Here's a poker theory question that I may have some trouble explaining but will do my best.

Pretend we have no idea of the preflop action. We were warped into this hand on the flop and in position. The only details we know are the blinds, the current size of the pot and the current size of the bet.

.50/1

Hero is on the button with 98o. Villain is CO.

Flop is 942r

Does the bet amount in BB matter when facing a decision, independent of the actions before it? In other words, does our decision differ or is it the same in these 3 following situations (all situations have the same characteristics as shown above)

1. Pot is $10 - villain bets $3
2. Pot is $100 - villain bets $33
3. Pot is $10000 - villain bets $3333

Notice that it's the same percentage of the pot in each situation but the bet amount in BBs are very different.

I guess my question is this - Do we need to have stronger holdings when more money is bet, even though the bet size relative to the pot has stayed the same?

You can’t separate the flop decision from he preflop action.

The relevant factors for a decision are the ranges of each player, the board state and the state of the action. Number of BBs is irrelevant except that it could give us info about the ranges if we are plopped into this situation without any other knowledge. Our own holding also gives a hint about ranges.

The example feels extremely contrived and it’s not easy to give a satisfactory answer. We always have knowledge of preflop actions. This is the best answer I think I can give.

I am not sure you can make a conclusion about whether you need stronger or weaker holdings, but I think you have to see that the preflop action affects the ranges your opponent holds, and therefore your continuing range.

If somehow you knew that your opponents range was the same in all 3 cases then with some caveats the answer is, yes, you call with the same ranges in all 3 cases.

The main caveat is, because the bet in #3 is so large compared to the BB, it's likely to be a very large percentage of your bankroll - it's 33 buyins. If you sat down with 1 buyin, with a 20 BI bankroll, and ran it up to 33 buyins, then your total bankroll is now 52 buyins and you're being asked to call 33, about 64% of your bankroll. The kelly criterion comes into play with bets that are this large of a percentage of your total bankroll.

Let's assume that calling the $3333 closes the action. So you're risking 3333 to win 13333
let's call your chance to win p
let's call your payout b, as in b:1

The kelly criterion says the fraction of your bankroll you should risk on a bet is
f = (bp - (1-p)) / b
so let's put some numbers in and solve
b = 13333/3333 = 4
f = .64
.64 = (4p - (1-p)) / 4
2.56 = 4p - (1-p)
2.56 = 4p - 1 + p
2.56 + 1 = 4p + p
3.56 = 5p
p = 3.56/5 = .712

So you'd need about 71% equity to "want" to make this bet.

Note that if your bankroll is very large, then the effect of the kelly criterion diminishes

I forgot to bring up the Kelly Criterion.

My only beef with your post is that the Kelly Criterion tells us the optimal bet sizing, not whether or not to accept a bet. We don't need as much as 71% equity to want to call. Any sizing up to double Kelly has expected bankroll growth.

That's a good point, the kelly criterion comes at it from the other direction. If we assumed that this bet had to be within the double-kelly envelope, then the single-kelly size would be for half that kelly fraction, 32%. Re-running the numbers you get about 46% equity needed.

Still, it's a lot more than pure pot odds suggest, which is 20%

kinda need to know stack depth

Thank you, Rusty. This is what I was looking for and kind of where my gut was leaning when proposing the question.

I ignored preflop for two reasons:

1. I didn't want anyone to focus on what happened preflop (I needed to properly set up the third situation which is so ridiculous and would never happen)
2. I didn't want anyone to focus on ranges (to isolate just the pot odds being offered)

Now I think your answer is very interesting as it shows there are other forces to consider outside of the current poker decision. These forces are either stronger or weaker depending on the game you're in.

To illustrate my experience - I took a shot at a bigger game and got myself in an uncomfortable 0EV spot in which I felt these "Kelly Criterion" forces acting on my decision (I was unfamiliar with this concept before your post so bear with me if I'm using it incorrectly). It felt right folding a larger part (maybe 4 or 5 hand combinations) of the bottom of my range in this spot even though an optimal strategy would have me calling those hands.

I want to set up one more situation on the other end of the spectrum to help illustrate and see if these forces still hold theoretically true.

You have a total bankroll of 20k and are taking a huge shot at 100/200NL for one last attempt at the big leagues.

HERO raise 87s in the CO to $600 and is 3bet by the SB to $2400. HERO calls.

Flop is 842r and SB continues for $4100. Is this now a fold based on our current bankroll status and the Kelly criterion?

That was the first thing I thought of when I read the OP. If hero only has 1bb, his decision is trivial. If he has 1000bb, it's less trivial.

I think inherent in the question is the notion that whatever the bet size your opponent makes is, you can call it.

I think it becomes very hard to reason about if the call doesn't close the action. And as I mentioned, unless you hand-wave away the differences in opponent ranges, there's not much you can do or say about it at all.

How you play one hand determines how to play others. I think it is very important to have a range and to understand where your current hand lies within that range. I base 99.99% of my post flop decisions on my range and where I'm currently at within the range.

Well tbh this seems like a silly situation. Playing 100/200nl with 20k total bankroll is silly. Also, at first glance I say I would have folded to the 3bet with 87s and after looking at the situation in snowie for full ring and 6max it agrees with me. I would have called with 76s/65s but not 87s but for this example I'll include 87s and not include 65s in my calling range. I'll explain why knowing your range and where you're at is very important too.

So lets define a fr CO raising range as:
44+, A2s+, K5s+, Q8s+, J9s+, T8s+, 98s, 87s, 76s, 65s, A9o+, KTo+, QTo+, JTo (360 combos)

Now that the villian 3 bet our calling range is: (I included 87s where it wouldn't normally be in my calling range there)
QQ-55, AQs-ATs, KTs+, QTs+, JTs, 87s, 76s, AQo+ (116 combos)

Now on the flop we rank our range as it relates to the flop:

Flop: 842

88, 44, QQ, JJ, TT, 99,87s,77,66,55,AKo,AQs,AQo,KQs,AJs,KJs,ATs,QTs,JT s,76s

1/(1+(4100/5000))=54.9% - 55%

The pot is 5k(2400+2400+200bb) and the bet is 4100. We need to continue with 55% of our range that beats a bluff to make our opponent indifferent to bluffing. I'm super likely to call my bottom bluff catcher 55-AKo here. I'll use AKo as the cut-off. That gives us 11x hands that beat bluff and we need to be continuing 9*0.55=6 hands or 88,44,QQ,JJ,TT,99. Continuing includes both calling and raising so some of these hands we're going to be raising. So to do that we use the following formula to determine how many bluff hands are optimal: (4100/5000)/(1+(4100/5000))=45%. So we need 6*0.45=2.7 of the best hands we would have otherwise folded(below our bluff catcher cutoff) to become our bluffs. Giving a bluff range of something like AQo,KQs and something like 76s like 70% of the time. We could argue about the last bluff hand but I'll just go with that for now.

As played I'm 100% folding 87s here(would have folded to 3bet preflop) and my continuation range(including raise, call and bluff range) is 88,44,QQ,JJ,TT, 99,AQo,KQs,76s

I went through this process for this hand to show that our range and where we are currently at within that range makes a big difference in how we play. Overall I think looking at specific hands in specific situations like this and ignoring our ranges is just bad poker. One could argue that this was a thought experiment but imho it wasn't thought out very well because our hand is a pretty clear fold to a sb 3 bet preflop unless you're raising a much wider range than I posted. Which would also probably be bad poker assuming it is wide enough to make 87s a call to the 3bet. Not to mention having your entire bankroll sitting on a table is a really bad idea.