How Is "Theoretical Win" Calculated?

Can you explain how "theoretical win" is calculated? Also, is it the same for slot play and table game play?

[Editor's Note: This answer is provided by one of gambling's greatest "theoreticians," Arnold Snyder.]

"Theoretical win," a term used primarily by casino accounting and marketing departments, also known as the "theo," is simply the casino’s “expected win,” which is determined by multiplying the house advantage times the amount of the player(s) bet(s).

Example: A pass line bettor at craps is bucking a house advantage of 1.41%. If he bets $100, the house’s theoretical win is $1.41.

Of course, craps has many bets other than the pass line, all of which have different house advantages, so calculating the theo will be more complicated for a player who is placing, laying, or buying numbers. In this case, we have to tally the total bet at each different house advantage and multiply by the house edge on each bet to come up the overall theo for this player.

The theoretical win for table games that have specific house advantages for any bet made — such as roulette or baccarat — is calculated the same way.

For a game like blackjack, it's easy to calculate the theoretical win for a basic strategy player, but so many players deviate from basic strategy, both optimally and mistakenly, that the basic strategy theo is fairly useless. Many casinos use software to estimate their theoretical win versus different players based on their analyses of players’ observed decisions at the table. But even this is often less than correct. Software that analyzes a blackjack player’s theo based on the assumption of card-counting deviations will provide bad data if the player is using hole-card information, shuffle tracking, or employing some other advantage technique that's not factored in.

Against slot players, the theo is figured out the same way as for table game players — by multiplying the house advantage times the amount bet. With machines that require no player strategy decisions, typical slots, the theoretical win is a straightforward calculation. With video poker of video blackjack where player decisions affect the results, the theo is based on the individual player’s skill (or lack of).

Theoretical win is used primarily by marketing departments to determine big players’ value to the casino in order to entice them with comps. Probably the best discussion of how casinos do this — and how smart players can game the system — is found in Max Rubin’s book Comp City (still available as an ebook). And Whale Hunt in the Desert has a detailed discussion of how marketing, particularly hosts, get a quick snapshot of a player's worth, and sometimes even earn their commissions, based on the theo.

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Kevin C Sep-06-2022

Tangentially relevant question.

From the article: "Against slot players, the theo is figured out the same way as for table game players — by multiplying the house advantage times the amount bet. With machines that require no player strategy decisions, typical slots, the theoretical win is a straightforward calculation. With video poker of video blackjack where player decisions affect the results, the theo is based on the individual player’s skill (or lack of)."
My question; what about a slot where the player has to choose options during a bonus round? Does that actually affect the bonus outcome or is that illusory?

Kevin Lewis Sep-06-2022

Then what...

So they've figured out what you're worth to them--what % of that are they willing to comp back to you? I'm given to understand that it used to be 40% of theo but nowadays, it's more like 20%. God knows, for me, it's been all over the spectrum.
Also...do they REALLY ever assess your VP skills? And if so, how? Or do they just assume you're playing optimally, although few players ever do? (That could be behind locals' casinos' "zero points on the good machines" policy.)

Brent Sep-06-2022

Kevin C

I have gotten somewhat conflicting explanations about the bonus rounds on slots, but I believe that the answer for most machines is that your "choices" on the bonus round are irrelevant. The outcome has been predetermined before you even start to "choose."

But even if you do actually affect the outcome, the overall odds will still be the same since you are selecting randomly. Ultimately, the house edge will be the same when spread out over all the players making random selections during the bonus round.

Janet Sep-06-2022

Theo Loss?

As I understand it, MGM's new...ahem..."Rewards" program, which they claim "members asked for and they listened" (NOT!) awards tier credits based on theoretical loss.  How does the calculation differ for theo loss differ from theo win?

Kevin Lewis Sep-06-2022

Janet

There are no games offered by MGM that afford a theoretical win, and if there were such games there, they certainly wouldn't comp you for playing them. That aside, "theo" can be a positive (expected win) or a negative (expected loss) number.

gaattc2001 Sep-06-2022

One major "off-strip" resort group, which shall here be nameless...

used to award tier credits for video poker according to a formula based on "perfect play" at each machine: one credit for $50 coin-in, divided by the house % advantage. For example with Airport Deuces, one tier credit would be $50/1.081 or $46.25 coin-in. I verified this empirically a couple of times.
It turned out the expected loss was constant across all VP games. You could play a looser machine for a longer time, or a tighter one for less, and get the same tier credits for the same expected loss. It wasn't difficult to play enough to get to second level and get comp offers; and the expected loss was about the same as two weekend trips' hotel charges and resort fees. 
Then all my hard-earned tier credits evaporated in the pandemic, and they didn't even offer an extension. I thought about rebuilding it, paying fees along the way, and decided not to bother. 
Al this was over two years ago, but I suspect they don't miss me in the current era of "unprecedented prosperity."
Cheers.

Question of the Day - 06 September 2022

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