[JNash]

For those who want to sse my logic for why I think you'd fold AA here, highlight the text below...

Option 1: Play the AA.
• In terms of bet-size, you might as well go all-in. If the maniac will call at all, he’ll call an all-in just as much as a smaller bet—might as well get maximum chip-EV out of it.
• In all likelihood, you’ll wind up heads up against the maniac (the 2 small stacks would be idiots to get involved since they can only benefit from #2 stack going heads-up against chip leader).
• In terms of $EV, what can you expect? AA vs. random cards is only about 4:1. That means you will go bust about 20% of the time, and wind up with 3900 chips 80% of time (300 from the blinds, your 1800 allin and his call)
• With 3900 chips, you’re pretty well assured of finishing at least second—i.e. you get at least $60.
• If you win, the chip-leader will have lost 1800 and will be down to 4900.
• A reasonably good approximation of how often you’ll come in first is the size of your stack vs. that of the chip leader, i.e. 3900/(3900+4900)=44%. That means you’ll win the extra $40 about 44% of the time—an EV of $17.6
• Thus the overall EV of playing the AA is about $77.6 if you win (prob. 80%) and $0 if you bust (prob 20%). Total EV is $62.1
• Note that your EV from playing the AA is just barely bigger than $60—the payout for 2nd place.

Option 2: Sit out the hand
• Avoid playing any hand against Mr. Big Stack, but play a normally aggressive game against the two short stacks (e.g. steal blinds, etc.)
• You are very likely to hang in there longer than the two short stacks. Once there are only 2 players left, you are assured of finishing at least second.
• Question is, how many chips will you have left when they bust out? I would argue that your most likely chip count will still be 1800 (depending on the cards, it could be higher if you can win some from the 2 short stacks, or lower if you have to give up a few blinds. A priori, I would think 1800-your current count--is most likely).
• This would give you an 18% chance of coming in first—in which case you’ll make an extra $40 (an EV of $7.2).
• The total EV of sitting out is $67.2.

By my math, you have a higher EV from sitting out than from playing the AA.

A different way of looking at this is that if you double up, you increase your chance of finishing #1 from small to modest. Your probability of AA holding up is about 80%, and if they do it’s worth about (44%-18%)*$40, so the EV pickup from improving your chances of finishing first is $8.3.

You have to weigh that against the probability of busting out, which means giving up $67 (the virtually assured $60 plus the slim chance of finishing first) about 20% of the time—an EV give-up of $13.4. The net is a difference of $5.1 in favor of NOT playing the AA.

Increasing your chances of coming in #1 isn’t always the most important thing—it depends on how steeply the payout structure increases for the final places. <font color="white"> </font> <font color="white"> </font>