Why Does SB Mix Bet and Checks Basically Every Value Hand?
When a hand mixes in GTO, it’s indifferent between its options, meaning both betting and checking make the same money. This implies that we expect BB to put in roughly the same amount chips (on avg) from here to river, regardless of whether we start with a bet or a check!
I call this betting volume indifference. It's everywhere.
Why?
- Imagine BB puts in more money facing a bet, and less facing a check. If that's the case, why should we (SB) check any strong hands? We should just lead value always.
- Now imagine BB puts in more money when we check. If that's the case, we should check all our strong hands.
Neither of those is a stable unexploitable state, so equilibrium pressure tends to make value hands indifferent on earlier streets.
The solving cycle looks something like this: BB puts in more volume when we check, so we check strong, which makes them want to invert play passive vs check and put in more betting volume vs a bet, which makes us want to bet stronger. Back and forth it goes until it reaches a stable equilibrium where ~every value hand is indifferent between betting and checking.
Exceptions
The primary exceptions to this rule are:
- Spots where one player is too weak to make the opposing player's value indifferent. This happens a lot when OOP is forced to range-check.
- Value hands with excellent unblocker properties (think middle set on some dry board). These hands, due to card removal, can get paid a smidgen more than average. But even they are pretty close EV-wise between betting and checking.
Is Folding +EV in Zoom Games?
An interesting consequence of zoom/fast-fold cash games is that you can fold marginal hands to increase your hourly.
If your win rate is 5 bb/hr, and you've got a breakeven 0bb hand, why play it when folding is higher EV/time?
People will adapt to your nittiness, but I think the equilibrium is tighter overall in fast-fold formats.
In general, you can apply this to any spot where folding meaningfully speeds up the game (e.g. chopping the blinds in live cash).
However, consider the corollary: If winners should play a bit tighter to speed up their play, then presumably losers should play a bit wider (?!) to slow down their losses. A wild idea indeed!
A subscriber called ngmcs8203 asked: "Are we talking true zero or close to zero? If the latter, how close? Are we folding if we know our opponent has on ?"
The line depends on your skill edge. If your win rate is 3 bb/100 then you'd fold hands with an EV < 0.03bb preflop to breakeven (or perhaps a touch less)
"Interesting," ngmcs8203 replied. "So in a config where UTG7 opens and you are in the LJ you really aren't 3! bluffing with much."
DRE_3000 wondered: "What about sacrificing skill progression by not playing/defending all the dusty cuspy 0ish EV hands – avoiding a lot of spots that challenge you as a player and force you to up your game by having to defend wider ranges."
Do these "cuspy" hands increase your skill level over time?
Undeterred, DRE_3000 continued to probe: "A specific example would be defending in BB vs pf open. I like to flat a lot of the 0EV hands, to get into post flop spots where I can work on my defense strategy (esp. x/r line, my favorite :) I would like to think that a lot of those hands will be >> 0EV in practice, as many regs in my games (100nl zoom) are not comfortable facing that sort of pressure and make mistakes compared to theory. (perhaps because they don't play those 0EV hands in the first place so don't have as many reps in as me).. I could also be completely wrong and just bleeding money long term, who knows!"
I feel like HU is great for this kind of thing. It really forces you to play wide and get into uncomfortable spots.
"I agree! Shame HU tables are not supported on Stars where I play (or GG)."
Editor – Fortunately, heads up games are supported on plenty of other sites, including CoinPoker.
Thoughts on the 72 Game
The 72 game is a variant of cash game poker where if anyone wins with 72, every other player at the table owes them a bounty (usually around 5bb per player).
This is a fascinating variant because it encourages people to play the worst hand in poker. There are no solvers, no solutions, but we can still examine 72 from a theoretical perspective.
1) Folding is -EV, So Bluff Less Often
When you fold, there's always a chance your opponent has 72, and you must pay a bounty. So folding is no longer 0EV, it's slightly -EV (proportional to how likely your opponent is to hold 72).
That means the aggressor should size up / under-bluff relative to sizing in order to make the defender indifferent between a -EV fold and a -EV continue.
2) Split 72 Into All Continuing Lines
Most players think you should always put 72 in your most aggressive lines. But I think an optimal strategy would split 72 into all the continuation lines. Calling sometimes, raising other times. Checking/betting. It's just such a heavy bluff (16 combos is a lot to carry all the way to the river) and its value is largely tied to its scarcity.
If you're representing a polarized range of 72/nuts, then your opponent should call super wide, such that you're indifferent to bluffing 72 and giving up. But if you have other bluffs in range, they can start to fold more, which increases the EV of your 72, and presumably your entire strategy.
3) Bet Way Thinner For Value
Final thought about 72 game: you should probably bet way thinner for value on runouts where 72 is a bluff.
You're incentivized to bluff a ton in this game, so you need to value bet more often to make those bluffs credible. Since you can't just wait for good cards, you need to consciously shift your thresholds to bet thin in spots you would normally check and go for showdown.
4) 72 > AA
Some people ask: How bad would it be to never play 72? Maybe you can just nit up?
I would argue that in this game, 72 is probably more valuable than AA in a regular game. It's an insanely valuable hand. But that doesn't mean you *always* need to go for stacks. If it's clear your opponent is not giving up you don't need to put in 100bb bluff.
5) Tight Ranges = More 72 = Stronger Adjustments
In general, tight ranges are more likely to have 72, because these bluffs are less inclined to give up compared to other bluffs. So as you narrow ranges they are more likely to contain this hand. But in wide vs wide configurations 72 is less common (because players rep more hands), so your adjustments (calling wider, bluffing less) ought to be less pronounced.
6) Value Bet More Hands With a 7 or a 2.
The lookalike principle tells us that optimal play involves disguising your value bets by sharing cards with bluffs where possible. So your should prefer thin value bets that contain a 7 or a 2, and the defending player should be less inclined to call with 7 or 2 (assume 72 is a bluff on this runout).
7) When 72 is a value bet, things get crazy.
This one I'm really not sure how to think about because both the aggressor and defender can have 72. But it adds a ton of value bet combinations and players may even run out of bluffs. As such, I think you need to bluff (non 72) hands considerably more often when 72 becomes a value hand.
What else?
I'm sure there are a ton of other considerations in the 72 game, these were just a few off the top of my head. I'm wondering what other adjustments you guys make or see in this variant?
Torchio14 put forward their own idea: "Speculative theory on bet sizing (could be wrong): Geometric sizing to get all-in results in the lowest overall fold % among all sizing plans that go all-in. For example, at SPR 13 over 3 streets using MDF as a baseline, geometric (PSB each street) gets 87.5% folds by river, while a 13x jam gets ~93% folds. Since so much of 72's EV comes from generating folds, maybe the theoretically optimal approach is the opposite of geometric: tiny bets early, massive shove at the end.
In practice: Having played this game a few times, I'd say many players massively overdefend, even in lines where 72 realistically can't be in range. Tombos' point 2 suggests 72 should be split into all lines, and I think that's right in theory. But: (1) almost no one actually does this, and (2) there are still spots where it shouldn't be, like river x/r with 7-high, where I would think you should just bet rather than risk it checking through. I've seen people still overdefend in spots where it felt super obvious 72 wouldn't be played that way.
Against most players, I think the exploitative adjustment is to never bluff without 72, always bluff with it.
Love the insights Tombos. The 72 game is fascinating, wish it ran more. It's hilarious, lots of fun, and I think most poker players are pretty bad at it."
Hot Take: Limping in Poker Is Fine in Principle.
The old poker adage "never open limp" is treated like gospel, but it misses the point. There's nothing inherently wrong with just calling preflop. The real issue is almost always the price.
Think about it this way:
When you limp for 1bb, you are calling 1bb to win a pot of 1.5bb (SB + BB). That means you need to win 1:1.5 = 40% of the pot after limping.
That's the same pot odds you'd get if you were facing a 2x pot overbet.
Would you feel comfortable calling a 2x pot overbet in a family pot with several uncapped players acting behind you? Probably not. Well the BB lays the same odds to you, so this is why you almost never see limping in preflop charts.
Watch what happens when we sweeten the pot.
Examples
With Big Antes: The dead money dramatically improves your price. Suddenly, a GTO solver is happily limping a large chunk of hands from the Cutoff, even 200bb deep.
In a 10bb Cash Drop it gets extreme. Now LJ pure-limps even in a raked game.
And of course, even without giant antes or cash drops, limping is pretty standard from the SB in MTTs:
There are also legitimate exploitative reasons to limp (e.g. if pool iso's too wide). On the other hand, limping adds a lot of needless complexity and increases the amount of rake you pay. So it's a mixed bag.
Look I’m not prescribing limps. I’m challenging the assumption that limping is inherently bad.
Needless Complexity in Solvers
It always amazes me how much needless complexity exists in GTO solutions simply because complexity is a free resource to a solver.
Compare this messy heads-up BB defense chart against my human-simplified version.
This simplified strategy loses less than 0.0% pot against the best possible response. It's orders of magnitude less complex (in terms of human-playability / k-complexity), yet it's virtually unexploitable!
Here I compare the EV at the root node. Keep in mind SB is playing a maximally exploitative strategy.
How Would a Solver Penalize Complexity?
It’s pretty clear you can produce strategies that are dramatically easier to execute with negligible EV loss against a best response. This got me thinking about how one might hypothetically build a solver that allows you to explicitly trade EV for simplicity.
But that’s easier said than done.
First, you need a definition of “complexity” that matches what humans experience. The most honest definition is basically K-complexity / description length: the minimum number of rules you’d need to memorize to play the spot well. That’s what “simplicity” really means. The problem is it’s computationally expensive to calculate this, so in practice I think we'd need a cheaper proxy.
Second, the way we solve poker (CFR) is inherently local: it updates strategies at the combo level. That means any complexity penalty has to be decomposable to the combo level. If a metric can’t be expressed as a sum of combo-level incentives, then it won't work very well in a solver.
Third, “simplifying a combo” isn’t the same as simplifying the strategy. A common idea is to penalize the entropy of each hand’s strategy so it prefers pure actions over mixing. But low entropy at the hand level can still produce a complex global strategy. You'd end up with a patchwork of pure actions where adjacent hands do different things for no human-readable reason. That could be far less intuitive than a mixed strategy that follows simple rules.
