If you are playing pot limit holdem where you each ante one dollar and both player's have more than enough chips and each bet or raise is required to be pot size, how large is the largest pot that can be observed if both of you are playing expertly, and the board doesn't allow a tie for the nuts (and excluding both players having AA)?

The answer, whatever it is, is not a matter of opinion.

I raise on the button and a raising war breaks out between my KK and your AA. You reraise so many times that I decide to call.

Flop AKK

You bet and another raising war breaks out. I reraise until you decide to call.

Then I think it’s close between an Ace on the turn and an ace on the river.

Maybe...

What are the blinds?

What is the raise cap? 4 or 5?

no cap.

If we assume stacks are equal and some arbitrary size S it seems like it would certainly be possible to observe a pot size of 2*S (i.e. both players end up all in).

I mean if the final bettor is bluffing correctly there should be scenarios were the bettor can bet the nuts and hands that block the nuts while still getting a call.

This may be a bad example but what about:

Board:
6h8h9hJhAh

Bettor: Qh7H
Caller: KhTh

Bettor's nut range can be 7hTh and QhTh and the QH7H blocks all potential straight flushes without having the TH. The KHTH blocks most straight flushes except for 5h7h, but that hand would have to worry about QHTH so there is likely a point where that hand shows down before getting all in.

Definitely a very rare event so we could argue if we would EVER observe that happening in a realistic sample size seems like it would prove there are instances were both players end up all in.

We should usually be folding out about half of our opponent's range when we make a pot-sized bet, though not quite because our hands retain some equity when opponent doesn't fold (if they always folded 50% of their range, they would be over-folding in most situations)... There are 1326 possible combos... Log2 (1326) is between 10 and 11. Therefore, I think there should normally be no more than about 10 or 11 pot-sized bets, where our opponent's range could be limited to a single nut combo (eg, QQ on a Q345Q board).

There's definitely a lot of issues with this approximation, but I think it gives a reasonable ballpark estimate, so I'll state my actual answer is something between 8 and 15 pot-sized-bets.

Another thought I've had about this...

It seems like in NLHE, it's fairly rare to see much more than 200BBs go in preflop when one of the hands is not aces. I think the 200BBs must be correlated to the fact that you get aces 1 in 221 hands. Maybe realistically, this "limit" should be more like 221*1.5=332 BBs (1.5 BBs = BB plus SB in pot preflop).

If there actually is justification to that correlation, maybe there is a similar limit for any one player to commit something around 1326*1.5=1989 BBs in any given hand.

I think that is quite interesting.

Perhaps looking at the examples up thread and plugging some situations into a solver, we could get an idea of the shape of the pot growth.

Thinking ahead about this number:

I think we should call this number win P^i, honoring the classics imo.

I think we need to figure out which hand progresses like this:

nuts vs 2nd nuts = raising war ends with 2nd nuts calling.

then the flop comes and improves the 2nd nuts to the nuts, and regresses the nuts to the 2nd nuts. raising war ends with 2nd nuts calling.

then the turn comes and improves the 2nd nuts to the nuts, and regresses the nuts to the 2nd nuts. raising war ends with 2nd nuts calling.

then the river comes and improves the 2nd nuts to the nuts, and regresses the nuts to the 2nd nuts. raising war ends with 2nd nuts calling.

et voila.

I wouldn't be surprised if pzeroe's numbers were close enough to perhaps provide an interesting cross sectional view of the geometric pot growth and distribution. If one were so inclined visually to imagine such a pot that increases in size, and one that sees such massive ev swings from street to street, in the context of the original post, then one could also imagine every other possibility of ev distribution among expert play will create a pot that is smaller in size than the one produced by the answer to the original post.

We could then identify the limits of ev distribution further by identifying the smallest possible pot:

(ante + blinds + bring in) = win P^ii

and maybe we could do some work from there. Maybe even get a program that shows your ev as a pie chart as the hand progresses, and the program could record the distributions individually. Then all identical situations could be combined to provide an (average ev) from any decision point resulting in a pot size somewhere between P^i and P^ii.

I couldn't think of a hand that fit all of those qualifications, but there's this hand:

AA vs KK preflop raising war, KK calls.

K22r

raising war, AA calls.

Ar

raising war, KK calls.

K

raising war, AA calls.

-----

The most important assumption I've made is that there are no deuces in the preflop callers range, thus allowing AA to be the effective second nuts on the flop.

I think after a preflop raising war that AA and KK should be such a big part of both ranges that there wouldn’t be a huge amount of raising postflop.

There’s also the fact that both AA and KK probably want to mix by sometimes flatting (or possibly check/calling) all streets preflop through turn earlier than they need to. Hard to say though - is it necessarily going to always provide an EV advantage to raise with the nuts, even though that means sacrificing that part of your range for flats? Anyway, the rarity of flatting with a strong hand, I think could perhaps allow more raises/money to go in later than if it were played fast from the start..

That's probably true, my guess as to how much money goes in on that particular instance:

preflop with 3x opening raise: KK is in position and calls a 5 bet.

flop with 162bb in the middle, note that 6 betting would exceed the limit given upthread by pocketzeroes. AA bets pot on flop, KK raises, AA calls.

turn with 1458bb in the middle. AA checks, KK bets, AA raises, KK calls.

river with 13,122bb in the middle. AA bets, KK raises, AA calls.

final pot: 118,098bb in the middle. I think.

I think the deciding factor here is the assumption from the original post was that both players are experts. Experts by definition should know that an expert opponent can hold nutty combos on any board. Thus it seems unlikely to me that the answer to the original post would involve a preflop slowplay.

It would be interesting to come up with hypothetical ranges/weights after each action. Eg, on river, will it be possible for KK player to have enough non-nut/bluff combos for its raise to be called? Is that player so heavy with KK that AA should take a x/c line or even b/f rather than b/c?

Maybe the solution is actually found in the bluffing part of each player’s preflop range - again (like I said about slow-playing), bc of the unlikeliness of a given bluff combo, perhaps the unlikeliness of a given combo permits additional raises postflop. I’m not sure which bluffs might choose to flat a preflop 5bet though - maybe some SCs or medium to small PPs? It seems bad for a suited wheel ace to flat, but maybe they’d occasionally show up as well?

All in all, interesting stuff to think about, but it seems unlikely we’ll ever - in our lifetime - have a definitive answer to this.

I think we can rule out bet folding the second nuts. Maybe check call is better than bet call, but that seems a bit extreme to me.

If the game was no limit holdem, then we could see the value of protection going way up, thus limiting the preflop caller's ability to draw profitably postflop, thus limiting the number of combos that may call the preflop 5 bet.

However, given the specs from the original post, this is not the case in a pot limit game; the value of draws goes up; the number of combos that may call the preflop 5 bet goes up.

Hmm.

Since each player has a range consisting of 1326 hands and Aces are not equal for the sake of simplicity, we can assign a range of [1-1326] to the hands. Then, the last bet placed expertly will have to come from a range consisting of exactly these hands:1,1325,1326 and will be called for value by exactly hand 1324.

Each player starts with 1326 possible hands, each raise folds out half these hands, and it takes N number of raises to go from 1326possible hands down to 3 hands.

1326/2^N=3

1326/3=2^N

442=2^N

Log442/Log2=N

8.787903=N

Round up to 9 bets/Raises before the atomic nature of the starting hands means stop

Pot starts at 2 dollars

2^9=512 so the last raise makes the pot 512+512=1024 plus the final call of 512
makes a total of 1536 dollars in the pot.

Can someone doublecheck the math, I am quite sure on the theory.'

Thanks!