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Quizzing the Anonymous
Ignoramus et ignorabimus
Coin flipping 
18th-Feb-2009 10:55 am
A post by a friend of mine reminded me of the following problem: coin tossing is considered to be the epitome of randomness. On the other hand it is not clear that this process is not deterministic if one considers a coin as a rigid body. Diaconis et al. famously showed that the probability that the coin (cought by hand) will rest on the side which is up at the initial moment is 0.51. They speculated that bouncing may even the chance, but that has not been shown. So, where the indeterminism comes from? Is it in the dynamic behavior of the coin itself (some kind of dynamic instability) or something else? The recent study ("Dynamics of coin tossing is predictable," doi: 10.1016/j.physrep.2008.08.003 suggests that it is fully deterministic.

...The analysis of the dynamical behavior of roulette goes back to Poincare. His results suggest that as the roulette ball is spun more and more vigorously the outcome number is independent of the initial conditions (initial conditions are washed out). For a large number of trials the numbers become close to the uniform distribution.

...All initial conditions are mapped into one of the final configurations. The initial conditions which are mapped onto heads configuration create heads basin of attraction while the initial conditions mapped onto tails configuration create tails basin of attraction. The boundary which separates heads and tails basins consists of initial conditions mapped onto the coin standing on the edge configuration. For an infinitely thin coin this set is a set of zero measure and thus with probability one the coin ends up either heads or tails. For the finite thinness of the coin this measure is not zero but the probability of edge configuration to be stable is low. An American 25 cents coin lands on the edge about one time in 6000 tosses.

...From the point of view of the dynamical systems the outcome of the tossing coin should be deterministic. As the initial conditions - final configuration mapping is strongly nonlinear one can expect deterministic unpredictability due to the sensitive dependence on the initial conditions or fractal basin boundaries. In other words one can pose the question, is anything chaotic in the dynamics of the tossed coin which can produce a random like behavior?

...A [Newtonian] mechanical model of coin tossing is constructed to examine whether the initial states leading to heads or tails are distributed uniformly in phase space. We give arguments supporting the statement that the outcome of the coin tossing is fully determined by the initial conditions, i.e. no dynamical uncertainties due to the exponential divergence of initial conditions or fractal basin boundaries occur.

...For the realistic coin in which the distance between the center of the mass and the geometrical center is small, it is sufficient to consider a simplified model of  the ideal thin coin. The air resistance causes the deviation of the trajectory of the mass center from vertical axis and damps the rotation of the coin. When the distance of the free fall is small the effect of the air resistance can be neglected. During the free fall the sensitive dependence on the initial conditions has not been observed. The process of the coin bouncing on the floor has a significant influence on the final state (heads or tails). It has been observed that the successive impacts introduce sensitive dependence on the initial conditions leading to transient chaotic behavior.

...The basins of attraction of heads and tails (the sets of the initial conditions leading to both outcomes) show that the boundaries between heads and tails domains are smooth. This allow us to state our main result; there exists an open set of initial conditions for which the outcome of the coin tossing is predictable. In practice although heads and tails boundaries are smooth the distance of a typical initial condition from a basin boundary is so small that practically any finite uncertainty in initial conditions can lead to the uncertainty of the result of tossing. This is especially visible in the case of the coin bouncing on the floor, when with the increase of the number of impacts the basin boundaries become more complicated. In this case one can consider the tossing of a coin as an  approximately random process.

...If the outcome of the long sequence of coin tosses is to give random results, it can only be because the initial conditions vary from toss to toss. In the previous section we show numerically that for each initial condition there exists the accuracy e for which the final state is predictable. In this section we try to explain why for practically small (but not infinitely small) e the coin tossing procedure can approximate the random process. A sequence of coin tosses will be random if the uncertainty e is large in comparison to the width W of the stripes characterizing the basins of attraction so that the condition e>W is essential for the outcome to be random. Uncertainty e depends on the mechanism of coin tossing while the quantity W is determined by the parameters of the coin.

So there is not much randomness in the coin tossing per se, even with bouncing off a smooth floor, as the coin, actually, is a mediocre amplifier of dynamical uncertainty. The randomness is introduced purely by the hand of a tosser. We are bad at controlling our motions.
18th-Feb-2009 07:40 pm (UTC)
In other words: absolutely fair coin tossing is unfair.
18th-Feb-2009 09:24 pm (UTC) - Persi's magic
This seems to be the case. Roulette is inherently fair, wereas coin flipping is inherently unfair. My biostatistician wife once attended a lecture by Persi Diaconis, where he demonstrated a machine that tossed coins. After so many flips in the air, for all appearances random, the coin always ended face up on the floor. He learned how to do it with 100% accuracy by hand with silver dollars (it is easier when it is heavier); for lighter coins he is less consistent, but still beating the odds by a wide margin. As they say in the paper, if e
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This seems to be the case. Roulette is inherently fair, wereas coin flipping is inherently unfair. My biostatistician wife once attended a lecture by Persi Diaconis, where he demonstrated a machine that tossed coins. After so many flips in the air, for all appearances random, the coin always ended face up on the floor. He learned how to do it with 100% accuracy by hand with silver dollars (it is easier when it is heavier); for lighter coins he is less consistent, but still beating the odds by a wide margin. As they say in the paper, if e<W that's what you expect. He showed these tricks for 10-15 minutes and then asked "What does it mean: a random trial?" Nobody volunteered to answer.

<img src="http://www.dartmouth.edu/~chance/chance_news/for_chance_news/ChanceNews13.02/tosser_300.jpg">
18th-Feb-2009 10:32 pm (UTC)
It is a kind of predictable result. I would be much more surprised if they were to find any kind of chaos there. A lot of mechanics is like that. However, the sensitivity to the initial/boundary conditions is often so pronounced that the accuracy of many (dynamic) mechanical experiments is not great at all.

I suppose, what I am trying to say, is that the determinism of the coin flight dynamics still does not make it determinate from the practical point of view. Chaos is sexy but life can be sufficiently interesting even without it.
18th-Feb-2009 10:35 pm (UTC)
Read your reply above: hmm, interesting. THAT determinate?
18th-Feb-2009 10:36 pm (UTC)
That determinate - if you are Persi Diaconis and dropped out of school at 14 to pursue math and magic...
18th-Feb-2009 10:40 pm (UTC)
Scanned through the paper. Seems good work; they simplified their life enormously by carefully controlling the coin position. I'd call this a very good experiment :)
19th-Feb-2009 01:39 am (UTC)
Случайность, это такая специально созданная справедливость,
каждый шанс должен иметь свой шанс.
Справедливость требует сложных процедур по нейтрализации всяких неравенств.
Нужны специально обученные люди для поддержания такой справедливости.
Случайность создают люди. Шлифуют монеты, разрабатывают алгоритмы, генераторы случайных чисел.
Или просто объявляют что-то случайным событием.
Но никто не ангел и вечно кому-то подсуживают.
19th-Feb-2009 01:46 am (UTC)
A wonderful concept, but how do people create randomness if nothing is truly random?
20th-Feb-2009 02:32 am (UTC)
Я не говорил, что ничто не случайно.
Я уже написал, есть два способа - создать генератор или просто назвать что-то случайным.

Случайность используется в тех случаях, когда мы смотрим на природу другого масштаба.
Когда разница между элементами нам неразличима или нам удобно этой разницей пренебречь.
Когда же понятие случайности переносится на масштаб нам близкий происходят ошибки.
Вы понимаете, какое неопределенное понятие случайная выборка при опросах.
И какая это сложная ментальная конструкция.

Случайность это синоним бессмысленности - отсутствие повторяемости, отсутствие намерения,
Когда мы создаем случайность, мы намеренно создаем отсутствие намерения.
Когда мы ищем случайности в природе, мы просто постулируем,
что где-то не хотим признать смысл и намерение.
20th-Feb-2009 04:35 pm (UTC)
Statistical sampling is a complex concept, but only because you want it to be uniform and representative. Randomness per se is not that hard, if neither of these are deemed important. So I'd class this as willful ignorance, too.

A generator is a deterministic machine, so what you provide is pseudorandom, it is not chance. As for postulating, I do not think there is an objective per se to profess ignorance of causes. We would love to know these causes, in many cases. It is the consequence of true ignorance of such causes. That was Darwin's position, by the way: "I have... spoken as if the variations... had been due to chance. This, of course, is a wholly incorrect expression, but it serves to acknowledge plainly our ignorance of the cause of each particular variation." To Darwin, randomness is not knowing the causes. To Leibniz, it is causes so complicated that reason cannot trace them to the origin (slight, multiple causes -> considerable consequences). To Aristotle and Darwin, it is lack of purpose; for Cournot it is intersection of chains of events (coincidence of events each having simple causes).

As for myself, I think that chance and randomness are the properties of logical constructions and propositions. It is not knowing, or general statements about repeatable events (the likely outcomes of a series of trials), or relations between statements. Perhaps all randomness, including the one in quantum mechanics, is truly of this kind, as no other kinds exists.
20th-Feb-2009 05:48 pm (UTC)
Мы с вами согласны.
Я только подчеркиваю роль личности.
Там, где вы пишете "случайность это свойство логических конструкций", я напоминаю,
кто делает эти конструкции.
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