How low would you go...

[Lloyd]

x=% of time you call and win

Total chips = 6,150,000
Total prize money playing for = $1M ($250k already wrapped up for each)

Fold = 825k
825k/6150k = 13.4% * $1M = $134,146

Call and win = 2,015,000
2015000/6150000 = 32.8% * $1M = $327,642

Call and lose = OUT

Value of calling = x($327,642)

$327,642x = 134,146
x=134,146/327,642
x= 40.9%

So theoretically, if you win 40.9% of the time the value of calling equals the value of folding. We obviously need to win more than that for it to be profitable. 50%? 60%? I'd probably lean toward the 50% number.

Put him on a range of 22+,A2s+,KTs+,QTs+,J9s+,T8s+,98s,87s,76s,A8o+,KTo+ ,QTo+,JTo. That sounds about right based on what you've said. Therefore, you could call with 66+, A9s+, KQs+, ATo+ and have greater than 50% equity. If you wanted to be very (perhaps too) conservative and require 60% equity you would need TT+, AQs+, AK. Obviously there's a middle ground there as well. I do think he could be pushing with a huge range of hands here and doubling up gives you a legitimate shot at 1st.

[Exitonly]

That equation is flawed, theres no way that if he folds worth he's that substantially less than 3rd place (250k).

[Lloyd]

Not flawed. Third place money is already locked up and not included. That's why there's only $1M of prize money being played for and not $1.75M. Everyone has locked up $250k.

[adanthar]

Lloyd,

Your equation *is* flawed, because the assumption is that the short stack will bust first. While you can't really count on it, it will happen ~90% of the time (as he'll be forced into at least 3 all ins by pot odds as a random hand) and therefore the hero has 500K locked up, not 250. You can consider it 450K if you'd like, but you still need waaaaay more than 40.9% to call.

BTW, this is from one of the, like, two WPT events I've ever seen on TV and the guy's call was *horrible*.

[Saborion]

[ QUOTE ]
Lloyd,

Your equation *is* flawed, because the assumption is that the short stack will bust first. While you can't really count on it, it will happen ~90% of the time (as he'll be forced into at least 3 all ins by pot odds as a random hand) and therefore the hero has 500K locked up, not 250. You can consider it 450K if you'd like, but you still need waaaaay more than 40.9% to call.

BTW, this is from one of the, like, two WPT events I've ever seen on TV and the guy's call was *horrible*.

[/ QUOTE ]

Yes, this was from a WPT event that I re-watched the other day. Personally I think the guys call was very interesting, not something I would do. Hence this post. The numbers may not be 100 % accurate, but they are close enough. If the BB would've had a little smaller stack I think the call would've been a must, but with 1M I think I'd want a bit better hand than the one he had. With more than 1M I'd want an even better hand since we're so much more likely to survive the short stack. But 1M is starting to get borderline with those blinds, since it's not impossible that the short stack will double/tripple up on the next hand, although unlikely, it will still suck big time.

Anyway, thought it would be interesting to know what the "math" says that we should do. For those interested it was from the Doyle Brunson blah blah, I think it was season 3? The BB called the SB (Carlos Mortensen) A5o push with TT without too much hesitation, if any at all, and doubled up.

[Lloyd]

Okay,

So let's say the value of folding equals $250,000 (the difference between the guaranteed 3rd place money and 2nd place). And what we are really playing for is the $500k difference between 1st and 2nd.

If we call and win we'll have 2,015,000 chips out of 6,150,000 total. That's 32.8%, multiplied by 500,000 equals 164,000. That makes sense. Essentially if we call and win we'll have 1/3 of the total chips and use that as an approximation of equity.

If we call and lose, we're LOSING $250,000 (what we have essentially locked up).

So the value of calling where "x" equals the percentage of the time we call and win:

x($164,000)+(1-x)(-$250,000) = 164000x -250000 +250000x
414000x = 250000
x=250000/414000
x=60%.

So we need to win 60% of the time for it to be a neutral decision. Let's say we need to win 70% of the time for it to be profitable enough.

We're then calling (versus my previous range) with AA-QQ. JJ would be slightly +$EV and TT would be neutral (and thus increasing variance with no reward).

I do agree that without taking into consideration the blinds my previous calc was flawed.

Edit: If we change his range of hands to any 2 cards (which I think is certainly reasonable) then a +$EV range would be 99+ so in that sense it could have been correct to call with TT.

[M.B.E.]

[ QUOTE ]
x=% of time you call and win

Total chips = 6,150,000
Total prize money playing for = $1M ($250k already wrapped up for each)

Fold = 825k
825k/6150k = 13.4% * $1M = $134,146

[/ QUOTE ]
I don't quite understand this -- aren't you overlooking second place money?

With a stack of 825k, we'd have a 13.4% chance at first plus an 82% chance (approx.) at second, with let's say a 4.6% chance at third. So our equity on folding is not $134,146 but $555,500.

(13.4% x $1M) + (82% x $500K) + (4.6% x 250K) = $555,500.

[ QUOTE ]
Call and win = 2,015,000
2015000/6150000 = 32.8% * $1M = $327,642

[/ QUOTE ]
Should be more like $658,500.

(32.8% x $1M) + (65% x $500K) + (2.2% x 250K) = $658,500.

So that agrees with Fiskebent's post: because of the prize-payout schedule and the very short third stack, the value of your stack increases by only 18% if you double up.

Here are my complete ICM calculations:

You fold (Stacks are 5M, 1M, 150K)
$EV = 1000000/6150000 * $1000000 + ( you take 1st )
5000000/6150000 * 1000000/1150000 * $500000 + ( BigStack takes 1st, you 2nd )
150000/6150000 * 1000000/6000000 * $500000 + (ShortStack takes 1st, you 2nd )
5000000/6150000 * 150000/1150000 * $250000 + (BS 1st, SS 2nd)
150000/6150000 * 5000000/6000000 * $250000 (SS 1st, BS 2nd)

$EV = $162611 + $353482 + $2033 + $26511 + $5081

$EV = $549718

If you double up (Stacks are 4M, 2M, 150K):
$EV = 2000000/6150000 * $1000000 + ( you take 1st )
4000000/6150000 * 2000000/2150000 * $500000 + ( BigStack takes 1st, you 2nd )
150000/6150000 * 2000000/6000000 * $500000 + (ShortStack takes 1st, you 2nd )
4000000/6150000 * 150000/2150000 * $250000 + (BS 1st, SS 2nd)
150000/6150000 * 4000000/6000000 * $250000 (SS 1st, BS 2nd)

$EV = $325203 + $302515 + $4065 + $11344 + $4065

$EV = $647192