What’s Your Play? Who Wants to be a Millionaire Results

Thanks for all the comments on What’s Your Play? Who Wants to be a Millionaire. I’m sorry it’s taken me so long to get results up, but I ended up playing two long days of poker on Friday and Saturday, so I’m just getting around to it now.

Estimating Ranges

Let’s start with first impressions of everyone’s ranges. Many of you did a good job with this. For example, ATrainBoston says that, “The last 2 players who called, as well as the UTG shover, don’t have to have great hands. But I’m concerned by the first caller, Mr.”tight but decent” who called with more than 10 percent of his 6k stack.” Here are my thoughts on each player’s range:

The guy shoving UTG has just 5 BBs. Some people will be on any two cards here, particularly, as was the case with this guy, if they just lost their stack on the previous hand. For now we’ll call him super-wide and assume KTo is well ahead of his range, and we’ll circle back to make a more specific assumption when it’s time to do math.

The first caller actually reveals a good bit about his hand. Granted he probably realizes how wide UTG’s range is, but still he is putting more than 10% of his stack into the pot with quite a few people left to act behind him. At the same time, he is not raising to isolate UTG. Against a larger shove that wouldn’t be necessary, but this one is small enough that other people could plausibly call or even raise trying to drive MP out of the pot and isolate UTG themselves. This suggests that MP has either a hand good enough to call UTG but with which he isn’t eager to commit his stack, or an extremely strong hand with which he’s trying to induce action. It’s important to note that the former is combinatorically far more likely.

The other callers’ ranges are functionally capped. With big pairs or strong Aces, they can be expected to raise virtually every time. They’ll have hands that they want to see a flop with given the size of the pot but that aren’t good enough to raise.

Roughing It

We’ll do some math at the moment, but first let’s talk about how to work out a solution to this problem at the table. With so much weak money in the pot, the first question I’m going to ask myself is whether I can shove. I know that I’m a favorite against UTG, gut given the size of my stack compared to the size of the pot, it’s not likely that I can get it heads up with him. There are three players, all of whom will realize that I don’t need a premium hand to shove and that they are getting a good price to call.

That by itself isn’t a reason not to shove, though. Even narrowing the field to one opponent in addition to UTG will trap 1500 in dead money in the pot. In such a large pot, you’ll be in fine shape as long as you have two live cards, so racing against AQ or 88 is not a problem. What will really tank your equity is putting 3K into the sidepot against a hand that dominates you, meaning AT, KJ, KQ, AK, or TT+. That could be offset if you thought your opponents would ever get it in with hands you dominate, but in this case that doesn’t seem likely.

Given the assumptions above, it sounds like these players would usually raise AK and the big pairs, and there’s a lot of other stuff in their ranges that, though we’d rather they folded, isn’t going to have Hero in terrible shape. A lot hinges on how likely you think you are to get everyone to fold, and, when they don’t fold, how likely you are to run into dominating hands. I determined the former was extremely unlikely and the latter likely enough to make a shove only marginally profitable.

Results

I folded, which was of course a disappointment to the UTG player who saw me tanking. “You’re supposed to shove there and push them out for me,” he joked.

“I had about the best hand I’d ever fold there,” I told him, which after crunching some numbers is even more true than I thought. It seemed like this would be a marginally +EV spot for me but that it would also come with a high risk of elimination and that in a soft WSOP event I could find a better, lower-variance spot for the last of my chips.

To be honest I don’t even remember who had what or who won the pot, but no one showed up with a hand I wouldn’t have expected to see, so in that sense the results were unremarkable.

The Math

The first thing that surprised me is that KTo isn’t much of a favorite against UTG. If we put him on 60% of hands, meaning any slightly above-average hand (a reasonable assumption given that with a truly junky hand he could fold and just take this chances with the random hand he got in the big blind), KTo has less than 55% equity. Of course that’s a lot when you’re getting a massive overlay, but once you start talking about contesting that main pot three ways, you’re not carrying over a ton of equity to offset the fact that you’ll be putting the rest of your stack at least slightly behind in the side pot.

For simplicity’s sake, I assumed that the first caller always calls Hero’s shove and that the other two players always fold. In actuality, I do think it’s extremely likely that Hero will get exactly one caller, though it won’t always be the first player. Since he’s the only one with any chance of showing up with AK or a big pair, treating the first player as the only caller is a conservative assumption – we should keep that in mind. I was also a little conservative in putting every combination of AA and AKs into his calling range, but these are really meant to be proxies for any of the dominating hands that he may have played this way for whatever reason. I ended up running Hero’s equity against the following ranges, Hand 1 is the UTG player and Hand 2 is MP:

Hand 0: 29.562% 27.25% 02.32% { KTo }
Hand 1: 29.758% 29.15% 00.61%  { 22+, A2s+, K2s+, Q2s+, J2s+, T3s+, 95s+, 85s+, 75s+, 64s+, 54s, A2o+, K3o+, Q5o+, J7o+, T7o+, 97o+, 87o }
Hand 2: 40.680% 38.18% 02.51%  { AA, 99-22, A8s+, KTs+, QJs, AJo-ATo, KTo+ }

Hero has 29.5% equity in the main pot of 3825, which comes out to an EV of 1128 in that pot. Then he gets 38% equity in the side pot of 6250, for an EV of 2375. Combined, those add to 3503, slightly less than the 3725 Hero risks by shoving.

Because we made some conservative assumptions, we can round those numbers up a little bit. The only thing that will drastically improve Hero’s equity is giving him a chance to get heads up with UTG, though. For example, pulling AA out of the caller’s range entirely improve’s Hero’s equity in the side pot by just 1%. So unless you think there’s a realistic chance of getting all three of those callers to fold, which I did not, then it doesn’t look profitable to shove, especially not if you value your survival in the tournament.

Gamboooool

I want to contest the popular notion that in a tournament with a shallow structure, you have to gamble in spots like this where you have little or no edge. It’s true that in such tournaments you’ll quickly up end up with fewer than 20 BBs if you don’t win chips early, or if you take a bad beat or something.

With 20 BBs or fewer, though, I’m pretty much always confident that I will find a +EV play to make. This is because shoving over a raise and open shoving are both extremely powerful plays, especially when you’re dealing with non-expert tournament players who may not realize how wide their calling ranges should be against some of your shoves.

Of course these are high-variance plays, and it’s the nature of having few chips that you could easily be eliminated from the tournament at any moment, but that doesn’t make it a -EV situation to be in. I certainly don’t feel like I have to throw my chips at the first break-even spot that I come across for fear of slipping below the 20 BB mark. Even if you were to win this pot, you’d still have to keep accumulating chips or you’d be at 20 BBs again in 2 or 3 hours.

You can’t fear playing a short stack, especially not in a tournament with a shallow structure. You have to see it as a profitable if high-variance opportunity. It doesn’t make sense to gamble with no edge now just to avoid being in a spot where you’re forced to gamble with an edge later.

Calling

I didn’t give this serious consideration at the time, because I’m not generally wild about playing KTo either out of position or in a multiway pot. This may be one of those cases where the combined harm is less than the sum of the parts, though. Because it’s a multiway pot and a dry sidepot, playing out of position may be much less difficult than usual, and it’s quite likely that when Hero has the best hand, he’ll be able to check it down. I don’t see much in the way of implied or reverse implied odds, though, as it’s unlikely that if Hero makes one pair he’ll ever catch a bluff or get a value bet paid off by worse.

Nate is seventy-four times the mathematician that I am, so I’m going to trust him when he says that Hero has 14-16% equity and needs 15% to call. Although those 600 chips don’t represent the last of Hero’s chips, they are nevertheless important. As Duggs points out, they may be the difference between getting a fold or a call when we shove over a raise. Thus, I still want a better than breakeven spot to invest them, though I won’t be as cautious as I will with the last of my chips.