I thought I knew ICM quite well, having read all of Moshman, Shaw & "Kill Everyone".

If all players have equal stacks then the Bubble Factor is identical for each of two players engaged in an all-in. The same for two identical stacked players going all-in when the other stacks have differing stack sizes (but both all-in players need >> 50% breakeven-equity due to ICM leakage to the other players).

BUT these books imply in ALL cases the caller must play much tighter than the shover even when they both have equal stacks.

Are the books utilizing the Gap Concept and Fold equity to require the caller to play much tighter than the shover when the two all-in participants have equal stacks? (but this has nothing to do with ICM)

The exact same question for short-stack 'GTO Equilibrium Play'.

I quess I'm missing something very fundamental. Can anyone clarify this? Thanks.

Not sure if I can explain this correctly, this are just my thoughts about it:

In very simple words:
Lets say opponent shoves 50%.

You call with 50%. So about half of the time you will be ahead, and half of the time you will not be ahead. The propability of being behind with the bottom of your range like T8o is very high. So you exclude this hand from your calling range.

You keep doing this with every possible hand vs the 50% range.

You exclude every hand that has below 50% equity.
After doing so, you got your smaller calling range.

If the opponent shoves 50% of all possible hands, the worst hands go out of the range.

Now only 50% is left.

Your new range now goes with 50% best hands OUT OF THE 50% that are left to be breakeven.

I know the numbers are not perfectly correct.

Gesendet von meinem ONE A2003 mit Tapatalk

Example:
* Prize pool: $100
* Prize structure: 50/30/20
* Players left: 4
* Everyone has a stacksize of 2500

Everyones ICM is now $25 ($100/4)

If I call an all-in and win I will eliminate a player, burst the bubble
and double my stack.
According to icmcalculator the ICM EV is now:
$38,33 for me and the other two remaining players' stacks have a $30,83 ICM EV each.
If I lose the all-in my ICM is $0 and I am eliminated.
Notice that the other two players gained $5,83 ICM EV without having to risk a single chip!

So I risk $25 to gain $13,33.
$25/$38,33 = I must be a 65% favorite to call the all-in.
The other player can push relatively wide on the bubble because he knows I must call relatively tight.
Even if I hold AT and he moves all-in and show me KQ I should fold in this simplified example. Having the best hand is not enough! Why? Because the chips one gains are not worth as much as the chips one lose.

Of course, the reality of blinds and ante make it so we should call wider, depending on how big those blinds and antes are.

When you call an all in you are guaranteed to have to go to showdown. In a cash game this is simple decision because your edge needs to be anything with positive chip equity. In a sng, icm requires that this edge be much greater so a caller will adjust his range to be very tight. This can be dramatically different.

The person who is considering shoving knows that anyone he shoves into should be tight and can therefore shove very wide because he is unlikely to see a showdown. Even if a caller knows that he is being shoved on with 100% of hands, there is no way to profitably exploit this as a caller in many instances. Remember that in sngs there are many situations where both the shove and the call can be negative $ev. The players who are not in the hand are the ones that gain the $ev.

It's not always tighter, for example blind vs blind 3BB can have 70-80% range for SB and any two for BB.

The EV of shoving takes into account how often you pick up the dead money, how often your opponents will call, and what equity you have against their calling range.

As a caller it's most important to have good enough equity. Some other factors such as how much players behind will overcall etc at play too.

All of this is true with or without ICM. With ICM you'll just have a more complex situation with more factors at play.

Long time since I've thought about poker so hopefully this makes sense

Consider this simple game:

Two players get dealt a real value between 0 and 1.

Player 1 shoves all-in with Q% of his range.

Player 2 can now either fold (and win/lose nothing) or call, in which case he wins $1 if his value is greater than player 1's, else he loses $1 if his value is less than player 1's.

What range, P%, should player 2 call with?

With the help of wolfram alpha, here's the solution:

Player 1's range: "UniformDistribution[{1-p, 1}]".
Player 2's range: "UniformDistribution[{1-q, 1}]".

PDF[UniformDistribution[{1-p, 1}], x] = 1/p
CDF[UniformDistribution[{1-q, 1}], x] = (q+x-1)/q

P(player 1 wins) = Integrate[PDF[UniformDistribution[{1-p, 1}], x] * CDF[UniformDistribution[{1-q, 1}], x], {x, 1-p, 1}] = Integrate[1/p * (q+x-1)/q, {x, 1-p, 1}] = 1-p/(2*q)

EV = p * ( P(player 1 wins)*(+1) + (1-P(player 1 wins))*(-1) ) + (1-p) * (0) = p*(P(player 1 wins)*1 - (1-P(player 1 wins)) = p*((1-p/(2 q))-(1-(1-p/(2 q)))) = p*(1-p/q)

So now to find the p that maximizes EV:

d/dp p*(1-p/q) = 1-(2*p)/q

1-(2*p)/q=0 ---> p=q/2

So player 2 maximizes his EV when he calls with exactly half the range player 1 shoves with - this is Sklansky's "Gap Concept".

--------------------

If you want then you can quite easily change the EV formula above to account for blinds or ICM equities.

If you don't get how this works then try redoing the steps with player 1 shoving 100% of his range... You should see then that the EV formula is a balancing act between how much you fold and how good a hand you have when you call; with the two extremes being folding every hand and calling every hand (both of which are obviously 0EV)... It turns out that your EV is maximized when 50% of the time you fold and 50% of the time you call (and win 75% of the times you call). You can see it's 75% of the time without any need for the PDF/CDF integral too: 50% of the time player 1 will have a worse hand than player 2 (and lose 100% of the time), and the other 50% of the time it will be a coinflip (and player 2 will lose 50% of the time).

Juk

Thanks for the replies.

jukofyork
I understand your great explanation. I already had a gut feeling that the "Gap Concept" was not explicitly integrated with the ICM discussion in all three books (to simplify things), but it's a big omission.
I think this is also omitted from "GTO Equilubrium Play (Shove & Call Charts)".

I guess I should have bought Chen & Ankenman's book for some rigour (I'm a mathematicion too).

No need to be confused cause it's not true

Yeah if pot odds overcome ICM tax, caller can be on wider range than shover.

And there are games where this isn't even a necessary condition.
E.G. if u have a much bigger stack than your opponent(s) in a game with high relative bounty value, your opt. calling range might be bigger than jammers range at all of his stack depth. Most of this is common sense combined with a feeling for how ICM/FGS works in certain spots. Some results look pretty counterintuitive at first glance tho, try overcalling spots in high rel. bounty games