Comments over on Foucault's brog made me want to post about shoving short stacks (or, almost equivalently, shoving when only short stacks are left to act) in donkaments. Say it's folded to you in the small blind, and you're holding some miserable hand like J3o. Let's say it's 200/400 with a 25 chip ante, and you've got 4k chips. Let's consider 4 options: folding, raising to 1100 and folding to a shove, raising to 1100 and calling a shove, and just open shoving yourself.
If you fold, you win zero. Simple enough.
If you open shove, your expected winnings obviously depend on how often your opponent calls you. But so does your equity in the hand when he calls. If he calls more often (bad for you), you win more often when he does call (good for you), since in order to call more often he calls with weaker hands. To give an idea, if he calls with top 10% of hands (as defined by PokerStove), your equity when called is 26%. Against top 20% of hands, it's 29%. Against top 30%, 32%. By the time he's calling 50% of all hands, your equity is 35%. Recall that your expected winnings in the hand are:
f*(400 + 200 + 225) + (1-f)*(-3775*(1-e(f)) + e(f)*4200)
Where f is the % of the time villain folds and e(f) is J3o's equity when called which, as I mentioned earlier, is a function of f. So for instance if villain calls only with top 10% hands, then this equals:
.9*825 + .1*(-3775*.74 + .26*4425) = 572
For f = .8, it's 368, for f=.7, 211, and finally for f=.5 (at which point the villain is calling with hands such as 96s and J7o), it finally goes slightly negative at -79. So for all but the very loosest of strategies by the big blind, shoving will be better than folding.
If you change the stack size to 15 bbs, however (replace 3775 in the above formula with 5775, replace 4425 with 6425), expected profit goes negative around the f=.7 area. Even a quite loose tourney player wouldn't call off 10-15 bbs in the bb much more than 20% of the time, as this is basically all pairs and broadways, so shoving here is probably still better than folding 15 bbs deep (although maybe not if raising from the button...the chances of one of the two players being dealt a top 20% hand is roughly (1 - (4/5)^2) = .36).
In order to evaluate raising a smaller amount and then either calling or folding, we have to guess how often the villain is reshipping on us. We'll just have to assume that he's either raising or folding, since 1) that's what most players do anyway, and 2) it's difficult to model our expected profit when he calls.
So let's say that we're raising to 1100, and our villain is folding f percent of the time, and re-shipping 1-f percent. If we're folding, EV is:
f*825 - (1-f)*875 = 1700*f - 875
For a given f, this generally gives a higher expectation than open shoving. The problem is that you're inducing more re-shoves, or at least calls, when you do this relative to shoving yourself. So, for instance, shoving when f=.8 yields a higher expectation than raise/folding with f=.7. However this is not the case with 15 bbs. Notice that the EV of raise/folding is invariant to stack size, whereas the EV of shoving goes down as stack size goes up, keeping f constant, since you lose more when called. The difference in f (i.e. the additional fold equity) has to be pretty large for shoving to be better than raise/folding with 15 bbs. For instance, with f=.8, shoving yields 213 chips in expectation, whereas raise/folding earns 315 even with an f=.7. Add in the fact that it's a lower variance play, which matters in tournaments, and I think raise/folding is clearly superior to shoving with 15 bbs.
Raise/calling is clearly going to be worse than raise/folding even with 10 bbs since in order to show a profit on the call, we need to be getting 2900/(2900 + 4425) = 40% equity in the pot, which we don't have unless the BB is re-shipping just about any two cards, in which case we should just be open folding anyway. Raise/calling with 15 bbs is even worse.
So in summary, I think 10 bbs is really the right point to start shoving to steal blinds and antes, rather than 15 bbs.
-BRUECHIPS
So let's say that we're raising to 1100, and our villain is folding f percent of the time, and re-shipping 1-f percent. If we're folding, EV is:
f*825 - (1-f)*875 = 1700*f - 875
For a given f, this generally gives a higher expectation than open shoving. The problem is that you're inducing more re-shoves, or at least calls, when you do this relative to shoving yourself. So, for instance, shoving when f=.8 yields a higher expectation than raise/folding with f=.7. However this is not the case with 15 bbs. Notice that the EV of raise/folding is invariant to stack size, whereas the EV of shoving goes down as stack size goes up, keeping f constant, since you lose more when called. The difference in f (i.e. the additional fold equity) has to be pretty large for shoving to be better than raise/folding with 15 bbs. For instance, with f=.8, shoving yields 213 chips in expectation, whereas raise/folding earns 315 even with an f=.7. Add in the fact that it's a lower variance play, which matters in tournaments, and I think raise/folding is clearly superior to shoving with 15 bbs.
Raise/calling is clearly going to be worse than raise/folding even with 10 bbs since in order to show a profit on the call, we need to be getting 2900/(2900 + 4425) = 40% equity in the pot, which we don't have unless the BB is re-shipping just about any two cards, in which case we should just be open folding anyway. Raise/calling with 15 bbs is even worse.
So in summary, I think 10 bbs is really the right point to start shoving to steal blinds and antes, rather than 15 bbs.
-BRUECHIPS
8 comments:
Please delete before the tourney donks find this post.
-Foucault
ROR! Thanks for the vote of confidence, but luckily we only get 40-50 hits a day max and there are many many tourney donks in donkland.
So Brue, what's with the apparent focus on donkaments lately? Are they providing a better yield these days or are you just practicing for bigger tourneys?
Nah, I just wander over to the donkaments for a bit every once in a while to mix things up from the constant cash game grind. And then a couple of weeks ago I went on that bad run in donkaments and wanted to grind until I got it back...no particular reason for playing more donkaments though.
I'm going to represent the tourney donks in this discussion as I'm a recent convert to the open-shipping philosophy. I'll start with a disclaimer: I'm a micro-menter, part of my argument gets less valid as you move up in buyin.
First I'd like to say I agree with your math and the majority of your thought process. However, I think there are two flaws in your logic.
Flaw 1: Ante Size. Your math is good for the ante size you post about, but I haven't seen this ante size at any site for quite a while. Even stars, who had been notorious for small antes in the past, now has this level at 200/400/50. This is not insignificant for your calculations, as it increases pf pot size by over 20% (though you could argue that it affects each calculation equally). Personally, I'd weight it slightly to moderately in favor of shoving pre.
Flaw 2: You discount the possibility that the BB flats. While in theory his play should involve 0% flats because of stack sizes, I would say that this is the case very seldom. In my experience, flat % is higher than shove %, but no doubt gets smaller as you move up. The issue is that you don't want to be flatted in this spot. This will give your opponent an extra decision point in the hand, which is fine if and when he misses, but contributes negatively to your equity when he connects. Since villain would be in position in this instance, I can't see how you would be +cEV postflop given preflop conditions. To paraphrase Bill Chen in "The Mathematics of Poker," strategic options have non-negative value. The fact is that villain is given an extra strategic option by having flatted preflop.
To counter my argument here, I do realize that his call bloats the pot, perhaps sufficiently that you can shove 100% of flops and be near neutral ev, but if that's the case I can't see how it outperforms just shoving pre.
Since Foucault agrees with your reasoning, I am sure that I'm missing something in the process. In either case, thanks for posting this, it really made me think about why I've been open shoving 13-16 bb's from the SB lately.
Greg -
Thanks for your comment. You're probably right about the ante size, I was going off memory. If you double the antes, it will change things a little but not that much. It will make shoving a little more attractive since you win the pot more frequently that way than be raise/folding, so the advantage of shoving relative to raise/folding with the larger antes would be 200*(f_s + (1-f_s)*e(f_s) - f_r) where f_s is the % of the time villain folds to a shove and f_r is the % of the time villain folds to the smaller raise.
On #2, the difficulty is in assigning some payoff to the hero when the villain calls. I'd like to think that I could at least break even when called by most players at the stakes I play, so if you let c equal villain's call %, f equal fold %, and r equal re-raise %, then payoff would be at least f*825 - r*875. So if the villain is callign with hands that he would otherwise be raising, that's good, but if he's calling with hands that he's otherwise folding, that's bad (under the assumptions I have made that I'm able to break even when he calls). If the villain is good enough that I really don't want him calling and playing post-flop, I'll probably just tighten up my opening range more, and maybe mix in some more shoves as well, depending on his stack size. If I have him covered and can jeapordize his stack so that he, as a good player, would pass up on a marginal spot to stay alive, I might shove more. But if he has a big stack and understands that he should be calling pretty light there, then I'll just be folding more.
Also, it's true that I'm giving the villain an extra "option" by allowing him to call preflop for less than his stack, I'm also giving myself more options, by allowing myself to fold before putting my whole stack in, c-bet flop if called, etc.. Basically I'm saying that I think I'll be able to use these extra options better than my opponent.
I agree with your reasoning for the most part. I'd just note that with the ante size of 50, if you think he's shipping top 30%, raise folding and raise calling j3o show only a difference of ~50 chips in equity when starting the hand with 4050 (favoring raise folding). Improve your hand by a fraction and raise calling becomes better.
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