One hundred dollar stacks. Two GTO expert players who know each other as very tough. One dollar on the button and two dollars on the big blind. Normally the button has the edge. Except for one thing. The big blind sees some of the cards on the bottom of the deck that are out of play but the button doesn't.


1. If the button doesn't realize this, how many cards must the blind see to have a positive EV that hand?


2. What if both players know of the extra info the blind has. How many cards must he see now?

I think its a decent amount. Blockers and info on drawing outs is very relevant but the fact that big blind is simple posting twice as much dead money pre as the small blind is just so huge (much more relevant than positional advantage of small blind).

I would say maybe around 12-15 cards if both players knew what's up.

If the sb is unaware that big blind has this extra advantage then its a case where you can lock in a very exploitable strategy and just calculate the nemesis. There also would be weird double-blocker opportunities. For example in a normal game if you shove 100 bb repping the nut flush, your opponent might call x% of the time with a king high flush and fold the rest, which might be enough to combat shoving the lone ace blocker. So if you have both blockers, from your opponent's perspective without correct game info it just means you actually have the nuts, with the understanding you can't just do this ever with pure air since you'll be running into king-high or ace-high flush too often. So your opponent might be folding 100% of their range in that spot, meaning every time there's one of these double-blocker scenarios you just win the pot automatically.

Wouldn't be surprised if in the hidden-advantage scenario BB needs as low as 4 cards or something.

It will depend on BB's hand and to a certain extent the cards exposed and how they interact with eachother.

It will also vary by the board texture and any prior information we have on villain's range because obviously some cards interact more with the hidden information than others so provide more information about the current game state than others.

For example there are clearly hands that are +EV when all information is hidden (QQ+, AK, etc) so the answer is 0 cards in that respect.

I guess if you just mean guaranteed in the context of that 1 hand we have to go back to the board/range interaction.




2. What if both players know of the extra info the blind has. How many cards must he see now?[/QUOTE]

Do both players know the cards that were exposed or the other player simply knows the player saw exposed cards?

I assume the former because it's more interesting. If the latter then the player should ask for a redeal.

If the information of exposed cards is shared then I think it becomes a more abouy the interaction of the board cards and the player's ranges. Whichever player's range is narrowed and capped the most by the cards already exposed would seem to be at the disadvantage.

I think David is asking about a "rule change" that is baked into the game. That is, it is in force for each and every deal.

I guess I am not sure how that changes the answer but it has been a long day so it's probably just me missing something.

You cannot say it depends upon the cards/board.

He asked a fairly simple question (which undoubtedly is very very difficult to answer). Exactly how many cards must the big blind see from the bottom of the deck, on each and every deal, for the EV of the big blind to become positive (where the EV is taken over all possible deals of HUNL).

As people have pointed out, the "benefit" varies with the exact cards dealt and the board. In theory, you could calculate the "benefit" for each possible number of seen cards (0, 1, 2, ... ) and calculate the big blind's EV at that setting and find at what number it turns positive.

As how you calculate the "benefit" for any specific deal is not easy to determine, I don't imagine David is looking for analytical solutions.

My guess is that it is around 5 cards in the first scenario and 10 cards in the second scenario. But those are pure guesses based on very little.

I guess I was merely trying to point out as you did below in a more concise and perhaps better way, that it varies based on what is already know. If I know the Ah is missing and I am in a spot where it's important then I only need to know the Ah is missing from my opponent's range.

Perhaps I didn't abstract far enough to the point that I should say it depends on the statistical distribution of our opponent's range and the distribution of the revealed cards in that range?

I thought that knowing the Ah is at the bottom of the deck is substantially more important on it's own than knowing 2h, 3c, 2s, 4d are on the bottom of the deck simply for the fact they don't often instersect well with our opponent's range or the board? So it could clearly vary just by know what cards you have seen which can include your own.

Perhaps we could ask at what number of cards can, ensure with a certain probability you have seen enough cards to guarantee a +EV play given the stastical distribution of your opponent's range?


Agree with all of the above. Hope I am not sounding too defensive this interests me quite a bit and it's clear you're more practiced and knowledgeable in these things. Thanks!

Nevermind I have thought about this more and convinced myself our opponent's strategy isn't necessarily relevant.

We could think of every preflop hand as having some EV in this game. Our opponent's strategy would just be those hands that have an EV greater than folding based on position. Same would hold for our strategy.

All other hands that aren't in the strategy because they don't meet the criteria above we could just consider EV 0 because they are folded.

For scenario 1 we could then loosely associate an EV value with a single card based on the number of preflop hands it belongs to that would appear in the hands that have EV greater than folding.

When we consider question 2 I would think we would have to also consider the intersection of exposed cards with hands that have an EV greater than folding for our own position.

As everybody know, the advantage of knowing the bottom N cards is not only the "blocker effect" (those cards cannot be in Villain's hand) but also knowing that those N cards cannot appear on the board.

I imagine both effects are significant (i.e., affect the overall EV).

Yes agree. Sorry I didn't mean to make the focus soley about preflop.

I meant to use that more of short hand for the beginning of the game tree, and the cards revealed could also be included in boards on the path from the start of the game tree to the terminal node where the EV would determined.

Thanks for your input!

Think both estimates are too high. In any case the most fascinating aspect to me is how the strategy changes when they both know (that the blind saw ten dead cards). Please program your computer, as you have many times in the past, to give me the answer.

I don't think you realize the complexity of the task. Answering the question you posed is orders of magnitude more complex than solving the entire game of HUNL, which nobody has come close accomplishing. This isn't something you can solve with simulations. We really can't do much better than make wild guesses.

The best someone might be able to do would be to solve a simplified version of hold'em with limited betting, for example BU can shove or fold, and BB can call or fold. Maybe allowing more pre-flop raises with pre-defined sizings would be doable, too, but even that is becoming a lot more work than anyone is likely to want to do without getting paid for it.

Wow.