22 highlights
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Some people believe that confronting problem gamblers with the ‘reality’ of mathematics – a kind of mathematical counselling, often called ‘facing the odds’ – can help them overcome it. After all, since our earliest school days, many of us have learned to trust mathematics as the provider of necessary and logical truths. But we also trust our senses, as well as the patterns we discern from our experiences and the words we use to communicate with one another.
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In my youth I was fascinated by games of chance, and loved to play them so I could to watch probability ‘at work’. But after studying mathematics and its philosophy in more depth, my interest in such games vanished. I came to see them as simply mathematical models wearing sparkling clothes.
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In mathematical terms, this guarantee is expressed through the fact that the house edge (HE) of a game is positive.
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The mathematics of the games, including their rules and payout schedules, assures the house will profit in aggregate, regardless of individual behaviour.
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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)
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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)
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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.
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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.’
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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. A particular instance of this arises when someone uses addition (rather than multiplication) to estimate the probability of two or more independent events.
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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.
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The near-miss effect involves incorrectly estimating probabilities, but is also linked to other conceptual inadequacies regarding conditional probabilities and time-dependence.
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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.
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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 – seeing the ball landing on the number we’ve bet on, or hearing the sound of the slot machine showing a winning payline.
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Problem gambling also 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.
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Overall, these studies have yielded contradictory, non-conclusive results, and some found that mathematical education yielded no change in behaviour. So what’s missing?
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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 did trust mathematics, she just misinterpreted it.
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The philosophical complexity of applied mathematics makes it clear that simply learning to ‘trust’ mathematics is a naive prescription for gambling addiction.
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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.
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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.
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Perhaps most problematically for gamblers, statistical models are grounded in probability theory, one of the fields in mathematics most open to philosophical debate.
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All probability theory is grounded in the concept of infinity, yet all our gaming experiences are finite
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So, getting to the bottom of our gambler’s problem is likely to require a conversation between the mathematician and the philosopher, who in turn need to guide the cognitive psychologist about how to talk to her client.