Mathematics for Gamblers and Investors
As author and poker champion Annie Duke wrote, “every choice is a bet into the future”. Indeed, if you want to improve your decision making, your investment process, test your cognitive distortions and stimulate your mathematical literacy, you might enjoy my summary of mathematician Catalin Barboianu.
Mathematics for gamblers (abridged article)
From a young age, mathematician Catalin was always fascinated by games of chance, to watch probability ‘at work’. But after studying mathematics and its philosophy in more depth, his interest in such games vanished: he saw them as simple mathematical models wearing sparkling clothes. Instead, he then felt the desire to help other people to see them in the same way.
For Catalin, all games of chance – whether casino games such as roulette, craps, blackjack and slots, or lottery and bingo, or card games such as poker or bridge – rely on certain basic statistical and probabilistic models. Uncertainty is built into them, which is what makes games ‘fun’ to play and also explains their continued existence: a Casino would never exist if ‘the house’ wasn’t confident that they’d always win in the end.
The mathematics of the games, including their rules and payout schedules, assures the house will profit in aggregate, regardless of individual behaviour.
In mathematical terms, this guarantee is expressed through the fact that the house edge (HE) of a game is positive. The expected value of a bet (EV) is defined as follows:
(probability of winning) × (payoff if you win) + (probability of losing) × (loss if you lose)
The HE of a game is defined as the opposite of the expected value calculated for all possible bets (HE = −EV). For example, in European Roulette, a wheel spins, and you have to decide where you think a small ball will land. There are 37 numbers (0 to 36). If you bet $1 on one number (called a straight-up bet), the payoff is 35 times what you bet, and the probability of winning is 1/37. So the EV of that bet is:
(1/37) × $35 + (36/37) × (−$1)
That is about −$0.027 or, as a percentage, 2.7 per cent of the initial bet. EV can be read as an average; in our example, you might expect to lose on average $2.70 at every 100 plays with that bet over the long run. This means that European Roulette has a house edge of 2.7 per cent. This is the house’s share of all the income produced by that game in the form of bets over the long run.
Cognitive distortions are recognised as important risk factors for problem gambling.
From a player’s point of view, a positive house edge should mean that she can’t make a living off that game: over the long run, the house will have an advantage. That’s why a pragmatic principle of safe gambling behaviour is: ‘When you make a satisfactory win, take the money and get out of there.’
However, there’s the so-called ‘gambler’s fallacy, where someone believes that a series of bad plays will be followed by a winning outcome in order for the randomness to be ‘restored’.
Then there’s the conjunction fallacy when the gambler estimates the probability of a combination of events to be higher than the probability of one of those events. For example, in sports betting, someone might bet once on several ‘almost sure’ outcomes occurring together, thinking that it’s likely that all the house’s favourite teams will win – ignoring the fact that the product of the probabilities of several wins is a number significantly lower than the probability of any individual win.
Another gambling misconception is the near-miss effect when an outcome differs just a little from a winning one, which induces the gambler to believe that she was ‘so close’ that she should try again. Here you might think of slots, scratch cards or lottery, where such events are the most frequent, but virtually any game of chance produces them. The near-miss effect involves incorrectly estimating probabilities but is also linked to other conceptual inadequacies regarding conditional probabilities and time-dependence. In such cases, the gambler mentally splits the winning outcome between the ‘matching’ and the ‘non-matching part – an action that’s mathematically irrelevant – and develops an overconfidence in a new occurrence of the ‘matching’ part in a future play. This ignores the actual probability of the occurrence of the ‘non-matching part, and the conjunction of the two predicted events which would happen at different moments; the ‘so close’ is actually ‘so far’ in probability terms and figures. All such cognitive distortions are recognised as important risk factors for problem gambling.
Gambling has existed since antiquity, but in the past 30 years, it’s grown at a spectacular rate, turbocharged by the internet and globalisation. It draws on a multitude of cognitive, social and psychobiological factors.
Problem gambling involves complex brain chemistry, as gambling stimulates the release of multiple neurotransmitters, including serotonin and dopamine, which in turn create feelings of pleasure and the attendant urge to maintain them. Serotonin is known as the happiness hormone and typically follows a sense of release from stress or fear. Dopamine is associated with intense pleasure, released when we’re engaged in activities that deserve a reward and precisely when that reward occurs.
Beginning a couple of decades ago, several studies were conducted to test the hypothesis that teaching basic statistics and applied probability theory to problem gamblers would change their behaviour. Overall, these studies have yielded contradictory, non-conclusive results, and some found that mathematical education yielded no change in behaviour. So what’s missing?
One problem is that interventions to promote mathematical literacy among gamblers generally push the message that gamblers should unconditionally trust mathematics. The problem for gamblers isn’t so much a lack of trust in mathematics as much as an incorrect application and interpretation. After all, the gambler trust mathematics, but she just misinterpreted it.
The limitations of mathematical counselling make sense when we recall that the mathematics of ‘real world’ events are far from pure numbers; rather, they take the form of descriptions, strategies, predictions and expectations, all mediated by language and meaning. By making a distinction between pure and applied mathematics, between truths that are necessary and those that are contingent, and noticing how often we mix mathematical and non-mathematical terms ourselves, we might steer ourselves on the right track to correct our cognitive distortions.
No single expert or guide can help us here. Indeed, some of the associated cognitive distortions tap into genuine philosophical debates, such as over the meaning of randomness, something that’s uncontrolled and proceeds without any rules. Mathematicians and philosophers struggle to agree on a rigorous and universally accepted definition, despite the centrality of the concept to probability theory.
The philosophical complexity of applied mathematics makes it clear that simply learning to ‘trust’ mathematics is a naive prescription for gambling addiction. Applied mathematics involves establishing an equivalence between the empirical, real-world context (the target domain) and abstract mathematical structures (the source domain). To do this, we need to idealise away from the messiness of the real world with models. After ‘doing the math’ with the help of the mathematical theory in the source domain, the derived mathematical truths are interpreted back in the target domain via our models, where we can create predictions, representations and descriptions.
The results of any mathematical model depend on language for interpretation and empirical validation.
When we put abstract, formal mathematics in empirical situations such as games of chance, we ultimately rely on language to express newly inferred relations as truths. However, these ‘truths’ are no longer necessary truths; they depend on meanings, interpretations and context. As such, they’re contingent truths. If we’re too zealous in abstracting or idealising our empirical context, or if we poorly interpret the mathematical truths in the target domain, such modelling can lead to erroneous results. When this happens, it’s not pure mathematics that’s to blame, but the whole setup.
All of this shows that any application of mathematics is a balance between relevance and convenience – a choice, a refinement, and finally, a cross-checking against the real world. All this relies as much on a mathematician’s or scientist’s intuition as it does on scientific or mathematical rigour. This analysis makes it clear that we shouldn’t blindly trust applied mathematics – which is not to say that we don’t trust the pure mathematics behind it, but simply that we need to be mindful of the place of language and interpretation.
The same goes for the mathematics of gambling. Whether we talk about describing games or making predictions about bets, the results of any mathematical model depend on language for interpretation and empirical validation. Take the statement that ‘the relative frequency of the die showing a 1 will converge towards 1/6 with the number of throws’. One interpretation is the gambler’s expression that: ‘I expect that number to come up about once every six throws.’ Here, ‘about’ means ‘on average’ or ‘approximately’ – but this wording doesn’t reflect all the mathematical meaning of ‘converge’, which assumes an infinite series of experiments for the limit to be approached. Moreover, it is the relative frequency (the ratio between the number of occurrences and the number of experiments) that approaches that limit and not the absolute frequency, which seems to be the reference of the gambler’s words. We might say in this case that we have a poor model due to interpretation and language, and it will be empirically invalidated when 1 comes up twice in a row in a certain period of time.
Perhaps most problematically for gamblers, statistical models are grounded in probability theory, one of the fields in mathematics most open to philosophical debate. ‘Probability’ carries various senses and meanings. You might think of it in terms of the recorded relative frequency of the occurrence of an event, like if a roulette ball landed on a red number in 38 per cent of the plays during a period of time. Or you might take it to be the payout percentage that the game offers (usually called ‘game odds’, like 3:2 in betting on the outcome of a match, which would translate to a 40 per cent probability of losing). Or you might perceive it as a physical feature of the game in question, its inner potential to produce a certain frequency of outcome – so if you rolled a pair of dice over and over, and noted that the pair 5 and 2 appeared 150 times in 3,000 plays, you might think of it as a property of the game, which then gives a game-dependent ‘probability’ of 5 per cent.
All probability theory is grounded in the concept of infinity, yet all our gaming experiences are finite
However, its standard mathematical meaning is so-called Kolmogorovian probability, a way of measuring likelihood. There are, however, other concepts of probability, such as inductive, propensic, subjective, frequentist or classical (Laplacian). All these versions are also mathematical in nature – and perhaps surprisingly, many gamblers’ subjective personal perceptions of the concept of probability more or less match some of these theoretical concepts. Other statistical terms can be imported and used in ordinary language too, such as ‘expectation’, ‘mean’ or ‘average’, with their usual, not-purely-mathematical meanings.
Perhaps the biggest difficulty with using mathematical concepts in the context of gambling is that all probability theory is grounded in the idea of infinity – yet all our gaming experiences are finite. This inconsistency lies behind many cognitive distortions. An illustrative example is the concept of an event: in mathematics, it’s a formal element of a set, having nothing to do with the complexity of what an event means in real life.
Overall, if we think that mathematics can provide any sort of ‘fix’ for problem gamblers, then we must be careful. It’s certainly an important cognitive asset for gamblers to know just how unlikely winning outcomes are – some with probabilities close to zero, which would mean they’d have to play for several lifetimes, sometimes in the order of thousands, to get close to a probability of one. Nonetheless, ‘facing the odds’ or learning to trust mathematics often isn’t enough. It’s not sufficient to send gamblers ‘back to school’. They also need to trust the role of representation and description in the mathematical models, yet to be careful when interpreting the real-life predictions obtained through these models. That requires a sophisticated grasp of which models and idealisations are adequate for the reality they try to describe, and which are irrelevant or misleading. Since this concerns the relationship between mathematics and reality, which also relies upon language, the issue falls within the philosophy of mathematics and mathematical modelling.
Sometimes truth is not as straightforward as being validated or invalidated by empirical evidence or even scientific facts. Sometimes it originates in the very nature of the arguments we make, including the language we use to express them. Mathematics has its own terminology, and the truths of applied mathematics are sensitive to the way we understand and express them. The cognitive distortions associated with gambling are a relevant example of such ‘sensitive’ truths. What’s remarkable is that fighting them reveals something about both the nature of mathematics and the nature of human understanding – and that knowing when not to trust mathematics is as crucial as knowing when to trust it.
Source: Catalin Barboianu, Aeon