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The Stupidity of Martingale Progressive Staking in Sports Betting

Posted 21st May 2013

A week or so ago I received an e-mail asking for my opinion about a sports betting advisory service, Bet Profit System, claiming to have an almost never ending sequence of winning months and large double digit yields for all of its advertised services. Needless to say, I was sceptical - I always am when things look too good to be true - took a closer look and then made a response. It did not make for pretty reading. I also posted it on the Verified Tipsters forum. The theme of the argument against the tipster was the stupidity of loss chasing as a money management tool in sports betting.

Bet Profit System was making its profits through progressive staking. This is really a euphemism for loss chasing or loss recovery staking. The word "progressive" might make such a money management strategy sound clever to the unsophisticated punter, but all it involves is trying to win back what one has previously lost before (plus a little extra). In the world of casino gambling, such money management, particularly when applied to even money games like red/black or odd/even in roulette is commonly known as the Martingale. Why is the Martingale, and other forms of progressive money management, so stupid? In this article we'll take a look.

In my first book Fixed Odd Sports Betting I reviewed and analysed at some length a number of staking systems, including the Martingale. In fact, it proved to be without a shadow of doubt the very worst possible money management strategy one could possibly devise. I won't replicate the findings here. Instead, I come at this question from the point of view of the expectancy of losing sequences. Typically, sports betting tipsters who advocate the use of Martingale-type staking plans will argue that because of the nature and odds of the things they are betting odd, long losing sequences, which cause stakes to spiral out of control, are rare. Such tipsters are either lying, or in denial, or just plain stupid.

In Fixed Odd Sports Betting I presented a concise argument for why classical Martingale is mathematically flawed. The key points are reproduced again below. Overlooking the influence of the house edge in a game of red/black roulette, the odds of either result are 1/1 or evens. 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, as the following sequence of wheel spins reveals.

Wheel spinBetStakeOutcomeProfitRunning total
1Red1Black-1-1
2Red2Black-2-3
3Red4Black-4-7
4Red8Red+8+1
5Red1Black-10
6Red2Red+2+2
7Red1Red+1+3
8Red1Black-1+2
9Red2Black-20
10Red4Red+4+4

Martingale might seem to offer the punter a chance of profiting even where he is unable to beat the bookmaker fairly. Although he may lose a bet more frequently than a more skilful bettor, each win will recover his preceding losses and add a little extra each time. It is obvious, however, that the Martingale progression is inherently a very dangerous strategy to follow, since any extended run of consecutive losses will soon increase the stake size to frighteningly high levels. 10 blacks in succession, for example, would require the 11th stake to be 1024 units, just to win 1. Quite possibly, this stake size might be beyond the accepted limits of the casino. Perhaps more importantly, the apparent ability of Martingale to turn losses into profits is quite simply an illusion.

Consider, for example, the first 3 wheel spins in the sequence above, again assuming no house advantage, i.e. fair odds. The 3 consecutive losing blacks represent just 1 of 8 possible outcomes, each of which is as likely as any other. The table below shows the profit expectancy for each of these 8 permutations, where R=Red and B=Black.

Perm.BetOutcomeStakesProfitTotalChanceProf. exp.
1R, R, RB, B, B1, 2, 4-1, -2, -4-70.125-0.875
2R, R, RB, B, R1, 2, 4-1, -2, +4+10.125+0.125
3R, R, RB, R, B1, 2, 1-1, +2, -100.1250
4R, R, RB, R, R1, 2, 1-1, +2, +1+20.125+0.25
5R, R, RR, B, B1, 1, 2+1, -1, -2-20.125-0.25
6R, R, RR, B, R1, 1, 2+1, -1, +2+20.125+0.25
7R, R, RR, R, B1, 1, 1+1, +1, -1+10.125+0.125
8R, R, RR, R, R1, 1, 1+1, +1, +1+30.125+0.375
Total  36 010

Overall, the profit expectancy is 0. In another words, with no edge over the roulette wheel, all we can hope for over the long term is to break even. A similar analysis for level staking returns exactly the same result – no overall expected profit expectancy.

Perm.BetOutcomeStakesProfitTotalChanceProf. exp.
1R, R, RB, B, B1, 1, 1-1, -1, -1-30.125-0.375
2R, R, RB, B, R1, 1, 1-1, -1, +1-10.125-0.125
3R, R, RB, R, B1, 1, 1-1, +1, -1-10.125-0.125
4R, R, RB, R, R1, 1, 1-1, +1, +1+10.125+0.125
5R, R, RR, B, B1, 1, 1+1, -1, -1-10.125-0.125
6R, R, RR, B, R1, 1, 1+1, -1, +1+10.125+0.125
7R, R, RR, R, B1, 1, 1+1, +1, -1+10.125+0.125
8R, R, RR, R, R1, 1, 1+1, +1, +1+30.125+0.375
Total  24 010

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 longer the losing sequence the greater the risk to one's bankroll. With a doubling of stakes after each loss, that risk rises at an exponential rate, or in lay speak, very, very quickly.

In the real world of sports betting there are essentially three things to consider with losing sequences:

  1. How many sequential losses can you suffer before you run out of money?
  2. Will you reach the bookmaker's maximum stake limit before you have a winner?
  3. Do you have the psychology to cope with long losing runs?

The answers to these questions will depend in no small manner on the betting odds of the bets you are placing. If odds are 1/1 or 2.00 then a classical Martingale staking doubling sequence will apply. If odds are longer, the exponential rate of stake size growth after losses will not need to be as great, since a win will deliver a greater profit. If odds are shorter, however, the progression in stake size will need to be larger. In fact, the rate of progression will be given by the following expression:

Martingale rate of progression = odds / (odds - 1)

where "odds" is expressed in European decimal format.

For odds of 2.00 the rate of progression is 2. For odds of 3.00, the rate is 1.5. For odds of 1.5, the rate is 3.

By betting at shorter odds we might hope to have more winners, and consequently shorter losing sequences, but the rate of progression will have to be bigger to compensate for the smaller profit with a win. Conversely, betting at longer odds will ensure that the rate of progression towards the really big stakes is slowed, but we can expect to see longer losing sequences. What we really want to know, then, is what sort of losing sequences we can expect to have over a specified period of betting, and whether these can be accommodated by a specified size of bankroll.

Suppose we decide, like a black/red roulette gambler, that we're just going to bet even-money sporting propositions. What is the chance of, say, having 3, 5, 8 or 10 losing bets in succession? Individually, this is straight forward enough to calculate using the expression:

{(odds - 1) / odds}n

where n is the number of consecutive losses.

So if our betting odds are 2.00, the probability of individually having 5 consecutive losses is 0.55 = 0.03125 or 3.125%.

But suppose we have 100 bets, what are the chances of seeing any sequences of 5 consecutive losses? Or to put it another way, how often might we see 5 consecutive losses during the course of 100 bets? Well, there are 96 possible ways we can have such a sequence, starting from bets 1, 2, 3, 4 and 5, right the way through to bets 96, 97, 98, 99 and 100. Consequently the number we would reasonably expect to see on average would be 0.03125 x 96 = 3. So despite an individual series of 5 bets having roughly a 3% chance of returning losses only, over the course of 100 plays we could usually expect it to happen 3 times. If we started our progression with a stake of £10, this would mean that on average there would be 3 occasions during 100 plays where we would be faced with having to bet a stake of £32 (for the 6th bet). Alternatively, if our starting stake was £100, out 6th bet would be £3,200. If our bankroll was £10,000 then such a sequence could reasonably be accommodated, although whether we might feel comfortable betting at such large stake after having started comparatively so small is quite another matter.

This analysis can be repeated for any betting odds and any specified bankroll, and I have created a simple Excel file available for download to do just that.

In Sheet 1, we can vary the betting odds, number of bets and bankroll to see what is the maximum losing sequence that such a bankroll could accommodate and how often we might expect that to occur during our specified number of bets. For this exercise it is assumed a) all bets are the same odds, b) if a bankroll is lost, it can be replaced, c) we don't care how much we lose, d) there is no limit to the stake size our bookmaker will accept, and e) any accumulated profits do not count towards the bankroll. [Given the findings of this exercise, including any accumulated profits as part of the bankroll would have only a limited influence at diminishing the risks.]

Suppose we decide that our bankroll should be 1000 times our starting stake size. If we then decide to bet even money propositions (odds = 2.00) then the most number of consecutive losses we can accommodate before losing it all would be 8. If our starting stake is £1, then our 9th stake, should 8 consecutive bets lose, would be £256. Should the 9th bet lose, accumulated losses plus the 10th stake (£512) would total £1,023, more than our bankroll of £1,000. The spreadsheet also shows that in a sequence of just 365 bets (that might be 1 bet per day), we could expect such a losing sequence at least once on average. Thus, in just the course of a year betting once per day, we could reasonably expect to reach our limits betting classical Martingale.

Suppose instead we decided to bet at odds of 10.00. This time, the maximum losing sequence we could accommodate would be 43, much larger because of the slower rate of progression in the stake size growth. However, over the course of 365 bets, we might see such a sequence 3 times. Clearly such a betting strategy would be far too risky with a bankroll of 1,000 times our initial stake. To reduce the sequence expectation to 1 or lower, we would need to more than double our bankroll at these odds.

Sheet 2 of the spreadsheet performs much the same calculations, but this time fixes the expected number of occurrences of the maximum permissible losing sequence at 1. Since anything longer than the maximum acceptable losing sequence will result in bankruptcy, any integer greater than zero for the value of the expected number of occurrences will effectively result in an end to our betting, assuming this time that we can't, or simply choose not to replace our lost bankroll. This worksheet allows us to quickly see how many bets, on average, we can have for a designated bankroll and betting odds, before everything is likely to go wrong. For betting odds of 2 and a bankroll of 1,000 for example, it might take just 263 bets. If we decided instead to use a bankroll of 10,000, then we should last a lot longer, with 4,107 bets and a maximum losing sequence of 12. But risking £10,000 to win just £2,053, it would appear hardly worth the bother. And unless we were betting many times per day, it would take forever just to accumulate such a profit, assuming of course that the worst case scenario of 13 or more consecutive losses does not occur. OK, so let's multiply up our stakes by 10. Would you really want to take a 50-50 risk of losing £100,000 to win just £20,530, over more than 4,000 bets? I certainly wouldn't.

Finally Sheet 3 in the Excel file charts the relationship between betting odds, or in this case the betting expectancy (= 1/odds) and the number of bets we might reasonably expect to be able to have before a terminal bankruptcy occurs, for a specified bankroll size. The chart for a bankroll of 1,000 is shown below.

The saw tooth nature of the relationship is simply a consequence of the step changes in the maximum losing sequence that is permitted as we vary the betting odds. For example, at odds of 1.25 (win expectancy = 0.8 or 80%), we might be able to have 628 bets during which a maximum losing sequence of 4 should occur once on average. At odds of 1.24, this figure rises to 715 bets. But at odds of 1.23, the maximum losing sequence drops to 3 on account of the faster progression rate in the stake size growth, with a consequent dramatic fall in the number of bets before such an event will likely take place.

Of course, fixing the odds for every bet is highly theoretical, as are the findings from this exercise. Nevertheless, at the very least they should adequately illustrate just how common long losing sequences really are (despite what some might tell you), how quickly they can come along and how dangerous these can be on a punter's bankroll. For bankrolls of anything up to 1,000, it's simply the case that disaster will not be that long in coming, whatever odds you are betting at. For much larger bankrolls, risk to the overall bankroll might be lower but it will take forever and a day to accumulate any meaningful profit at all. And that's without worrying about what betting such massive stakes might do to our mind set or wondering whether the next bigger stake will be refused by the bookmaker. There is really no getting away from it. Progressive money management is stupid, and anyone who promotes it is either deceitful or stupid.

All of the above assumes that the punter is betting to fair odds. For most punters, and particularly those who decide in desperation to turn to Martingale strategies, that will not be the case. Where the bookmaker has an edge over the punter, the risks from progressive staking will increase even further. Bet Profit System, incidentally, was not able to gain an edge over the bookmaker. Perhaps that's why it turned to this type of stupidity in the first place. Incidentally, sometime during the last week their website has been emptied of content. Perhaps it ran into that bankrupting sequence of losses that is inevitable in the end. Or perhaps with sufficient subscription money taken from equally unsophisticated and greedy punters, its owner just decided to do a runner.

Progressive staking: you have been warned!!!