I’m surprised by the lack of consensus on this spot, so I’m going to take a deeper look. It’s impossible to come up with remotely accurate $EV answers without knowing average chip stacks, specific stack depths of everyone at your table, payout structure, and how many people actually get paid. I’ll run a few different scenarios.

A few general guidelines:

– The deeper your stack is relative to the rest of your table/the field, the more inclined you should be to fold when facing an (effective) all-in from someone that has you covered.

– The larger the remaining field, the more inclined you should be to fold. I’d probably want to fold anything less than KK here if 1000+ players were getting paid. If something like 27 players cash, I’d be readily looking to gamble with less than AKs. I assumed ~27 would cash due to buy-in and no mention of the event name, leading me to believe this is likely a daily/weekly casino tournament and not a massive field. Doubling your stack on the money bubble has a lot less value if it barely increases your chance of making a final table, but it’s worth a ton when it puts you in a great position to build a deep stack nearing the big prize jumps. Accepting risk when ~100 players cash is a murkier proposition, but I think I’d still be looking to get 20bb in with at least AKs, JJ.

– I’d be more willing to gamble in a 15-minute blind structure than a 30+ minute blind structure.

– If the min-cash is very significant to you, whether you’re playing higher than your normal buy-in or you really hate the short-term pain of busting so close to the money (more than you hate not maximizing your long-term EV), it’s totally fine to fold 99.1% of hands here and wait to mix it up after the bubble bursts.

If you expect CO to have <6% of hands in their ISO range, AKs starts to look like a fold. However, it still outperforms AKo and TT against strong ranges, so you’re playing a pretty narrow range if you’re folding it. Against 4.22% (99+,AQs+,AK), AKs has 45.9% equity. AKo has 43.1% while TT has 41.3%. JJ is much stronger than AKs against this particular range (48.6%), but as ranges get wider they tend to include more combos of dominated Aces than middle pairs.

Widening CO range to 5.88% of hands (88+,AJs,AQ) takes AKs to 51.8% and JJ to 53.2%. AKo sits at 49.3% while TT brings up the rear with 47.9%. Worth noting, 99 and AQs are miles behind these hands with only 42% and 41.4% equity, respectively.

Even 53% equity sounds pretty bad for risking tourney life on the bubble but Villain can and should be ISO’ing much wider than 5.88%. They can exert a lot of fold equity against any shorter stack behind and a 4bb range really can’t be very strong when they’re posting half their stack 2 hands from now (this looks like a BB ante game?). This does change if either BTN or SB have CO covered, and AKs starts to look pretty marginal (although I’m not convinced it isn’t +$EV) in that case.

Against a slightly tight (~6bb Nash) UTG+1 shove range (22,A2s,K6s,Q9s,J9s,T9s,98s,A7,KT,QJ), CO is >0.25bb profitable isolating with all combos of 66,A8s,KJs,AT if everyone behind only gets in JJ+,AK. Adding TT to the ranges of players behind narrows their >0.25bb range to 66,A9s,KQs,AT. These calculations use a 30bb stack for BTN and 25bb for SB. If both players cover CO (this is only for chipEV), their >0.25bb range narrows to 77,ATs,KQs,AJ with both 77 and KQs being just above the threshold.

The equity of AKs against the tightest reasonable range of 88,ATs,AJ only increases to 55.5%. Adding 77/KQs barely changes anything (55.51 instead of 55.49%). JJ vs the same ranges has 55.2 and 56.8% respectively. Kind of amazing how 6 combos of lower pairs skyrockets JJ but 3 combos of KQs barely improves AKs.

 As we add more combos, AKs steadily improves but it is worth noting that it will never be a slam-dunk favorite.

             AKs                                   JJ

57.3% vs 66,A9s,KQs,AT            60.0%

58.1% vs 66,A8s,KJs,AT             60.4% 

Regardless of CO range, we are accepting significant risk by getting our stack in here. Busting and missing a guaranteed min-cash >40% of the time seems like a disaster. However, in a relatively small field, I still think it is going to be a hugely profitable play if you’re comfortable accepting the variance. As important as min-cashing is for our bottom line, we make most of our long term gains by final tabling and especially finishing in the top 3. We can’t do that nearly as often without being willing to accept a fair amount of risk.

Finally, we split the pot 9.7% of the time vs. the 66,A8s,KJs,AT range so our actual risk of busting is closer to 37.1% than 41.9%. We split the pot slightly more often as CO range gets tighter. We tie a whopping 13% of the time vs 88,ATs,AJ meaning that we only bust 39% of the time against the strongest range I looked at.

ICM will demand a much tighter range but, facing the 66,A9s,KQs,AT range and the 23.6% U1 range shown earlier, we’re winning >2bb with all of 99,AKs,AKo while AQs(1.8bb), 88(0.78bb) and AQo(0.55bb) are also profitable.

Equities vs. the CO change slightly due to card removal from the U1 shoving range. More to follow about that and some $ICM analysis for a few different ranges, chip distributions, and payout structures.

I’ll start by analyzing 31 players remaining with 27 cashing. The following tournament setup is arbitrary but deliberately chosen to slightly inflate ICM pressure.

Average Stack of Field: 145161 (representing 25k starting stack in a 180 person field)

136040 UTG (17bb) 

33200 U1 (4bb)

84720 MP (10.5bb)

216000 HJ (27bb)

327000 CO (41bb) 

147100 BTN (19bb)

124000 (16bb) 

164000 BB (20.5bb) – Hero, after posting BB Ante

Using ICMizer’s randomized stacks for other tables, HERO is 8th in chips, CO is 3rd in chips, U1 is 29th. Tourney shortstack is at 17130 (2.2bb) while the chip leader has 450600 (56bb).

This stack distribution is somewhat extreme but not unrealistic for a small live tournament with a relatively fast structure. The average stack is only 18bb, there are 22 players with <20bb and 6 players with <10bb. Deeper average stacks and/or fewer players with <20/10bb would incentivize looser ranges for U1/BB but not CO.

Payout Structure:

1st – 25.14%              6th – 4.47%              10 – 18th: 1.26%

2nd – 17.3%               7th – 3.27%              19 – 27th: 0.972%  

3rd – 11.61%              8th – 2.35%

4th – 8.15%                9th – 1.7%

5th – 5.92%

If there were 180 runners and the entire £140 goes to the prize pool, the final prize pool would be £25,200. 1st place wins £6,285 and a min-cash is worth £245. I used a relatively large min-cash and a relatively small 1st place prize. I think a ~28% first prize (£7,056) and ~0.8% min-cash (£201) would be a more common structure.

I’ll start with a slightly tighter range for U1 and CO than ICM Nash Ranges.

U1 shoves 44,A2s,K9s,QTs,JTs,A7,KT or 19.76% (Nash = 22.62%).

CO shoves 55,A8s,KJs,A9,KQ or 12.37% (Nash = 13.42%).

Hero wants to call TT,AKs (2.56%). AKo is borderline profitable (+0.01%) but should probably fold.

If both players shove the Nash ranges than AKo is a clear call and 99 becomes borderline (+0.00%).

Looking at the very tight ranges of 9.8% for U1 and 5.88% for CO:

JJ is actually a fold! AKs is a very clear fold. QQ is still very profitable.

Interestingly, U1 can profitably shove any two cards ($EV not just chipEV) if no-one other than the blinds call with >5.88% of hands and SB plays 7.54% while BB calls 21% (max$EV calling ranges for blinds vs a 9.8% range).

Main takeaways so far:

– AKs is still profitable with three factors that inflate ICM pressure. Short stack depth throughout the field, larger payouts for min-cashing than I would typically expect, and tighter than Nash ranges for both players.

– Even JJ prefers to fold on the bubble against very tight opponents, at least with these payouts and stack distributions.

I’ll play around with the tournament conditions and also look into the implications of a larger field soon.

Lol, I appreciate the offer Riceman but it would be hard to find a compelling mathematical/strategic reason for folding AKs in this scenario, no matter how much you pay me. Besides, I’m learning a lot as I go, so the eventual ROI of doing this kind of work should more than pay for itself, and I enjoy doing it (when I can get over the hump of actually starting a deep dive).

It’s also worth mentioning that all the calculations above use FGS1 (the only FGS option with >18 players remaining), which slightly lowers EV. If we look at this hand in isolation, the $EV of AKs improves by ~0.02%.

That is not to say that you are giving up a ton by folding AKs. Given the slightly tighter (19.76 / 12.11*) than Nash ranges and the tourney conditions above, you’re only passing on a +0.12% ICM play by folding AKs, which might equate to ~ £20 (I’m pretty sure it’s between £10 and £30) in EV. It’s a stretch to argue that lower variance strategically justifies passing on that much EV in a £140 tourney, but it’s totally justifiable based on life/psychological EV if the positive result of at least cashing is worth slightly more to you than absolutely maxing long term $EV. Even playing professionally in games that are <0.2% of my bankroll, I’m not above folding +$EV hands when I’ve had a rough day/week/month and could really use a win, whether that be surviving the bubble or increasing my chances of sneaking my short stack onto the FT.

If you’re looking for a similar scenario where you can easily justify folding AKs, all you need to do is double the U1 stack to 8.3bb. Their range changes a lot, and everyone else needs to play much tighter:

Nash for U1:  15.34% (44,A3s,ATo,K9s,KJo,QTs,JTs)          >0.02% EV for U1:  10.26% (77,A9s,AJ,KTs,KQ,QTs)

>0.02% for CO:  7.54% (77,ATs,AJ)                                      >0.02% for CO:  4.68% (TT,AQs,AQ)

>0.05% for BB:  1.81% (JJ)                                                   >0.05% for BB: 0.9% (KK)

I’m using >0.05% EV for BB because we should be looking to err on the side of too tight rather than too loose when we have no fold equity. QQ is +0.03% profitable vs the tighter ranges but really doesn’t gain much by getting it in here.

——————-

Going back to the original scenario (U1 has 4.2bb), it’s paradoxically more profitable for Hero to get it in with AKs if this were the stone bubble (28 remain and 27 cash). I can’t really find a logical explanation for that, especially since both the U1 and CO play slightly tighter ranges.

The only theory I can come up with is that keeping the same average stack brings a lot of players in the field very close to Hero’s stack. Using ICMizer’s random distribution, Hero is now 7th in chips, CO is 2nd and U1 is dead last (next shortest stack has nearly 7bb). However, the difference in chips between 5th and 15th place is now less than 4bb. This might incentivize Hero to gamble slightly more since their stack is much closer to a large portion of the remaining field, which I guess means that picking up chips has more of an effect on Hero’s theoretical chances of finishing in the top 3. This is a pretty strange result. I was expecting to find another scenario where folding AKs was more justifiable but it actually increases by 0.03% EV on the stone bubble. I’ll have to play around with stack distributions to see if I can figure out a more precise explanation of why that is.

EDIT: I think there’s a fairly simple explanation for why $EV increases on the stone bubble. The only scenario that has Hero busting without min-cashing is when U1 wins the main pot and CO wins the side pot. If CO scoops, U1 bubbles and Hero still makes the money. Even in the rare cases where U1 > CO > Hero, one of the 15 players with fewer chips will occasionally bust on another table, so Hero still cashes some % of the time (assuming hand-for-hand play).

I’m still planning to look at larger field scenarios and a few more stack distributions / prize pools. I will report back with more findings.

I solemnly vow to find other situations where AKs is a $EV fold.