The Church of Betting

Staking and the St. Petersburg Paradox

Staking and the St. Petersburg Paradox

In the long run we’re all dead

John Maynard Keynes

I have previously written about the Closing Line Value, the benefits of taking positive expected value (EV+) bets and how to find out if your bets contain value. Indeed, if you ever want to make money in sports betting, placing EV+ bets is the only way to do it. Having established that EV+ bets are the first step in your quest for profit, what is the second one? The answer is: a staking plan.

A staking plan, money management or bankroll management are terms that refer to the same concept. They all try to answer the question, how much money should you bet on a selection with a certain positive expected return (a.k.a. an EV+ bet). 

Those are the terms that also make the majority of punters yawn just at the sound of them. Believe it or not, that is not why I haven’t covered them in my blog so far. The real reason is that talking about money management makes no sense if you don’t have EV+ bets.

Staking doesn’t matter without EV+

Finding value on the market is a serious challenge. Betting events represent markets and market forces wipe value out of those markets pretty quickly. Therefore, you will always struggle to consistently identify and play value bets.

Most punters indeed never manage to get there. For them each and every stake plan will lead to the same destination – bankruptcy. In case you are unsure if the above applies to you, measuring the Closing Line Value of your bets would be a good test. If your bets have no value, it’s back to the drawing board and no discussions about staking plans are necessary at that point.

But maybe you are one of the very few who have built a successful betting model. Or perhaps you chose to play the cat and mouse game with the soft bookies and purchased a classical tech value betting software such as RebelBetting, WinnerOdds, BetBurger or Trademate Sports. You have your EV+ bets. You know how to calculate, as precisely as possible, the expected value of your bet. Now comes the question: 

How much to stake

This is the question staking plans aim to answer. And this is a question you need to ask yourself but only after:

  1. You have an edge: You have secured a steady supply of EV+ bets
  2. You know your edge: You know can estimate with reasonable certainty the expected yield of each bet

Hopefully the first point has been made by now. Let’s clarify the second one.

Knowing your edge

Staking plans are all about risk management. You can make money in the long term with EV+ bets, but the enormous variance typical for a value betting strategy will always stand in your way. You need to place a stake big enough to be worth your time plus any costs of acquiring the bet, but small enough to protect yourself from the volatility and remain solvent. 

To make this trade-off, you need to know what downside risk you can expect. The downside risk is directly correlated with two factors. First, the size of your edge. Second, how long are the odds you are betting on. While you can clearly see what odds you are betting on, the size of your edge might be hard to find out.

Edge size

All other things equal, the bigger your edge, the lower your downside risk. Also, the bigger your edge, the higher your expected profit. So the more edge you have, the more money you stake. If you know you have an edge, but are unsure about its size, you would probably just do best with level stakes. However, if you can measure it reasonably well, this will be a main factor in determining your stake.

Staking with unknown edge size

Two of pyckio’s shareholders, Barge-Gil and García-Hiernaux, have published a work on the optimal staking if you edge size is unknown. If you know you have an edge but are unsure about its size and you don’t assume your edge is correlated with the odds length, your theoretical optimal strategy would be to use a unit win strategy.

However, at least for the pyckio tipsters, the records of whom the study is looking at, the edge seemed to indeed be slightly positively correlated with the odds length. In this case, a unit impact staking seems to be the optimal staking plan.

The following article explains the findings above in more detail.

How long are the odds?

This second point is a bit less intuitive. But it basically goes like this – the longer odds you take, the bigger volatility you should expect in the long term, regardless of your edge. This automatically means you should wager less on bets with longer odds. This second factor is sadly often ignored, which may have unfortunate effects on your bank, especially on multiplayer sports where long odds are typical, such as horse racing, golf or Olympic events.

To get the idea of how odds size reflects on return volatility, I have made 30 simulations of 2000 1-unit bets with 0 vig (10 simulations for odds 3.00, 10 for odds 2.00 and 10 more for odds 1.50). No vig means all those series must trend towards 0. You can see the results below:

Betting series with different odds

Kelly criterion

The two factors above (edge & odds) are the basis for most staking plans, including the famous Kelly criterion. Here is the Kelly criterion formula for a simple two-outcome bet:

… where

  • f* is the proportion of your bank you should wager
  • b is the amount to win divided by the wagered amount (for example that would be 2 for European odds of 3)
  • p is the probability of winning
  • q is the probability of losing (or 1-p)

As you see edge size and odds length are well accounted for by the Kelly formula.

Why not just use Kelly staking?

The Kelly criterion is highly regarded by sharp bettors as it has been mathematically proven to maximize bank size in the long run. It finds a good balance between profit maximizing and risk management. If you are serious about your betting, just use Kelly and you can’t go wrong. But before that, two important things to keep in mind.

Kelly tends to be too risky for most people

Running a few simulations, you will see that full-Kelly stakes would still be too risky for most punters. I will give a short example to illustrate.

Let’s assume you are doing a Technical Value Betting, using the arbing software of RebelBetting or BetBurger. You see a bet at a soft book with odds 3.0, which is forming an arb with an opposing bet at Pinnacle with odds 1.546. Your ROI for this arb would be 1-(1/3.00 + 1/1.546) ~ 2% (as shown by the software as well). However, you decide to value bet it. 

To calculate your stake you need to know your edge. More specifically, you need to know the real probability of winning your bet. Your arb is 2% but your edge is higher. To make the calculation you first need to determine the fair odds for your bet. When value betting, the usual assumption is that the sharp book has accurate odds, so you just need to remove the margin to calculate the fair odds.

Let’s assume the sharp book has your bet at 2.48. The overround at the sharp book will then be 1/1.546 + 1/2.48 – 1 ~ 5%. Using the formula for calculating overround accounting for the favourite-longshot bias, we arrive at fair odds of 2.643, which corresponds to a 37.83% chance of striking your bet. Now you have all the factors you need to input into the Kelly formula.  

f* = (0.3783*3-1) / 2 = 0.06745

So, the Kelly formula advises you to stake 6.745% of your bank on that bet. 

Monte Carlo staking simulation

Let us make a few simulations of making that bet repeatedly to see how it would affect your bank after 100 bets. I have done 10 simulations of placing the above bet 100 times (always adjusting for the current bank level). The starting bank in all 10 cases is 1000. How do the results look like?

Kelly Staking with long odds

All or nothing

As you see with a full Kelly it is a bit of an all-or-nothing scenario. You can make great returns if things go your way, but you also have a good chance to lose almost everything (almost, since with full Kelly, as risky as it is, you never risk complete ruin as you are betting a portion of your current bank).

Due to the above most punters prefer to use some kind of fractional Kelly to smooth the returns a bit. If you do Technical Value Betting, another option is to combine full-Kelly value betting with arbing, as I have written already. 

In any case, Kelly remains the most mathematically sound staking plan, but in case the volatility of the original Kelly formula would be too high for you (and for most punters it is), betting according to a fractional Kelly (half- or quarter-Kelly) is probably the better choice.

Kelly staking for simultaneous bets is complicated

Keep in mind that using Kelly for many bets simultaneously is a bit tricky. The above formula won’t serve you well in that case. It relies on the assumption that your bets are placed one at a time. If you play many bets simultaneously, the formula for the simple case will encourage you to bet more than you should. 

PlusEVAnalytics, a regular writer for Pinnacle’s blog has issued a great article on that topic a few years ago, which I highly recommend. If you like to dive deeper into quantitative problems it is recommendable to try and understand the correct way of applying Kelly if you are placing many bets simultaneously. Otherwise, you might just remember that you need to reduce your stake in these cases and use some simple procedure to do so. Which brings me to the last point.

Practical limitations

The Kelly criterion might be the optimal staking method. But in the real world optimal methods are not always applicable. For example, if I am doing value betting I might have to place more than 50 bets in a day and inputting the data every time to calculate the Kelly stake would perhaps cost me more time than the added ROI would justify. So to save time I would often try to quickly calculate an appropriate stake in my head (along the lines of the Kelly formula) and just work with that.

Furthermore, if you are value betting at soft books, you will mostly be unable to bet what Kelly advises you to due to bet limits on the given event. In that case there are some other factors to consider – like how much should you bet on an event of this type in order to stay off the bookies’ radar.

Plugging in all those factors into a formula probably won’t work. So if you are doing Technical Value Betting in soft books it is probably best to develop a habit to calculate stake in your head based on your edge, your odds and the event you are betting on (and the bookmaker’s limits). You can master this only through practice, but it is useful to try to start thinking in “Kelly terms”. Once you have seen the calculations and have gotten used to them, you can run them in your head much more quickly and precisely.

Alternative staking methods

There are so many alternative staking methods out there that there are books delving into just the most famous ones, their benefits and drawbacks. “Fixed Odds Sports Betting” by Joseph Buchdal comes to mind, with a big section dedicated to staking plans principles and simulations.

The better known staking methods range from level staking, through level profit staking, to esoteric bankroll management strategies like Martingale or Fibonacci. The later are often being ridiculed in the betting community, for a good reason. In an EV- setting, those staking plans are characterised by long series of artificially sustained constant low profits, followed by a single huge drawdown, which more often than not wipes the whole bank of the poor punter following such staking scheme. As such, they are usually used in loss-chasing strategies by EV- punters, which gave them a terrible reputation. 

However, funnily enough, even with those staking plans you still have a good chance of making a healthy profit at the end of the day if all your bets are EV+. Your profit size would perhaps be suboptimal, but you would have a relatively steady and predictable profit line.

EV+ at any price

In general, in the long run and with an infinite bank, EV+ bets will always return you a profit. However, in the real world your bank is never infinite. In other words, even with EV+, your greed can lead you to financial ruin, from which you might not manage to re-bounce. 

According to the efficient market theory, even after you have lost everything, you should still be able to find someone to finance you, provided that you have a positive expectation. Sadly, such optimistic presumption does not account for less quantifiable factors like your reputation ruin in such a scenario, or the psychological damage that a huge loss may inflict on you. 

All in all, it could turn out extremely dangerous to apply model theories to the real world, which is why EV+ should be enjoyed responsibly. Here is another illustration of the same point:

The St. Petersburg Paradox

St. Petersburg: Beautiful architecture and mathematical paradoxes

The St. Petersburg Paradox describes a game of chance thought out by the Swiss mathematician Daniel Bernoulli while he was residing in the Russian city.

Imagine you are invited to play the following game: A fair coin is being flipped until tails appear. You get a minimum of 2 units back, which doubles for each time heads has appeared before you saw tails. At the end you get 2 to the power of the number of times heads appear before the coin landed tails.

The question is, how much would you pay to play this game? To answer this, let’s calculate the EV of this game.

You have a 50% chance of flipping tails on the first throw, which would return you 2 units. For turning two heads in the row the chance reduces to (½)^2 = 25%, however your return in this case increases to 2^2 = 4. For three heads in a row your probability is (½)^3 = 12.5% with a return of 2^3 = 8 and so on.

If you calculate the expected value of this game you get the following equation:

You easily see that the expected value of this game is infinite. However, experiments have shown that most people would only play something in the region of 10-15 units for the chance to play. At least according to EV theory this seems strange. But is it really?

Thou shall not blow thy bank

The purely theoretical version of the game is blind to some very valid considerations, which come to mind when this game is played in real life. These are all being accounted for in the various proposed solutions to the paradox – here is just one example.

A lottery ticket

The components that largely contribute to the infinite expected value of the game are basically very unlikely events that promise to deliver a huge profit. This essentially means you play a gamble at incredibly long odds. As a consequence of that fact, the volatility of the possible outcomes increases so much, that it would take a huge amount of iterations before you can expect, with reasonable certainty, to “hit it big” with this game. 

In summary, if you buy yourself into this game too expensively:

  1. You must employ a huge amount of capital just to survive the inevitable downswing before going into positive return territory. 
  2. The number of times you need to play this game to have a reasonable chance for profit might require longer than an average lifetime.
  3. Finally, by the time you hit your huge profit, which will finally justify paying a high entry stake for playing the game, you must be sure the casino will be able to honor its promise of paying out your winnings. Considering the fact that the winnings are theoretically unlimited, this won’t always be the case. 

As it turns out, the infinite EV of the game seems to be just a secondary factor here. The people taking part in experiments about the St. Petersburg paradox make quite a reasonable valuation of the game ticket, considering the “real world” factors listed above, regardless of the game’s infinite EV. Heuristics 1 : 0 Statistics.

Objective 1: Remain solvent. Objective 2: Grow your bank

I believe no further evidence is needed to conclude that the first objective you should always have when betting is to remain solvent. Only after you are confident about that, should you consider ways of growing your bank. Therefore it is advisable, before betting your whole savings on a single good bet, to consider using the (fractional-) Kelly formula instead. In the short term betting success is pure luck, regardless of whether you make good or bad bets:

This is a lesson worth remembering. 

Conclusion

Only with a decent money management system can you profit from betting in the long term. Your first objective should always be not to go bankrupt, the second one to grow your bank. To achieve those, getting used to applying Kelly stakes is certainly helpful. After sufficient practice you might even manage to make pretty accurate stake estimations by gut feeling alone. 

You will hardly ever need anything else, as far as bankroll management is concerned. However, it is worth remembering that full-Kelly is way too risky by most measures. So it should be enjoyed with caution and adjusted accordingly or combined with arbing

This was all I wanted to share on the topic of staking plans and I believe it covers the most important points quite well. Therefore I promise not to torture you with that topic in the foreseeable future.

What comes next?

Since the League of Legends Worlds Championship 2019 is (sadly) over, I will turn again to reviewing some tools that might help your betting action. A review of RebelBetting’s Value Betting tool, bet management software as well as an extensive article about Live Betting are all on the list. I have also planned articles on props, betting angles and tipsters for when the time is right. Stay tuned by following my Twitter page and/or subscribing to my newsletter on the upper-right corner of this page to never miss out on new material. Thanks for reading and see you around!

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