The aim of any punter is to make money from betting. If he is serious about doing this methodically, he will start by setting aside a bankroll of money – a betting bank – which he knows he can afford to lose should everything go wrong. To decide what is reasonable to set aside for the purposes of betting, we should pose a simple question: If we lose all of it, will this detrimentally affect our way of living, and others around us? If the answer is yes, then we should reduce the size of the bankroll, or in the extreme, don’t bet at all.
The next step is to decide how much of our betting bank we are going to stake each time we place a bet. Naturally, this might vary from bet to bet according to our staking preferences and the types of odds we like to bet on. For example, it is quite reasonable to bet more on things that have a better chance of winning as defined by the odds. However, we’re thinking here in terms of averages. We could choose to stake our whole betting bank on one bet. Should it win, job done, and we can retire from gambling for a month, a year or longer. Should it lose, well, oh dear, that was pretty stupid wasn’t it, we’ll have to quit until we’ve had long enough to raise another bankroll. Of course a more sensible, less risky and probably more rewarding approach would be to place much smaller bets, with our average stake size perhaps 1%, 2% or at most 5% of our bankroll. In this way, we may not increase the size of our bankroll as fast when we win, but neither do we run the risk of losing so much of it in one go in such a short space of time. How big or small our stakes will be will entirely depend on our attitudes to risk. And how long we can look after our bankroll will depend on whether or not we can find an edge over the bookmaker. This is the business of money management.
The most common staking plan in use is known as the fixed or level staking plan. Here, every bet placed has a stake of the same size. For example, if the bankroll we start with is 100 units we could choose to stake 1 unit on everything we bet on, that is 1% of the starting bankroll. There are perhaps 3 advantages to this approach: 1) it’s simple; 2) the bookmakers may be less inclined to think you are a professional (and plenty of bookmakers are only too happy to take away your custom if they think that you are); and 3) it’s the best staking plan to find out what edge you have over the bookmaker.
Detractors of level staking point out, firstly, that it’s not such a sensible approach to stake the same on a 10/1 underdog as on a 1/10 hot favourite. In terms of risk management, there is some merit in that, and in fact staking to win similar profits no matter what the odds, rather than placing similar stake sizes is actually a more risk-efficient strategy. However, provided we bet on things where the odds don’t vary too much, betting level stakes is a perfectly reasonable money management strategy to adopt.
Secondly, it is argued that we should actually be changing the size of our stake as the size of our bankroll changes, that is to say betting stakes as a percentage of our current bankroll rather than the one we started out with. So if our first £1 bet at 2.00 is a winner, the next one will have a £1.01 stake, since our new bankroll is £101. Alternatively, if it lost, our second stake would be £0.99. Again it is true to say that such rolling percentage bank staking is more risk-efficient, since it has the compounded benefit of growing the bankroll faster when we are winning, but reducing the rate of losses when we are losing. Proponents of level staking, however, might reasonably argue that it will take longer betting rolling percentage stakes to recover from previous losses, and that is certainly true. Additionally, it’s not always possible to calculate precise stake sizes because usually we will be placing more than one bet at a time.
Some punters advocate a half way house: use a fixed stake size until your bankroll has changed by some pre-specified amount, typically 25%. When your bankroll has grown by 25%, increase your fixed stake size by 25%. On the other hand if you’ve suffered a 25% loss in your available betting funds, reduce stakes by 25%. Such a strategy is unsurprisingly called the Plateau system. Ultimately, it will come down to the risk preferences of the individual punter what staking plan they prefer to choose.
Perhaps the most theoretically risk-efficient money management strategy of all is what is known as the Kelly staking plan. Developed by John Kelly while working at AT&T’s Bell Labs in 1956, the mathematics of the Kelly Criterion are rather complex, involving what are known as utility functions. Utility is really just another word for benefit, and in the context of sports betting, the benefit to the punter of winning something relative to the risk he has to take to win it. By taking into account the expected rate of return and the risk, the Kelly utility function provides an economically justified and mathematically precise way to compute optimal bet sizes that maximise the overall growth of a bankroll, rather than profit over turnover. (We’ll learn more about profit over turnover, or yield, in the lesson on analysing a betting record.) In sports betting, the Kelly stake size is given by the following relationship:
Kelly stake percentage = (betting edge – 1) / (betting odds – 1),
All 3 variables are expressed as decimals. For example, if we are offered 2.50 for Liverpool to beat Manchester United, and our rating system has calculated that the fair odds should be 2.00, then our betting edge is 1.25 (that is 2.50 divided by 2.00) and our Kelly stake (decimal) percentage is 0.1667 (that is 0.25 divided by 1.50). If our current bankroll was 100, then the Kelly Criterion would suggest we stake 16.67 units on this bet, if 200 then 33.33 units. The stake percentage is dependent upon both the odds and crucially also the edge a punter believes he has found.
Although the Kelly money management strategy appears to offer a stronger rate of return than any other staking plan, some economists have argued against it, mainly because an individual’s specific wagering constraints and risk preferences may override the desire for optimal growth rate. In other words, the utility function prescribed by the Kelly Criterion (a logarithmic one) may not necessarily be anything like the punter’s preferred utility function. For example, imagine a bet with odds of 1.50 and with an edge of 1.20. The prescribed Kelly stake would be a whopping 40%. Would punters conceivably be happy to stake nearly half their bankroll just because theory suggests that this was the optimum stake size to maximise the rate which their bankroll could grow?
Even Kelly supporters recognise the limitations with this staking strategy, and argue for the use of fractional Kelly stakes to reduce volatility, risk and the influence of random errors in their edge calculations. The latter of these presents possibly the biggest headache for those using Kelly. If you are accurately assessing your edge over the bookmaker, Kelly offers a useful means of accelerating bankroll growth beyond what is possible with other money management strategies. If you are not, however, the consequences of betting too much on things that actually have little or no edge at all will be entirely predictable. As with any money management plan, if you don’t have what it takes to beat the bookmaker, bankruptcy will be inevitable in the long run.
In addition to fixed-size and percentage staking, there are a handful of gamblers who swear by a system know as progressive staking. The word “progressive” is really just a euphemism for loss-chasing. The idea behind loss chasing is to bet progressively bigger stakes after every loss until a winner is achieved, thereby recovering all previous losses and earning the profit targeted from the original bet. Advocates of this type of money management strategy will have you believe that it’s fail-safe, because you will always land that winner at some point that you need to make this system work. This thinking is completely flawed. If you know you are going to land a winner, there is hardly much point chasing losses. Just bet when you know you will win. Of course, in truth we can’t possibly know for sure when we will win, if we did, betting would be easy and the bookmakers would be out of business. The stark reality is you don’t know when or whether you will have that winner, and more specifically whether you will achieve it before you either run out of money or the bookmaker tells you that you have reached their staking limit. The problem with progressive staking is that stake sizes after losses increase at an exponential rate, meaning that even after a few losses they can be much, much larger than the initial first stake of the series. Of course, if we had infinite financial resources, an infinitely generous bookmaker and an infinite amount of time, progressive stake might have some merit. Naturally, infinity has nothing to do with the real world of sports betting.
It is also possible to illustrate the flaw in the logic of progressive staking (and any other staking plan for that matter where the punter doesn’t have an edge) through a very simple mathematical proof which analyses the theoretical profit expectancy. Here, we have done this for a well known progressive staking strategy, the Martingale. The Martingale staking plan comes from the world of casino gambling, and in particular the game of roulette. A popular game at the roulette wheel is red-black, where the gambler must decide whether the ball will land on either a red or a black number after each spin. Overlooking the influence of the house edge, the odds of either result are 2.00. The idea behind Martingale is to double the stake size after each losing wager, and return to the starting stake after every win. In this way, previous losses are recovered after each successful result plus the original expected profit.
The table below considers the 8 possible outcomes from 3 wheel spins where for each spin we have bet red (R) starting with a stake of 1 unit and doubled it after loss. For example, the first row shows 1 of the 8 possibilities, 3 blacks (B) with stake losses of 1, 2 and finally 4 for a total loss of 7, a probability of 12.5% (0.125) and therefore a profit expectancy of -0.875 (that is -7 x 0.125).
|1||R, R, R||B, B, B||1, 2, 4||-1, -2, -4||-7||0.125||-0.875|
|2||R, R, R||B, B, R||1, 2, 4||-1, -2, +4||+1||0.125||+0.125|
|3||R, R, R||B, R, B||1, 2, 1||-1, +2, -1||0||0.125||0|
|4||R, R, R||B, R, R||1, 2, 1||-1, +2, +1||+2||0.125||+0.25|
|5||R, R, R||R, B, B||1, 1, 2||+1, -1, -2||-2||0.125||-0.25|
|6||R, R, R||R, B, R||1, 1, 2||+1, -1, +2||+2||0.125||+0.25|
|7||R, R, R||R, R, B||1, 1, 1||+1, +1, -1||+1||0.125||+0.125|
|8||R, R, R||R, R, R||1, 1, 1||+1, +1, +1||+3||0.125||+0.375|
The total profit expectancy for all 8 possible outcomes is 0. In another words, with no edge over the roulette wheel, all we can hope to achieve over the long term is to break even. A similar analysis for level staking returns exactly the same result.
|Permutation||Bet||Outcome||Level stakes||Profit||Total||Chance||Profit expectancy|
|1||R, R, R||B, B, B||1, 1, 1||-1, -1, -1||-3||0.125||-0.375|
|2||R, R, R||B, B, R||1, 1, 1||-1, -1, +1||-1||0.125||-0.125|
|3||R, R, R||B, R, B||1, 1, 1||-1, +1, -1||-1||0.125||-0.125|
|4||R, R, R||B, R, R||1, 1, 1||-1, +1, +1||+1||0.125||+0.125|
|5||R, R, R||R, B, B||1, 1, 1||+1, -1, -1||-1||0.125||-0.125|
|6||R, R, R||R, B, R||1, 1, 1||+1, -1, +1||+1||0.125||+0.125|
|7||R, R, R||R, R, B||1, 1, 1||+1, +1, -1||+1||0.125||+0.125|
|8||R, R, R||R, R, R||1, 1, 1||+1, +1, +1||+3||0.125||+0.375|
All Martingale has achieved is an increase in the number of times we can expect to make a profit, in this example from 4 with level staking to 5. Unfortunately, this is at the expense of one large loss, which is essentially the source of the inherent risk associated with Martingale staking. The more permutations we include into the analysis the bigger the potential loss for the worst case scenario.
Despite the presentation above, there will still be those who insist that it’s just a matter of time before you bag a winner, either because they believe they have an edge over the bookmaker, or far worse they have committed the gambler’s fallacy of assuming that a winner becomes more probable after a series of losses according to the law of averages. If you understand that each bet is completely independent of the proceeding ones, you will understand why such thinking is fallacious. If you don’t then unfortunately it’s just a matter of time before you go bust. And if you genuinely do have an edge over the bookmaker, you don’t even need to go chasing losses in the first place. Progressive staking preys on greed and stupidity and has absolutely no place in the world of intelligent and responsible sports betting. Smart punters don’t chase losses; ever!
So chasing losses cannot turn a losing system into a winning one. Indeed, without a betting edge, no money management strategy can do that. We could repeat the profit expectancy calculation for any staking plan we want, fixed profits, rolling bank, plateau system, D’Alembert, Pyramid, Oscar’s Grind, Steady Drip, Kelly and so, the end result will always be the same. With zero edge your profit expectancy is zero. The only thing a different money management strategy will do is change the relationship between risk and reward. If you want to win bigger and faster you are going to have to take more risk to achieve it. But if all you have is luck, you will never win enough in the long run to make any level of risk worth it.