Lately I've been looking at a game-theoretical model to analyse optimal play in the blinds if the rest of the table has folded. In brief, the model is this: 2 players are dealt randomly some real number on the interval [0,1]. If it goes to showdown, the low number wins. While it's obviously simpler than poker and doesn't capture the drawing aspect on later streets, it does allow you to analyse what hand strengths are required for raising, re-raising, calling, etc. (including optimal bluffing frequencies). And it really should be accurate for that.

What I've concluded is that very few players play their blinds anywhere close to optimally.

For defending one's BB, here's the game I looked at: Blinds are 1/2 and stacks are each 24. SB is allowed only to fold or raise to 8. If SB raises, BB can fold, call or shove.

In this [0,1] game, SB should be raising top 42.7% of all starting hands and folding the rest. BB should be at least calling on top 26.7%, re-raising top 6.7% and bluff re-raising some 3.3% of hands that he would otherwise fold to the raise. SB should be calling the shove on top 13.3% of starting hands.

For an overview:

SB raises [0,42.7%]
SB calls shove [0,13.3%]
BB value shoves [0,6.7%]
BB just calls [6.7%,26.7%]
BB bluff shoves [26.7%,30.0%]

That's optimal play for this game.

Now I'll attempt to apply these values to the situation where you're in BB, and it's folded around to SB.

If SB is pretty tight and not attacking your blind much, then there's nothing to worry about. If he raises <42.7% of the time, he's effectively giving you money. So, just let him do it would be my advice. Plenty of players steal only something like top 30%, so I say just fold non-premium hands and re-raise only really big hands.

But if he raises 42.7 or more, as many do, you're bleeding money if you don't do something about it. Also worth noting is that if a player is stealing that often, you generally don't have kicker problems on your A6o if you hit. Any pair in that kind of battle is already a pretty decent hand.

Now for pure value, you're theoretically supposed to be re-raising top 6.7%, and that's actually quite a few hands. If you take 99-AA, AJ+ (suited or not) and ATs, that works out to be almost exactly top 6.7% (there is a bit of a subjective component, as one can also consider KQs if you like that better than 99). Anyhow, my suggested re-raise range is 88 hands total (88.4 is 6.7% of all starting hands).

And you're also supposed to be bluff raising half that many hands--and specifically hands that aren't good call hands. Here it gets even more subjective, since the hands that hold up best hot and cold are largely determined simply by the biggest card, lots of AX hands, the better of which are suited, etc. But against a player whose raising range is 50%+, I actually do think A5o is a perfectly fine call with position.

Hands that actually don't hold up as well hot and cold are the very small pairs and SCs. If you look at HU hand rankings, 22/33 are folds here as well as all the SCs: They're not in the top 26.7%. Moreover, these are the types of hands that can hit big but hit rarely. So, if you get called, you may still have a chance even though 80% or more of the time, you're probably best off just giving up on the flop.

Taking 22-33, 54s-T9s and 53s-T8s, I get a total of 60 candidates for a bluff re-raise here against a loose SB raiser. That's actually significantly more than the supposedly optimal number of 44.

The real question there is how inclined SB is to call a re-raise. If he'll call a lot, you probably want to stick with only the premiums and forget about these bluff re-raises altogether. Just bear in mind that there really should be absolutely nothing your opponent can do to take away your value re-raises. Those hands are genuinely WORTH re-raising in this spot.

If villain will fold a lot to your re-raise, then imo you should definitely re-raise the really low pairs and SCs--and perhaps even throw in some others on occasion if you feel like the time is right. But there, too, one should presumably be choosing these bluff re-raises from hands that aren't really playable for a call here.

Here, too, I think the game theoretical aspect should provide some helpful orientation: SB should theoretically be calling top 13.3% of hands if you're making both value raises and bluffs. That's roughly 1/4 of the time after raising if he raises somewhere around 50%. If he calls MORE than that, stick to the value re-raises. If he calls less, then throw in quite a few bluff re-raises.

Opinions?

Have you read Mathematics of poker?
I think he analyzed this type of game too. Although I am too lazy too look it up, and to dumb to remember.

I actually haven't read it, but I know it's in there because I got the game from one of the authors before the book came out. I kind of doubt that he analyses this specific variation, but it could easily be in there. I just don't know how far in-depth he goes on it.

hmm... way theoretical. ever just go with the flow?

Blinds are 1/2 and stacks are 24? (ie 12 BB's).

Sounds like you are in my territory

Any 3 bet here is a shove as noted.

The thing is you are assuming an evolutionary stable strategy to have arisen ab initio. My first reaction on seeing someone stealing 40%+ would be to increase my push range. I would never have a push range against a 42% raiser who only calls 30% of pushed of only 10% even with stacks twice as deep. I also would never flat call 2.5x as much as I pushed when so short. What sort of post flop play are you assuming to calculate that as optimal?

If SB is calling a shove with his top 13% and is raising top 42% (which you say is optimal) then BB calling 2.5x more than he shoves just looks very wrong. My gut tells me that (except perhaps to get trappy with a monster) this is fold or shove time.

Let's look at always shoving.
You pick up 10BB's whenever he folds to a shove which he does 70% of the time.
The other 30% of the time you have 2 random vs his top 13% which is 33% equity.

So as BB as always push strategy results in:
7 times out of ten you win 5 BB
1 time out of ten you win 12BB
2 times out of then you lose 12BB

Net EV if he plays (43% of hands) and you always push is +2.3BB/hand unless he starts to adjust his calling range. Of course he is losing 0.5BB/hand to you on the 57% of hands he chooses not to play too, so over 100 hands he is losing 127.4BB's.

So if villain starts playing 'optimally' when he sits down at your table he is easy to exploit until he adjusts.

Now of course his best adjustment is to call all if you are pushing all which gives him a 60/40 edge. - that means over the 100 hands SB is +309.6, -206.4, -28.5 = +74.7 up.

But a more standard strategy from BB would be to push say top 25% vs a 43% raiser and fold the rest. The maths now get more difficult.

Out of 100 times SB folds 57 times (SB is -28.5)
BB folds 32 times (SB is +32)
So SB is +3.5BB/100 going into the interesting hands:

The other 11 times SB raises to $8 (pot is 5BB) and BB pushes over the top.

So how often does SB call BB's push?
Calling with top 13% of his 43% (30% of the time) gives 57% equity when there is a call, so of the 11 times he loses 5BB 70% (when he folds) (SB -38.5), and 30% of the time he has a 1.7BB/hand edge (SB+5.5).

Overall calling off with top 13% means SB is losing 29.5BB/100 hands (too tight)


Let's try SB calling off with top half of his 43%. That only gives him a 51/49 edge and he is still just giving up 5BB in half those 11 hands.
Folding (SB -27.5).
Calling (SB + 1.3)
So that strategy loses less (-22.7BB/100)

OK if SB calls off with 75% of his raising range (top 32%)
Folding (SB -13.75BB)
Calling SB is 48/52 behind (SB -4)

So that is SB at -14.25BB/100 (getting better and demonstrating that the biggest mistake most folks make vs the push from a shorty who has adjusted to their stealing range is to fold)

OK let's look at call all.
SB is 45/55 dog.
SB is -13.2BB/100 in the called hands, overall -9.7BB/100.

OK so if SB turns up raising 43% he can be beaten by a simple push top 25% fold the rest strategy from BB no matter what his calling range is. That isn't necessarily optimal vs a 43% raiser, but it clearly does work.
However if SB starts out only calling with top 13% (a terrible mistake vs a shorty) he can be better exploited until he adjusts by pushing many more hands (any two will do to start).

Any 20BB shorty knows that someone stealing more than 30% of the time is a prime candidate for a shove with top 15%. Not lowering your steal % vs a shorty is usually one of the worse leaks an aggressive player has. To claim that it's optimal to raise it suggests to me that your maths or assumptions are wrong somewhere..

To excession (commenting chronologically):

I'm really not doing this for purposes of short-stack play (although it's applicable there) but rather to get an idea of "correct play" from the blinds, actually with full stacks. Also, this game doesn't have any post-flop play, hence the model definitely isn't poker. The main thing is that hand values don't change after the flop in the [0,1] game.

Now, on the evolutionarily stable play aspect: I really don't think this model tells us that we should play according to the optimal state. In my opinion, one should always be playing either too tight or too loose, depending on what your opponent is doing. Where I think the model is useful is in defining what too tight and too loose actually are.

Example: Villain is stealing around 50%. That's too loose. Consequence imo is that you should be re-raising this guy a lot. Or... villain is stealing around 30%. This guy is playing too tight and leaving you too often with your blind--so do nothing unusual, since he's not going after your money.

Ok, the rest of it (which I'm going to have to digest a little at a time) relies on several things. First, as noted, the model doesn't capture postflop changes of hand values at all.

So, on the shoving all idea, in the model it wouldn't increase BB's EV because on top 13%, SB wins 87% of the time. But again, SB can really screw up BB's strategy by calling more often (namely all of top 42%) because BB is bluffing more often.

Also, I wasn't really concerned here either about short-stack play (despite appearances) or about optimal SB strategy, which I've also been looking at. I was more concerned about what BB needs to do in full stacks. Here, I assume SB can only raise or fold, but there's of course also the possibility of limping, and I think that option tends to be a bit undervalued (more on that in a different thread once I've organized my thoughts a bit more).

For the moment, though, leaving the limp option out still and also factoring in the dimension of the change in hand values after the flop, let's look at your proposed BB strategy of shoving top 25% vs. the raises. And I'll just accept your numbers 55/45 if SB calls all.

It's costing SB 16 here for a chance at 32, so SB must call with any hand that has >1/3 equity in the pot.

Over 100 hands, here's what I get for SB:

BB folds 75, and SB wins 3 each for a total of 225.

On the remaining 25, SB has 45% equity for stacks. So, SB wins .45*25 and loses .55*23 each time. .45*25 - .55*23 = 11.25-12.65 = -1.4

That happens 25 times for a total of -35.

So, SB's EV over 100 hands is 190. How is that bad for SB? I don't see where you got the negative EV for SB here, as my numbers are completely different.

And over 100 hands, BB folds 75 for EV 0. Then on the remaining 25, we get .55*26-.45*24 = 3.5. So, BB gets 87.5 out of it. (I'm counting blinds as dead money).

Anyhow, this still looks to me like a better proposition for SB than for BB if BB is going to play that way.

Ok, let me try to re-focus this a bit: My intention here (mainly because I don't consider the possibility of a SB limp) is to get an idea of optimal play for BB. I was really thinking of it with full stacks behind and not as a short-stack strategy, but I'm definitely down for considering short-stack strategies as well.

I'm interpreting the brunt of excession's argument as being that in short-stack, BB can gain more by shoving a lot more than top 7%. And that may well be the case given the fact that in poker, underdogs can catch up, whereas they don't in this model.

I'm not convinced that the 25% shove strategy works, though. I get BB losing, as argued in the above post, if SB is willing to call the shove on anything he raises.

In practice, wouldn't it really look like this: You notice SB is raising you a lot, like somewhere around 50%. So, you shove on a top 25%. Now you see if he calls or not. If not, you stick with shoving at least that number of hands, because it's not going to take much fold equity to make you money on his raises. If he calls, you see what he has and take it from there.

Are you of the opinion that BB should almost never flat call here in short-stack?

Actually, I'd be very tempted against most of these guys to go with flat call on something like QQ-AA, maybe most trouble hands (shoving AQ and AK), then shove another 10% or so, including most pairs, aces, and decent kings, then bluff shove the SCs.

In any case, if there's no such thing as a flat call from BB, then whatever your shove range is, you might as well just pick the top. But if there's a flat call in there, then you can have some mix of bluff shoves and value shoves.

In summary: I'm not seeing any critique up to now for my suggestions for re-raising in full stack. I tend to agree with excession that it's too tight for a short-stack shove, but I don't think just shoving top 25% is very good in short-stack either unless SB is a kind of bully type (who loves to attack but backs down too much if you fire back).