I think you’ve got it–the bettor wins when he has the best hand and a portion of his bluffs, while the caller only wins when he has the best hand. An optimal bluffing frequency would result in an optimal calling frequency having an EV of zero, I think, while the optimal bluffing frequency would still have a positive EV due to the dead money.
This seems counterintuitive to the idea that expectation for bettor rises as bet size rises (e.g., at a pot size of zero, the optimal bluffing frequency vs optimal calling frequency should both have EVs of zero, right?). I guess it could be possible that the reduction in the optimal calling frequency (i.e., the increase in how often the bettor picks up the pot) would result in an increased EV for the bettor, since the caller’s EV is still zero when he calls, which he does less often.
If I understand correctly, when you talk about a condensed calling range, you are talking about a capped range without many strong hands. Without too many nutted hands in the calling range, it’s an overall weaker range than the bettor’s value hands.
Imagine a paired board with a flush and straight possible, facing an all-in bet on the river. How does the bettor’s advantage increase if he can eliminate full houses from your range, condensing it? Is coming up with an example like this what you mean by visualizing it?
The question comes from a high card game model: Jake is always dealt A or Q. Janet is always dealt K, and they both know this. They each ante $1, then there’s a deal, and a single $1 betting round. Equilibrium for Janet is to always check. The Nash equilibrium for Jake is to make 3 value bets for every bluff. He bets all his aces and 1/3 of queens. Janet’s eq is to then call a portion of the time. If the game stopped after the ante, both players would have 0 expectation. But the opportunity to bet $one 66% (I think) of the time gives an EV to Jake of .33 each round. The best Janet can do (equilibrium) is to limit his EV to .33. Jake can even tell her he’s betting all aces and 1/3 of queens and she cannot do better. If she deviates her calling frequency he can gain by bluffing more or less. So even though the game seems at first glance to be a coin flip (A or Q vs. K), Jake can leverage his polarized range into a solid advantage. But I have been unsuccessful is “seeing” how he gets that .33. How come Janet cannot recover the .33 by adjusting her fold/call frequency? Does it make sense to ask if the .33 comes out of the ante or the bet?
As you suggest, Janet must always check in any sensible strategy profile. If she bets and he has a Queen, her stack will increase $1 (his ante), but if he has an Ace, her stack decreases by $2 (her ante and her bet). So it makes no sense for her to bet. Similarly, it makes no sense for Jake to check an Ace.
Given this, there are only two parameters to the strategy profile. The rate y at which Jake bets a Queen, and the rate z at which Janet calls facing a bet.
You ask “why can’t Janet adjust” if y=1/3, and why is Jake’s EV 33 cents. Both can be seen in an EV calc for Jake.
In 1/3 of all hands, Jake will check down his Q and lose his $1 ante.
In the hands where Jake bets and Janet folds, he wins her $1 ante. Since he bets 2/3 of the time, and she folds 1-z of the time, this represents (2/3)*(1-z) of all hands.
In the hands where Jake bets an Ace and Janet calls, he wins $2, her ante and her call. This represents (1/2)*z of all hands.
In the hands where Jake bets a Queen and Janet calls, he loses $2, his ante and his bet. This represents (1/6)*z of all hands.
As you see Jake’s EV is 33 cents, and this was INDEPENDENT of the value of z chosen. Meaning no matter what Janet’s fold/call frequency, Jake will always have this EV provided he bets 1/3 of Queens.
ETA: Cliffs, when y=1/3, Janet can’t adjust because all the z’s (her strats) cancel out in Jake’s EV formula.
Thanks for laying that out. Puzzling through your formulas I can see that the math works out, but I guess the dazzling clarity I’ve been wishing for probably cannot be accessed by any means but the dry equations. This is probably the closest I’ll get: he can already win half the time with his aces. Since she cannot tell which of the wagers are bluffs, she is forced to call a portion of bets that turn out to be value bets. It’s the uncertainty that gets her, while he has perfect information. She is forced to call a certain proportion of hands, otherwise he wins with both his aces and his Q bluffs. This is another reason why aggression is so often correct – we know the proportion of bluffs we make, the caller/folder does not.
I think you do understand the big picture, as you explain it well. If you want some intuition for why Jake’s equilibrium should be y=1/3 as opposed to y=1/2, 3/8, or some other number, consider that when Jake bets, Janet will be getting 3:1 on the call, so she needs 25% equity. In order to make Janet indifferent to calling or folding, Jake’s range has to contain one bluff for every three value hands. Since he is always value betting the Ace, this means he needs to bluff a third of the Queens.
And now I realize a simpler way of explaining the $0.33 figure is that since, by the above argument, she’s indifferent to calling or folding at y=1/3, we can just imagine that she always folds, so that Jake wins her $1 ante the 2/3 of the time that he bets, and loses his $1 ante the 1/3 of the time that he checks (he always has Q when he checks), thus averaging $0.33.
BTW, thanks for sharing this problem. I like it a lot.
The Hand
It was a high-stakes no-limit hold’em cash game in Montenegro, played at HK$10,000/HK$20,000 (i.e., around $1,275/$2,550 USD) with everyone deep with around HK$3 million in their stacks.
Action began with Cates opening to HK$50,000 from the button with {5-Hearts}{3-Hearts} and Ivey calling from the small blind. The big blind then reraised to HK$200,000 and both Cates and Ivey called.
The conversation begins with Cates and O’Kearney discussing the potential profitability of his calling the raise with 5-3-suited, with all agreeing the circumstances were such that the call was recommended.
The flop came {a-Hearts}{3-Clubs}{4-Diamonds}. Ivey checked, the big blind bet HK$200,000, and Cates and Ivey both called.
I have PokerStove on my tablet, but Flopzilla on my laptop. I love Flopzilla and running hands through it after sessions is helping me feel confident in my play.
I have played 23.5 hours of $3/$300 spread limit since May 4 and am making $74.51/hr over that stretch. I think if I can avoid playing poorly I can overcome some run bad. My overall stats are 105 hrs at $35.39/hr, so small sample, but I was playing super weak tight for 75 hours or so and stopped playing until I could study and fix that flaw (Read Poker’s 1% and buy Flopzilla). I feel like I’m in control of most games and I’m on a nice run atm.
One hand that bugged me tonight was HU versus an early raise to $20, folds to me on the BB and I defend with QTs. I have him well covered, effective stacks are $250ish. I check flop of 8J4r and he bets $30, I call with my gutshot. Turn brings a 9 and I check-call $50. River J pairs the board and action goes check-check.
I feel like I played it bad. This guy had not gotten out of line and I feel like he was fairly strong and probably would have called $50 or $60 if I had bet the river. My image was super tight and aggressive, but I had raised in EP three times and check folded when I whiffed the flop against two callers each time. Also, I caught myself looking at his stack as I checked the river. I was hoping to bluff catch and feel like he was going to bet but that stopped him. Kind of pissed at myself for that.
As I racked up to leave, a guy I recognized from a local blog announced that I had won every hand I’d played and I corrected him that I had actually won every showdown, but I had bet folded a few times (EP with KQs, KQo, and KJs). Probably should have just agreed with him. Also open raised AJo to $15 from the hijack and folded to a $50 re-raise from the BB.
Why didn’t you check raise the turn? Check calling the turn is probably fine sometimes for deception but I think the standard line should be raising with the nuts on the turn to try to get all the money in on the turn and river. But if you check call the turn you should be checking the river with your whole range. The jack is probably better for the player in position’s range than it is for your range. Now if the eight or nine pairs you could probably lead the river because that card is probably better for your range than villian’s.
While my image was TAG, I did open KQs, KQo, and KJs from EP only to whiff and check-fold the flop. In the past this has led to some people floating my flop and turn bets with the intention of stealing the pot on the river, allowing me to check-call and realize three streets of value. That was incorrectly in the back of my mind when I check called the nuts on the turn. I don’t know what the proper CR frequency is, but it’s safe to say mine is nowhere close to that number. It’s super rare in the games I play and adding some as bluffs would be +EV. Guessing a ratio of 1:1 on the turn would be close to correct, but I’m not confident in my assessment.
What ratio are you talking about? CC 50% of your nut straights on the turn? That’s way too much. Probably something like 80% CR and 20% CC. Mostly you want to be betting and raising your very strong hands while slow playing your nut hands only sometimes. And the shorter the effective stacks are the less important deception is. When you’re 100 blinds deep playing the nuts deceptively sometimes is less important than when you’re 500 blinds deep.
And if you’re talking about CR the turn with 50% of the hands you continue with and CC 50%, well that’s way too much. You’d either be check raising way too much or folding way too much.
So you need to come up with what hands in your range you’re check raising for value on the turn (straights and maybe like 99 or a slowplayed set). Then add in some bluffs. Depending on the size of the bet relative to the pot and the amount of equity the hands you’re bluffing with have (on this board you could have a lot of high equity straight draws) then the bluffs might make up half your range.
One caveat: the bet most of the time with your very strong hands only applies when card removal doesn’t play a major role. When you have AA and the flop comes AA2 you check the flop every single time.
The proper c/r ratio almost certainly doesn’t matter at the level you are playing. It’s more important to profile your opponents and know whether exploiting them means bluffing more often than GTO or never bluffing.
You should probably value-bet the river because most players are checking behind anything with showdown value against that scare card and your opponent likely has a strong range with a lot of ace-highs and pocket pairs. Bet an amount that you think will get your opponent to make a crying call with an overpair.
I meant bluff:value, with some % being CR bluffs or CR for value. People generally snap fold to a CR in my experience, so it seems like a potential gold mine if I can get my frequencies optimized and figure out how to tell when my opponent’s range is capped.
I regretted not betting $50 or $60 on the river. He probably would have folded TPTK to a min-raise on the turn though.
There was a 3/5 player slumming with us yesterday and he was awful at bluffing. I raised QQ to $15 pre-flop UTG+1 and he called, K high dry board, I continue for $25, he calls. Turn is a blank, I check, he bets $25, I call, river blanks out and I check and he up-bets to $28, so I have to call $28 to win $154. I drop in the call, he throws his cards in the muck, and I drag the pot without having to show my hand. Trying to think what I raise in EP, C-bet the flop, call a turn bet, and then fold on the river while getting 5.5 to 1 on a call. I’m drawing a blank. I think $7 got raked from the pot, so the pot was $126 going to the river. I’m betting at least $80 in his shoes, but $105 sounds better.
I’ve played 5 $65-$220 (with rebuy) live tourneys in the last couple weeks.
I love Jonathan Little’s videos. They’re probably what’s inspired me to play a little bit again. But holy cow do they not apply to this level of player.
I saw so many things I can’t even explain if I tried. Players calling 4x pot shoves with AK - no pair, no draw - just overs. Players accidentally putting in 2/3 of their stack on the turn then folding getting 10-1. Like just slightly above not sure of the order of hands.
It’s very interesting playing against players who basically overplay every hand, are VPIP 50/20 or something, will happily call 1/5th of their stack facing an UTG raise with KTo preflop, go broke with TP3K, etc. One weird thing that threw me is they seemed to like to donkbet with draws. Or call flop, then lead turn with draws.
Yes obviously these are very profitable games. But I find myself wanting to massively expand my preflop and postflop range so I can take all their chips. You can expand your ranges some - but it’s tricky to know just how much. And no one is making videos about how to play against these kinds of players.
Obviously just play tight and wait for premium hands is profitable. But is it the most profitable strategy?
They seem unbluffable, so it feels like seeing flops for cheap and betting huge for fat value when you flop good is the right strategy. If it is hard to get it heads-up, I’d consider limping a lot early in the tournament if no one punishes you for it.
Yeah I did a lot of limping - open limping and limping behind. I also called some raises with suited connectors that wouldn’t be profitable vs. one raise - but I assumed a lot of callers would come along behind and usually they did.
Definitely suited connectors and suited aces become a lot more valuable - whereas AK becomes less valuable because you’re likely to be in a 5-way pot no matter how much you raise. And people will happily go broke with TP if your connectors hit.
It’s tough though when you get ATo in UTG+1. Raise, call, fold?
I don’t think game conditions would ever allow for playing this hand, but certainly not in the conditions you describe. If AK is down in value AT is unplayable early.