The Poker Theory of Poker Night

Poker seems nice as a hobby, but terrible as a job as discussed on the motte.

Also, if all bets were placed before the flop, the equilibrium strategy would probably be to bet along some fixed probability distribution depending on your position, the previous bets and what cards you have. Instead, the three rounds of betting after some cards are open on the table make the game much more complicated. If you know you have a winning hand, you do not want your opponent to fold, you want them to match your bet. So you kinda have to balance optimizing for the maximum pool at showdown with limiting the information you are leaking so there is a showdown. Or at least it would seem like that to me, I barely know the rules.

Role playing groups have a similar conundrum. In some way, it is even more severe because while you can have switching members of a poker night, having too many switching members in a role playing game does not work great. On the other hand, typical role players don't have 56 things they would be rather doing. (Personally, I think having five people (DM plus four players) is ideal because you have a bit of leeway to play even if one cancels.) So far, my group manages ok without imposing penalties on players found to be absent without leave.

[-]Lukas_Gloor1y*20

If you know you have a winning hand, you do not want your opponent to fold, you want them to match your bet. So you kinda have to balance optimizing for the maximum pool at showdown with limiting the information you are leaking so there is a showdown. Or at least it would seem like that to me, I barely know the rules.

This is pretty accurate.

For simplicity, let's assume you have a hand that has a very high likelihood of winning at showdown on pretty much any runout. E.g., you have KK on a flop that is AK4, and your opponent didn't raise you before the flop, so you can mostly rule out AA. (Sure, if an A comes then A6 now beats you, or maybe they'll have 53s for a straight draw to A2345 with a 2 coming, or maybe they somehow backdoor into a different straight or flush depending on the runout and their specific hand – but those outcomes where you end up losing are unlikely enough to not make a significant difference to the math and strategy.)

The part about information leakage is indeed important, but rather than adjusting your bet sizing to prevent information leakage (i.e., "make the bet sizing smaller so it's less obvious that I've got a monster"), you simply add the right number of bluffs to your big-bet line to make your opponent's bluff-catching hands exactly indifferent. So, instead of betting small with KK to "keep them in" or "disguise the strength of your hand," you still bomb it, but you'd play the same way with a hand like J5ss (can pick up a 1-to-a-straight draw on all of the following turns: 2,3, T, Q; and can pick up a flush draw on any turn with a spade if there was one spade already on the flop).

To optimize for the maximum pot at showdown and maximum likelihood of getting called for all the chips, you want to bet the same proportion of the pot on each street (flop, turn, and river) to get all-in with the last bet. (This is forcing your opponent to defend the most; if you make just one huge bet of all-in right away, your opponent mathematically has to call you with fewer hands to prevent you from automatically profiting with every hand as a bluff.)
So, if the pot starts out at 6 big blinds (you raise 2.75x, get called by the big blind, and there's a 0.5 small blind in there as well). Your stack was100 big blinds to start. If you were to bet 100% of the pot on each street, this would be 6+18+54, which is 78, so slightly too small (you want it to sum to 100 or [100 minus the preflop raise of 2.75 big blinds or whatever]). So, with your very best hands, it's slightly better here to bet a little larger on each street, like 110% pot. (So you get something like 7 into 6 on the flop, 22 into 20 on the turn, and 70ish into 64 on the river for the rest of your stack.)

Let's say the board runs out very dry AK469, no flushes possible. You have KK, your opponent has A2 as a bluffcatcher (or AT, doesn't make a difference here because you wouldn't want to take this line with anything worse for value than AQ probably). If you get the ratio of bluffs-to-value exactly right, then your opponent is now faced with a choice: Either forfeit what's in the pot right now (zero further EV for them), or look you up with a bluffcatcher (also zero EV – they win when you're bluffing, but they lose when you've got it). If they overfold, you always win what's already in the pot and your bluffs are printing money. If they overcall, your bluffs become losing plays (and if you knew the opponent was overcalling, you'd stop bluffing!), but your value hands get paid off more often than the should in game theory, so the overall outcome is the same.

Of course, what is a proper "bluff-catching hand" depends on the board and previous action. If someone thinks any pair is a good bluffcatcher, but some of your bluffs include a pair that beats their pair, then they're committing a huge blunder if they call. (E.g., they might consider bluffcatching with the pocket pair 88 on this AK469 runout, but you might be bluffing J9s on the river, which is actually reasonable since you get remaining Kx to fold and some Ax to fold and your 9 is making it less likely that they backed into an easy-to-call two pair with K9 or A9. So, if you bluff 9x on the river, they're now giving you a present by ever calling with 88 [even if you have some other bluffs that they beat, their call is still losing lots of money in expectation because the bluff-catching ratio is now messed up.]) Or, conversely, if they fold a hand that they think is a bluffcatcher, but it actually beats some of your thinnest value hands (or at least ties with them), they'd again be blundering by folding. So, for instance, if someone folds AQ here, it could be that you went for a thin value bet with AQ yourself and they now folded something that not only beats all your bluffs, but also ties with some value (again changing the ratio favorably to make the call fairly highly +EV). (Note that a call can be +EV even if you're losing a little more than 50% of the time, because there's already a fair bit of money in the pot before the last bet).

The above bet-sizing example had some properties that made the analysis easy:

Your KK didn't block the hands you most want to get value from (Ax hands).
[KK is actually a less clear example for betting big than 44, because at least on flop and turn, you should get value by lots of Kx hands. So 44 prefers betting big even more than KK because it gets paid more.]
The board AK4 was pretty static – hands that are good on the flop tend to be still good by the river. (Compare this to the "wet" board 8s75s, the "ss" signifying that there's a flush draw. Say you have 88 on this board for three of a kind 8s. It's unlikely your opponent has exactly 96 or 64 for a straight, so you're likely ahead at the moment. But will you still be ahead if a 6, a T, or a 4 comes, or if the flush comes in? Who knows, so things can change radically with even just one more card!).
If we assume that you raised before the flop and got merely called by the big blind (everyone else folded), you have the advantage of top hands on this board because the big blind is really supposed to always raise AK, KK, and AA. "Trapping" them for surprise value would simply not be worth it: those hands win a lot more by making the pot bigger before the flop. Having the advantage in top hands allows you to bet really big.

These three bullet points all highlight separate reasons why a hand that is great on the flop shouldn't always go for the sizing to get all-in by the river with three proportional big bets ("geometric sizing"). To express them in simple heuristics:

Bet smaller (or throw in a check) when you block the main hands you want to get called by.
Bet smaller (or throw in a check) when a lot can still change later in the hand because of how "wet" the board is.
Bet smaller (or throw in a check) if your opponent has the advantage of top hands – this means they should in theory do a lot of the betting for you and bet big and include bluffs. (Of course, if your opponent is generally too passive, it's better for you to do the betting yourself even if it's not following optimal theory.)

There's also a concept where you want to make sure you still have some good hands in nodes of the game where you only make a small bet or even check, so your opponent cannot always pounce on these signs of weakness. However, it's often sufficient to allocate your second tier hands there, you don't need to "protect your checking range" with something as strong as KK. This would be like allocating Achilles to defending the boats at the beach when the rest of your army storms against the walls of Troy. You want someone back there with the boats, but it doesn't have to be your top fighter.

Lastly, one other interesting situation where you want to check strong hands is if you think that the best way to get the money in is "check-in-order-to-check-raise" rather than "bet big outright." That happens when your "out of position" (your opponent will still have the option to bet after you check) and your opponent's range is capped to hands that are worth one big bet, but are almost never worth enough to raise against your big bet. In that situation, you get extra money from all their bluffs if you check, and the hands that would've called your big bet had you bet, those hands will bet for you anyway, and then have to make a though decision against your check-raise. So, let's say you're the big blind now on AK4 and you hold 99. This time, your opponent checks back the flop, indicating that they're unlikely to have anything very strong (except maybe AA that can't easily get paid). After he checks back, the turn is a 6. You both check once again. At this point, you're pretty sure that the best hands in your opponent's range are one pair at best, because it just wouldn't make sense for them to keep the pot small and give you a free river csard with a hand that can get paid off. The river is a 9 (bingo for you!). In theory, your opponent could have 99 themselves or back into two pair with K9, 96s, or A9 that weirdly enough didn't bet neither the turn nor the flop. However, since you have two 9s, you're blocking those strong hands your opponent could have pretty hard. As a result, if you were to bet big here with 99, your most likely outcome is getting called at best. That's a disaster and you want to get more value. So, you check. Your opponent will now bet KQ for value (since most of your Ax would've bet the river) and will bet all their weak Ax for value that they got there in the check-flop, check-turn line. Against this bet, your 99 now (and also A9 to some degree, but having the A isn't ideal because you want to get called by an A) can jam all-in for 10 times the size of the pot to make all your opponent's hands except rivered A9 indifferent. Any smaller bet sizing wouldn't make a lot of sense because you're never getting raised, so why not go for maximum value (or maximum pressure, in case you're bluffing). Your bluffs to balance this play should all contain a 9 themselves, so you again make it less likely that your opponent holds 99 or A9/K9. (Note that 9x can also function as a breakeven call against a river bet after you check, catching your opponent's river bluffs. However, since 9x never beats value, so you're not really wasting your handstrength by turning it into a bluff some of the time.) If you have the right ratio of bluffs-to-value for your 10x-the-pot overbet, your opponent is supposed to fold all Kx and most Ax hands, but some Ax has to defend because it blocks the A9 that you also go for value with. So, the opponent has to occasionally call this crazy big raise with just an A. If they never do this, and you know they never do it, you can check-jam every single 9x on the river and it makes for a +EV bluffs. (You probably cannot just check-jam every single hand because if you don't have a 9, the times your opponent has A9 or 99 or K9 are now drastically increased, and you'll always get called by those.)
Of course, if your opponent correctly guesses that you're jamming all your 9x there, they realize that you have a lot more 9x-that-is-just-one-pair than 99/A9/K9, so they'll now exploitatively call all their Ax. Since you now bet 10x the pot as a bluff with too many bluffs in expectation, you're losing way more than you were standing to gain, so you have to be especially careful with exploits here when betting many times the pot on the river.)

[-]quiet_NaN1y10

Thanks, this is interesting.

From my understanding, in no-limit games, one would want to only have some fraction of ones bankroll in chips on the table, so that one can re-buy after losing an all-in bluff. (I would guess that this fraction should be determined by the Kelly criterion or something.)

On the other hand, from browsing Wikipedia, it seems like many poker tournaments prohibit or limit re-buying after going bust. This would indicate that one has limited amounts of opportunity to get familiar with the strategy of the opponents (which could very well change once the stakes change).

(Of course, Kelly is kind of brutal with regard to gambling. In a zero sum game, the average edge is zero, so at least one participant should not be playing even from an EV perspective. But even under the generous assumption that you are 50% more likely than chance to win a 50 participant elimination tournament (e.g. because a third of the participants are actively trying to lose) (so your EV is 0.5 the buy-in) Kelly tells you to wager about 1% of your bankroll. So if the buy-in is 10k$ you would have to be a millionaire.)

[-]Lukas_Gloor1y20

Yeah, you need an enormous bankroll to play $10,000 tournaments. What a lot of pros do is sell action. Let's say you're highly skilled and have a, say, 125% expected return on investment. If you find someone with a big bankroll and they're convinced of your skills, you can you sell them your action at a markup somewhere between 1 and 1.2 to incentivize them to make a profit. I'd say something like 1.1 markup is fairest, so you're paying them a good prize to weather the variance for you. At 1.1 markup, they pay 1.1x whatever it costs you to buy into the tournament. You can sell a large part of your action but not quite all of it to keep an incentive to play well (if you sold everything at $11,000, you could, if you were shady, just pocket the extra $1,000, go out early on purpose, and register the next tournament where you sold action for another round of instant profit).

So, let's say they paid you $8,800 to get 80% of your winnings, so they make an expected profit of ($8,000 * 1.25) - $8,000, which is $1,200. And then you yourself still have 20% of your own action, for which you only paid $1,200 (since you got $800 from the 1.1 markup and you invest that into your tournament). Now, you're only in for $1,200 of your own money, but you have 20% of the tournament, so you'd already be highly profitable if you were just breaking even. In addition, as we stipulated, you have an edge on the field expecting 125% ROI, so in expectation, that $1,200 is worth $2,000*1.25, which is $2,500. This still comes with a lot of variance, but your ROI is now so high that Kelly allows you to play a big tournament in this way even if your net worth is <$100k.

(This analysis simplified things assuming there's no casino fee. In reality, if a tournament is advertized as a $10k tournament, the buy in tends to be more like $10,500, and $500 is just the casino fee that doesn't go into the prize pool. This makes edges considerably smaller.)

Regarding busting a tournament with a risky bluff: In the comment above I was assuming we're playing cash game where chips are equivalent to real dollars and you can leave the table at any point. In tournaments, at least if they are not "winner takes it all" format (which they almost never are), there's additional expected value in playing a little more conservative than the strategy "maximizing expected value in chips." Namely, you have to figure out how "having lots of chips" translates into "probabilities of making various pay jumps." If you're close to the money, or close to a big pay jump when you're already in the money (and at a big final table, every pay jump tends to be huge!), you actually make money by folding, since every time you fold, there's a chance that some other player will go out (either behind you at your table, or at some other table in a tournament where there are still many tables playing). If someone else goes out and you make the pay jump, you get more money without having to risk your stack. So, in tournaments, you gotta be more selective with the big bluffs for multiples of what is already in the pot, especially if you think you have an edge on the field and if the pay jumps are close.

[-]Alexei1y48

I like the idea and would consider doing something like that in the future. Thanks! FWIW, I found the explanation of poker completely extraneous to the main point.

[-]omark1y10

Do you mean the explanation of Poker details is too lengthy or that you don't see value in making an analogy between how the blinds in Poker incentivize the game and how a similar social rule could incentivize a group gathering?

[-]Alexei1y41

The former. I think you can just explain the blinds without explaining the entire poker game.

[-]Daniel V1y10

I agree, I was thinking more generally this isn't a "poker" theory specifically, just one about rules and buy-in. But it's about poker night, so I'll let it slide. The main game rules, though, remain extraneous. Loved the post still!

[-]Dagon1y36

I've played a lot of poker, both socially and in cardrooms. I would probably not join a game that needs legible commitment mechanisms. If it's not sufficiently high-value to succeed on it's own, I probably have better things to do most of the time.

Mostly my recommendation is to find/create games that people like enough and feel safe enough to tell each other their expected attendance. Also, get good at short-handed play - it's useful if a few people drop out, but even more useful to be able to play while some people are still en-route.

[-]Kenoubi1y01

I started a book club in February 2023 and since the beginning I pushed for the rule that if you don't come, you pay for everyone's drinks next time.

I'm very surprised that in that particular form that worked, because the extremely obvious way to postpone (or, in the end, avoid) the penalty is to not go next time either (or, in the end, ever again). I guess if there's agreement that pretty close to 100% attendance is the norm, as in if you can only show up 60% of the time don't bother showing up at all, then it could work. That would make sense for something like a D&D or other tabletop RPG session, or certain forms of competition like, I dunno, a table tennis league, where someone being absent even one time really does cause quite significant harm to the event. But it eliminates a chunk of the possible attendees entirely right from the start, and I imagine would make the members feel quite constrained by the club, particularly if it doesn't appear to be really required by the event itself. And those don't seem good for getting people to show up, either.

That's not to say the analogy overall doesn't work. I'd imagine requiring people to buy a ticket to go to poker night, with that ticket also covering the night's first ante / blind, does work to increase attendance, and for the reasons you state (and not just people being foolish about "sunk costs"). It's just payment of the penalty after the fact, and presumably with no real enforcement, that I don't get. And if you say it works for your book club, I guess probably it does and I'm wrong somehow. But in any case, I notice that I am confused.

[-]omark1y10

I'm very surprised that in that particular form that worked, because the extremely obvious way to postpone (or, in the end, avoid) the penalty is to not go next time either (or, in the end, ever again).

This is definitely based on two assumptions that I mention in the article:

Let's assume everyone who joins generally enjoys it.

Just like in Poker the negative impact should be small. There is a reason why blinds and antes are small amounts compared to your entire stack.

If people don't really want to attend or the costs of the "blinds" are huge then things are different.

That being said, you raise a good point. I can elaborate a little about the book club:

You can cancel "for free" if you do it sufficiently in advance (in theory 7 days, in practice 5 seems ok). This allows postponing if too many people cancel.
You can completely skip any book you don't find interesting (books are chosen via voting so only books that are generally popular make the cut).
There are now 10 attendees so paying for drinks is getting expensive. We are discussing how to keep it simple (e.g. collecting money to later spend it seems annoying) but also reduce the costs.
In practice everyone ends up paying occasionally so it evens out.
Some attendees feel ambivalent about the rule because it's constraining as you wrote. As I mentioned, it's important to be careful (and communicate well) about such things.

[-]Kenoubi1y10

Ah, okay, some of those seem to me like they'd change things quite a lot. In particular, a week's notice is usually possible for major plans (going out of town, a birthday or anniversary, concert that night only, etc.) and being able to skip books that don't interest one also removes a major class of reason not to go. The ones I can still see are (1) competing in-town plans, (2) illness or other personal emergency, and (3) just don't feel like going out tonight. (1) is what you're trying to avoid, of course. On (3) I can see your opinion going either way. It does legitimately happen sometimes that one is too tired for whatever plans one had to seem appealing, but it's legitimate to say that if that happens to you so often that you mind the cost of the extra rounds of drinks you end up buying, maybe you're not a great member for that club. (2) seems like a real problem, and I'm gonna guess that you actually wouldn't make people pay for drinks if they said they missed because they had COVID, there was a death in the family, etc.?

[-]omark1y10

I'm gonna guess that you actually wouldn't make people pay for drinks if they said they missed because they had COVID, there was a death in the family, etc.?

This is a tough call. How do you determine what is a "legitimately bad enough" case to miss the event? The examples you mention are clearly bad enough but there are other situation where it's much more personal. If I'm feeling low on energy is that a choice I am making or an unavoidable fact about my metabolism? You would have to set up some kind of tribunal or voting for deciding on these cases. That's a lot of effort and would only create bad vibes. So no, if you don't come you pay, no matter the reason. However, enforcement is lax. Mostly it's up to the people themselves to say "Yeah, today is my turn since two weeks ago I couldn't make it". If someone considers their case to be special they can easily get away with not paying and in all likelihood nobody would even notice let alone question it.

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29

The Poker Theory of Poker Night

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Aside: Poker Basics

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Returning to Poker nights

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