able to subtly influence rolls of the dice by “setting” the intial conditions of the dice. [4] Though actual success sounds unlikely, it cannot be ruled out *a priori*, and surprisingly, the physical analysis suggests a detailed strategy for a particularly common bet. First, however, I will mention generic “odds bets”, where a single outcome from 2 to 12 is bet on. These offer no advantage to the house or player for uniformly random outcomes, though a line bet must also be made, which is designed to confer a slight net advantage to the house. By the simple expedient of choosing stable axes, however, these net odds might conceivably be perturbed enough to give the player a slight advantage.

A more complicated but in principle nontrivial strategy for so-called “pass line” bets is not altogether implausible. [5] In such bets, the losing or “craps” outcomes are 2, 3, and 12; the winning outcomes are 7 and 11. Any other outcome—4, 5, 6, 8, 9 or 10—is a “point”, and throws are repeated until either the point is duplicated (the winning outcome) or a 7 is rolled (the losing outcome). By setting the dice about a single stable axis, but with opposing orientations, you might expect a slight increase in the number of 7 rolls, independently of the details of successive throws. The nominal determinant of such an effect would be the average extent to which the two dice follow similar trajectories in the course of a single throw. [6] The *z*-principal axis would be preferred over the *y*-principal axis for our pass line strategy, because craps would then be slightly less likely—pure rotations about *z*-axes parallel to the table cannot produce them—and the winning outcome of 7 slightly more likely (but not, at least to first order, the winning outcome of 11).

If a point occurred, the proper thing to do would be to attempt to duplicate the point in the succeeding throws by reproducing the detailed throw mechanics. Even given the usual requirement to hit the back wall of the table, a reasonable degree of skill might thereby suffice to improve point duplication to a statistically significant degree. Though it seems the strategy probably offers no practical utility, it nevertheless appears that the argument is at least reasonable in principle. That said, any truly successful series of results from our pass line strategy would probably also require giveaways in the player’s throwing mechanics that would result in her identification as a dice controller, and therefore almost certainly entail expulsion from the casino. Thus ends our *Gedankenglückspiel*.

One can imagine a more detailed die design where the moments of inertia about the center of mass are exactly the same, but such a die would be difficult to manufacture, and imperfections in that manufacturing process might even negate the supposed remedy. The bottom line is that the detailed dynamics are operationally intractable, but even so, regularities might arise if the ensemble of initial conditions has some special property.

A similar argument could be made for almost any sufficiently complex system. There is very little truly random (in the colloquial sense) about most complex phenomena, but almost all the details are operationally intractable. This intractability is at the heart of a definition of randomness that is due to Kolmogorov and elaborated by Chaitin. It also highlights a point of considerable importance: the fact that the detailed specification of the initial state of a complex system is an intractable task is the normal justification for invoking the traditional methodology of statistical physics. Arguments that lead from nonequilibrium initial conditions to paradoxes in the context of thermodynamics and statistical physics (e.g., arguments based on spin-echo experiments) usually fail to recognize that the nonequilibrium states involved in these arguments are relatively easy to describe, and that such states are usually outside the scope of the theory.

There might well be a gambler in the real world that generally wins at the craps table, having carefully refined her tossing motion and studied the effects of the table geometry and material. [7] This gambler would be making a living by exploiting our naiveté regarding the scope and applicability of ergodic theorems or even our understanding of what randomness is. If any of us knew of such a person we would probably envy her. If you ran a casino or sat on a gaming commission, you would do well to see to it that such a person would be barred from gambling. By the same token I would do well to ensure that perpetrating any such shenanigans with an implementation of [a security system such as Equilibrium's] should be prohibitively difficult. This “pit boss” frame of mind is helpful when selecting [configurations for such a system].

**Footnotes**

[1] “It is impossible for a Die, with such determin’d force and direction, not to fall on such a determin’d side, only I don’t know the force and direction which makes it fall on such a determin’d side, and therefore I call that Chance, which is nothing but want of Art…” From Arbuthnot’s preface to “Of the laws of chance” (1692), as reproduced in Grimmett and Stirzaker

[2] Performing the experiment might produce a distribution that exhibits a “statistically significant” (say, as measured by some *ad hoc* protocol such as a chi-squared test, or even simple common sense) deviation from the uniform one. In fact just such an experiment was performed and used in an attempt to criticize maximum entropy methods: the uniform measure turned out to deviate statistically significantly from the measured distribution. Jaynes turned the criticism on its head and provided a convincing demonstration that the physics of this particular die-tossing experiment was improperly characterized. (See Jaynes, E. T. “Where do we stand on maximum entropy?” In *The Maximum Entropy Formalism,* Levine, R. D. and Tribus, M., eds. MIT (1979).) He detailed and justified *Ansätze* (equivalently, constraints) expressing data-skewing reflecting the possible effects of the die’s pips on the center of gravity and of imprecise milling: he then applied MAXENT to obtain a fitted distribution with no statistically significant deviation from the experimental one. It seems plausible that a careful characterization of the die used in the experiment would have supported Jaynes’ theoretical result. Certainly careless applications of MAXENT can encounter problems.

[3] Recall that the rotation of an asymmetrical top is stable about its principal axes with greatest and least moments of inertia, and unstable about the intermediate principal axis. See, e.g., Landau, L. D. and Lifshitz, E. M. *Mechanics*. 3^{rd} ed. Butterworth-Heinemann (1976).

[4] Techniques often advertised for controlling dice throws in craps are dependent on the player’s supposed ability to dictate the detailed geometry of the throw by setting, establishing a consistent throwing motion through keeping a rhythm or other methods, attempting to have the dice impact the table at 45-degree angles, etc. (See, e.g. books by Frank Scoblete or http://www.goldentouchcraps.com/, where $1295 buys the inclined punter two days of coaching.) Setting is of at best questionable utility, and the other techniques are still more dubious. It should also be mentioned that while some casinos disallow even setting, the rest all require setting to be done quickly so as not to disrupt play.

[5] The pass line is the most common example of the mandatory line bet: the house advantage for it is roughly 1.5%. Most other line bets are considerably worse choices.

[6] The motion of the dice appears not to be chaotic in any strong technical sense. The dice trajectories should diverge most strongly at their termini, where, for instance, a little extra kinetic energy could result in a die just barely tipping over one more time. The motion therefore does not even seem to be intractable in principle, but merely in practice; and our arguments do not seem to rely on anything so foolish as “beating Lyapunov exponents”. The dice are certainly correlated with each other, even for a typical throw, and it is not unreasonable to suppose that these correlations remain localized in a small enough region of phase space during a time evolution so that they might be made manifest along the lines described. Of course, whether they actually are, or even can be in practice, is another issue entirely.

[7] Gamblers that appear to fit this description were profiled in the television feature *Breaking Vegas: Dice Dominator*. The History Channel (2005). Thanks to Brian Hearing for alluding to this reference.

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The truly ironic thing about this is that I happen to know a PhD physicist who is also a former craps dealer. He hasn’t read this, but I’ve spoken about it with him, and he assured me that the pip densities are sufficiently carefully controlled to defeat any sort of stability-oriented “attack” such as this. Not that I was actually going to try the experiment…

**Update:** typos fixed and formatting improved.

Also of interest regarding moments of inertia and principal axes for dice:

http://maths.dur.ac.uk/~dma0cvj/mathphys/supplements/supplement2/supplement2.html

and note that any three orthogonal axes are principal axes for a cube, but for a die it’s not clear that the principal axes aren’t substantially different from the x-, y- and z- axes. But it seems likely that they are close to them: see, e.g.

http://skepticsplay.blogspot.com/2008/07/home-experiment-spinning-box.html

The key ideas of this post hold either way, though.

… “pip densities are sufficiently carefully controlled”…

don’t most die have painted dots nowadays?

Seems like it. But as long as the paint density doesn’t match the body density then these arguments still apply…it’s just a difference of the degree of bias

“It is shown and emphasized that, from the dynamical point of view, outcomes are predictable, i.e. if an experienced player can reproduce initial conditions with a small finite uncertainty, there is a good chance that the desired final state will be obtained.” See http://books.google.com/books?id=kf-OOB7EMyYC

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