Reid Young

GameTheory, you miscounted value combinations for both players quite a bit. The math, in the way it is set up, looks correct. The variables are not correct.

You need to account for card removal and for post-flop play. The re-raiser's range gets discounted for all possible Jx hands in several ways that Hero's range does not. We need to account for these when considering the polarity difference between the two players. 

Without abstracting several streets back from the river, creating 100% accurate variables for input in your math is difficult. I used multipliers and logic-based assumptions about post-flop play, rather than altogether ignoring the issue - that is why your results are different than mine.

When we are talking about a value range that is essentially one card, the idea that "For simplicity of argument I'm ignoring card removal effects, also since they are a small effect in this spot" is not correct. Card removal accounts for a very large portion of the combinations and skews the result of your math quite a bit. Ignoring card removal makes game theory calculations much easier to do, but the answers, especially in the case of narrow value ranges like these, are going to be inaccurate. 

I think you're correct given your own assumptions, they're just assumptions that overlook much of what is the case and prevent an accurate description of this hand.

The proportion is to the value combinations. It's not that they're any more complex to discover (I hope!). Once we agree on the value combinations, then let's find the bluffing proportions. 

I just wanted to make sure we can come to an agreement there and move forward as a group with more information. Our solution should be able to hold any number in place of any variable to account for differing assumptions anyway.  

I used the multiplier to avoid abstraction back from several streets. If we continue to go back (beyond the turn IMO) then the number of variables to quantify is pretty nuts as far as optimal play goes. I'm also going to say that I'm much more of the opinion that solving for when Jx can can call and other hands will not call is much more important that how often either player has strong or weak bluff catchers. Again, this highlights a difference between solving for the optimal answer and the maximally exploitive one. The good part is that the maximally exploitive answer should be much more simply because of our knowledge of our jack.

I think you are thinking of optimal with information beyond the game conditions. You can solve for the best strategy given information. For example, optimal strategy of RPS is 33.33% for each throw, but if a player throws rock 40%, the best strategy is to play paper always. 33.33% is optimal, paper is best given information. Can post more later, but check out MoP for a good game theory primer.

Not trying to attack anyone's assumptions so much as clear up what was said about my own. I'd rather just have a discussion; otherwise, ego tends to get involved where there should be logical discourse (not a quip directed at anyone, just what I notice happens in poker forums a lot and something I try to avoid). Thanks to GameTheory for organizing spots for unknowns. Let's fill it out together and solve for the inflection point at which the value of betting a smaller size is equivalent to the value of betting all-in, and accounting for the respective frequencies of our ability to do each. That would be a great exercise. For anyone following along, I am discussing what is in Chapter 14 of MoP. Page 149 discusses some of the points I'm referencing.

Before I get into estimations of ranges for each player, I wanted to note a fairly common outcome in hands like this hand: the opponent is not check-calling with worse than one of a particular absolute strength. I realize that idea has the potential to get us away from a purely GTO solution, but it's worth considering because its by far the most likely in-game response of most players at this buy in level, and probably most buy in levels. Perhaps then we need two answers, what's optimal, and what's maximally exploitive if he's never calling worse than Jx when we bet, though I don't believe that the answers will be different for our relatively "small" over bet.

There's an equilibrium amount of Jx combinations to be reached based on the probability and adjustments of either player's "ability" to hold a jack. Ability accounts for likelihood a player holds Jx based on rational play up until the river decision. I'll discount both players Jx combinations just a bit based on their need to capture the most value on average, and not simply to balance for this scenario (Hero has Jx, Villain might have Jx - the reality is both players have Jx with some probability and make oscillating adjustments to the other's strategy until equilibrium is reached) as I don't anticipate it being an overly strong 'force' on the amount of Jx combinations, I'll use a 15% discount multiplier. If anyone can make a more accurate multiplier or show math about creating this equilibrium, that would be great to see and very much appreciated! So in that way, the solution for us knowing that we hold a jack is different than the actual solution, if that makes sense. Because we know we have a jack, we can remove several of villain's Jx combinations. To assume that Villain is balanced with perfect knowledge of us holding a Jx hand is not correct, again giving credence to shoving being a better play because further discounting Villain's checking range (fewer combinations of Jx) increases Hero's distinct polarity advantage. Anyway, on to the combination estimation based on the fact we hold a jack. Maybe there should be two multipliers? One for Hero, given knowledge of his jack, and one for Villain given no knowledge. I'm just using the 15% across the board for now.

I've said "Varied as needed in proportion to XYZ" a few times because those combinations should be predicated on the value combinations. Both players, at equilibrium, have some balancing of their bluffs to do, so sometimes I say "Varied as needed in proportion to XYZ" to account for that.

Villain - Checking River Range (11.05 combinations of Jx)

A - AJ combinations: 4 * (1-0.15) = 3.4

B - Jx combinations: 4 KJ (discounted given flop), 1 JJ, 4 QJ (if he checks all of them on flop) * (1-0.15) = 7.65

C - non-Jx made hands: Varied as needed in proportion to Hero's turn calling range

D - air: Varied as needed in proportion to Hero's turn calling range

Hero - Value betting River Range (35.7 combinations of Jx)

A - AJ combinations: 10 * (1-0.15) = 8.5            <--- maybe we raise turn some of the time for value

B - Jx combinations: [4 J7s, 4 J8s, 4 J9s, 2 JT, 12 QJ, 4 KJ] * (1-0.15) = 27.2

C - non-Jx made hands: Varied as needed in proportion to value range

D - air: Varied as needed in proportion to value range

I used the top 40% of hands that may call the 3bet, as GameTheory suggests. We could have a few more Jx combinations, like J9o; but I just omitted them. What do you guys think so far?

Hey everyone! I just wanted to come on here for the first time and make a few things clear about my point of view, since it looks like a lot of what was said and quoted are a bit out of context. It's a cool problem and a fun spot. BTW, the cat spot was joking around with another 2p2er. I'm not an asshole :) We are just having fun!

I did not purport that we had to bluff with all our air, just that we should bluff in proportion to our Jx/AJ combinations. At the very start of the thread where the original poster asked about value betting, I commented that it might just be because he isn't floating the turn and/or turning weaker made hands into a bluff often enough (what most would call floats anyway since few hand combinations correctly check back flop that call turn with any decent equity against Jx). Obviously, if we can get away with it, turning something weaker into a bluff that has no showdown value probably works out better than having an overly widened turn calling range. My point is that we take into account river possibilities when calling the turn.

The river ranges listed above seem quite a bit off to me. As I'm interpreting the spot (and others are welcome to share different opinions), our range is much richer in Jx than our opponent's. I believe that was another above listed assumption of mine that isn't the way I laid out the issue and believe that it should be laid out. There are several reasons for discounting our opponents range including pre-flop play (we call more Jx than our opponent 3bets), flop play (we probably check back more Jx than does our opponent), the small affect of slow plays on the turn (our opponent may check some straights to us on the turn in order to avoid exploitation), the mixed strategy of betting and checking Jx on the river (our opponent wants to check some 'nuts' to us to avoid being so exploited), and card removal, which in this case is a very valuable source of information given the narrow value ranges. For that reason, I'm fairly certain that our value range is polar enough (necessarily best if called by bulk of bluff catchers) relative to our opponent's possible calling range (and possible range as the hand plays) such that we can value bet Jx and bluff in proportion (optimally speaking) in order to win the pot with a high frequency and steal the most ex-showdown equity. For more, definitely check out The Mathematics of Poker.

Of course, those ideas refer to optimal play. Others have mentioned that optimal isn't necessarily maximum value winning. If your opponent calls a small bet size with a worse hand often enough, then it may very well be the case that you can get away by betting a smaller size and enticing a weaker range to pay off. Because OP is worried about value betting Jx, I did mention on 2p2 that was probably a bit optimistic of an assumption. That's when the cat gifs came into play! Probably a better way of exploiting an opponent (instead of a smaller size than all-in with these stacks) would be to bluff the river with a higher frequency than optimal. Frequency manipulation is more difficult to read and the bluff catching player is already in an impossible spot wherein check-calling an over bet. His best response, if both players are playing optimally is to fold to hero's all-in. He is exploited in a way, but the minimum amount he can lose is by folding. Again, there's a nice example of why this is the case and why smaller bet sizes are exploitive in MoP. 

To clear up the combination of Jx points (mentioned on TwoPlusTwo http://forumserver.twoplustwo.com/56/medium-stakes-pl-nl/400nl-hu-there-value-1359663/index4.html), I would estimate that after all the discounting of our opponent's range based on the best plays he can make given the board run out from earlier streets that our opponent may have around 3-7 combinations of Jx (NOT 63!). Also, I would certainly think that given the pre-flop odds that we can have more Jx than some discounted combinations of KJ, some QJ, and JT. At least throw in J9s! There's definitely points on either side of the small bet/shove argument where someone might attempt to skew these combinations to prove their point and I think I've listed out each opportunity to do so, but I certainly think ignoring discounting the bluff catcher's combinations of Jx at all these points (or any of them) isn't a good way to go about creating the best approximation of the bluff catcher's river checking range. I've tried to be fairly pessimistic with my discounting for that reason. 

So yes, there's an inflection point whereat betting smaller with Jx/AJ makes sense. For instance, it's almost certainly not wise to shove 70 times the pot with ranges as estimated. Wherever that point is, and I would love it if someone wants to take the time to grind the math; but, I'm fairly confident it's well beyond a bet size of 2.437 times the pot.