It's a math geek thing. People who don't understand what I do often think that everything is simulated. I can't tell you how many times my work has been misunderstood as entirely simulations - that I let a random number generator play some very large number of hands to figure out everything from the house edge and game strategy, to counting systems and the edges from various types of advantage play.

Here's the truth: simulations are my last resort. I could not imagine delivering an approximate result to my clients, or posting an approximate result here, if a full and complete solution was available. A full solution means to use "combinatorial analysis," which I discuss in this video. Combinatorial analysis gives an answer that is 100% exactly correct.

In particular, I use combinatorial analysis for every card counting analysis that I've presented in this website. My article on card counting the Dragon 7 baccarat side bet used combinatorial analysis (the article that inspired this blog).

Besides getting exact results, the other reason that I lead with combinatorial analysis is that I can use the results to verify and audit my simulations. After I use combinatorial analysis to figure out card counting systems, I run simulations to get win-rates. I verify that the simulations are working by comparing certain outputs generated by these random simulations to the exact results.

Where is all of this leading? Well, obviously this is leading to a math geek pet-peeve. If I was explaining this to my dog Rosie, she wouldn't know the difference between combinatorial analysis, random simulations and a dog biscuit. But in this crazy game protection business, getting it 100% right is more important to me than anything else. Any errors in assessing game protection concerns can potentially harm the game inventor, the gaming company and the casino. More than once, I've been threatened with lawsuits for claims I've made about vulnerabilities in games. Getting it right has saved me.

This recent article by gaming mathematician Elliot Frome, Card Counting can Occur in Other Casino Games, contained a few sentences that rankled me.

Frome wrote,

Each rank of card has a certain “value” to the payback relative to the others. To figure out this “value,” we simply run a random simulation of blackjack with ONE card of that rank missing from the deck. We compare the payback of the simulation to the payback of a normal simulation.

Maybe it was the word "we" that pushed me to feel the need to reply to his apparent methodological aspersions. I sent an email to Frome and asked for a comment on what I considered to be his "wrong" way to do this type of analysis. Frome replied, in part,

The point of the article was to convey to the lay person some of the basic concepts of what goes into a counting model and the focus of this is the idea that each rank has a value associated with it that impacts the payback.  Whether that is achieved via math model or computer simulation is, in my opinion, not as relevant to the lay person.

Yeah, he's right. But since the lay person doesn't understand the difference anyway, what is the harm in explaining the right way to do this work?

Frome subsequently stated,

I believe the methodology I described would result in the same conclusions even if most analysts performing a vulnerability analysis would do so via a math model. 

Yeah, close enough. I agree that for practical purposes, advantage play win-rates are not an exact science. Risk analysis falls into broad categories. But there is always the chance that for some obscure advantage play opportunity, sometime, somewhere, the extra accuracy might make the difference between wildly different conclusions about the game's risk to a method of advantage play.

Question: Who cares about a measly tenth of a percent?

Answer: George Bush, State of Florida, 2000 U.S. presidential election.

My desire to get the exact answer to math problems is roughly equal to my dog Rosie's desire to get a biscuit. If it's even remotely possible, then for a few moments Rosie and I want it more than anything else in the whole world. The difference is that my dog Rosie has yet to figure out anything else to do with a biscuit other than eat it.

Rosie the dog


Received his Ph.D. in Mathematics from the University of Arizona in 1983. Eliot has been a Professor of both Mathematics and Computer Science. Eliot retired from academia in 2009. Eliot Jacobson