Although people have played casino games like Roulette for centuries, some of the elementary facts about betting systems are unknown. For example, most people will insist that your chances at Keno are much much worse than your chances betting a color at the roulette wheel. As we will see, this isn't completely true. I couldn't find the information shown on this page anywhere by Googling, so I wrote this page myself!
Let's start with some elementary definitions about a casino bet, which we call "bet A".
Here the payoff is called "wA for 1." It can also be called "(wA-1) to 1." (Sometimes the "for" formulation is more convenient. Sometimes the "to" formulation is more convenient.) vA may also be called the "House vigorish." Although we show three parameters (pA, wA, vA), any two of them define the third so we will call wA and vA the parameters of the bet "A."
For example, if you bet a single number on an American Roulette wheel:
The House vigorish here is greater than zero, and we will assume vA > 0 for the remainder of this webpage. Some people think that since vA is constant at the roulette wheel, it doesn't matter how you bet: you could sprinkle your money on the table randomly. This is not true however. As an extreme example, suppose you bet 1/38 of your stake on each of the 38 numbers! The combined effect of these 38 bets is a single bet "C" with the following parameters:
This section is so basic, I'm afraid many readers have gotten bored already! But things will get more interesting. We will introduce the idea of fashioning your own bet from two or more spins of the roulette wheel. Throughout the remaining discussion we will assume that casino allows you to bet any non-negative amount of money you want to whether it be $1,000,000 or 39.3 pennies. (This assumption simplifies the math and the discussion, even though it may not correspond with real casinos.) The only restriction is that the casino will not offer credit: You must have a bankroll large enough to actually place the wager.
Please note that throughout this page we consider only the refashioning of a single bet. The simplest way to focus on this assumption is to suppose you have $Zstart of risk capital and that that is all the money you have; that you want $Zgoal and that if you can't achieve that goal, you don't care if all your money disappears. (We might suppose $Zgoal is the cost of some special emergency medicine.)
Without loss of generality, we may assume Zstart = 1.
Let's start with a very simple example. You want to quadruple your money at the roulette wheel. Your choices include betting on a group of nine numbers (e.g. 1 to 9), or betting on a color and, if you win, letting your chips all ride one more time on that same color. (As we will see later, there are better options, but let's start here.)
We already know the house vigorish on the first option; it's .052632. Let's look at the second option (back-to-back bets on a color):
Anyway, what we have done with this example is to fashion a compound bet "T" from a simple bet "A" of parameters wA and vA. The compound bet we fashion can have whatever wT we choose it to have. Because you are now fashioning your own bet, with your own goal target, rather than directly applying a bet the casino offers, we will call this bet "T", "T" for target. The question is: What is the efficacy of the compound bet?
The remainder of this page considers just one question: how do we construct the compound bet "T" to minimize vT and what is that minimal vT?
We will derive this optimal vT as a function of wA, vA and wT.
If you want to increase your $1 to $1024 and the casino offers you no choice but an even-money bet, your strategy should be obvious: Bet everything, and continue to let it ride, hoping that you win ten times in a row.
More generally, when
wT = (wA)K
for a positive integer K, you let your money ride K times. Since
pA = (vA - 1) / wA
it follows that
pT = (vA - 1)K / wAK
which leads to
vT = 1 - (vA - 1)K
vT = 1 - (vA - 1) (log wT / log wA)
Proving that "Bet Entire Bankroll" is the correct strategy in this case is straightforward: it minimizes the total action you expose to the House vigorish. The final equation will be exactly valid only when K = log wT / log wA is a positive integer.
If you have $1023 but need $1024, and your only opportunity is a casino which offers only an even-money bet, your best strategy is to bet $1 and of course walk away if you win. If you lose, bet $2; losing again, bet $4 and so on. This is the famous Martingale but please think carefully before laughing! I'm not saying the Martingale is a casino-beating idea, just that it's the best strategy (and indeed obviously so) with the stated condition: That you need $1024 and don't care if, when unsuccessful, you lose your entire $1023 savings. As before, we have fashioned our own bet "T" from casino's bet "A." In this case, wA = 2 and wT = 1024/1023.
More generally, you will bet G1, G2,
G3, G4, ...
Gs = (wT - 1) · Sum(k=1,2,...,s) (wA - 1)-k = (wT - 1) · wAs-1 / (wA - 1)s
You will have the opportunity to make exactly M bets when
1 = G1 + G2 + ... + GM = (wT - 1) · ((wA / (wA - 1))M - 1)
When we solve for M, we get something complicated, but that matters little as we intend just to plug this into computer programs. The formula for M is:
M = log (wT / (wT - 1)) / log (wA / (wA - 1))
(For those following along carefully, in our very simple example wA = 2, wT = 1024/1023 and this equation does indeed yield M = 10 as expected.)
The probability of success on our compound bet "T" is
pT = 1 - (1 - pA)M = 1 - (wA + vA - 1)M · wA-M
so the net vigorish is
vT = 1 - wT + wT · (1 - pA)M
vT = 1 - wT + wT · (wA + vA - 1)M · wA-M
Since M was derived above as a function of wT and wA, once again we've derived vT as a function of the three parameters wA, vA and wT. And once again, the formula is valid only when M is a positive integer.
In general, we should bet precisely what we need to achieve our
goal. These bets are the so-called Martingale bets
G1, G2, G3, G4 ...
we saw in the last section.
The exception is when that bet would exceed our total bankroll;
then we bet our entire bankroll instead.
The analysis of efficacy (which can be denoted by the effective
vigorish vT) thus uses both the formulae
vT = 1 - (vA - 1) (log wT / log wA)
vT = 1 - wT + wT(wA + vA - 1)M · wA-M
But these formulae are almost absurdly different! I said I was going to give you a general formula, but I lied! What I have is a C program you can compile and run to find vT for any input values of wA, vA and wT. That program will also print out the vigorishes estimated by each of our two formula. It displays "need" (the number of successive winning bets needed in the first formula) and "opps" (the number of winning opportunities in the second formula). You will see that the first vigorish estimate is fairly close to the actual effective vigorish when "need" is large, the second estimate close when "opps" is large. (An estimate will be exact only when one of these numbers is an integer.)
Given that you've restricted your options to a single bet "A", with associated parameters, is it always optimal to follow the advice in this section? I believe so, though cannot show a rigorous proof. It will sometimes be the case that an alternate betting regime yields a success rate identical to the described regime, but I don't think it can surpass it.
Throughout this page we've assumed the casino offers only one bet, the one described by wA, vA. How should you choose among choices when the casino offers a variety of bets? (Say bets "A" and "B.")
There are exceptions where the lower-payoff bet of equal vigorish outperforms the higher-payoff bet slightly. This usually arises when wA is very close to your target wT; even then bet "B" is better when wB is much larger than wA. To give just one other example, when wT = 8/7, a wA = 2 bet may outperform 2 < wB < 2.4, since it gives you an exact number (M = 3) of Martingale chances.
While passing time watching a roulette wheel, a millionaire, just for fun, places $9,000 on Red and says "It's all your sif it wins!" (He then walks away.) You realize this "ticket" is worth $8526 on average, but you'll probably finish with zero. So you decide to place bets of your own: $9000 on Black and $500 each on 0 and 00. You've just laid out $10,000 but whatever happens the croupier will be shoving $18,000 your way. A sure $8000 seems better than the highly uncertain $8526 "ticket." But by the Kelly Criterion, whether you take this hedge is a function of your bankroll B. You want to maximize the weighted geometric mean of (B + r), where r is your win or loss on this bet. Instead of $10,000 what's the total (X) you should bet on Not-Red to maximize that weighted geometric mean? Dust off your calculus and see if you get the same answer as I do.
Solution: X = ($144,000 - B) / 15.2 (Or zero, if your bankroll exceeds $144,000 and the casino doesn't allow negative bets!)
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A write-up for another wagering puzzle.