ABSTRACT

Through empirical analysis and Monte Carlo simulation, a proposed method for finding an optimal balance between risk and reward in sports betting is presented. The model assumes that the implied odds, as represented by the betting line, and the actual outcome odds of any given proposition are known. Risk, in this case, is not the risk associated with improperly assessing these odds. Rather, risk here is the risk of experiencing "gambler's ruin", a mathematical concept which states that, given a finite bankroll, a gambler playing against an opponent with an infinite bankroll, i.e. the house, will eventually lose his entire stake. The balance is finding a small enough betting size to minimize the risk of gambler's ruin without making the bet size so small that the money that is won becomes insignificant. 

In making any investment, there are two choices that have to be made. The first is what the investment should be (pick a side). The second is how much to invest. The same is true in sports gambling.

Selecting a bet size is all about managing your bankroll, which is another way of saying that its all about managing risk. Gambling is a constant struggle against a pervasive (but awesomely named) mathematical enemy: gambler's ruin.

The casino has two tools working against you. The first is one that everybody understands: the odds are tilted in their favor. The second is gambler's ruin. It's a very simple concept to understand. Let's play a coin flip game. Heads you win $1, tails I win $1. The only difference is I have infinite money, and you only have $10 bucks. Even though the coin flip is a 50/50 chance, this game will end with me having all of your $10. The reason is, once you lose your last $1 on the mathematically inevitable streak of bad luck, you don't have any more dollars to bet against me. That, in a nutshell, is gambler's ruin*.

(*This is why the idea of playing the roulette wheel and doubling your bet each time you lose doesn't work. Eventually you will go on a losing streak long enough to squash you - and because your bet doubles after each loss, the losing streak doesn't even need to be that long. Gambler's ruin!!)

Our example might seem trivial, but the implications are pretty staggering. To illustrate why, I'm going to use a technique called Monte-Carlo simulation. I'll select a starting bankroll, a standard betting unit, the odds of winning, and the payoff for that bet. Then I'll make that bet 6000 times in a computer (roughly equivalent to one season of baseball bets.) And I'll do THAT 1000 times, also in the computer (i.e. 1000 different baseball seasons). Then, out of those 1000 seasons, we'll see how many times I go broke before the season is over.

In the initial example, I started with betting $1 with a $10 bankroll, which is a 10% betting unit. But what the chart above shows that, even if I dropped down to a 1% bet, ten cents, I'd still have a 20% chance of being flat broke before I got to the end of a season.

So, how do we bet on sports while avoiding gambler's ruin? Well, first of all, we don't make bets that don't have any value. We only want to make bets with positive expected outcomes. So let's change the scenario. I'm not going to change it much. We'll keep the payout at even money, but change the odds of winning from 50% to 51.5%. This has what I call a betting advantage (calculated or known odds of winning / implied odds* - 1) of 0.03.

*implied odds is what the odds of winning would have to be for the betting advantage to be zero. They are calculated as: 1 - (money won) / (money risked + money won). If the odds are +100, the implied odds are 1 - ($100) / ($100 + $100) = 50%. In the case of -200, the implied odds are 1 - ($100) / ($200 + $100) = 66%. In the case of -110, the implied odds are 1 - ($100) / ($110 + $100) = 52.3%. And so on.

This relatively small change in winning % has a huge impact on going broke. A 1% bet standard goes from an 18% of busting after 6000 bets to less than 1 in 100. However, it is also true that reducing my betting unit will reduce my opportunity to make money. What we want to do is balance risk and investment.

Reducing our betting unit reduces our risk, but it also reduces our return, since we get paid less money on smaller bets. It follows that there should be an optimal betting unit based on risk, return, and betting advantage. Let's take another swing at our monte carlo simulation, with one modification: instead of keeping the absolute value of the betting unit constant, we will keep the percentage we bet constant instead. If we lose money, we reduce our betting unit, and if we win money, we increase our betting unit, such that we are always betting the same % of our bankroll with each bet. While this change technically eliminates the risk of "gambler's ruin" because our bankroll will never go to zero, it doesn't eliminate the risk of being left with no more than a few nickels to rub together with the wrong luck and too large of a betting unit.

In this case, we want to use the median value, NOT the average value, to evaluate the outcome. When evaluating bets that we will be making many times over, the average value is the right number to use. However, when evaluating a bet that can be made only once, the median value is a better reflection of the expected outcome - it is not skewed upwards or downwards by one or two highly lucky outcomes in the simulation.

The above figure shows the expected outcomes for our starting proposition: a 50/50 coinflip that pays even money. This graph shows that the maximum median outcome is with a standard bet of 0%, i.e. we should not make this bet. This result makes intuitive sense - there is no value to us here, so why would we ever make this play? Let's change the odds to be more and more in our favor and see what happens.

Boy, some of those numbers got very big! In reality, we would never get there, because no casino would take a 5% bet of a $100M bankroll. However, theoretically we see where the maximum median value occurs. We also see one of the variables that impacts betting size emerge: how much advantage we have over the implied odds. There is a black line here which charts maximum median value, betting advantage, and % stake. We can combine these three variables into a two variable chart: optimum % stake and betting advantage*. Here's that chart.

*betting advantage is % odds / % implied odds - 1, i.e. (51.5%/50%) -1 = 0.02. In each of the median value graphs the lines charted will represent 0, 0.02, 0.03, 0.04, 0.05, and 0.06 betting advantage.

Another variable that impacts this calculation is what the implied odds are. Let's look at another version of the previous two charts, except instead of 50% implied odds, let's consider a +150 bet, i.e. 40% implied odds, and see how the figures change.

Compared to a +100 bet, a +150 bet with the same betting advantage requires a smaller bet for optimum risk/value balance - nearly half the size. This is because an underdog bet is inherently riskier; there is a higher chance of going on an extended losing streak that can wipe you out. We can do the same thing for a whole range of bets. In fact, let's just skip ahead to that right now.

  Each of these is a roughly linear equation, with a different slope depending on the implied odds. We can plot the value of these slops as a function of the implied odds. This will allow us to find a correlation that can be used to calculate the optimum bet size once the implied odds and betting advantage are known over the entire range of values.

The result is a roughly polynomial curve over the range of interest (it is very rare in baseball to see odds outside of the range of +200 to -200). We now have all the tools in place to create a mathematical formula that tells us how much to bet to optimize risk and value capture.

  Recall, the implied odds are calculated as: CONCLUSIONS

A methodology for calculating the optimal betting unit based on balancing risk and reward for betting moneylines is presented. Inputs to this formula are the implied odds as calculated from the moneyline, and the calculated odds, i.e. the betting advantage anticipated by the bettor. The analysis is backed up by some intuitive results that occur from examination of the fomrula:

- The bigger the favorite, the more money should be risked. The bigger the underdog, the less money should be risked.

- The larger the betting advantage, the more money should be risked.

- If no betting advantage exists, the only way to win is not to play.

The risk discussed herein is the risk of experience "gambler's ruin", i.e. going on a long losing streak such that your stake is wiped out. There is a separate risk; that is, the risk of mis-evaluating your betting advantage. This is a very real risk: just ask anybody who used to work for Lehman Brothers. This will be discussed in a future post.