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

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.

Also, if you're banking every other hand, you come out ahead. So saying it's "optimal" is not necessarily wrong.

If your goal is to lose as little as possible, try the game under the sign that says "change."

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

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

Quote:docbrookif 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.

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

Quote:docbrockIncidentally, 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. :)

Whether to pay commission up front or not...(as I understand it you bet $105, they put a marker somewhere, and pay $100 if you win)

(i) $420 with prepay will return $400.

(ii) $420 with commission will return $399 (as 5% of 420 is 21).

If you're aiming for minimum cost per hand rounding means you're better to bet $25 (the minimum) and pay commission rather than $26.25 pre-paid (since you're less than 50% to win the additional $1.25 commission-free element).

btw best of luck trying to memorise a viable strategy as playing optimally requires remembering a lot of exceptions, so in practice you'll be even less than 50% chance of winning vs losing any hand.

That's exactly what I'm seeing when I do the simple math to calculate EV with the goal of minimizing cost per hand/expected loss per hand...

Basically I used the Wizard's calculator and assumed a random hand, optimal strategy, facing the Taj house way, a non-prepaid 5% commission and the dealer always banking (I don't bank when there are other players at the table out of courtesy to them so they don't have to sit out, and to the casino, as I don't want to book their action and I don't want to deprive the casino of it either, especially when little old me is playing for minimum bets and the other players may be playing for hundreds or thousands a hand).

To go on a short tangent, it's not Vegas, but I do play in AC, and the table max at some casinos is 10k, and I rather frequently see players playing for 1k and up per hand, and pretty much always on the weekends playing for at least several hundred a hand. It's really a double benefit for me, because it keeps the casino's lights on, allows them to spread the game and cover all the overhead that comes with that, and I get to enjoy the game at my own level at the same time.

But anyway, we're talking fractions of a penny, but if my math is right I'm better off, as you said, betting $25 to win $23.75 than $27 to win $25.75 despite the fact that the outright commission remains the same ($1.25) and the percentage commission is lowered to about 4.63% by betting the $27 IF the dealer is always banking. The reason, I'd speculate, is that the house edge on the extra $2 outweighs the percentage reduction in commission and so still results in a lower EV even though the commission is cut down from 5% to just shy of 4.63%.

So, consider these two examples:

Betting $25 to win $23.75

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

= 7.05092375 - 7.450675

= -0.39975125

Betting $27 to win $25.75

= (0.296881)(25.75) + (0.298027)(-27)

= 7.64468575 - 8.046729

= -0.40204325

Now, of course, I'm not playing the optimal strategy because as you alluded to its very complex and technically would have to be customized to whatever particular house way one would be playing against. So it'd stand to reason that any strategy that is less efficient than optimal will result in an even greater expected loss per hand, assuming the dealer is always banking.

BUT, if I'm heads-up with the dealer and banking every other hand (and let's assume for the moment that I'm playing the optimal strategy), it IS to my benefit to bet $27 to win $25.75, because, if I'm seeing it right, I have a greater chance of winning than losing (because neither I nor the dealer have the advantage given by banking because we cancel each other out, and I'm playing a more efficient strategy than the house way). So it's in my best interest to minimize the percentage commission and therefore minimize expected loss per hand.

So, consider these examples, assuming optimal strategy, facing the Taj house way and that the dealer and I are alternating banking playing heads-up:

Betting $25 to win $23.75

= (.300084)(23.75) + (.292387)(-25)

= 7.126995 - 7.309675

= -0.18268

Betting $26.25 to win $25

= (.300084)(25) + (.292387)(-26.25)

= 7.5021 - 7.67515875

= -0.17305875

Betting $27 to win $25.75

= (.300084)(25.75) + (.292387)(-27)

= 7.727163 - 7.894449

= -0.167286

Betting $5000 to win $4998.75 (which obviously the casino would never allow)

= (.300084)(4998.75) + (.292387)(-5000)

= 1500.04490 - 1461.935

= 38.1099

Now, again, I'm not playing optimal strategy in practice, but consider this: Let's say both I and the house were playing the exact same strategy (in this case, the Taj house way) and also that we're still taking turns banking and playing heads-up. Interestingly, the expected loss of any size bet so long as the outright commission remains $1.25 is exactly the same, because the probability of a win and a loss are identical because both the house and I are playing exactly symmetrically. (I also verified this with the Wizard's calculator.)

So, if we let "p" be the probability of a win and loss, "b" be the amount bet and "v" the expected value per hand, then:

v = (p)(b-1.25) + (p)(-b)

v = (p)[(b-1.25) + (-b)]

v = (p)[(b-b) - 1.25]

v = (p)(-1.25) for all b

So, what I conclude just intuitively is that, assuming a constant outright commission (in this case $1.25), that the house and I are taking turns banking heads-up, assuming that I play a strategy (however heuristic) that is more efficient than the house way, even if it's not optimal, and assuming just for the sake of simplicity I have an infinite bankroll and don't have to worry about managing variance, I should bet as much as I can so long as the outright commission doesn't change - because my EV will be somewhere between "v" at the lower bound (playing the Taj house way at the worst) and would increase until it reaches the EV given by playing optimally.

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

LOL. That's hilarious.... I didn't play often and I haven't played for a long time, but when I played I played with three discs, I think like a "driver," a mid-range disc, and a "putter." But I see these guys walking around that literally have a bag full of, I don't know, 40, maybe 50 discs?

I'm terrible at disc golf, but I mean, 50 discs? Seriously? LOL.

I myself bought about ten discs and have been trying them out since I just recently started playing again. It's nice that it's free to play!

Quote:MrGoldenSunYeah, I am not great and only throw a few myself, but I know the guys you're talking about. I imagine it's just so easy to convince yourself the problem is the DISC, not you, and what if THIS ten bucks gets you the disc you really need...

I myself bought about ten discs and have been trying them out since I just recently started playing again. It's nice that it's free to play!

Nice....

Yea, it's definitely cool that it's free to play. I hear you too about the 'blaming the discs' thing-- But I like to think I know better bc I've seen my father buy this putter (for real golf), than that putter, and on and on and all the while his putting didn't really improve, or maybe just a little bit.

I guess in a lot of sports it's always easy to 'blame the gear' especially when you're early on in the learning curve.

I've only thrown a handful of different types of discs, but it's pretty clear at this point that no matter what disc I throw, it kinda flies up and to the right but tilted to the left, and starts to curve over more and more until it eventually hits the ground vertically about 25 feet in front of me. LOL. Pathetic.

But these guys with the 50-disc bags, I mean, yes, it looks kinda silly to see that, but when you watch them play they're incredible -- the distance they get, how the disc tracks horizontally and straight, their accuracy. And I guess if you're at that level you can tell the difference between the 50 discs and take advantage of the specific purpose they all have.

I kinda feel like it's a shame though that it's not really a sport that brings you riches and glory or recognition if you want all that, because a lot of these guys are incredibly talented. But disc golf is just one of those sports that you have to watch on "ESPN 8, the 'Ocho'" or read about in "Obscure Sports Quarterly". LOL.

Pros definitely can make use of dozens of different discs, but for me, I think I play fine with two midranges and one putter. I have the ten discs because I'm experimenting to figure out which ones I like best (e.g., I have four putters). Plus I have a driver that my arm isn't big enough to control...one day... :)

There are some pretty sweet disc golf videos. The pros make it look so effortless to throw 350'. Like ball golf, it's all about technique and timing, not simply swinging your arm real fast.

But yea, I think you're right, its about technique more than brute strength. I try to throw the disc as hard as I can and put my whole body behind it, but I still throw like 25' and the disc ends up probably rolling farther than it flew through the air.

Maybe the disc even generates a little lift when it's thrown properly, because when I watch the guys that are really good they'll throw a disc and midway through its flight it'll seem to rise up a little bit and then maintain its height until it finally starts to descend slowly and eventually hit the ground. My discs just plummet vertically when I throw them.

But I still enjoy it! Regardless of whether I'm playing with friends who are actually decent or who are as awful as I am. :)