I have looked at a lot of deep-stack preflop sims of both HU games and 5-handed/6-handed games and almost all the same principles apply (adjusting for range width), except for one very important one:

At least for the sims I've looked at, when facing a 3/4p raise in a multiway game, the BB always defends WIDER than the raiser's range (eg: ~50% against a BTN's 40% RFI, ~35% against a CO's 30% RFI, and so on), but for the HU sims I've looked at, the BB defends TIGHTER than the raiser's range (~60-70% against BTN's 80% RFI).

I'm having trouble understanding this from a theoretical perspective. K2o has 47% equity against the top 80% of hands but is a fold, whereas K8o has 46% equity against the top 45% of hands, but is a call. I can't imagine K8o has an easier time realizing equity against a tighter range than K2o does against almost any two cards. If anything, K2o has more showdown value unimproved and so should make it to the turn more often?

3b and other frequencies are all pretty congruent (once adjusted for the different range widths), so I don't think it has anything to do with how capped ranges are.

The reason it annoys me so much doesn't even have to do with preflop. So the preflop chart looks a little different than I'd expect, whatever. But it has big effects postflop. In most spots, the IP preflop raiser has a tighter range, so a much bigger range advantage and bets a higher frequency across flops, but if I run postflop sims based off the supposed equilibrium preflop ranges I'm looking at, then IP actually has a wider range than OOP, so the advantage of their uncapped range is mitigated considerably. It also means ranges for value betting and showdown value for both players is much closer to Button vs BB in a multiway game than I'd expect given the PFR has almost twice as wide of a range!

Is something wrong with the preflop sims or is something wrong with me?

NOTE TO MODS: I hope this is the right place for this discussion. Since there is no longer a Heads Up strat forum and this question is very broad and theoretical in nature, I think this is where we'll get the best discussion.

Maybe it has to do with the small blind being in already for the 3+ max games.

EX.

BB facing 2.5bb open from BTN in 3max+ game, when he calls the pot in middle is 2.5+2.5+0.5

In HU it's just 2.5+2.5

I thought about this, which is why I'm being careful comparing raises that or of equivalent size relative to the pot, rather than comparing 2.5x raises directly with each other. So a 3/4p raise multiway is 2.9x but for HU it's 2.5x, but for multiway you're calling 1.9bb to win 6.3bb (~30%) and HU it's 1.5bb to win 5bb (30%). Same pot odds.

I would think comparing your pot odds against your equity adjusted for your equity realization would wholly account for the EV of a call, so I don't see how the dead SB changes the decision.

So looking at more ranges more closely, I think I notice a trend that BB's defense does not actually grow in direct proportion with the PFR's range width.

In an extreme example, Modern Poker Theory points out that facing a min-raise 15bbs deep in a MTT setting, BB continues over 70% of the time at very similar rates regardless of whether the opener is the BTN or UTG at a 10-handed table (the only thing that changes is they 3b jam over a BTN raiser more often, whereas they largely just flat an UTG open).

Even in a deep-stack setting facing larger raises, the gap between how wide the BB's defense range is and the PFR's range gets narrower as you get closer to the button, so I guess it'd make sense that eventually it hits an inflection point where PFR's range is SO wide and BB's defense range's returns on expanding its range are SO diminished, that BB becomes narrower than PFR.

This makes a lot of sense to me when it comes to hands that realize their equity well. BB quickly defends a lot pocket pairs and suited connectors, and so it quickly runs out of those hands to expand with as ranges get wider, until you get to the point that BB has basically run out of ALL suited hands facing a BTN open in a 3-handed game, so the amount of extra hands to call with getting good odds are diminished.

Where it makes less intuitive sense to me is with the offsuit, unpaired hands. Usually BB defends about as wide with these hands as PFR raises, so why would it suddenly DEcrease.

Even this might have to do with equity realization though. I can see how 98o realizes its equity better than 95o. I can even see how Q9o is going to fare better against a tighter 40% range than Q5o will against an 80% range both because Q9o is semi-connected, but also because Q-high isn't going to x/c the flop unimproved even against an 80% range preflop. So I guess that just leaves me struggling to understand K3o and K2o specifically...

I like these

BB's defense doesn't scale linearly, because the EV distribution is not linear.

This chart represents the EV of every hand in BB's range (blue line) facing a 2.5x HU open, relative to folding. Every hand above 0 defends.



The intersection of the blue line with the 0 EV axis determines how wide BB defends. Changing the pot odds is akin to shifting the 0 axis up or down. Changing the PFR opening range or SPR affects the curvature of the blue line. As you can see, BB defense is not a simple linear function.

----

In HU, whether or not IP or OOP plays more hands largely comes down to the blind structure. I'll use GTO Wizard's HU Preflop solver to test this. For example, in a 100bb rakeless cash game:



If I change the blinds to something absurd like 0.1/1, then BB defends wider than SB.

I haven't really studied heads up poker in depth, but unless I'm missing something I think there might be a simple explanation.

The reason the BB defends wider than the button opens in 6-max poker is because the BB player is getting a discount/ better pot odds due to the already committed big blind.

In heads up poker the button/small blind player is also getting a discount on their raise with their already committed small blind. So they can raise wider than if they didn't have a blind in play.

Another way to think about it is to imagine what would happen in multi way poker if there was a button straddle in play. In that case the button would have pre-committed chips analogous to a blind, so the button would end up playing more hands than the BB.

I think this is what Tombos was alluding to. The IP player is generally incentivised to play more hands due to their positional advantage, but this is counteracted by the BB OOP player having pre-committed chips. In heads up poker the button has also pre-committed chips so the positional advantage predominates over the difference in the relative blinds. Does that make sense or am I missing something?

All those hands that are just slightly -EV for the button to open multi way when the button has not pre-committed a blind become +EV if they already have a SB committed. As the IP player raises more hands the OOP player also expands their defense range, but not to the extent that they're defending more hands than the discounted SB IP opener is playing.