In his seminal book on gambling, ‘Getting the Best of It!,’ David Sklansky covers a wide variety of gambling-related topics. He gives the equivalent of a course in basic probability, Bayesian thinking, and maximizing your expectation as an advantage gambler in many pursuits, such as poker, blackjack, and sports betting. The common theme of the different examples in the book is, as the title implies, figuring out how to get the “best of it.” This term has been used for many years in gambling circles and it simply means “finding an edge” or, technically, identifying and exploiting situations where the net expected value (EV) of your play is positive. In other words, if you have to put up $100 on a proposition and the expected value of your return is $110, then you have a positive EV of +$10 and you’re getting the best of it.
In David’s book, there’s a small and simple example that caught my eye while rereading it. In his discussion of expectation and percentage edge in Part 1, he tells us that we’re given a choice between two bets on a 50/50 proposition. For Bet 1 we must risk $10 to win $20, but for Bet 2 we’d have to risk $50 to win $70. As he rightly points out, while Bet 1 has a bigger percentage edge (50% vs. 20%), Bet 2 has a bigger money expectation since we’ll win $10 on average rather than just $5. He also wisely warns us that taking Bet 2 would be incorrect sometimes if we don’t have a large enough bankroll to comfortably cover the larger bet. My question here is: how big should our bankroll be to prefer Bet 2, and how do we figure that out?
The answer is that instead of trying to get the best of it, let’s try to get the rest of it. In other words, let’s try to optimally grow our whole bankroll such that we can wager more and more over time and take maximum advantage of the edges we find. It doesn’t matter that this is only a single bet; our goal should be the same as when making a million bets, just like it would be if we were trying to maximize our EV over the long run. So, how do we decide between Bet 1 and Bet 2 to optimize the expected growth (EG) of our bankroll? We can define a critical bankroll size “r” for which the EG of both bets is the same. That way we’re indifferent to taking either one.
This critical bankroll amount is a tipping point. If our bankroll is greater than this equilibrium size, then it’s big enough to risk the $50 to win $70 like with Bet 2, and to take advantage of the larger positive expectation it brings. To work out how large this critical bankroll size is, we need to do a lot of complicated math. Luckily, I worked it all out for you and we can go straight to the final answer. Well, almost. The equation to figure it out boils down to:
10r – (10*20) = 20r – (50*70)
10r – 200 = 20r – 3500
-10r = -3300
r = 330
So, the critical bankroll size for which we’re indifferent to taking either bet is $330. That shouldn’t seem very large to most of us, so I think most readers would agree with David that we should take Bet 2. What if we have to risk a lot more to get that +$10 expectation, though? If we had to risk $500 to win $520 on Bet 2 instead, how much do you think the critical bankroll size would be? To figure it out more easily, we can take a shortcut. Notice that the constant terms in my equation are the product of the “risk” amount and the “to win” amount for each bet you can choose from. Therefore, when comparing our larger Bet 2 we can skip to this step and set up the following equation:
10r – (10*20) = 20r – (500*520)
10r – 200 = 20r – 260,000
-10r = -259,800
r = 25,980
Now our critical bankroll size is almost $26,000. Whether or not that seems large to you should tell you a lot about how often you should be passing up bigger edges in pursuit of larger expectation. While the edges in this example may be ridiculously large, and you don’t always have to choose between two bets like this offered at the same time, this same concept lends itself to analyzing a common sports betting question: should I bet openers?
Openers (or the earliest posted lines for a betting market) typically offer the biggest edge you’ll see for a given market, barring surprising news coming out. But, many sharp bettors will pass on them, waiting until closer to game time to bet into the market after the betting limits are increased. Their theory is that if they can get down 5-10 times the amount on their bet by waiting, then that will more than make up for the likelihood of getting a smaller edge. For example, if the limit to win on the opener is $500 (Bet 1) and the max to win closer to game time is $5,000 (Bet 2), theoretically you’d get a bigger mathematical expectation by waiting if the later line had at least 1/10th the edge of the opener. Say you estimate your edge on the opener is about 7%, but expect it will be reduced to 1.2% by the time the $5,000 limits kick in. If it’s a typical spread bet at -110 on each side, then that’s equivalent to beating the opener 56% of the time, and beating the later line 53% of the time. We can calculate your mathematical expectation for Bet 1 like so:
EV = 0.56 * 500 – 0.44 * 550
EV = 280 – 242
EV = $38
Whereas, you’d earn a higher expectation by waiting and making Bet 2:
EV = 0.53 * 5000 – 0.47 * 5500
EV = 2650 – 2585
EV = $65
Betting pro Spanky often talks about the fact that when limits go up your edge goes down, and has called deciding when to bet here a fine line to walk. To choose most wisely while on this journey, don’t just assume that the higher expectation is better, but instead calculate your critical bankroll size for this pair of options. Then, if your bankroll is large enough, you’ll know it’s a better play to wait till the limits are raised and go for the higher expectation. To use our shortcut from above to do the math, the bets must be a 50/50 proposition, but we can work out a pretty good approximation in this spot since the win probability for each bet is only a little more than 50%. Jumping right to the end of the calculation gives us a critical bankroll size of about:
(2*38)r – (500*550) = (2*65)r – (5000*5500)
76r – 275,000 = 130r – 27,500,000
-54r = -27,225,000
r = $504,167
The conclusion then is that, for those of us with bankrolls of less than half a million, hitting the opener for $550 is a better choice than waiting for the higher limits, even though the expectation on your bet will be much lower. The reason this is true is that your relative risk is much higher when betting on a 1.2% edge than on a 7% one, even if your bet on the 1.2% edge isn’t technically an overbet (since in this example, a full Kelly stake size is about $6,500 for a $500k roll and you can only risk $5,500). By factoring in the element of risk, you can follow the goal of the Kelly Criterion by maximizing your EG and do even better than getting the best of it.