Hold vs. House Advantage

This link shows NV gaming statistics.

Knowing that the house advantage is a fixed number based on the rules of the game and the hold is the percentage of money won/money dropped, would we not expect a game with a higher HA to have a higher hold after millions of trials?

According to the link above:

Pai Gow Poker held 20.48%
Let It Ride held 23.24%
Roulette held 17.57%

Yet their HAs are 1.75%, 3.51% and 5.26% respectively. How does Roulette hold less that Pai Gow Poker when the HA is 3xs greater? Over the millions of "hands" represented by the data, the HA should have the greatest impact on the hold and should lessen the effect of all the other variables on the hold? I would more expect to see holds of 20-23-26 respectively. At least that would appear to be in line with the HAs.

The same holds true when comparing craps and blackjack. Craps HA is roughly 2xs that of BJ yet the holds are very close.

What about 3 Card Poker vs. Keno? C'MON MAN!! (Keyshawn Johnson - Sunday NFL Countdown)

Can one of you math "Wizards" please help me with this mental block I seem to have?

Hold = win/drop, but action/drop is not equal for all games. Pai Gow poker players play through their buy-ins many, many times more than roulette players. For a $100 buy-in on roulette, a player might play $300-400 in action. For the same $100 buy-in on Pai Gow, it's more like $1000-1200 in action, or 3x as much. So the same buy-in generates very different action. Since win approaches HA * action, longer sessions means your win as a percentage of drop goes up.

If you want to look into this more in-depth with your specific property numbers, PM me.

Quote: Larrymac

Can one of you math "Wizards" please help me with this mental block I seem to have?

Until one of those math types comes along, why don't you just look at it as two totally separate concepts and therefore you will feel no discomfiture at what you only think is a discrepancy.

The problem is that some of this is computed over time... a shift, a day, a week, a year ... etc. And will be computed for a certain table, a certain device, a certain pit, a certain game... etc.

Lets just do that roulette thingie. We will do it for THREE stints at the roulette wheel.

Mr. Player shows up and Buys In for FIVE GRAND. He ain't bet any of it, he just bought in!
What is the DROP? Heck it better be FIVE GRAND. He got chips that little Drop Box got his money!
What is the Handle? (or Action). Heck, I just told you he ain't made no bets yet.
What is the HOLD? Heck, the table just sold him five grand worth of chips but he ain't walked away yet much less walked away with some of what he bought or more of what he bought.

So lets say he plays one session and he loses the exact expected loss.
He goes to have lunch and comes back for more... loses the exact expected loss.
He goes to have dinner and comes back for more... loses the exact expected loss.

Now compute the figures from PLAYER's perception as to each of his sessions and his overall loss.
Now compute the figures from CASINO's perception.

The problem is that, particularly at craps or blackjack tables, people often arrive with chips purchased at a different table or leave and take some chips with them. This never changes the House Advantage.

Here is a quick summary from an earlier post somewhere:
House edge is ratio of average loss to the amount of the initial bet. (Not all games are decided on just the initial bet).
Hold is ratio of chips the casino keeps to the total chips sold. (Generally measured over a shift and will be affected by amount of chips players may bring to a table but not buy from it).

If E is equal to the House Edge for a game such as 5.26% roulette and S is the number of weighted average sessions within a 24 hour period per player, then the formula to derive the Hold % is 1-(E) raised to the S. If the average number of sessions is 3 then, 1-(5.26%) =.9474 raised to the 3rd or .8504. Mr. Roulette left with $8,504 after starting with $10,000 and playing 3 sessions within the 1 day Hold period that casinos use.

Hold % is equal to 1-(1-E)s.
EV = Expected Value = (+5)(18/38) + (-5)(20/38) = -0.263 (Player's Expected Value is MINUS 0.263)
(House Advantage = Expected Value divided by Amount of Initial Bet = 0.263/5 = 5.26%)
Hold % = Win/Drop
Win % (actual) = Win/Handle

Quote: Larrymac

Knowing that the house advantage is a fixed number based on the rules of the game and the hold is the percentage of money won/money dropped, would we not expect a game with a higher HA to have a higher hold after millions of trials?

I wouldn't think HA would be very significant. The significant factor would be how many times the player plays through his winnings.
Baccarat in particular has very high win percentages the same month as Chinese New Years. I assume that is because Asian players have lots of stamina in staying at the table. At other months the drop is just as high, but the win percent is very low. I have always assumed that the non-Asian players do not spend as much time at the table.

I appreciate the feedback from all above. I have to say that Flea lost me near the bottom of the page though.

OK here's a realistic scenario:

Two guys go to Vegas with their wives. They say to meet in the casino at 4pm to play for 4 hours before dinner with the wives. One guy plays Roulette, the other Pai Gow Poker. They each have $1000 to play and both bet $25/hand. Roulette guy will play more spins than PGP guy as there are typically more hands per hour in Roulette. So Roulette guys plays through his $1000 buy in at a rate of 40 spins per hour making a total of $4000 in wagers. The house expects to win (in a perfect world) about $210 for a hold of 21%. Now PGP Pete gets about 140 hands in over his 4 hours wagering a total of $3500. PGP Pete is expected to lose $61 (in that same perfect world). The house holds only 6% of his buy in. Even if PGP Pete goes back after dinner for another 4 hour session at the same table using the chips he had from before, the house only holds 12% of his buy in after 8 hours of play (the arguement above about going through their buy in more times on PGP fails). No matter what the buy ins, the ratio will still be the same - so again, How does PGP hold more than Roulette?

The case about coming up to the game with chips doesn't hold much water with me as I see players do this all the time at all types of tables - my statement that over millions of trials those factors' impact on the hold should be lessened as they will tend to balance out over a period of an entire year and every NV casino's figures being combined as in the data in the link. The ratio of the figures should get closer and closer to that perfect world scenario as we consider more and more trials - combine all of NV casinos over a 5 year period and the holds would likely be very close to the one year figures. Now we are taking billions of decisions into consideration. All those little chips in, marker redemptions, huge wins, oversized buy ins, marathon sessions, 10 minute hit-and-runs etc will have a negligible impact (or should).

Quick question for the "Wizards" - What is the HA on Fortune Bonus Pai Gow Poker where player banking is not allowed? Consider a player betting $25 on the Pai Gow hand and $5 on the Fortune Bonus using the typical pay table. My (un)educated guess would be about a 2.7% HA.