Tag Archive for: Monte Carlo simulation

‘Money Without Work’ 6: Bond, Bloom, Benham and Buffett – Variance and EV+

In part 6 of his outstanding series, Russell Clarke looks at the bigger picture through the prism of short-term hiccups. Russell's previous articles can be found here.


Value betting as an accepted modus operandi is a relatively recent concept in the mainstream betting world. The Pricewise column in the Racing Post during the 1980’s was the flag bearer and the continued success of that column has seen the concept of value become uppermost in the minds of most intelligent backers. We accept that value is subjective and that the market is accurate but we believe we are shrewd enough to spot the anomalies, otherwise why bother betting at all?

What image comes into your mind when you hear ‘professional gambler’? A James Bond type, suave and handsome, standing at a roulette wheel, martini in hand and a gorgeous blonde draped over his shoulder? He pushes forward a huge pile of chips onto a number, and watches with smug certainty as the ball falls into the right slot. Yep, that’s me...

This is all absolute nonsense of course. For a start, James Bond’s favourite game was baccarat not roulette. Secondly, if you shake a martini you chip the ice and just get a watered down drink. Thirdly, those lazy gender stereotypes went out of fashion a very long time ago around these parts. And fourthly, neither James Bond nor anyone else in the history of human civilisation (fictional or real) has ever been able to accurately predict where the ball will finish on a roulette wheel. A roulette wheel is an efficient random number generator, and the only way to beat it is by having the odds on your side.

So how do successful gamblers win? How do some investors make loads of money, when most investors lose? Every successful professional gambler/investor in history has something in common: they bet with Positive Expected Value (EV+). An EV+ is having the odds in your favour. Over time, if the odds are in your favour then you will win. How to calculate EV was covered previously in the article on Price Boosts (Episode 4).

So the reality of professional gambling is somewhat less glamorous than the James Bond fantasy. A pro gambler is much more likely to be found reading a newspaper, perusing a website, or playing with numbers in a spreadsheet than standing in a casino, drinking and flirting with the opposite sex. The reality of professional gambling is mostly a little dull, and unfortunately there’s no way of explaining the basics that will be a roller-coaster ride of page-turning excitement. So, I won’t even pretend to try.

But if you master the basics it could just be possible for you to become filthy stinking rich through professional betting. Witness Messrs. Bloom and Benham, and just look at Warren Buffett. The latter leads a frugal, slightly eccentric life, with his head buried in a newspaper most of the time. But, depending on what the US stock market has done in the previous couple of days, he may very well be the richest man in the world as you read this (he isn’t, but it read quite well, so I have left it in!).

All Warren Buffett has done his whole life is practice ‘value investing’. He’s a professional gambler. He bets on the share prices of companies. He buys them for less than they’re worth and then sells them for more than they’re worth. That’s it. He understands randomness, he recognises value and he has developed investing methods to turn that value into EV+.


VARIANCE (or Randomness)

The first step to becoming a successful investor is to understand variance. Variance is a theoretical concept that you need to ‘get’ before you can move on. It is invisible, but you have to know that it’s there, and how it works. Like a physicist has to believe in, and understand, gravity even though he can’t actually see it.

Understanding variance/randomness is the opposite of believing in fate. Events are not preordained. Events are chaotic and random. Nothing happens ‘for a reason’. Things just happen because events that take place, no matter how small, have an effect on everything around them (sometimes known as 'the butterfly effect').

The influence of the laws of cause and effect are at play all around us, every second of every day, everywhere in the universe, from the moment of the big bang (if you believe in that). Anything that can happen, might happen. Indeed, it will happen if you wait long enough. Everything that happens in the universe does so within a framework, the ‘laws’ of how the universe works. These are the rules of the game. Our best way of describing these laws is with:

  1. The standard model of particle physics
  2. Einstein’s general law of relativity

Essentially, the force of gravity and the speed of light are fixed. Everything that happens in the universe conforms to these laws, but what actually happens within the framework that these create is random and chaotic. There is loads of stuff in the universe, moving around, so it is interacting all the time with lots of other stuff. Even the tiniest event, the briefest collision between the most tiny and insignificant of these can set off a chain reaction that leads to a radically different outcome than would be observed if the tiny event hadn’t taken place. By the way, if you are a physicist you will know that some of what I have just said is not strictly true. I know this because my youngest daughter is a physicist and she has pointed this out... daughters, eh?

OK, enough already with the physics. What on earth has all this got to do with gambling? Everything, is the answer because games of sport, hands of cards and the economies of the world all work in the same fundamental way as the universe: there are rules and there is randomness. That’s all.

Take a football match. The rules are fixed. There will be 22 players, a referee, a rectangular field and 2 sets of goals. The referee will blow his whistle and the players will start to play. What happens over the next 90+ minutes on that rectangle is random. There is a discernible and predictable pattern to the randomness for sure. We can know that it’s likely that the better players will play better. The team with more of the better players is more likely to win. The number of goals scored is most likely to be between 2 and 4. Et cetera.

We can know these things, these ‘likelihoods’, by observation and research, considering data on previous similar occurrences, i.e. other football matches, especially those involving these teams and these players. But what we cannot do is predict exactly what will happen. From the moment the referee blows his whistle to start the match there is a virtually infinite number of possibilities of how the game might play out. Every decision a player makes, every spin and deflection of the ball, every instruction given by the coach, each breath of wind, every noise from the crowd that the players hear, every decision by the officials; they all come together to create a narrative, a story on a timeline across the 90 minutes that describes exactly what happened. And if you played the match a trillion times, the story would never be exactly the same twice.

This is because every variable is multiplied by every other variable to come up with the total number of possible story lines. In the infinite number of story lines a percentage of them will result in the score ending nil-nil. A different percentage will lead to 1-0, 2-0, 3-1 etc. A much smaller percentage will result in the score ending 12-7. But if it is possible that it can happen, it will happen, eventually, even if it’s a tiny percentage of the time.

Every possible outcome will be included in the percentage distribution of different scorelines that result from our near infinite number of story lines. We can look to this distribution to observe the implications of the rules of the game, the framework within which it operates. None of the story lines will end up yielding a score of 5000-0. The rules of the game are that you play for 90 minutes (plus a bit more) and that after a goal the ball gets placed back on the centre spot. The clock continues to run while the ball is returned to the middle. So there isn’t enough time for a team to score 5000 goals in a football match. That possibility exceeds the framework of the game established by the rules, so it will never happen. Nothing will ever travel faster than the speed of light.

So what are the practical implications for understanding this randomness theory? First, you understand that, fundamentally, predictions are useless. It is impossible to predict exactly what will happen because the number of actual possible story lines is almost infinite. But it is possible to guess at the pattern of likelihoods in advance. That is the best we can do, and it is what we must do.

We know that, within the framework, all the things that are possible will occur a certain percentage of times. The job of the professional gambler is to discern the pattern in the randomness; to say ‘how likely’ something is to happen. Not to say what ‘will happen’. And then to compare those perceived likelihoods against market prices.

Where the subject involves animate objects, like players, officials, fans, the pitch and weather of a football match then the pattern in the randomness cannot be projected precisely. It involves an element of guesswork. Observation, such as watching previous matches involving the teams, or analysis by looking at a league table can make the guesswork more accurate than a guess plucked from thin air. Modelling the relative strengths of the teams and the players using sophisticated analysis, and then feeding that into an engine which works out a distribution of possible scorelines can get you pretty close to projecting the percentage distribution within the infinite story lines. But it is still guesswork, even when it is very informed guesswork using a computer model.

When two boxers get into a ring the better fighter will normally win. But the rules of the ring dictate that either fighter could win. So there doesn’t have to be a ‘reason’ why Buster Douglas knocked out Mike Tyson. Randomness means that it was inevitable that it would happen at some point, if you iterated that fight over enough times. It just happened to be that night.

But where the subject involves an inanimate object such as a roulette wheel or a drum of lottery balls then we can be absolutely precise in discerning the patterns in the randomness. So long as the roulette wheel (let’s use a European wheel here with a single 0) is well made and working properly then the distribution of the ball falling into each slot will be 2.703% over an infinite number of spins of the wheel.

When a roulette wheel spins it is randomness that governs which slot it falls into. There is no memory to the wheel, no number is ‘due’ to come up just because it hasn’t come out for ages. In 1913 in the Monte Carlo Casino, the ball in a roulette wheel landed in a black slot 26 times in a row. The odds of that happening were over 67 million to 1. So while it was surprising to the onlookers (and ruinous to the ‘red backers’) the sequence was actually no more surprising than any of the other 67 million possible story lines that the 26 spins could have produced.

So the point of learning the theory of randomness is to realise that predictions are useless to a professional gambler, because they are impossible. It is impossible to see into the future. It is one of the immutable laws of nature. It is part of the framework. We need to understand that our job is not to predict, but to discern patterns in the randomness; to express how likely something is to happen, not to say what we think will happen. Once we understand this principle we can move on to Expected Value.


Positive Expected Value (EV+)

Positive Expected Value means finding investment opportunities where the odds are in your favour. It is, for instance, backing something with a 50% chance of happening at odds of 11-10.

If anything can happen (and we cannot know what is going to happen), how can we profit from betting on something that is going to take place in the future? The answer is that all you need is to be armed with an idea of how likely something is to happen, and then to know that the chance of it happening is greater than the odds being offered when you make your investment.

It’s all about the odds.

An investment is risking something in the hope of a profitable return. The profit you make when you win, divided by the amount you risked are the odds. So if you bet £100 on a horse, and it wins, and you get £400 back then your profit was £300. 300 over 100 is 3 over 1. Your odds were 3/1.

On this occasion the horse won. But how likely was it to win? If we ran the race a million times, on how many occasions would our horse win? What is the pattern in the distribution of the randomness? Let’s say out of a million races our horse wins 200,000 times. The pattern in the randomness is that our horse’s true chance of winning the race is 800,000 over 200,000. Thus the horse's true odds were 4/1.

If we bet £1 a million times on our horse at 3/1 we would lose money. We would get back £800,000 having staked £1,000,000. Our loss would be £200,000. £200,000 is 20% of £1,000,000 (apologies if this is labouring the point). 3/1 is ‘bad value’ for that horse, to the tune of 20%: the EV was only 0.8.

But if we could get 5/1 about the horse the sums become £1,200,000 return on our £1,000,000 stake. The horse becomes value, at 20%. An EV of 1.2.

When I say the ‘horse’ becomes value, I don’t really mean the horse. I mean the odds of 5/1 are value. Odds of 3/1 are not. The horse is, effectively, irrelevant. What matters are the odds that you get, not the horse itself. Any horse, no matter how slow, has a chance of winning any race that it lines up for. Those are the rules. That is the framework within which we are operating. What happens in the race on any single occasion doesn’t make the bet a bad bet. Single results don’t prove whether something was value or not, whether it was a good bet to make or a bad bet.

The truth of value investing only reveals itself over time.

There’s a paradox that gamblers have to get their head around. The difference between short-term and long-term. The only thing that matters is winning overall, in the long term. But winning on any one single occasion barely matters at all. Value investing is a war waged though a series of many, many battles. Winning or losing any single battle does not really matter. Looking back on all the battles, from a position of triumph having prevailed in the war, the fuss that you made about the loss of any single battle will seem ridiculous. Value investing is nothing to do with trying to win every battle. The only thing that matters is having the odds on your side consistently as you fight the battles, so that as the results of a great number of battles become known your superiority becomes apparent.

Even great football teams lose games. The best poker players regularly lose loads of hands. The best investors buy shares in companies who go bust. The best golfers make bogeys. Champion jockeys lose far more races than they win. Short-term losses are ultimately irrelevant. All that matters is long term overall victory.

There is a neat, simple mantra for any professional investor to adhere to:

Decisions Not Results

If you keep making the right decisions, keep betting with the odds in your favour, keep finding Positive Expected Value (EV+) then, as long as you stay in the game for the long-term, you will end up a winner.

So how do we know if odds are value? When we’re dealing exclusively with an inanimate object like a roulette wheel then we can tell for certain. While randomness dictates that the ball could land in any of the 37 slots on any given spin, we know that pattern to the randomness will play out to reveal an even 2.703% distribution in each slot over a long period. There is exactly a 36/1 chance of each slot being the winner on each spin. So to get value in betting on a single number on a roulette wheel we would need to get odds of greater than 36/1. The casino actually offers 35/1. So we can say that roulette is bad value. If you play for long enough you will lose. It is inevitable. The only exception would be if you stumbled across the equivalent of the run of 26 blacks in a row, and kept betting black. That would be the equivalent of a lottery win. Don’t hold your breath.

Poker is different. Although the cards are inanimate the other players are human, meaning that betting on hands of poker is very much chaotic and random. For a top professional player like Phil Ivey his ability to win overall at poker comes from his ability to discern the patterns in the randomness of the betting on the hands. It has nothing to do with the hands he gets dealt. Over a long term the strength of the hands he has are exactly the same as they are for any other player. It’s what he does with the betting on the hands that makes him successful.

Part of the job of understanding the randomness of poker hands comes from an understanding of the likelihood of any particular card or type of card being turned over on the flop. But it also comes from understanding opponents. How likely they are to have certain hands. Phil Ivey doesn’t know which card is going to get turned over on the flop. Nor can he know for certain which cards the other players hold. But he is able to discern enough from the hard and soft evidence at his disposal a good estimate of how likely he is to win the pot. His decision to bet or not bet is then based entirely on value. If the odds of return (the amount of money in the pot) exceeds the chance that he will win it then he bets. The exact same principle as betting on the horses. He bets when the odds are in his favour. What happens on any single hand is irrelevant. The only thing that matters is winning in the long run, winning the war. To accurately compute odds, we must take a trip to the Cote d’Azur...


Monte Carlo or Bust

Monte Carlo simulations are “computational algorithms that rely on repeated random sampling to obtain numerical results”, according to wikipedia. The further definition is even more off-putting for non-mathematicians, so I will spare you that! Instead, I will attempt to explain, in layman’s terms, how a Monte Carlo simulation (MC Sim) is a useful tool in determining accurate percentages and thus, true odds.

Monte Carlo Simulation: THE 2010 WORLD CUP

A good example of how you can utilise an MC Sim was how myself and the betting syndicate I advise used it for the 2010 World Cup. First, we handicap the teams in terms of goal superiority. For example, if we top rate Spain, they were on 0.00. Germany may have been 0.2 (0.2 goals inferior to Spain), Brazil 0.3, England 0.6 and so on. We then used previous World Cup data to calculate the average goals per game in first group game, second group game, third group game, quarter-final, semi-final, and final. This gave us a prediction for every group game in terms of superiority and total goals. Finally, we programmed the World Cup draw into the MC sim.

The next stage is to simply decide how many iterations (the number of times the simulation plays the tournament) and hit the Start button.  Playing the World Cup 10,000 times according to the inputs gave us the percentage chance of any criteria we wished (winner, group winner, number of goals etc). As the tournament progressed, we had actual results to input as well as altered handicap marks. After every game I would generate another 10,000 iterations, get the updated percentage chances and bet accordingly. I must have played half a million World Cups!

Without an MC Sim to help you calculate EV, you can utilise the “sharp” bookmakers as the “true odds”. An MC Sim is undoubtedly ideal to predict variance, but without such an aid, your safest option is to trust your EV+ and ‘accept’ the inevitability of variance.

A real world example of this can be demonstrated by “The Best Racing System in the World” (well, possibly). This is an each-way system. The brief ‘rules’ are to back horses each-way, where the place element of the bet is EV+. These are highlighted by a simple piece of software that surveys bookmaker sites searching for odds and comparing them with the current offer with the exchanges. It is slightly more complex, but those are the bare bones. Lots of bets (sometimes hundreds) can be found in a day.

In a recent sample I saw 3,770 bets had been placed (1 pt e/w) for a profit of 1900. The system works because it calculates that there is EV+ in all of the bets at the time they are placed. In fact, the system can be improved further by laying back the win element of the bet on the exchanges (because it is the place part of the bet that holds the EV+). But, before you start clamouring for more precise details, there are notable drawbacks, primarily the logistical drawback of getting the bets on without restrictions and/or account closures.

But, more importantly, the variance with these bets is extreme and I have seen examples of 1,000+ bets making a loss despite the implicit EV+ of the proposition. That becomes psychologically demanding and leads me nicely (this hasn’t just been thrown together you know!) to the vital subject of psychology, which we will cover in detail in the next episode.

Until then...

- RC