The other thread seems to have died out, so I don't feel as much like I'm just stepping on somebody else's toes. There seemed to be some enthusiasm for me doing an AMA about game theory in poker, so here I am. Fire away.

(Also if you have PMed me hard game theory problems in the past, maybe you should post them in this thread. Then even if I don't have time to help you with them, maybe someone else will.)

I have a question that arose in a different thread a while back. The topic was concerning a river shove in NL in a heads up pot. It's my limited (and perhaps wrong) understanding that if the player first to act makes an optimal pot sized shove first to act, that his EV is a pot sized bet, while the player second to act has an EV of 0. No matter if he calls or folds with any varying frequencies, his EV is still 0.

If this is correct, then I perceive this to mean that the player who shoves optimally first to act has the advantage on the river, and the player in position can do nothing to improve his EV. But several other posters advised me that the player in position always has the advantage no matter what, in holdem, in toy games like in MoP, etc. Hopefully this question makes sense. I would appreciate your thoughts on this. Thanks.

First let's be clear about what's happening. There's some river, and the effective stack is one pot. Now you say X (the first player) makes an "optimal pot sized jam." This I assume means "jam as part of an optimal strategy." So maybe he doesn't do it with all his hands or even all his value betting hands. But he does it with SOME set of hands, and now we want to investigate his EV conditional on him jamming.

Well, it's clear that X's EV can't be MORE than a pot, since if that were the case, Y (the second player) could unilaterally improve by folding all the time to his jams. Since this is part of some optimal strategy pair, this is impossible. Hence X's EV is capped at the size of the pot. In fact, it can achieve this bound if we are allowed to screw around with the distributions and are not constrained to truly optimal strategies throughout the hand.

Suppose that X has 90% the nuts and 10% nothing, and Y has some in-between hand all the time. Then X just jams the pot with everything, and Y folds. So X's EV is the size of the pot. But this is clearly a degenerate case. Consider a more typical case where, say, X has quite a few hand types and the distributions are roughly similar. If X shoves with some of them, and balances those shoves with appropriate bluffs, and Y calls optimally, X's EV will be a lot less than the pot. This is the more typical thing.

Now it's possible that "shove optimally" means something different to you than this, and if so, you should clarify that.

Yes, by shove optimally I mean "as part of an optimal strategy". Thanks for explaining about the EV of the shove in more typical scenarios, that makes sense. I may have gotten the "EV for X=pot sized bet" from a toy game example that resembles the "90% nuts 10% bluffs" example you gave. My overall question though is regarding who has the advantage in this spot. Does X, as the player first to act, shoving with an optimal strategy ever have the advantage over the the player in position? Or does the player in position have the advantage always no matter what?

Thanks for your replies. Seems like the more I study these things the more confused I get.

how can i incorporate the fact the equities change into a flop or turn startegy?

i.e. i want to find a near optimal flop cr range as the pf raiser.

all the ideas i had about this are extremely hard to realize without some serious programming prowess to create ones own simulator.

Nevermind this question.......the more I think about it the more ridiculous it seems.....back to the drawing board for me.

As you say, solving this to any reasonable threshold of exactness is a computer science problem. But a couple of useful rules of thumb are:

1) Correlate the size of the pot to the strength of your hand on the river. In other words, grow the pot with value hands (where you will still have a pretty strong hand on the river naturally), and with hands that will make strong hands on the river with some frequency; ie draws. Act passively with hands of medium or mediocre strength; don't take every decent hand out of your passive distributions so that you don't have anything when you call.

2) The hands that don't make it (ie missed draws and the like) turn into bluffs. Balance your value bets and bluffs through some alpha-like ratio so that you don't have too many or too few of these hands on later streets. If being aggressive with all your draws results in too many low-showdown-value hands on later streets, mix them back in with your passive hands.

3) Balance your distributions with an eye to strengthening them on flops/turns/rivers. So for example, take a very dry board K82 rainbow. There aren't any draws here, so are we really playing a value only game? Consider hands like T9s of one of the suits, or 76s. A hand like that has a ton of outs on the turn to become a semi-bluff on the turn.

3-handed NLHE, with an option to straddle the button 2xbb. thoughts on whether this is a winning play (as opposed to just playing the button normally), and adjustments to make in the blinds? intuitively the blinds should have higher 3bet % with a merged value range, and should flat-call the straddle infrequently since button has the option to then raise and play a big pot in position. of course the second adjustment depends on button's raising range in this spot.

Do you think limping button heads up could be part of a gto strategy for anything under 25bbs? My gut tells me that giving up getting a fold precludes it and the only flops can occur when calling a villain who has room to profitably raise/fold. Thoughts.

Hard to say anything definitive about this; it seems that for deep stacks, straddling can't be a lot worse, as it's analogous to playing blind v blind in position for double the stakes. I suspect that for shallow stacks (like 30), it's probably worse because you just don't want to raise the button that often. I agree that flat-calling the straddle is probably going to mostly be bad, because flat-calling a raise there is often marginal and the button has a worse distribution and an additional option to raise again.

Whether the advantages of the straddle for deep stacks outweigh its cost, I don't know. Probably NL experts would know more than I about this.

I expect limping is part of optimal strategy at most stack sizes above like 10 or 15. Proving that is beyond the scope of this thread (and my meager abilities), but armed with the strength of playing superaccurately in the future and the relatively large amount of button hands that are close to zero EV for raising, it seems pretty unlikely that there isn't some appropriate mixture of hands that can be limped for better EV than can be raised.

But I mean, to show whether this is the case requires knowing how to play with stacks of 24 and a pot of 2 with arbitrary distributions. Best of luck with that.

In [0,1] games the assumption is that players are facing the same range regardless of the hand they hold. Is there a good way to represent both the distribution of the strength of the hand and the fact that every hand removes some hands from our opponents range?

On a similar note, 1 street limit [0,1] games are solvable in paractice, right? Even if the ranges are not exactly [0,1] but some arbitrary distribution over [0,1].

As long as ranges are fairly well defined they should also be solvable exactly using computers. Has anyone tried this and compared it to the result where the ranges were mapped to [0,1] and solved that way?

I don't know if this has been asked or if it belongs in GTO:
But, do you think that in split-pot big-bet games like PLO8/NLO8, heads-up, position can be a serious disadvantage? Assuming frequent situations arise where one player has a strong-to-nut low and another has < nut high.

This may be out of left field, but do you know how to deal non-iterated poker? And what would be the effect on strategy if poker was played that way? (I think I know the answers, btw.)

Is there a way of extending [0,1] games to model a community card or closed draw?

I spoke to you in a PM about the AKQJ game. A thread on my solution is at http://forumserver.twoplustwo.com/15...j-game-959686/. Is it correct?

How did you get involved with Bill Chen and Mathematics of Poker?

Where should one go after Maths of Poker? Could you provide a list of toy games to attempt to solve?

Do you have a list of favourite poker maths papers?

And finally, thank you so much for doing this, as well as writing MoP.