50-50 Final Table Hand Revisited

On Thursday I wrote about the silly suckout that knocked me out of the 50-50 on the final table bubble on Wednesday night, netting me about $900 cash but raping me of an expected value from the tournament that was probably closer to $3000 or so before the suckout occurred, and significantly higher once I was a better than 4-to-1 favorite to become the chip leader with ten players remaining in the tournament with just one river card to come. In the comments to Thursday's post it was suggested that my opponent did not make a bad play by pushing in his entire chip-leading stack on a massive overbet to the size of the pot when he flopped 743 rainbow with his holding of 75o. This strikes me as a particuarly bad play -- in fact as the very worst of the chipleader's options in this spot -- but I always focus on keeping an open mind about things so I spent my entire train ride home on Thursday with a pen and some paper trying to work this question out and see if my math instincts, which do not usually lead my astray, might have done just that on this one. As I worked through the problem it occurred to me this might be a fun blog post. So here we are. And keep in mind, I am well aware that solving this sort of problem in a poker context requires one to make about 8500 assumptions, all of which are challengeable if you want to, but without assuming many things there is simply no way to compare the worth of one play versus another. So I'm going to solve this the way I believe Sklansky, Harrington, etc. would approach it, having read pretty much all of their books.

OK so to review and give you the whole setup, there are 10 players left in the 50-50. This was a larger than usual 50-50 with just under 1100 runners, a good 10-15% more entrants than this thing has been attracting most weeknights these days, and this number of players equates to approximately 2.1 million chips in play. The approximate payouts for the top 10 spots range from $900 for 10th place, $1000 for 9th, $1200 for 8th, $1600 for 7th, $2100 for 6th, $2700 for 5th, $3500 for 4th, $4600 for 3rd, $7000 for 2nd place and up to just about $11,000 for first place, for a total distrubtion to the final 10 finishers of $35,600. For lack of any better way of determining the players' relative Expected Value of the Tournament in cold, hard cash (TEV), I will add up all the money available in the remaining prize pool ($35,600), and I will assume that each player's TEV share of that $35,600 prize pool is equal to their proportion of the total chips in play with 10 players remaining, multiplied by the total prize pool left to be distributed. As I mentioned above, this obviously is not an exact science, but I can't think of a more accurate way than this of determining what each player is likely to win given their current chip stack and the money available to the remaining players in payouts in this event.

One important caveat I will have to follow about using this TEV formula, however, is that it very quickly gets out of whack once one player amasses more than, say, 20% of the outstanding chips with still a full table of players remaining. This is because the first prize of 11k is itself only less than a third of the total $35,600 prize pool to be paid out to the top 10 finishers in the 50-50, so if one player has, say, half the chips, it's not like his Tournament EV is actually half of $35,600, since the maximum he could possibly win by taking down the entire tournament is only the 11k first prize. So, once we get above around 20% of the chips in play with ten players remaining, I can't use this simple TEV formula anymore since it leads to TEV outcomes that are highly skewed to the upside. So for those calculations in the below solution, I will explain the alternative formula I have opted to use instead as I go through it.

So here is a chart of the approximate chipcounts of the 10 remaining players and the respective Tournament Expected Value (TEV) of each player in the tournament given their current chip stack when this problem occurred:

Player     Chips  TEV
Player 1   440k  $7120 (Villian)
Player 2   290k  $4692
Player 3   270k  $4369
Player 4   250k  $4045
Player 5   240k  $3883 (Me)
Player 6   190k  $3075
Player 7   170k   $2750
Player 8   140k  $2265
Player 9   120k  $1942
Player 10  90k   $1456

Again, these TEV figures come from simply taking each player's proportion of the total 2.2 million chips in play, and calculating an equal proportion of the $35,600 in total prize money available.

So, Player 1, the Villain in this hand, has an expected value from this tournament of $7120 right now before the hand in question, due mainly to the fact that he has almst exactly one-fifth of the total chips in play, giving his an excellent chance of nabbing one of the top few cash payouts in the prize pool. I should note that his nearly 20% of the outstanding chips is right up against the point I mentioned above where my simple TEV formula starts to skew, but even if this $7120 figure is skewed by his 20% of the outstanding chips, it is only skewed by a little bit, a couple hundred chips maximum in any event. So I'm sticking with the $7120 as Villain's starting TEV for this problem, although if you prefer to think of it as $7000, or even as low as maybe $6800 or so, I can't really say that is any more or less accurate than the $7120 figure I am using. Anyways, my TEV at the time this hand occurred was $3883, as I had just over 10% of the chips in play and thus my TEV is very close to the average of all payouts remaining in the pool.

OK on to the hand in question. Villain, the significant chip leader, is in the small blind and the action folds around to him. He looks down to find 75o. He should probably fold here, but I'm not gonna kill him for trying to pad his chip lead at a time when everyone else is probably going to be playing tight, scared of missing the final table. So he raises the 5000-chip blind around the size of the pot to $18,500, bringing his chip stack down to 421k (these numbers are not necessarily exact, but they're all very close). I wake up in the big blind with pocket Kings, my first premium pair of the entire tournament. No point in slow-playing this, I don't want to go up against the big stack here on the literal final table bubble with a hand that's going to be hard to lay down, so I bump it up by the size of the pot again, to 55k, bringing my chip stack down to 185k. Villain calls -- with 75o!! -- which was a horrible, horrid, putrid play that stinks so bad I can still smell it here more than 30 hours later, making his stack now 384k.

The flop comes down 743, rainbow. Villain has flopped top pair and a 5 kicker. He also has an inside straight draw. As I even just begin to ponder how much I love this raggy flop in this key spot for me, Villain instantly pushes allin for 384k into a 113k pot. I think for about 2 seconds and then of course I call for all my chips, the turn is a raggy 9 but the river brings a 7, giving the Villain trips and sending me home in 10th place for the $900 final table bubble payout. I know I played this hand optimally, so I don't have any doubts at all about my own play, but my feeling when I posted on Thursday was that Villain's allin insta-overpush on the flop against my preflop reraise was a bad move that clearly cost Villain TEV over time. Some comments said otherwise. So let's work it out.

Remember, the starting point here is that Villain comes into the hand with 440k in chips, and thus an TEV of this tournament of $7120.

The first, and really the central, issue in this hand is what is my hand range for me to pot-reraise the chip leader preflop for over 25% of my stack here on the final table bubble. The answer is that this is a hugely tight range of only the absolute very best hands. He is the chip leader, and we're right on the final table bubble. Now, with ATs or something like that, it is unlikely but I suppose conceivable that I might push allin on a reraise. I actually would fold a hand like that in this spot, but trying to get into Villain's head, maybe he could think that way. But to only pot-reraise as opposed to pushing allin, with about 27% of my stack into the pot preflop, against the chip leader, on the literal final table bubble with the payouts just about to start really escalating, a reasonable hand range for me is AA-JJ, and AK. That's it. Perhaps he could think I would make this move with AQ as well, although of course in reality I would not. I might reraise allin, and I might (more likely) fold to his preflop raise given that this is the final table bubble and he is the chip leader, but no way in hike I pot-reraise with AQ there. So it's AA-JJ or AK, and maybe he would throw AQ in there as well. That's it.

The next assumption is to predict what I'm going to do with each of those hands to his massive insta-allin overbet on the flop. This one is easy. I'm folding the AK (or AQ), and I'm calling with all the big pairs. He was making a move from the small blind, I had been playing fairly tight while he had been an uber-calling donkey for the last hour or so, and I felt more or less positive that I was well ahead of anything he might have been holding as soon as he did not re-reraise me before the flop. His allin massive overbet insta-push on the flop absolutely iced it for me. AA or a flopped set would never, ever do that in this spot for fear of losing me, and would surely require at least a few moments of thought as to how to extract the most chips from me with such a flop. So I'm calling 100% of the time with AA-JJ, and folding 100% of the time with AK, or AQ if you think that's in my range.

So let's look at the structured hand analysis of how likely each of those hands is for me. There are 6 ways to make each pocket pair of JJ, QQ, KK and AA, and there are 12 ways to make AK (and AQ). So there are either 36 hands in my total range (if AQ is not included), or 48 hands in the range (if AQ is included), and in both cases 24 of them are pocket pairs which call 100% of the time in this spot.

So, moving on to the math, let's see what happens to his stack and with what probabilities, and thus what happens to that $7120 TEV he started the hand with, once he pushes allin on the flop. He should figure, if he is behind (he always will be if I call), he has 9 outs (two more 7s, four 6s and three 5s), discounted a bit for my redraws to trips with my pocket pair and better two-pair hands:

If AQ is not in my hand range, so it is just AA-JJ and AK:

66.7% of the time, I have a pocket pair (24 out of 36 possible hands), which I call with every time given the circumstances. In this case:

(1) Villain will win 36% of the time with his 9 outs twice, discounted slightly for my redraws. If he wins the 483k pot, his stack jumps to 682k, giving him around 31% of the chips in play and a very high likelihood of nabbing one of the top 3 payouts. Unfortunately here we cannot use my Tournament EV formula, because it would equate with him having a TEV of around $11,100, or higher than the first place payout, which of course makes no sense whatsoever since the maximum he can possibly win by winning this tournament outright is the $11,000 first prize, and there is still an entire final table of players to get through. This is exactly the skew I was referencing earlier with my TEV formula once someone starts amassing a huge portion of the remaining chips with more than a few players remaining. So for Villain if he beats me here, we will have to devise a different method of computing his Tournament EV. To do this, let's instead just intuit that, with 31% of the chips outstanding and 10 players remaining, he has just a 10% chance of missing the top 3 payouts, paying him an average of $2000 if he does, and a 20% chance of ending in 3rd place ($4600), 30% of second place ($7000) and 40% of ending in 1st place ($11,000). This equates to a Tournament EV of $200 + $920 + $2100 + $4400 = $7620.

At first glance this may not sound right because his TEV with 440k in chips was already $7120, and now with another 245k in chips on top, the TEV only rises another $500 or so. But in reality this makes perfect sense. With ten players left, there is only so close to the $11,000 first prize that one's Tournament EV can ever come. So much can happen, so much is still dependent on luck, that with 10 players left it's going to be nearly impossible to nab a TEV much higher than 2/3 or so of the first prize payout. So $7620 for Villain's TEV if he wins my stack seems about right to me.

(2) Villain will lose 64% of the time to my higher pocket pair, in which case his stack drops to 199k. At 199k, his TEV drops to $3220 from the $7120 it began the hand with.

So, to summarize, 66.7% of the time when he pushes, I have a pocket pair and will call. When that happens, 36% of the time he wins anyways and his TEV climbs to $7620. 64% of the time he loses and his TEV drops to $3220.

What happens the other 33.3% of the time, when I have AK? I fold, so Villain automatically wins the 483k pot, bringing his stack up to 517k. Once again my original TEV formula is going to produce a TEV value skewed too high with Villain then holding nearly a quarter of the total chips in play with still 10 players remaining, so let's intuit again for a bit. With 517,000 out of 2.2 million chips, let's figure he has a 15% chance of missing the top 3 payouts for an average payout of $2000, an 18% chance of 3rd place for $4600, a 28% chance of 2nd place for $7000 and a 39% chance of first place for $11,000. This equates to a Tournament EV of $300 + $828 + $1960 + $4290 = $7378. Again, not much higher than his TEV with 440k chips and ten players remaining before the hand even started. Of course these numbers are all so inexact and you could adjust them here and there however you want, but my point is, his TEV does not rise much by getting me to fold and bumping his stack up another 113k with still ten players left.

OK, so for the grand summary, when Villain pushes allin on this flop, 66.7% of the time, I have a pocket pair and will call. When that happens, 36% of those times he wins anyways and his TEV climbs to $7620. 64% of those times he loses and his TEV drops to $3220. In the other 33.3% of cases, I have AK and I fold, raising his TEV to $7378.

Now we can easily figure out his overall TEV from the above probabilities and values. Remember, he started with a TEV of $7120 before this hand began.

66.7% (I call) x 36% (he wins anyways) = TEV of $7620.
66.7% (I call) x 64% (he loses) = TEV of $3220
33.3% (I fold) = TEV of $7378

Adding these three mutually exclusive outcomes up, we have $1829.71 + $1374.55 + $2456.87 = $5661.13. Villain's Tournament EV from pushing allin on the flop against a guy with a range of AA-JJ or AK who will call with any of the pairs and fold the AK, drops from $7120 to $5661.13. That right there is a hugely bad play for Villain, which is what I felt at the time. As chip leader with these payouts now all within reach, making that overpush is a major mistake. Now of course there are a million different assumptions and guesses thrown in to the above calculation, but hopefully it is clear that my results are so far below the $7120 TEV that he started with, that it is obvious the play is a bad one regardless of my being a little bit off here or there. He simply cannot increase his TEV very much no matter what he does in this particular hand with ten players still remaining in the tournament, but he has an overall 42% chance of decreasing his TEV to $3220 if he doubles me through. This is a gamble he should never make given that he is in position to make a serious run to the hefty top few payouts in the tournament given where he is already situated.

For interest's sake, let's throw AQ into my range as well. Like I said, there is no way I pot-reraise the chip leader with AQ in this spot, absolutely none (I might reraise allin, or I might more likely fold, but never a pot-reraise for 27% of my stack), and I refuse to even consider my doing that with a shithand like AJ or KQ or worse because that is just plain looptidin this spot. But let's throw in AQ, because it makes a whole new 12 hands in the structured hand analysis that I would fold to his flop push. I will try to simplify some of these calculations now since I've already been through how it all works once above.

So, if AQ is in my range, so it's AA-JJ, AK and AQ:

(1) Now only 50% of the time do I have one of the pairs (24 out of 48 possible hands). So I call 50% of the time, and once again when that happens, 36% of the time he wins anyways and his TEV climbs to $7620. 64% of the time he loses and his TEV drops to $3220.

(2) Now the other 50% of the time, I have AK or AQ, and I fold, raising Villain's TEV to $7378.

Again, we can easily figure out his overall TEV from the above probabilities and values, given a range of AA-JJ, AK and AQ for me:

50% (I call) x 36% (he wins anyways) = TEV of $7620.
50% (I call) x 64% (he loses) = TEV of $3220
50% (I fold) = TEV of $7378

Adding the TEV's of these three outcomes up, we have $1373.04 + $1030.40 + $3689 = $6092.44.

So, in sum, if I would pot-reraise Villain before the flop in this spot with AA-JJ or AK only, Villain's TEV by overpushing allin on the 743 flop drops from $7120 to $5661. If I would pot-reraise preflop with AA-JJ, AK or AQ, his TEV drops from $7120 to $6092. As I said, I won't even consider adding AJ, KQ or weaker hands to the mix, because that is a ludicrous assumption given that we're on the final table bubble and I'm going up against the chip leader. I suppose we could add TT to my hand range as well, but hopefully it is intuitive to you that adding another big pair to my range is only going to lower Villain's TEV from the move, because that is another hand that I will call with and be ahead of him on the flop. And I wouldn't pot-reraise with TT anyways preflop in this spot. Again, as with AQ, I might reraise allin preflop with it, and I might more likely fold it in this spot, but pot-reraise with a vulnerable and hard-to-play hand on the flop like TT? Not happening. And equally unthinkable is sinking 27% of my chips into this pot before the flop here on the final table bubble, and then folding an overpair to hsi insta-overbet allin push. Not in a million years.

So there it is, in all its boring and overcomplicated glory. It is definitively a bad play for Villain to push allin on that 734 rainbow flop with his 75o, only because the range of hands I would need to have in order to have reraised the chip leader the size of the pot on the final table bubble is so strong that his 75 is behind more often than not. If he wants to push way earlier in the tournament with 9 outs twice and possibly being in the lead in the hand in any event, there is some math to at least make that not an objectively poor play. But once you're down to the final table like this, the cash payouts take on utmost importance and the relevant calculation for many allin moves like this becomes not EV but Tournament EV -- at least the way I view it, it does.

By the way, once the guy flopped top pair, you know he could have considered that he had only top pair shit kicker and an inside straight draw, and that it was about 2-to-1 likely that I had a higher pocket pair in my hand, and he could have gone ahead with a bet of approximately half the pot,say 60k into the 113k pot. This would have brought his chip stack from 384k down to 324k, and I would doubtless have raised him allin, and he could have insta-folded there. With 324k chips and still a solid 2nd-place stack, this would have left his Tournament EV at $5196. So he could have made this half-pot bet on the flop, and I'm still going to fold my AK (33.3% of the time) or my AK/AQ (50% of the time). So his TEV of betting half the pot instead of the foolish massive overbet allin looks much better:

With a hand range of AA-JJ and AK, I have the big pair 66.7% of the time, I raise allin on the flop, and he folds, leaving him with a TEV of $5196. The other 33.3% of the time I have AK and I still fold to his half-pot bet, giving him a TEV of $7378. So this equates to an overall TEV of $3465.73 + 2456.87 = $5922.60.

With a hand range of AA-JJ and AK or AQ, I have the big pair 50% of the time, I raise allin on the flop, and he folds, leaving him with a TEV of $5196. The other 50% of the time I have AK or AQ and I fold to his half-pot bet, giving him a TEV of $7378. So this equates to an overall TEV of $2598 + 3689 = $6287.

As you can see, whether my range includes AQ or excludes AQ, the strategy of Villain betting just half the pot on the flop and then folding to any push by me dominates his silly allin overpush, leading to a higher TEV for him in either case. So even if you think him check-folding to me after flopping top pair and an inside straight draw (admittedly, that seems a bit weak to me as well) is too wimpy for the chip leader in this spot, the allin overpush on the flop was truly the worst of all of his available options.

Flame away.

Have a great weekend everyone. I should be in the FTOPS Main Event on Sunday night at 6pm ET, looking to make a big run in the first ME I have played in I think this entire year in the FTOPS.