# Love Island

A while back Harry.Hill@smartodds.co.uk gave a talk to the (then) quant team about trading strategies. The general issue is well-known: traders have to decide when to place a bet. Generally speaking they can place a bet early, when the price – the amount you get if you win the bet – is likely to be reasonably attractive. But in that case the liquidity of the market – the amount of money you can bet against – is likely to be low. Or they can wait until there is greater liquidity, but then the price is likely to be less attractive. So, given the option of a certain bet size at a stated price, should they bet now or wait in the hope of being able to make a bigger bet, albeit at a probably poorer price?

In general this is a difficult problem to tackle, and to make any sort of progress some assumptions have to be made about the way both prices and liquidity are likely to change as kick-off approaches. And Harry was presenting some tentative ideas, and pointing out some relevant research, that might enable us to get a handle on some of these issues.

Anyway, one of the pieces of work Harry referred to is a paper by F. Thomas Bruss, which includes the following type of example. You play a game where you can throw a dice (say) 10 times. Your objective is to throw a 6, at which point you can nominate that as your score, or continue.  But, here’s the catch: you only win if you throw a 6 and it’s the  final 6 in the sequence of 10 throws.

So, suppose you throw a 6 on the 3rd roll; should you stop? How about the 7th roll? Or the 9th? You can maybe see the connection with the trading issue: both problems require us to choose whether to stop or continue, based on an evaluation of the risk of what will subsequently occur.

Fast-forward a few days after Harry’s talk and I was reading Alex Bellos’s column in the Guardian. Alex is a journalist who writes about both football and mathematics (and sometimes both at the same time). His bi-weekly contributions to the Guardian take the form of mathematically-based puzzles. These puzzles are quite varied, covering everything from logic to geometry to arithmetic and so on. And sometimes even Statistics. Anyway, the puzzle I was reading after Harry’s talk is here. If you have time, take a read. Otherwise, here’s a brief summary.

It’s a basic version of Love Island. You have to choose from 3 potential love partners, but you only see them individually and sequentially. You are shown the first potential partner, and can decide to keep them or not. If you keep them, everything stops there. Otherwise you are shown the second potential partner. Again, you have to stick or twist: you can keep them, or you reject and are shown the third possibility. And in that case you are obliged to stick with that option.

In summary: once you stick with someone, that’s the end of the game. But if you reject someone, you can’t go back to them later. The question is: what strategy should you adopt in order to maximise the chances of choosing the person that you would have picked if you had seen all 3 at the same time?

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As well as giving a clearer description of the problem, Alex’s article also contains a link to his discussion of the solution. But what’s interesting is that it’s another example of an optimal stopping problem: once we’ve seen a new potential partner, and also previous potential partners, we have to make a decision on whether to stop with what we currently have, or risk trying to get an improvement in the future, knowing that we could also end up with something/someone worse. And if we can solve the problem for love partners, we are one step towards solving the problem for traders as well.

The Love Island problem discussed by Alex is actually a special case of The Secretary Problem.  A company needs to hire a secretary and does so by individual interviews. Once they’ve conducted an interview they have to hire or reject that candidate, without the possibility of returning to him/her once rejected. What strategy should they adopt in order to try to get the best candidate? In the Love Island version, there are just 3 candidates; in the more general problem, there can be any number. With 3 choices, and a little bit of patience, you can probably find the solution yourself (or follow the links towards Alex’s discussion of the solution). But how about if you had 1000 possible love partners? (Disclaimer: you don’t).

Actually, there is a remarkably simple solution to this problem whatever the number of options to choose from: whether it’s 3, 1000, 10,000,000 or whatever. Let this number of candidates be N. Then reject all candidates up to the M’th for some value of M, but keep note of the best candidate, C say, from those M options. Then accept the first subsequent candidate who is better than C in subsequent interviews (or the last candidate if none happens to be better).

But how to choose M? Well, even more remarkably, it turns out that if N is reasonably large, the best choice for M is around N/e, where $e \approx 2.718$ is a number that crops up a lot in mathematics. For N=1000 candidates, this means rejecting the first 368 and then choosing the first that is better than the best of those. And one more remarkable thing about this result: the probability that the candidate selected this way is actually the best out of all the available candidates is 1/e, or approximately 37%, regardless of the value of N.

With N=3, the value of N is too small for this approximate calculation of M to be accurate, but if you calculated the solution to the problem – or looked at Alex’s – you’ll see that the solution is precisely of this form, with M=2 and a probability of 50% of picking the best candidate overall.

Anyway, what I really like about all this is the way things that are apparently unconnected – Love Island, choosing secretaries, trading strategies – are fundamentally linked once you formulate things in statistical terms. And even if the solution in one of the areas is too simple to be immediately transferable to another, it might at least provide useful direction.

# Ernie is dead, long live Ernie

Oh no, this weekend they killed Ernie

Well, actually, not that one. This one…

No, no, no. That one died some time ago. This one…

But don’t worry, here’s Ernie (mark 5)…

Let me explain…

Ernie (Electronic Random Number Indicator Equipment) is the acronym of the random number generator that is used by the government’s National Savings and Investments (NSI) department for selecting Premium Bond winners each month.

Premium bonds are a form of savings certificates. But instead of receiving a fixed or variable interest rate paid at regular intervals, like most savings accounts, premium bonds are a gamble. Each month a number of bonds from all those in circulation are selected at random and awarded prizes, with values ranging from ￡25 to ￡1,000,000. Overall, the annual interest rate is currently around 1.4%, but with this method most bond holders will receive 0%, while a few will win many times more than the actual bond value of ￡1, up to one million pounds.

So, your initial outlay is safe when you buy a premium bond – you can always cash them in at the price you paid for them – but you are gambling with the interest.

Now, the interesting thing from a statistical point of view is the monthly selection of the winning bonds. Each month there are nearly 3 million winning bonds, most of which win the minimum prize of ￡25, but 2 of which win the maximum of a million pounds. All these winning bonds have to be selected at random. But how?

As you probably know, the National Lottery is based on a single set of numbers that are randomly generated through the physical mechanism of the mixing and selection of numbered balls. But this method of random number generation is completely impractical for the random selection of several million winning bonds each month. So, a method of statistical simulation is required.

In a previous post we already discussed the idea of simulation in a statistical context. In fact, it turns out to be fairly straightforward to generate mathematically a series of numbers that, to all intents and purposes, look random. I’ll discuss this technique in a future post, but the basic idea is that there are certain formulae which, when used recursively, generate a sequence of numbers that are essentially indistinguishable from a series of random numbers.

But here’s the thing: the numbers are not really random at all. If you know the formula and the current value in the sequence, you can calculate exactly the next value in the sequence. And the next one. And so on.

Strictly, a sequence of numbers generated this way is called ‘pseudo-random’, which is a fancy way of saying ‘pretend-random’. They look random, but they’re not. For most statistical purposes, the difference between a sequence that looks random and is genuinely random is unimportant, so this method is used as the basis for simulation procedures. But for the random selection of Premium Bond winners, there are obvious logistic and moral problems in using a sequence of numbers that is actually predictable, even if it looks entirely random.

For this reason, Ernie was invented. Ernie is a random number generator. But to ensure the numbers are genuinely random, it incorporates a genuine physical process whose behaviour is entirely random. A mathematical representation of the state of this physical process then leads to the random numbers.

The very first Ernie is shown in the second picture above. It was first used in 1957, was the size of a van and used a gas neon diode to induce the randomness. Though effective, this version of Ernie was fairly slow, generating just 2000 numbers per hour. It was subsequently killed-off and replaced with ever-more efficient designs over the years.

The third picture above shows Ernie (mark 4), which has been in operation from 2004 up until this weekend. In place of gas diodes, it used thermal noise in transistors to generate the required randomness, which in turn generated the numbers. Clearly, in terms of size, this version was a big improvement on Ernie (mark 1), being about the size of a normal PC. It was also much more efficient, being able to generate one million numbers in an hour.

But Ernie (mark 4) is no more. The final picture above shows Ernie (mark 5), which came into operation this weekend, shown against the tip of a pencil. It’s essentially a microchip. And of course, the evolution of computing equipment the size of a van to the size of a pencil head over the last 60 years or so is a familiar story. Indeed Ernie (mark 5) is considerably faster – by a factor of 42.5 or so – even compared to Ernie (mark 4), despite the size reduction. But what really makes the new version of Ernie stand out is that the physical process that induces the randomness has fundamentally changed. One way or another, all the previous versions used thermal noise to generate the randomness; Ernie (mark 5) uses quantum random variation in light signals.

More information on the evolution of Ernie can be found here. A slightly more technical account of the way thermal noise was used to generate randomness in each of the Ernie’s up to mark 4 is given here. The basis of the quantum technology for Ernie mark 5 is that when a photon is emitted towards a semi-transparent surface, is either reflected or transmitted at random. Converting these outcomes into 0/1 bit values, forms the building block of random number generation.

Incidentally, although the randomness in the physical processes built into Ernie should guarantee that the numbers generated are random, checks on the output are carried out by the Government Actuary’s Department to ensure that the output can genuinely be regarded as random. In fact they apply four tests to the sequence:

1. Frequency: do all digits occur (approximately) equally often?
2. Serial: do all consecutive number pairs occur (approximately) equally often?
3. Poker: do poker combinations (4 identical digits; 3 identical digits; two pairs; one pair; all different) occur as often as they should in consecutive numbers?
4. Correlation: do pairs of digits at different spacings in bond numbers have approximately the correct correlation that would be expected under randomness?

In the 60 or so years that premium Bonds have been in circulation, the monthly numbers generated by each of the successive Ernie’s have never failed to pass these tests.

However:

Finally, in case you’re disappointed that I started this post with a gratuitous reference to Sesame Street which I didn’t follow-up on, here’s a link to 10 facts and statistics about Sesame Street.

It’s sometimes said that a little knowledge is a dangerous thing. Arguably, too much knowledge is equally bad. Indeed, Einstein is quoted as saying:

A little knowledge is a dangerous thing. So is a lot.

I don’t suppose Einstein had gambling in mind, but still…

March Madness pools are a popular form of betting in the United States. They are based on the playoff tournament for NCAA college basketball, held annually every March, and comprise a so-called bracket bet. Prior to the tournament start, a player predicts the winners of each game from the round-of-sixteen right through to the final. This is possible since teams are seeded, as in tennis, so match pairings for future rounds are determined automatically once the winners from previous rounds are known. In practice, it’s equivalent to picking winners from the round-of-sixteen onwards in the World Cup.

There are different scoring systems for judging success in bracket picks, often with more weight given to correct outcomes in the later rounds, but in essence the more correct outcomes a gambler predicts, the better their score. And the player with the best score within a pool of players wins the prize.

Naturally, you’d expect players with some knowledge of the differing strength of the teams involved in the March Madness playoffs to do better than those with no knowledge at all. But is it the case that the more knowledge a player has, the more successful they’re likely to be? In other words:

To what extent is success in the March Madness pools determined by a player’s basketball knowledge?

This question was explored in a recent academic study discussed here. In summary, participants were given a 25-question basketball quiz, the results of which were used to determine their level of basketball knowledge. Next, they were asked to make their bracket picks for the March Madness. A comparison was then made between accuracy of bracket picks and level of basketball knowledge.

The results are summarised in the following graph, which shows the average relationship between pick accuracy and basketball knowledge:

As you’d expect, the players with low knowledge do relatively badly.  Then, as a player’s basketball knowledge increases, so does their pick accuracy. But only up to a point. After a certain point, as a player’s knowledge increases, their pick accuracy was found to decrease. Indeed, the players with the most basketball knowledge were found to perform slightly worse than those with the least knowledge!

Why should this be?

The most likely explanation is as follows…

Consider an average team, who have recently had a few great results. It’s possible that these great results are due to skill, but it’s also plausible that the team has just been a bit lucky. The player with expert knowledge is likely to know about these recent results, and make their picks accordingly. The player with medium knowledge  will simply know that this is an average team, and also bet accordingly. While the player with very little knowledge is likely to treat the team randomly.

Random betting due to lack of knowledge is obviously not a great strategy. However, making picks that are driven primarily by recent results can be even worse, and the evidence suggests that’s exactly what most highly  knowledgable players do. And it turns out to be better to have just a medium knowledge of the game, so that you’d have a rough idea of the relative rankings of the different teams, without being overly influenced by recent results.

Now, obviously, someone with expert knowledge of the game, but who also knows how to exploit that knowledge for making predictions, is likely to do best of all. And that, of course, is the way sports betting companies operate, combining expert sports knowledge with statistical support to exploit and implement that knowledge. But the study here shows that, in the absence of that explicit statistical support, the player with a medium level of knowledge is likely to do better than players with too little or too much knowledge.

In some ways this post complements the earlier post ‘The benefit of foresight’. The theme of that post was that successful gambling cannot rely solely on Statistics, but also needs the input of expert sports knowledge. This one says that expert knowledge, in isolation, is also insufficient, and needs to be used in tandem with statistical expertise for a successful trading strategy.

In the specific context of betting on the NCAA March Madness bracket, the argument is developed fully in this book. The argument, though, is valid much more widely across all sports and betting regimes, and emphasises the importance to a sports betting company of both statistical and sport expertise.

Update (21/3): The NCAA tournament actually starts today. In case you’re interested, here’s Barack Obama’s bracket pick. Maybe see if you can do better than the ex-President of the United States…

# Who wants to win £194,375?

In an earlier post I included a link to Oscar predictions by film critic Mark Kermode over the years, which included 100% success rate across all of the main categories in a couple of years. I also recounted his story of how he failed to make a fortune in 1992 by not knowing about accumulator bets.

Well, it’s almost Oscar season, and fabien.mauroy@smartodds.co.uk pointed me to this article, which includes Mark’s personal shortlist for the coming awards. Now, these aren’t the same as predictions: in some year’s, Mark has listed his own personal favourites as well as what he believes to be the likely winners, and there’s often very little in common. On the other hand, these lists have been produced prior to the nominations, so you’re likely to get better prices on bets now, rather than later. You’ll have to be quick though, as the nominations are announced in a couple of hours.

Anyway, maybe you’d like to sift through Mark’s recommendations, look for hints as to who he thinks the winner is likely to be, and make a bet accordingly. But if you do make a bet based on these lists, here are a few things to take into account:

1. Please remember the difference between an accumulator bet and single bets;
3. Please don’t blame me if you lose.

If Mark subsequently publishes actual predictions for the Oscars, I’ll include a link to those as well.

Update: the nominations have now been announced and are listed here. Comparing the nominations with Mark Kermode’s own list, the number of nominations which appear in Mark’s personal list for each category are as follows:

Best Picture: 1

Best Director: 2

Best Actor: 1

Best Actress: 2

Best supporting Actor: 3

Best supporting Actress: 1

Best Score: 2

In each case except Best Picture, there are 5 nominations and Mark’s list also comprised 5 contenders. For Best Picture, there are 8 nominations, though Mark only provided 5 suggestions.

So, not much overlap. But again, these weren’t intended to be Mark’s predictions. They were his own choices. I’ll aim to update with Mark’s actual predictions if he publishes them.

# Pulp Fiction (Our Esteemed Leader’s cut)

The previous post had a cinematic theme. That got me remembering an offsite a while back where Matthew.Benham@smartbapps.co.uk gave a talk that I think he called ‘Do the Right Thing’, which is the title of a 1989 Spike Lee film. Midway through his talk Matthew gave a premiere screening of his own version of a scene from Pulp Fiction. Unfortunately, I’ve been unable to get hold of a copy of Matthew’s cut, so we’ll just have to make do with the inferior original….

The theme of Matthew’s talk was the importance of always acting in relation to best knowledge, even if it contradicts previous actions taken when different information was available. So, given the knowledge and information you had at the start of a game, you might have bet on team A. But if the game evolves in such a way that a bet on team B becomes positive value, you should do that. Always do the right thing. And the point of the scene from Pulp Fiction? Don’t let pride get in the way of that principle.

These issues will make a great topic for this blog sometime. But this post is about something else…

Dependence is a big issue in Statistics, and we’re likely to return to it in different ways in future posts. Loosely speaking, two events are said to be independent if knowing the outcome of one, doesn’t affect the probabilities of the outcomes of the other. For example, it’s usually reasonable to treat the outcomes of two different football matches taking place on the same day as independent. If we know one match finished 3-0, that information is unlikely to affect any judgements we might have about the possible outcomes of a later match. Events that are not independent are said to be dependent: in this case, knowing the outcome of one will affect the outcome of the other.  In tennis matches, for example, the outcome of one set tends to affect the chances of who will win a subsequent set, so set winners are dependent events.

With this in mind, let’s follow-up the discussion in the previous 2 posts (here and here) about accumulator bets. By multiplying prices from separate bets together, bookmakers are assuming that the events are independent. But if there were dependence between the events, it’s possible that an accumulator offers a value bet, even if the individual bets are of negative value. This might be part of the reason why Mark Kermode has been successful in several accumulator bets over the years (or would have been if he’d taken his predictions to the bookmaker and actually placed an accumulator bet).

Let me illustrate this with some entirely made-up numbers. Let’s suppose ‘Pulp Fiction (Our Esteemed Leader’s cut)’, is up for a best movie award, and its upstart director, Matthew Benham, has also been nominated for best director. The numbers for single bets on PF and MB are given in the following table. We’ll suppose the bookmakers are accurate in their evaluation of the probabilities, and that they guarantee themselves an expected profit by offering prices that are below the fair prices (see the earlier post).

True Probability Fair Price Bookmaker Price
Best Movie: PF 0.4 2.5 2
Best Director: MB 0.25 4 3.5

Because the available prices are lower than the fair prices and the probabilities are correct, both individual bets have negative value (-0.2 and -0.125 respectively for a unit stake). The overall price for a PF/MB accumulator bet is 7, which assuming independence is an even poorer value bet, since the expected winnings from a unit stake are

$0.4 \times 0.25 \times 7 -1 = -0.3$

However, suppose voters for the awards tend to have similar preferences across categories, so that if they like a particular movie, there’s an increased chance they’ll also like the director of that movie. In that case, although the table above might be correct, the probability of MB winning the director award if PF (MB cut) is the movie winner is likely to be greater than 0.25. For argument’s sake, let’s suppose it’s 0.5. Then, the expected winnings from a unit stake accumulator bet become

$0.4 \times 0.5 \times 7 -1 = 0.4$

That’s to say, although the individual bets are still both negative value, the accumulator bet is extremely good value. This situation arises because of the implicit assumption of independence in the calculation of accumulator prices. When the assumption is wrong, the true expected winnings will be different from those implied by the bookmaker prices, potentially generating a positive value bet.

Obviously with most accumulator bets – like multiple football results – independence is more realistic, and this discussion is unhelpful. But for speciality bets like the Oscars, or perhaps some political bets where late swings in votes are likely to affect more that one region, there may be considerable value in accumulator bets if available.

If anyone has a copy of Our Esteemed Leader’s cut of the Pulp Fiction scene on a pen-drive somewhere, and would kindly pass it to me, I will happily update this post to include it.

# How to not win ￡194,375

In the previous post we looked at why bookmakers like punters to make accumulator bets: so long as a gambler is not smart enough to be able to make positive value bets, the bookmaker will make bigger expected profits from accumulator bets than from single bets. Moreover, even for smart bettors, if any of their individual bets are not smart, accumulator bets may also favour the bookmaker.

With all this in mind, here’s a true story…

Mark Kermode is a well-known film critic, who often appears on BBC TV and radio. In the early 90’s he had a regular slot on Danny Baker’s Radio 5 show, discussing recent movie releases etc. On one particular show early in 1992, chatting to Danny, he said he had a pretty good idea of how most of the important Oscars would be awarded that year. This was actually before the nominations had been made, so bookmaker prices on award winners would have been pretty good and since Radio 5 was a predominantly sports radio station, Danny suggested Mark make a bet on the basis of his predictions.

Fast-forward a few months to the day after the Oscar awards and Danny asked Mark how his predictions had worked out. Mark explained that he’d bet on five of the major Oscar awards and they’d all won. Danny asked Mark how much he’d won and he replied that he’d won around ￡120 for a ￡25 stake.  Considering the difficulty in predicting five correct winners, especially before nominations had been made, this didn’t seem like much of a return, and Danny Baker was incredulous. He’d naturally assumed that Mark would have placed an accumulator bet with the total stake of ￡25, whereas what Mark had actually done was place individual bets of ￡5 on each of the awards.

Now, I’ve no idea what the actual prices were, but since the bets were placed before the nominations were announced, it’s reasonable to assume that the prices were quite generous. For argument’s sake, let’s suppose the bets on each of the individual awards  had a price of 6. Mark then placed a ￡5 bet on each, so he’d have made a profit of ￡25 per bet, for an overall profit of ￡125. Now suppose, instead, he’d made a single accumulator bet on all 5 awards. In this case he’d have made a profit of

$\pounds 25 \times 6 \times 6 \times 6 \times 6 \times 6 -\pounds 25 = \pounds 194,375$

Again, I’ve no idea if these numbers are accurate or not, but you get the picture. Had Mark made the accumulator bet that Danny intended, he’d have made a pretty big profit. As it was, he won enough for a night out with a couple of mates at the cinema, albeit with popcorn included.

Of course, the risk you take with an accumulator is that it just takes one bet to fail and you lose everything. By placing 5 single bets Mark would still have won ￡95 if one of his predictions had been wrong, and would even make a fiver if he got just one prediction correct. But by not accumulating his bets, he also avoided the possibility of winning ￡194,375 if all 5 bets came in. Which they did!

So, what’s the story here? Though an accumulator is a poor value bet for mug gamblers, it may be an extremely valuable bet for sharp gamblers, and the evidence suggests (see below) that Mark Kermode is sharper than the bookmakers for Oscar predictions.

Is Mark Kermode really sharper than the bookmakers for Oscar predictions? Well, here’s a list  of his predictions for the main 6 (not 5) categories for the years 2006-2017. Mark predicted all 6 categories with 100% accuracy twice in twelve years. I guess that these predictions weren’t always made before the nominations, so the prices are unlikely to be as good as in the example described above. But still, the price on a 6-fold accumulator will have been pretty good regardless. And he’d have won twice, in addition to the 1992 episode (and possibly more often in the intervening years for which I don’t have data). Remarkably, he would have won again in 2017 if the award for best movie had gone to La La Land, as was originally declared winner, rather than Moonlight, which was the eventual winner.

Moral: try to find out Mark’s predictions for the 2019 Oscars and don’t make the mistake of betting singles!

And finally, here’s Mark telling the story of not winning something like￡194,375 in his own words:

# Bookmakers love accumulators

You probably know about accumulator, or so-called ‘acca’, bets. Rather than betting individually on several different matches, in an accumulator any winnings from a first bet are used as the stake in a second bet.  If either bet loses, you lose, but if both bets win, there’s the potential to make more money than is available from single bets due to the accumulation of the prices. This process can be applied multiple times, with the winnings from several bets carried over as the stake to a subsequent bet, and the total winnings if all bets come in can be substantial. On the downside, it just takes one bet to lose and you win nothing.

Bookmakers love accumulators, and often apply special offers – as you can see in the profile picture above – to encourage gamblers to make such bets. Let’s see why that’s the case.

Consider a tennis match between two equally-matched players. Since the players are equally-matched, it’s reasonable to assume that each has a probability 0.5 of winning. So if a bookmaker was offering fair odds on the winner of this match, he should offer a price of 2 on either player, meaning that if I place a bet of 1 unit I will receive 2 units (including the return of my stake) if I win. This makes the bet fair, in the sense that my expected winnings – the amount I would win on average if the game were repeated  many times – is zero. This is because

$(1/2 \times 2) + (1/2 \times 0) -1 = 0$

That’s the sum of the probabilities multiplied by the prices, take away the stake.

The bet is fair in the sense that, if the match were repeated many times, both the gambler and the bookmaker would expect neither to win nor lose. But bookmakers aren’t in the business of being fair; they’re out to make money and will set lower prices to ensure that they have long-run winnings. So instead of offering a price of 2 on either player, they might offer a price of 1.9. In this case, assuming gamblers split their stakes evenly across two players, bookmakers will expect to win the following proportion of the total stake

$1-1/2\times(1/2 \times 1.9) - 1/2\times (1/2 \times 1.9)=0.05$

In other words, bookmakers have a locked-in 5% expected profit. Of course, they might not get 5%. Suppose most of the money is placed on player A, who happens to win. Then, the bookmaker is likely to lose money. But this is unlikely: if the players are evenly matched, the money placed by different gamblers will probably be evenly spread between the two players. And if it’s not, then the bookmakers can adjust their prices to try to encourage more bets on the less-favoured side.

Now, in an accumulator bet, the prices are multiplied. It’s equivalent to taking all of your winnings from a first bet and placing them on a second bet. Then those winnings are placed on the outcome of a third bet, and so on. So if there are two tennis matches, A versus B and C versus D, each of which is evenly-matched, the fair and actual prices on the accumulator outcomes are as follows:

Accumulator Bet A-C A-D B-C B-D
Fair Price 4 4 4 4
Actual Price 3.61 3.61  3.61 3.61

The value 3.61 comes from taking the prices of the individual bets, 1.9 in each case, and multiplying them together. It follows that the expected profit for the bookmaker is

$1-4\times 1/4\times(1/4 \times 3.61) = 0.0975$.

So, the bookmaker profit is now expected to be almost 10%. In other words, with a single accumulator, bookmakers almost double their expected profits. With further accumulators, the profits increase further and further. With 3 bets it’s over 14%; with 4 bets it’s around 18.5%. Because of this considerable increase in expected profits with accumulator bets, bookmakers can be ‘generous’ in their offers, as the headline graphic to this post suggests. In actual fact, the offers they are making are peanuts compared to the additional profits they make through gamblers making accumulator bets.

However… all of this assumes that the bookmaker sets prices accurately. What happens if the gambler is more accurate in identifying the fair price for a bet than the bookmaker? Suppose, for example, a gambler reckons correctly that the probabilities for players A and C to win are 0.55 rather than 0.5. A single stake bet spread across the 2 matches would then generate an expected profit of

$0.55\times(1/2 \times 1.9) + 0.55\times (1/2 \times 1.9) -1 = 0.045$

On the other hand, the expected profit from an accumulator bet on A-C is

$(0.55\times1.9) \times (0.55\times1.9) -1 = 0.092$

In other words, just as the bookmaker increases his expected profit through accumulator bets when he has an advantage per single bet, so does the gambler. So, bookmakers do indeed love accumulators, but not against smart gamblers.

In the next post we’ll find out how not knowing the difference between accumulator and standard bets cost one famous gambler a small fortune.

Actually, the situation is not quite as favourable for smart gamblers as the above calculation suggests. Suppose that the true probabilities for a win for A and C are 0.7 and 0.4, which still averages at 0.55. This situation would arise, for example, if the gambler was using a model which performed better than he realised for some matches, but worse than he realised for others.

The expected winnings from single bets remain at 0.045. But now, the expected winnings from an accumulator bet are just:

$(0.7\times1.9) \times (0.4\times1.9) -1 = 0.011,$

which is considerably lower. Moreover, with different numbers, the expected winnings from the accumulator bet could be negative, even though the expected winnings from separate bets is positive. (This would happen, for example, if the win probabilities for A and C were 0.8 and 0.3 respectively.)

So unless the smart gambler is genuinely smart on every bet, an accumulator bet may no longer be in his favour.

# Borel

Struggling for ideas for Christmas presents? Stuck with an Amazon voucher from your employer and don’t know what to do with it? No idea how you’re going to get through Christmas with the in-laws? Trying to ‘Gamble Responsibly‘ but can’t quite kick the habit?

You can thank me later, but I have the perfect solution for you:

Borel

This is a new Trivial-Pursuit-style board game, but with a twist. Players are given a question involving dice, coloured balls or some other experimental apparatus, and have to bet on the outcome. There’s not enough time to actually do the probability calculations, so you just have to go with intuition. You can make bets of different sizes and, just like in real life, should make bigger bets when you think the odds are more in your favour.

This is part of the description at Amazon:

The game combines the human mind’s difficulty to deal with probabilistic dilemmas with the strategic thinking of competitive gambling.

And:

It is designed to reward probabilistic reasoning, intuition, strategic thinking and risk-taking!

In other words, it’s just like Smartodds-world, but without models to help you.

Disclaimer: The description and reviews look great, and I’ve ordered a set for myself, but I haven’t played it yet. I’ll try it on my family over Christmas and let you know how we get on. If you want a set for yourself or your loved ones, it’s available on Amazon here.