The opponent is world class, so I'm going to try to stick to quasi gto frequencies here. This requires J9s to be folded on the turn.

Tipton outlines the formula in his books. He talks about a polar range vs traps/bluffcatcher and finding the right betsizings using calculus against those ranges. We can use that to deduce appropriate betsizing for different equity levels. I prefer to use my average value bet equity to keep from being super exploitable by a good hand reader.

If you want me to follow up with more information, let me know.

This would be a really neat theory video.

The exploitablity is straight forward. If we assume that we bet different amounts for hands with different equities, then hands with lower equity would be bet smaller and hands with large equity would be bet bigger. Whenever we bet smaller, our opponent could shove any hand better than our low equity hand. This would make betting smaller with our big hands correct to induce loose shoves and (corresponding bluffs), which violates the original premise that we should bet big with big equity hands. Showing that our strategy is exploitable.

Making one betsize with the entire range is less exploitable though still could be less valuable than very complicated mixed betsizing strategies.

It could be, I'd often choose to raise this on the flop, because I could jam many turns that I would have to fold if he continued betting.

How much would you raise, and how do you balance it?

1/2 pot raise I'd balance with some aj and bottom set.

you c/r AJ and jam turn on j84 sb v btn when he has qq-aa (even if slightly discounted?)

Hi FBB,

You can raise aj occasionally on the flop to then check the turn and call river.It should be marginally okay.

Thanks, I appreciate your kind words!

I'd use Ax. Having small suited connectors in your range is a clear mistake. If your opponent tries to make 54s indifferent to bluffing, then bluffing with Axs will be very very profitable. Since Ax makes up 30% of your button opening range, the villain will always make those hand indifferent to 4-betting, which forces weaker hands like 54s negative.

I was talking with a friend about this idea. As the preflop 3bettor, how do we go about deciding what subset of villain's range we should be making indifferent? Should villain be aiming to make the worst hand in our opponent's range indifferent to 4betting or should he be trying to make whatever the most frequent hand class in his range indifferent? What if the most frequent hand class in his range is also a quite high in his range?

I'd don't understand this concept fully either but here's the basic bounds:

Upper bound:
1. We need to make him indifferent to a hand range smaller than the hand range he needs to make our bluffs go negative by using a 4-bet or fold strategy.

Example: We 3-bet 54s. Assuming no postflop play, we need button to fold approx 66% of the time to show a profit. If button 4-bets > 34% of the time, 3-betting 54s is negative. If button 4-bet all Ax+ (30%ish) and 99+ (6%ish), 54s would be negative to 3-bet. In fact any 3-bet hand we folded to his 4-bet would be negative! We know that the bounds for making a 4-bet indifferent needs to be tighter than Ax otherwise our villain would always 4-bet Ax and we would lose money with any hand we folded to his 4-bet.

So we've established an upper bound on the hands we need to make indifferent it needs to stronger than Ax.

Lower Bound:

1. If we make all hands that he folds to a 5-bet negative to 4-bet, then he will never bluff. He will gain too much money with his value hands making this a worse strategy than optimal.

Example: He 4-bets AA. We shove 100% of the time against a 4-bet. He wins roughly 66bbs. If we folded 55% of the time to a 4-bet and jammed 45% (roughly indifference to 4-betting), he now only wins 34bb. Clearly making his bluffs zero ev rather than massively negative does better against his value range.

My educated guess is hands like ATo, KTo, KJo, A9o are the threshold hands we should make indifferent. They comprise roughly 10% of our button opening range and are roughly breakeven call 3-bets. (Hands that are positive to call 3-bets should also be positive to 4-bet, otherwise 3-bet call > 4-bet). Because they comprise10% of button range, we've made a small enough range that the 4-bet/Fold game no longer is +EV against us.

.I think we've found good material for a video if I can ever find the time to shoot a power point presentation :).

@ 28.25 There's some math here. If he jams every pocket pair and AQ+, AQo+ over my 4-bet. Then I could profitability 4-bet call off a range of AJs+, KQs, 66+, AQo+ (4-bet Range. This is 15.5% of my button opening range. To make him indifferent to jamming 22 (rather it being negative) I'd bluff around 33% of the time.

22 has 32% equity against a 4-bet Range: It loses 27bbs whenever its 5-bet jam is called. The pot is roughly 30bbs, so to make his bluff breakeven I could fold 47% of the time. I could actually 4-bet/fold with 14% of my range. However I found that players in practice tend to call and to make him indifferent to calling I need a lower %, which I like to be around 33%.

@Quite often, because I can't really semibluff very effectively on this turn. I'd have too many in general and a check/jam by villain would crush my equity realization.

Range that calls my jam. Math 38bb - 80bb * .4 = - 6 bb. I'd risk 6 bbs to win 9bbs so I'd need him to fold > 40% to show a profit.

Thanks Cloud!

@11:10 We balance against unknowns too, because we don't know how to exploit them yet. Unknown can be a murky term here, because a player I played 5 hands with might be "unknown" but his stack size/betsizing might actually make something about him known.

@22:50 Doesn't change the math, but makes players change strategies. Playing bigger pots with equity edges = more money. I've used it for years. Its possible he borrowed it from me. :)

@29:50 That makes sense, I'd be curious if you would tell me the EV that pio assigned here rather than the action. You are right that as a defensive (gto) strategy here calling makes sense, but I expect this reg to under cbet this board texture, which should shrink my defending range slightly (muck bad one pairs/gutshots).

@37:26 Calling here makes more sense against 40% ranges of button opens/Small blinds opens. Against a 13% range, calling here is too loose. His range is simply too strong to begin with. We have a big range disadvantage on this board texture and I don't want to start calling too lightly on the flop, because it cascades into too loose calls on the turn and river. And being a calling station late in a hand is a good way to lose money.

Thanks for the answer. Regarding the hand at 29:50 I remade the piosolver calculation with the following assumptions:
Input summary: http://i.imgur.com/00SOBlF.png
OOP preflop: http://i.imgur.com/2qKPMnK.png
IP preflop: http://i.imgur.com/b7YMiIs.png

IP current calling range is slightly wider than I did in my previous sim so some of the QTo without heart get called in this simulation (although they have BE or slightly negative EV) and calling 8s6s has an EV of 13$ against OOP strategy of cbetting 60% range.
Result: http://i.imgur.com/V8Zy0s0.png

Also I stopped the sim at 0.3% pot exploitability

Interesting so its worth almost 10% of the pot to call 86s here. It must be due to the bluffing potential on the Tx, 9x, and other some hearts (Qh and Kh) come to mind. Do you mind looking at how pio plays 86s on later streets?

Honestly this is a monster under the bed thinking. We need an 8 or 2 perfect cards to kill our action. And if villain is so lousy as to c/f all scare cards, then I'll make my money elsewhere.

1. Raising 33 allows villain to valuebet more slimly on later streets when we don't raise (the other 92% of the time). Every bluff catcher loses value when we raise 333 on the flop. We have way more bluff catchers then value bets.
2. Very hard for villain to stack off lighter than AJ here, if he assumes we are balanced and plays balances himself. This of course goes out the window if hes an unbeliever (losing player). Its also very hard for villain to fold AJ or better here no matter what comes out, if he is consistently then we should be bluffing him aggressively. So if raising/calling is the same for the AJ+, then that leaves other hands. It seems more likely to me that he'll barrel off QJ then stack with it when we raise. His marginal bluff catchers will be called down on the turn and river at similar frequencies to raising the flop.

Cliffs: If opponent is bad raise flop. If he is balanced wait.

Thanks Will!

I'd assume that he's 3-bet AJs, QJs, ATs, JTs and the rest of suited connectors very frequently. The offsuit varieties of those hands are more likely to be flatted. Flush draws make a big difference in equity.

Tipton outlines the formula solvign for the relationship between equity and betsizing in his books. He talks about a polar range vs traps/bluffcatcher and finding the right betsizings using calculus against those ranges. We can use that to deduce appropriate betsizing for different equity levels. I prefer to use my average value bet equity to keep from being super exploitable by a good hand reader.

If you want me to follow up with more information, let me know.

No not yet, there some good reasons for this sizing on this board though. It has to do with the correct defending frequencies to these small bets are roughly 80% of my range, which on this board implies things like QJo and 98s with a backdoorflush draw are calls. He thinks the average tag isn't going to be willing to play this lloose on this board texture, so this sizing makes him maximum profit.

Hi Dddog,

I generally don't 3-bet my Axs preflop against a range that is as wide as a sb 3-bet. i'd reserve that for tighter ranges.

As to range play, I'll usually be calling AQ, AJ,, A8, A4 here because they are profitable calls, by raising them I'm turn them into 0ish Ev bluffs which would make me lose money in the long run (++ EV > slightly +EV) .. In addition, we have a small value raising range on this board texture, so its very easy to add too many bluffs. Even if I raised a range of 88, 44, J8s, A9s, ATs, QTo, I would easily exploitable.