The Price of Gambling

by vonNudge

My high street in south London recently featured in a Channel 4 documentary on gambling (12.06, see video). In a short space of road there are six bookmakers: two branches of William Hill, two branches of Betfred, a Paddy Power, a Coral and a Ladbrokes. Placed conveniently nearby are a number of payday lenders and pawn shops.


The documentary points out that bookmakers cluster in this way partly because of the regulations over “fixed odds betting terminals”, high-speed betting machines where large sums can be gambled frighteningly quickly. Law restricts these machines to four per betting shop. But since they are such high-earners for the bookmakers, they simply open extra branches, usually in relatively deprived areas (9.30, see video).

Gambling is, clearly enough, a high-revenue industry. Although there are around 450,000 problem gamblers in the UK (19.50, see video) – a scary number – the gambling industry must always find new ways of increasing revenue. And if creating new gamblers is too difficult, then the only other way to increase revenue is to put up the “price” of gambling.

The price of gambling is hidden over time. Gamblers do not feel the price on every trial, since sometimes they will win and hence pay a negative “price”. Everyone knows the joy of getting something for free – at a price of zero. Perhaps the thrill of gambling is a souped up version of the same reward process.

But most gamblers do pay the price over time. Especially in mechanical games such as roulette, with no skill element, the house will always win in the long run. The industry is currently exploring two main ways to increase the price of gambling., decomposed as follows:

Price of gambling = expected loss per bet * amount wagered per hour

You can either get people to accept bets that are more statistically unfair, or you can get them betting higher amounts per hour. And if you can’t get them betting more per bet (because of loss aversion), you can increase the number of bets per hour. The online gambling industry was the first instance of how modern technology could exploit this latter point, but it is now moving into mobile devices and the high street with fixed odds betting terminals. The terminals are an issue the government is aware about, but there are some other features of the price of gambling that are less well known.

If the speed and stakes of gambling are already maxed out, then the only other way to increase the price of gambling is to get gamblers accepting bets that are more statistically unfair. This is an especially new and sophisticated way of increasing the price of gambling. Traditionally, bookmakers strove to have an “even book”, where the odds are initially set, and then adjusted so that a roughly equal amount is wagered on each outcome of a sporting event. In American football the main bet is whether a team can “beat the spread”, a bet with two mutually exclusive outcomes (beat the spread/not beat the spread) that should be equally likely. The spread is used to adjust for skill differentials between teams, so if team A is a marginal favourite, beating the spread might see them winning by a margin of more than three points. If an equal amount is bet on each side, then the bookmaker earns a risk-free profit of 5% of the total book, since they only offer odds of of 10-to-11 on each event with p=0.5 (bet 11 to win 10). This is pretty good, since it requires little knowledge of the psychology of gambling and probability, or the event being bet on (some sophistication is required to set the initial odds on a skill/luck domain such as a sporting event).

But bookmakers are starting to realise that a little more savviness can can produce bigger, although riskier, profits by exploiting biases in judgement and decision making. The football season started today with the community shield between Manchester United and Wigan. Almost all bookmakers promote football bets in their windows, although only three of the shops on my street were geared up this early in the calender. These bets follow a simple and almost universal pattern. They pay-off only if two or more events occur (a conjunction), and these are usually likely events (the best team winning, or a famous player scoring).

So in the match today, Manchester United are clearly the favourite as Premier League champions against a newly relegated side, and Robin van Persie is the most famous goalscorer, with 30 club goals last season. Here are the bets I found most highly promoted:


Notice how the bets follow the same pattern. Manchester United need to win, and a scoreline of 3-0 wouldn’t flatter them in many people’s minds given the skill difference. And Robin van Persie has to score the first goal. In the last bet Manchester United have to win by the specific scoreline 3-2. In actual fact, they won 2-0, with van Persie getting both goals. Notice how this creates the sense of a near miss, with individual events that are quite likely to happen, there’s a good chance of at least one occurring. When an event occurs, but other events of the necessary conjunction for the bet do not, this creates the sense of a “near miss”. The gambler did not lose; she “nearly won”. But in order to win these bets, a number of events must co-occur. Probabilities of all events happening quickly shrink as more events are added to the conjunction. If the events were independent (not the case here), two 0.5 probabilities would co-occur with 25% of the time, three events 12.5% and four 6.25%. Large conjunctions create many near wins, without a lot of actual ones. This is perhaps why accumulator bets, which offer seemingly high odds if a large number of specific teams win, are so popular.

But there is something even sneakier with the promoted bets. There is a large literature in judgement and decision making on the psychology of probabilistic reasoning, with conjunctions being a key topic of interest. This paper by Tversky and Kahneman (1983) kicked things off, with the finding of a “conjunction fallacy”: where, contrary to the axioms of probability, participants rated the probability of a conjunction (P(AandB)) being higher than one of the constituent parts (P(A)). This can’t, of course, happen, even if B happens with certainty. Of particular relevance here, they found this effect in a sporting context (p.10). Bjorn Borg was at the time the reigning Wimbledon champion, and participants rated the probability of him “losing the first set and winning the match”, as being higher than the probability of him “losing the first set”. The high perceived likelihood of him winning the match, an event that most participants had strong memories of, led to the conjunction fallacy.

This paper led to a large debate in the field, much of which isn’t especially relevant here. A recent paper by Khemlani, Lotstein & Johnson-Laird shows, however, that you can be irrational about conjunctions without committing the fallacy. Asking participants to estimate P(A), P(B), P(AandB), is enough to fix the “joint probability distribution”. Probability estimates can still be irrational, even if they do not commit the conjunction fallacy, if the joint probability distribution implies a negative probability for one event or more. And this happened frequently in their experiments (although it was not in a sporting context), especially if the probability of the conjunction was estimated before the individual event probabilities.

Bookmakers are clearly exploiting our poor ability to comprehend the probability of two or more events co-occuring. Bookmakers are no longer striving to earn risk-free profits with an even book. They do not promote, or even offer, bets that are the complements of their highly-advertised bets, such as “Robin van Persie not to score the first goal, and Manchester United not to win 3-0”. Bookmakers would rather risk making a loss when their promoted bets do happen, simply because the likelihood of these events happening is so much worse than the odds they offer.

These bets are a sophisticated way of increasing the price of gambling. Not only do they look attractive on the surface, Robin van Persie scores lots of goals, Manchester United often win, hence attracting more bets, but they are really overpriced compared to a true appraisal of the relevant probabilities. And bookmakers would rather take a risk on these bets not happening, than hope to earn risk-free profits from a balanced book. Exploiting errors in probabilistic reasoning is a clever way to raise the price of gambling.

Now, of course the gambling industry has an answer to every criticism. They are just offering the bets that people like, and they are bringing jobs to areas that need them. But jobs in this industry are only paid by a zero-sum transfer of wealth away from gamblers, something that the industry is becoming more efficient at. Take away the gambling, and consumers will have more money, which they can spend on other things and which in itself can create as many jobs as the gambling industry. Plus consumers can now actually spend their money on things with intrinsic value.

In most industries, consumers happily pay for high-price products that they highly value. A brand new iPhone makes a lot of people very happy. There is little evidence that the same thing works in gambling. Increasing the price of gambling makes most gamblers worse off, since they lose more money and at a faster rate. The price of gambling is hidden, both because it is only paid over time, and because a number of psychological effects are being used to add extra opacity. How much does this latter point add to the price of gambling?