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High Hand Jackpot: Hand Buying Strategy (ATTN: stats/math people) (1 Viewer)

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jbutler

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I've mentioned before that my local club takes $1 per hand for a high hand jackpot to be awarded at the end of the night. Typically it's a one table game, but it very often goes very late, so there is typically $200-300 in the jackpot by the end of the night (would be more, but the games are Stud 8 on Fridays and PLO and limit Tahoe on Sundays, so the hands take a bit longer than NLHE).

It's very common for people to offer and take offers to purchase or buy another player's high hand. Sometimes players are motivated because the offer is instant money rather than having to wait until the end of the night (or the next day if you don't play until the game breaks). Sometimes players are motivated simply because they're very conservative people and would rather have the bird in hand.

In any case, I'm known as a buyer of high hands, so I'm frequently asked if I'm interested. I haven't spent a significant amount of time trying to work out the exactly mathematically correct buying strategy, so I usually lowball the hell out of people and they sometimes take it.

I'd like to try to put together some kind of basic formula for working through how much I should be willing to pay for certain strengths of hands and when. The below are what seem to be to me the relevant variables. For the purposes of the exercise, let's just assume I'm always playing the PLO/Tahoe night. So the facts:

Games: PLO (4-card); Tahoe (5-card; must play either exactly two in your hand or all five).
Players: 8-max; typically full.
Hands per hour: Generally 25ish.

If there are other facts you think I should have that would help, let me know.

Certainly if anyone with a math/stats background wants to give me a quick and dirty guide to determining the price, it would be appreciated. Or if anyone has a place to direct me to look to figure it out myself that might work as well so long as it's a resource that might be understood by a layman.

You can also bookmark this thread for when one of your kids or students asks you when they're going to need this math bullshit anyway. Uh, when you're a degenerate gambler, kid.
 
This isn't going to be very amenable to a mathematical solution because the players hand selection and willingness to see a turn and river will make a difference, perhaps quite a difference. Missing data - how long is the session? What is the mix between Tahoe and Omaha? As the session winds down, does the game become short handed?

I think your table sense is going to serve you better than math. But if I were going to use math, I think a sampling technique is better than calculations. That means I would keep track of the big hands, quads or better but maybe aces full too (if that ever wins). Not just the winning hands but the contenders as well so that you have a table for what types of hands are out there.

As a buyer of high hands, I would be looking for big mistakes. Thin margins aren't worth the mental anguish unless you are trying for some meta-game effect.
 
DrStrange said:
This isn't going to be very amenable to a mathematical solution because the players hand selection and willingness to see a turn and river will make a difference, perhaps quite a difference. Missing data - how long is the session? What is the mix between Tahoe and Omaha? As the session winds down, does the game become short handed?

I think your table sense is going to serve you better than math. But if I were going to use math, I think a sampling technique is better than calculations. That means I would keep track of the big hands, quads or better but maybe aces full too (if that ever wins). Not just the winning hands but the contenders as well so that you have a table for what types of hands are out there.

As a buyer of high hands, I would be looking for big mistakes. Thin margins aren't worth the mental anguish unless you are trying for some meta-game effect.
Click to expand...

Length of game/# of players: Full table of 8 players for approximately 6 hours with the next 2-6 hours being between 4 and 6 players.
Mix: One orbit each. Occasionally at the end of the night (last 2 hours) one of the games will be dropped.

Virtually every hand in the Tahoe round is played to the river, almost all multi-way. If anyone has a legitimate shot at the high hand, they aren't folding until they see the river. PLO is played slightly more aggressively (a lot more aggressively when I or another particularly player are in the pot), so it ends on the flop or turn fairly often.

As far as my gorilla math method of gauging what holds up, prob 30% of the time it's a straight flush, 55% of the time it's quads, and 5% of the time it's a boat. I'm probably being overly liberal on the boat percentage. I can only remember once in probably 60 or more sessions that a boat won it. On the other end of the curve, I hit a king high straight flush one week and had to chop it three ways. Unreal to see three king high straight flushes in one night.
 
In a perfect world, there are four values we'd be able to work with:

J = jackpot amount at the end of the night
T = time in hours remaining in the session
H = average hands played per hour
P = probability (decimal/fraction form) of current high hand being beat on any individual hand (e.g., prob. of JJJJx or better if HH is TTTTA)

Given these values, the probability that the current high hand will stand up through any individual hand is 1 – P.

The number of hands left in the session is TH, so the probability it will stand up the rest of the night is (1 – P)^(TH).

Multiply that expression by J to find the EV of the standing high hand at that moment: J(1 – P)^(TH).

However, as DrStrange noted, all of these variables are estimates, and P is probably the toughest one to nail down. I agree that you're going to tend to make your money in the high-hand market by buying the high hand from players who grossly underestimate its value.

Also agree with sampling as a more realistic way to approach this problem, since those figures are much more available to you. Record the prevailing high hands over all of your sessions. When a buying opportunity comes up, write the fraction of the sessions where the current high hand would have stood up.

This fraction is a LOLsamplesize approximation of (1 – P)^(TH). Multiply the fraction by J to estimate your price, and go with your low estimate of J ($200). Offer a price substantially lower than that. Kick me and DrStrange 1% each when you cash. Profit.
 
Actually, I take back that whole part about using the fraction on the end. I'll leave it there as food for thought, but in retrospect, it's wrong.

The reason it's wrong—and why the prevailing high hand at the end of the night is not useful for this calculation—is because it completely ignores the number of hands remaining in the session.

Consider: Current high hand is AAAAK, first hand of the session. Compare this to when it's the second-to-last hand of the night. Pretty dramatic difference, obviously. Using the fraction would actually be your best approximation for the first hand of the session, but it would get worse and worse as time wears on.

Also, your percents only add up to 90%.
 
DrStrange said:
As a buyer of high hands, I would be looking for big mistakes. Thin margins aren't worth the mental anguish unless you are trying for some meta-game effect.
Click to expand...

Re: metagame of HHJ buying, about three months ago, one guy sold his high hand for $100, literally the next hand made a better high hand, sold that one for $100, then an hour later beat it again, didn't sell his third one, and ended up winning. So he made whatever the jackpot amount was plus $200 that night. Too bad it was one of the biggest assholes in the game, but I was still a little happy for him.

Jimulacrum said:
Consider: Current high hand is AAAAK, first hand of the session. Compare this to when it's the second-to-last hand of the night. Pretty dramatic difference, obviously.
Click to expand...

You'd love to hear some of the discussions at the table there. Even as a math ignoramus I can't believe some of it. Last night two guys had this exchange after someone made quad kings at about 11pm (game starts at 8pm):

Thing 1: "That'll hold up."
Thing 2: "It didn't last week."
Thing 1: "Yeah but Roy hit it at 8:30 last week."
Thing 2: "What difference does that make?"
Thing 1: "We had more time to beat it."
Thing 2: "That doesn't matter."
Thing 1: "Yeah you get to see more hands."
Thing 2: "It doesn't matter how many hands you see, it's just as likely every hand for a bigger hand to come out."
Thing 1: "Yeah I guess that's right. It just seems like it's easier to beat the longer you play."

I didn't insert myself into the conversation.
 
I'm in agreement with the issues put out by @DrStrange, @Jimulacrum, you're never getting an analytic solution here.

Based on your own observations, never buy boats.

How low are you getting away with low balling these guys? How often are you holding up?
 
I once at @MikesDad's cash game(I think properly) bought a $100 HHJ for $97 from the biggest nit at the table with an orbit left of NLHE. He had quad 8's.
 
BTW, I'm thinking about how to use that fraction (which I'm going to call F). It is certainly useful, and if you're not already, you should be recording those high hands each night you play.

Pretty sure you'd raise F to the portion of time remaining in the session. This makes sense because, assuming F is accurate, the probability of a hand standing up at the very beginning of the session should be equal to F, but at the very end of the session, it must be equal to 1 (because the current HH can't be beat).

So take F and raise it to (time remaining in session)/(total session time). Then multiply that by J. This is about as good an estimate as you're going to get using the LOLsamplesize method.

Since the game runs short-handed toward the end of the night, this calculation will be an underestimate of the hand's EV as the night wears on, since the probability of it being beat decreases with fewer players at the table.

We like underestimates, since we're trying to make money by buying low.
 
Chicken Rob said:
How low are you getting away with low balling these guys? How often are you holding up?
Click to expand...

The worst hand I've bought was JJJJK and it didn't hold up. My recollection is that I bought it for $10 or $20 at 11pm or midnight. Otherwise I've bought quads maybe half a dozen times, never for more than $40, and won twice. Once I bought a guy's 10 high straight flush for $120 when there was $200 in the HHJ at 2am and won that after it built up further to $300 before the end of the night.

These guys are almost all way too conservative in their view of the HHJ, so it's pretty easy to start low (literally $5) and eventually "begrudgingly" pay $40 for quad queens.
 
kirchhausen said:
I once at @MikesDad's cash game(I think properly) bought a $100 HHJ for $97 from the biggest nit at the table with an orbit left of NLHE. He had quad 8's.
Click to expand...

If you take the worst case scenario, 10 players at the table, and assume anyone who is dealt a pocket pair will see it all the way to the river, I believe the odds are actually around 5% (3% for 8 players as this reduces the number of players and the number of remaining hands). However, the assumption about all pocket pairs seeing their hands all the way to showdown is unrealistic and makes the calculation a bit too conservative. So with that in mind, I think the $97 buy-out was fair (though I personally wouldn't have done it just to win $3, because I myself am a nit).



edit: this assume you need to be holding a pocket pair to qualify for a quad high hand.
 
Chicken Rob said:
Based on your own observations, never buy boats.
Click to expand...

I could consider buying a boat, given the right price and little enough time remaining in the session. From Jack's examples, these guys love their insurance, even if they're taking a really short price. No sense in failing to take full advantage of that.
 
The main parameters appear to be:

1) the high hand

2) number of attempts to beat it (number of hands/hour x numbers of hours)

3) size of jackpot : cost to buy in

4) game - 4 cards in PLO, 5 cards in Tahoe; % PLO/% Tahoe

This might be a 3-dimensional graph, with jackpot/cost on one axis, # of attempts on another axis, and PLO/Tahoe game mix on the other axis.
 
At the risk of looking like an idiot, I took a few minutes to run some numbers for just PLO. Consider this directional as it has been a shitty day at work and my mind is mush :D

Assumptions:
  • Combinations below are accurate (I pulled them off a site and didn't have time to validate)
  • High Hand qualifier is assumed at being minimum 4 of a Kind (may not be fair, but I needed to keep the combinations reasonable)
  • VPIP is assumed at 25% (so not all 8 players will see every street in every hand)
  • >= Four of a Kind will occur one out of every 176 hands (see math)
    • At 8 players seeing every card, that means >= Four of a Kind will hit every 22 hands
    • Assuming 2 players each hand, this is adjusted to every 88 hands (4 x 22)
    • In a 10 hour session, roughly 3 qualifying hands should statistically occur (could be more or less in reality)
  • Assumed a straight flush is a guaranteed high hand winner (not necessarily true - but no time to run full combinations)
upload_2016-8-29_13-3-57.png


Thoughts:
  • CONSIDER TIMING!!! If about 3 qualifying hands occur per 10 hour session, there's a good chance lower quads won't win. However, the value greatly improves at hour #10.
  • Don't buy the HH for more than the EV unless the timing factor heavily favors you (I have a feeling that this won't be a problem based on the low $'s spent buying HH in the past)
If my math is jacked up, I'm sorry. I took Combinatorics in college like 13 years ago :cautious:
 
Last edited:
In the end, this is very much like an options trade. You have to look at the greeks.
  • 32769037c408874e1890f77554c65f39c523ebe2
    , the rate of change of the theoretical option value with respect to changes in the underlying asset's price. If you buy the hand now, or in an hour, what is the overall increase in cost?
  • Vega, the derivative of the option value with respect to the volatility of the underlying asset. More hands to be played changes the chance it will be cracked, but it also increases the value as it goes up a buck every hand.
  • bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b
    , the measure of the sensitivity of the value of the derivative to the passage of time. This is commonly referred to Time Decay.
    bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b
    is the reason I no longer invest in derivatives. If I could wrap my head around it, I would still trade options.
  • 1f7d439671d1289b6a816e6af7a304be40608d64
    , measures sensitivity to the interest rate. You're probably (hopefully) not taking out a loan to pay for your gambol, but if you shorten your stack you are losing jam-equity. In short, if you buy a high hand, it best be out of your wallet, or you are putting some rho at risk.
  • b43d0ea3c9c025af1be9128e62a18fa74bedda2a
    ,     is the percentage change in option value per percentage change in the underlying price. As we discussed, the value of the HH goes up each hand by $1, but the percentage value the HH increases is less and less with each hand (though still increasing). It's likelyhood of being cracked also goes down with each hand as there are now fewer hands left to try and crack it, and the percentage of remaining hands that can attempt to crack it increases with each hand - the second to last hand of the night has 50% of the chances to crack the current HH, and the last hand of the night has 100% of the chances to crack it.
These variables are not going to be nailed down in context of a game. You don't know how many hands are going to be played. You probably cannot even put a value on the Delta without stats that covered each individual and the price hand they are selling.

I have a love-hate relation with this level of economics. This post was not as much fun to write as I thought it was going to be.
 
Special thanks to @Jimulacrum who emailed me what I imagine will be a pretty useful spreadsheet to track the relevant data and which will produce recommendations to make better buying decisions. Looking forward to using it. And thanks to @MegaTon44 for the work put toward the above chart and to everyone else for their thoughts as well.

I'll update this thread after I get sufficient data to begin implementing the recommendations in Jim's spreadsheet. Should be interesting, but will take a while to get enough data for it to be useful. I'm sure you'll all be waiting on pins and needles ;)
 
jbutler said:
Special thanks to @Jimulacrum who emailed me what I imagine will be a pretty useful spreadsheet to track the relevant data and which will produce recommendations to make better buying decisions. Looking forward to using it. And thanks to @MegaTon44 for the work put toward the above chart and to everyone else for their thoughts as well.

I'll update this thread after I get sufficient data to begin implementing the recommendations in Jim's spreadsheet. Should be interesting, but will take a while to get enough data for it to be useful. I'm sure you'll all be waiting on pins and needles ;)
Click to expand...

I would have considered contributing greater, but I actually had to earn my paycheck today. :(

I'd love to see the jimulacrum recommendations when the data starts coming in.
 
jbutler said:
I didn't mention any names in the OP. I'm disappointed in you; nerds are typically much more pedantic.
Click to expand...

By your own admission in the second post, you meant to tag the people in the OP. As such, I think it's fair to consider post 1 and 2 to be the combined OP for the purposes of this thread. Any (other) lawyers want to chime in and help a brother out?
 
bentax1978 said:
By your own admission in the second post, you meant to tag the people in the OP. As such, I think it's fair to consider post 1 and 2 to be the combined OP for the purposes of this thread. Any (other) lawyers want to chime in and help a brother out?
Click to expand...

"Because someone else made a mistake I can now use words to mean things that they don't mean."

Okay, nerd.
 
Chicken Rob said:
I'd love to see the jimulacrum recommendations when the data starts coming in.
Click to expand...

It's really just an Excel sheet that has Jack input the variables' values described below, and and applies the formula J × F^[(time remaining)/(total time)]:

Jimulacrum said:
Record the prevailing high hands over all of your sessions. When a buying opportunity comes up, write the fraction of the sessions where the current high hand would have stood up.
Click to expand...

Jimulacrum said:
So take F [the fraction described above] and raise it to (time remaining in session)/(total session time). Then multiply that by J.
Click to expand...

ScreenHunter_623 Aug. 29 18.21.jpg


Output is a very rough approximate EV of the standing HH.
 
kirchhausen said:
I once at @MikesDad's cash game(I think properly) bought a $100 HHJ for $97 from the biggest nit at the table with an orbit left of NLHE. He had quad 8's.
Click to expand...

And I would gladly donate that $3 to you over and over again just to see your reaction the one time it doesn't hold :)
 

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