Here is one for fun.

In the game of heads-up no limit holdem, if both players play perfect GTO, then what is the value of the game to the player on the button, in terms of BB?

Any valid action within the rules of holdem are allowed, and each player has infinitely large stacks of chips. There is no rake.

I know -> minus Villain value.

Would finding the EV not require determining the GTO strategies for infinite stack holdem? Is this somehow simpler than with finite stacks?

The infinite stacks help with regard to stack-to-pot ratio being eliminated. The answer would not change with finite stacks, but discussion and equations would be needlessly complicated, in my opinion.

With tiny stacks, the asymmetry of the blinds would become a nuance for strategy as well.

yes it would.

With infinite stacks, why would anyone ever fold? You could just keep raising an infinite number of times and you'd still have an infinite number of chips behind. The hand would never end.
[ ] Fun game.

Just keep raising and drink more coffee than the other guy!

Easy: the value is 0bb because even if the BTN wins every hand, they make $0 profit because infinity + anything = infinity.

With much respect to your insights, Brokenstars, can you give an example of how finite stacks EV is different from infinite stacks EV?

The question of infinite reraises requires a valid behavioral model that includes a stop condition. This is at present undefined, but for the purpose of the question can be conjectured.

Assuming a perfect GTO call or fold will eventually happen should suffice for discussion, imo.

Fortunately for the purpose of discussion, betting a billion-billion dollars into a pot of 1.50 is very sub-optimal.

The optimal bet size has a local maximium and there is not a blast off to infinity.

You asked what's the Button's EV. Under the conditions stated, the EV is either 0 or -INF (depending on whether losing an all-in results in being felted). If they're not going all-in, then the EV is 0 regardless of whether they're betting $1 or a googolplex dollars.

I've even solved the game: Hero can open-fold AA and the EV is $0, which is also the value of winning the pot. So open-folding every hand would be an optimal fixed strategy. There are infinitely many optimal strategies.

Imo change the scenario from "infinite stacks" to "a gazillion big blinds" and then there can be a more meaningful discussion.

I’m not going to try to prove it mathematically but it seems intuitive that this theoretical EV that you’re searching for will vary based on stack sizes; because optimal strategy will vary based on stack size.

Also Infinity is probably not a good size to use for this hypo..

If hero on the button open-folds full range, then the EV of this pure strategy is -0.5 BB. While this is an answer, it being negative would seem highly unlikely.

If that leads to meaningful discussion, then by all means let that be a given. If the question of infinite stacks is a hindrance to discussion then an amount of a gazillion should suffice to remove SPR. The strategy of betting all-in from a GTO perspective requires SPR.

I agree that many people would agree with you intuitively. However, I affirm the idea the EV does not change with stack sizes, other than possible with very tiny stacks.

Remember, stacks are equal and both players are GTO.

I'm not sure how to prove it, but the strategy is going to be different at 5bb vs. 100bb. The power of position will increase, so as the stack depths increase the winrate of the player in position will likely asymptotically approach some value.

Trivial example, but suppose both players have 1bb. Then they will both get it in with 100% and thus each player has a 50% of winning. 0EV

It should be obvious that at higher stack sizes (including for example infinite), the EV of BTN will be greater than 0.

Very tiny stacks is granted to skew the results, ITT this is agreed thus far.

Pretty hard question to answer without solving the game first. But I would imagine the answer is more than 0 BB (since the button has an inherent advantage) and less than 1 BB (since otherwise the BB would do better by folding every hand).

All right! This is logical and gives a range to start with. Somewhere between 0 and 1BB. I don’t see how it could be anywhere outside this range.

So, even with 100k effective stacks, the EV will be at most 1BB.

Only if infinity - 0.5 < infinity, which would defy current mathematical theory.

Infinite stakes/bankroll turns just about any gambling question into a silly one. It can turn a -EV strategy into a "winning" one, for instance. This isn't the only thread I recall starting off goofy thanks to infinity (and I suppose wise-ass posters like myself deserve half the blame).

I'm pretty sure at even large stack sizes positional advantage still increases as stack sizes increases, which would be reflected in the winrate of the small blind player. I'm sure it would approach a limit though. There would be a larger difference between 100bb and 50bb than between 30,050bb and 30,000, but in the latter case its still non-zero.

In any case as long as there are finite stack sizes and a positional advantage for the sb, the only way you can know the advantage is to solve the game.

Yeh, pretty sure this is correct (or at least partially correct). A HU specialist chiming in might be helpful (WCGrider?!?!)

In honor of heehaww, let the effective stacks be 1 trillion BB.

The problem with OP's problem is there will always be finite stack sizes as an infinity of anything doesn't exist in reality. Infinity is simply a concept of method existing to be used as a mathematical tool.

I don't even think using stacks sizes as "small" as a bazillion or whatever is useful as such high dollar amounts will never be wagered let alone possessed by two individuals in reality.

If using a very high number of buy-ins is useful in answering the question, something like 20 to 30 should be at least somewhat realistic.

TBH, this reminds me of the silly "moral" questions of what should someone do with a track switcher where in one case everyone on a train will die vs some small number of people standing on another set of tracks, which completely drops a huge amount of context and also drops the fact that morality requires choice and in emergency situations choice doesn't exist and therefore neither does moral choice.