docbrock
docbrock
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May 15th, 2016 at 6:45:44 PM permalink
Hi everyone,

Basically, I'm looking for some clarification as to the mathematical advantage of "prepaying the commission" in Pai Gow Tiles, as is suggested by a brief discussion of the topic on Wizard of Odds.

From the Wizard's site:

"Prepaying the Commission

Some casinos let the player prepay the 5% commission. For example, betting $105 to win $100. This lowers the overall commission to 1/21, or 4.76%. The effect on the house edge is a reduction of 0.07%. This is an option the player should always invoke when available, yet many don't."

Doing the calculation myself, however, I seem to arrive at a higher expected loss per trial if the player bets $105 to win $100 as opposed to $100 to win $95.

Said another way, I understand how the commission is lowered to 1/21 from 1/20 betting $105 to win $100; however, it seems to me that the logic neglects to recognize that, while the commission is certainly reduced on winning hands, the player when losing a hand loses more money - $105 as opposed to $100.

I'm sure my calculations are flawed as I'm not very well versed in gambling math, but I'd love to know why.

This is how I'm seeing it:

Using the Wizard's calculator and assuming optimal strategy versus the 'traditional house way' and the dealer banking, the probability of a win given a random deal of four tiles is about 0.294193; a loss 0.296076 and a push 0.409731.

Thus the expected value of a bet of $100 to win $95 would be

(0.296076)*(-100) + (0.294193)(95) + (0.409731)(0) = -1.659265

and the expected value of a bet of $105 to win $100 would be

(0.296076)*(-105) + (0.294193)(100) + (0.409731)(0) = -1.66868

So, it seems as though betting $105 to win $100 would cost slightly more, on average, per trial, than betting $100 to win $95, presumably because, though the commission on a winning wager is lower percentage-wise when betting $105, the player also loses more when he loses a $105 wager versus a $100 wager.

By that logic, then, it would seem as though the player ought NOT to choose to prepay the commission if they wish to minimize their expected loss.

Can anyone explain to me why it's more optimal, then, to prepay the commission?

Thank you kindly for the help and for the wonderfully diverse and informative site.

Ben
Wizardofnothing
Wizardofnothing
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May 15th, 2016 at 6:57:53 PM permalink
I'm not a math wizard but I play tiles and it's definitely to your advantage to prepay commission and to bank every other hand if they let you
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phendricks
phendricks
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May 15th, 2016 at 7:09:08 PM permalink
You did your calculation incorrectly, hence your logic is wrong. We are comparing $100 to win $95 and $99.75 to win $95.

Without pre-paying 5% commission: Bet $100 to win $95. House edge of 1.66%.

With pre-paying 5% commission: Bet $99.75 to win $95; you save $0.25 on a loss, hence a reduction in the house edge. House edge of 1.58%.

---

You then give another example where you want to win $100 on a hand.

If you want to win $100, normally you'd have to bet $105.26, but on a pre-pay table, you only put up $105, saving you $0.26 on a loss.
Last edited by: phendricks on May 15, 2016
ahiromu
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May 15th, 2016 at 7:17:28 PM permalink
This is another "why are odds good in craps?" question. That extra $5 is the best bet you're going to get in the game, it's better than the first $100 you put up. Nonetheless, you're right, the game has a house edge before commission is taken into account.

Also, if you're banking every other hand, you come out ahead. So saying it's "optimal" is not necessarily wrong.
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Wizard
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May 15th, 2016 at 7:38:20 PM permalink
You need to divide the expected loss figures by the total amount bet.

If your goal is to lose as little as possible, try the game under the sign that says "change."
It's not whether you win or lose; it's whether or not you had a good bet.
docbrock
docbrock
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May 16th, 2016 at 8:27:01 AM permalink
Thanks to everyone for the replies.... Now you can see how terrible I am at gambling math. :)

So one of the flaws in my logic was that I was comparing the expected loss of betting $105 to win $100 with the expected loss of betting $100 to win $95, when I should've been comparing the expected loss of betting $100 to win $95 vs. betting $99.75 to win $95, or $105.26.... to win $100 vs. $105 to win $100.

And, not dividing the expected loss figures by the total amount bet, which would give a percentage loss that would be higher in the case of betting $100 to win $95 than betting $105 to win $100.

Incidentally, my goal isn't just 'to lose as little as possible' - if that were my goal I wouldn't set foot in a casino and I'd take up disc golf.

Barring being able to become some advantage player in one of the handful of cases where a game, even played fairly, can be beaten.

Instead, my goal is to lose as little as possible while still being able to enjoy playing the game.

Mastering advanced strategies is challenging; the game is cerebral and combinatorial and I like the fact that it's 'off the beaten path': When I go to a casino with my girlfriend or buddies everyone wants to shoot dice, play blackjack, roulette, slots, and all that between trips back and forth to the club or to the bar -- and I do it on rare occasions and we have a great time. But honestly, I enjoy tiles a lot more than any of those other games, so, when I have the opportunity to play, that's what I do.

And, when I play, I realize that I'm paying a price for that entertainment by virtue of playing, and I gladly and willingly pay that price.

But I do play my fair share of the game and I'm not extraordinarily wealthy, so yes, I want to minimize my entertainment expenses.

Not exactly the whale that the casino would rather have in my seat, but I can't afford to, and frankly I just don't want to, play a losing game for bigger money.

Thanks again for the help.

Ben
Last edited by: docbrock on May 16, 2016
docbrock
docbrock
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May 16th, 2016 at 8:37:47 PM permalink
Hi again everyone,

I did a little more analysis and I was hoping to run it by you all to see if I'm correct in my thinking here-- Would be very grateful for a few helpful responses.

This has kind of become a fun little mathematical experiment for me, and I know it's splitting hairs for very small amounts of money; after I leave the table the next time I play I'll probably get up and pay six bucks for a cup of coffee and blow all my theoretical "savings" that I garnered playing tiles -- it's more about the mathematical principles and the idea that the player should always invoke their right to prepay the commission when given the opportunity.

Now, say we want to absolutely minimize our expected loss per hand at a $25 minimum table at the Taj. The Taj allows you to bet up to $27 and still charges only $1.25 in commission, lowering the commission on a $27 bet to 1.25/27, or just shy of 4.63%.

For the purposes of the exercise let's assume we're playing optimal strategy, facing the Taj house way and that the dealer is banking.

According to the calculator, given a random draw of 4 tiles, the probability of winning in that scenario is 0.296881; losing 0.298027 and pushing 0.405092.

So, we have a couple of options.


First, we could bet the table minimum, $25, for which a win would pay $23.75. We wouldn't be prepaying a commission and the house would charge the standard 5%.

In this case, our EV would be:

= (0.296881)(23.75) + (0.298027)(-25) + (0.405092)(0)

= 7.05092375 - 7.450675 = -0.39975125


Second, we could try to prepay the commission to win $23.75, which would mean a commission of:

(0.05)(23.75) = 1.1875

and so would mean a total bet of

= (0.05)(23.75) + 23.75

= 1.1875 + 23.75 = 24.9375

which beside just being impractical would be below the table minimum, so it wouldn't be allowed.


So then, we could bet a variety of amounts between $25 and $27, still paying $1.25 in commission no matter what the amount bet.

We could prepay the commission by betting $26.25 to win $25, as opposed to betting the roughly $26.32 we'd need to bet to win $25 if the commission were not prepaid and was a full 5%.

That $26.32 is given by:

x - (0.05x) = 25
20x/20 - x/20 = 25
19x/20 = 25
19x = 500
x = 500/19
x = 26.32

So, betting $26.25 to win $25 lowers the commission to 1.25/26.25 = 1/21 = roughly 4.76%.

In this case our EV would be:

= (0.296881)(25) + (0.298027)(-26.25) + (0.405092)(0)

= 7.422025 - 7.82320875 = -0.40118375


We could also bet $26 to win $24.75, in which case our EV would be:

(0.296881)(24.75) + (0.298027)(-26) + (0.405092)(0)

7.34780475 - 7.748702 = -0.40089725


And we could go all the way and bet the full $27 to win $25.75, lowering the commission even further, to 1.25/27, or just shy of 4.63%.

In that case our EV would be:

(0.296881)(25.75) + (0.298027)(-27) + (0.405092)(0)

7.64468575 - 8.046729 = -0.40204325


Overall, what we see in terms of our EV in each of these betting scenarios, then, is:

25 to win 23.75 (5% commission, not prepaid) => -0.39975125

26 to win 24.75 => -0.40089725

26.25 to win 25 (prepaying a 4.76% commission to win 25 as opposed to betting 26.32 to win 25, a 5% non-prepayed commission) => -0.40118375

27 to win 25.75 (a 4.63% commission) => -0.40204325

So, we see an increase in our expected loss as we increase our bet, despite the fact that the outright commission stays the same ($1.25) and the percentage commission decreases to as low as 4.63% as our bet runs up to $27.


The conclusion that I make from this is that, if a player intends to make a bet to win a certain amount, y, it behooves them to prepay the commission for that intended win as opposed to betting the amount that will win them that money when the house charges a non-prepayed 5%.

However, if we want to minimize our losses at this theoretical $25 minimum Taj table, we're better served by betting the table minimum and no more -- that is, by not prepaying the commission and not betting any amount more than $25, even if the outright commission ($1.25) does not change for bets up to $27.

That would seem to me to mean that the player should not always necessarily invoke his right to prepay a commission if the option is available, if for no other reason than betting the table minimum and paying the full 5%, in this case, has the lowest expected loss - despite the fact that, betting more (given a steady outright commission) can lower the percentage commission to, at least in the case of this Taj table, under 4.63%.

Am I correct or am I still missing something?

Thanks, and cheers,

Ben
Last edited by: docbrock on May 16, 2016
phendricks
phendricks
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May 17th, 2016 at 3:43:31 AM permalink
Quote: docbrook

if a player intends to make a bet to win a certain amount, y, it behooves them to prepay the commission for that intended win as opposed to betting the amount that will win them that money when the house charges a non-prepayed 5%.



Again Ben, you ask for a certain amount y but give three different amounts you would like to win: $23.75, $24.75, $25.75. Which one is y?

---

It's very simple: the additional $2 Taj lets you bet at 0% commission still has a house edge.

If you never bank, and only care to minimize losses, you should always bet the table minimum of $25 (expected loss: $0.41).

Suppose the Taj lets you bet up to $5,000 on top of your $25 and still pay $1.25 commission. The effective commission is now 0.00024. You are still winning 29.68%, losing 29.8%, and pushing the rest. The expected loss is now $6.03.

Just enjoy tiles- it's a beautiful game, and bank whenever possible. Instead of calculating tenth of a penny savings, spend some time reviewing when to play 6/Gong and when to play 8/8.
docbrock
docbrock
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May 17th, 2016 at 7:30:07 PM permalink
Thanks phendricks--

Tiles is a great game and I really enjoy playing it for its own sake, regardless of stakes; and I certainly don't want to waste my energy quibbling about tenth of a cent savings, trying to pinch every little fraction of a penny that I can.

It just became a fun little mathematical exercise to try to understand the rationale behind prepaying the commission and also which bet size would technically minimize expected loss in the theoretical Taj scenario. I'm obviously not very knowledgeable when it comes to gambling math and often I find the correct statistical point of view for this or that gambling situation counterintuitive.

Just like tiles, it's a challenge I enjoy taking on, even though I realize I have a long way to go before I come even close to mastering it, and that, in all likelihood, I'll probably keep on trying to master it indefinitely.

But I like taking on the challenge with an attitude of continuous improvement.

I also like having friendly debates with dealers when I play (h6 h6 11 h2) as 7/8; (h8 L8 L2 L4) as 6/6; (5 5 H2 L2) as 7/7 [rarely]; (H10 L10 H4 5) as 0/9, etc. - when each of those plays clearly deviates from the typical house way but is more efficient.

I've had dealers say anything from "did you see my tiles?" jokingly when I win with a hand like those, all the way to "you're never supposed to split sixes, don't you know that?" and on and on when the dealer rolls over a 1/Gong and I push with the 7/8 when I could've won by playing the high sixes together.

I joked with a dealer at Resorts (right next to Taj) a few weekends ago after 5 or so minutes of back and forth (we were playing heads-up) after I pushed with the 7/8 against his (H2 L8 L7 5) => 2/Gong:

I made the comment that our debate was a debate between "ancient wisdom vs. modern technology", and we both laughed. He was insisting that '95% of experienced players would play 3/Pair,' or something to that effect, while I was trying to counter that the 7/8 was the more efficient setting.

I learned tiles by computer, not by word of mouth, and I enjoy trying to make plays that have been demonstrated to be optimal, especially when it results in a win instead of a push, a push instead of a loss, etc.

And I like studying strategy to better the chances that I'll know what play is most efficient for a given hand, especially the less obvious ones.

It's just a lot of fun. It's much more enjoyable than betting on roulette or whatever and just standing there waiting for the ball to land on a winning or losing number all the while knowing that you're being paid at 35:1 (or whatever it is) but you're drawing at 37:1. When it comes down to it, I just don't enjoy playing that game. Other people do, of course, but it's just not for me.

Anyway, thanks for the help-- I feel like my understanding is much better and I genuinely appreciate it.

Cheers-

Ben
MrGoldenSun
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May 23rd, 2016 at 6:44:19 AM permalink
Quote: docbrock

Incidentally, my goal isn't just 'to lose as little as possible' - if that were my goal I wouldn't set foot in a casino and I'd take up disc golf.



I like both and I think I've spent more money on the latter. :)

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