The nature of uncertainty in betting

Betting dates back thousands of years and while how we bet and what we bet on has changed dramatically, the nature of uncertainty remains. Bettors obviously need to be aware of what uncertainty and probability mean, but can the classic probability of an event occurring be reduced to a quantum one? Read on to find out.

For as long as we can trace history back, people have been fascinated by games of chance. Archaeological evidence from prehistoric sites across Europe, Asia and into North America has uncovered cube-shaped ankle-bones called astragalus, some dating back as much as 40,000 years.

The purpose of these bones is a matter of speculation, but accompanying cave drawings hint at the possibility they were used as some form of entertainment and a means of prophecy or divination.

The ancient Chinese, Greeks and Romans all played games of chance, involving dice as well as betting on the outcome of sporting events. For the ancients, gambling was viewed as a metaphor for life.

If you could predict what the future might bring, you could control it. And if you can control it, arguably you make your life considerably less uncertain and easier to live. Markets abhor uncertainty; so do people, who of course make the markets in the first place.

The birth of expectation

However, it wasn’t until the 17th century that chance, uncertainty and probability were formalised mathematically when two French mathematicians, Blaise Pascal and Pierre de Fermat, collaborated to solve a gambling dispute concerning a game of dice.

In formulating a general theory of probability they introduced to the world the concept of mathematical expectation or expected value, still used today by bettors to estimate how much profit they are likely to win.

What is chance, really?

If something is said to be subject to chance, that is to say random, what does that actually mean? Informally, if we do something the same way each time with the same starting conditions, for example, roll a dice, but we get different outcomes that is said to be random.

For something like dice, however, it would be virtually impossible to replicate the starting conditions exactly each time. Slight differences in the way we hold and throw them are what lead to the variance in outcomes. According to this model, randomness is simply a manifestation of the sensitivity to initial conditions. As Blaise Pascal once famously remarked:

“Cleopatra’s nose, had it been shorter, the whole face of the world would have been changed.”

Incomplete knowledge

Thus, uncertainty in the outcome must represent not some fundamental property of the system but merely an incomplete knowledge about it. If you could know exactly all the forces applied to the rolling of the dice and the directions in which they were applied, you could predict with absolute certainty how it would land.

Albrecht explains that anyone tossing a coin is engaged in a kind of Schrödinger’s cat experiment where the final state of the coin toss is both heads and tails at the same time.

This is the business of determinism: everything is inherently predictable given enough information, and for every specific set of initial conditions there is only one outcome. The fact that many things aren’t is simply due to a lack of data. In 1814, another French mathematician articulated the following thought experiment which became known as Laplace’s Demon:

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect [the Demon] which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”

Presumably, Laplace’s Demon would clean up at the bookmakers, although most of them (unlike Pinnacle) would close his account. Sadly none of us can be as intelligent; there will always be errors in our measurement of the initial conditions. Hence, there will always be some level of uncertainty in the outcome; it is this uncertainty which we call randomness.

The Uncertainty Principle

Around the turn of the 20th century, the philosophy of determinism began to unravel with the realisation that the world of the very small – atoms and the subatomic particles which make them – don’t behave in the same way as everyday objects.

Quantum mechanics – the physics of the very small – began to reveal that Laplace’s ‘items of nature’ didn’t actually assume fixed entities, but appeared to behave more like waves whose position in time and space could only be described by means of a probability (wave) function. How can you predict where something is going to be in the future when you don’t even know where it is right now?

In 1927 the German physicist Werner Heisenberg published his now famous Uncertainty Principle. Put simply, you can’t know precisely the momentum and position of a particle, and the more you know about one, the less you will know about the other.

It has typically been assumed that we needn’t worry about the Uncertainty Principle when it comes to matters of classical probability which are relevant to the betting world

Crucially this ‘uncertainty’ did not arise because of some constraint imposed by the physical limitations of practical observation and lack of information, as Laplace might have postulated. On the contrary, it was a mathematical impossibility imposed by the very nature of matter itself.

Albert Einstein was troubled by such a proposition, stating; “I… am convinced that He does not throw dice.” Einstein’s conviction, however, was wrong. Quantum mechanics is arguably mankind’s crowning scientific achievement, with predictions that have been made and validated on innumerable occasions, no matter how weird and mind-boggling they appear to be.

It turns out that even Laplace’s Demon is bound by the Uncertainty Principle and cannot know both the position and the speed of a particle. As Stephen Hawking said; “all the evidence points to Him being an inveterate gambler, who throws the dice on every possible occasion.” Moreover, He doesn’t even know what the outcomes will be.

Understanding quantum probability

It has typically been assumed that we needn’t worry about the Uncertainty Principle when it comes to matters of classical probability which are relevant to the betting world, because the sorts of things we like to bet on – football, playing cards, spinning a roulette wheel – take place at scales many orders of magnitude larger than the subatomic world. The physical stuff of reality is much too large to be discernibly influenced by quantum mechanics.

While the Uncertainty Principle requires a completely different interpretation of cause and effect in the quantum world, causality in the macroscopic world and the business of determinism might be thought of as emergent, exhibiting properties that the subatomic entities from which these phenomena are constructed do not. As the saying goes, the whole is greater than the sum of the parts.

Not so fast, says Andreas Albrecht, a theoretical physicist and one of the founders of the inflation theory of the universe. Investigating the influence of quantum uncertainty on the behaviour of colliding water molecules and their subsequent influence on the random Brownian motion of neurotransmitters in the nervous system, Albrecht has argued that the uncertainty in the outcome of something like a coin flip (which will be dependent on the activity taking place in the flipper’s brain neurons) can be accounted for entirely by the amplification of the original quantum fluctuations affecting the water molecules.

This means, according to Albrecht, quantum uncertainty completely randomises the coin flipping, and that the classical probability for the outcome of a coin flip can be reduced to a quantum one.

Quantum ignorance

Since the uncertainty of such a system increases non-linearly with every subsequent Brownian collision, once that uncertainty becomes large enough, its quantum origins become the dominant influence in the outcome, not classical mechanics.

For a game of snooker, for example, Albrecht has calculated that it could take just 8 collisions between balls for quantum uncertainty to dominate. Indeed, it would appear that any random system driven by neural processing, including rolling a dice, striking a snooker ball, kicking a football or playing a poker hand will have an underlying ‘quantum ignorance’.

What if a coin toss is both heads and tails?

In keeping with the weirdness of quantum mechanics, Albrecht explains that anyone tossing a coin is engaged in a kind of Schrödinger’s cat experiment where the final state of the coin toss is both heads and tails at the same time. It is only once the final outcome is observed that the system assumes a defined value of either heads or tails.

If one was to bet on the coin toss (or football game, tennis match, election result or anything else involving human behaviour for that matter) that bet would be both won and lost at the same time until the outcome is observed.

I don’t know what will happen or I can’t know what will happen?

If causality, determinism and classical probability are merely illusory, emerging from and yet reducible to quantum uncertainty, the implications could be considerable. Essentially we have moved from Laplace’s proposition ‘I don’t know what will happen’ to Heisenberg’s proposition ‘I can’t know what will happen.’

One might argue that at the macroscopic scale for bettors, this doesn’t really change the analysis. Yet from a philosophical perspective, the idea that the final outcome of a game involving chance cannot intrinsically be predicted until it has actually happened is a very disconcerting picture for human beings designed to think deterministically and bi-modally in terms of only either/or.

The consequence is that there may be no physically verifiable fully classical theory of probability at all, just a quantum one, where a multitude of possible (betting) histories may all happening at the same time.

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