The ICM, TopCat and those Double or Nothing SnGs

I've read a few blogs about the Double or Nothing Sit'n'Gos. Probably the best of them is TopCat's showing how these can be a real money spinner. I have also undertaken my own challenge .. how many of these can I play simultaneously and still make a profit.

However, I've yet to see any analysis of how these tournaments should be played from a mathematical perspective. There are big differences in the structure (50,30,20 payouts as compared to 5x20) which should result in a shift in how these tournaments should be tackled.

I turn once again to the ICM (Independent Chip Modeling). There are many issues surrounding ICM ( not least that it assumes all players play with equal skill ) but it will suffice to provide some insight into the most common situations you will find yourself in during these SnGs.

All of the situations I will analyze are based on an ALL-IN situation at some point in the hand. This is essentially to make my calculations easier, but it also reflects the fact that at any stage in a no-limit tournament all your chips could come into play.

The scenarios that are in the table below show the following.
(i) The number of players still at the table
(ii) The size of your stack
(iii) the size of your opponents stack
(iv) your Tournament Expected Value at the start of the hand (tEV - which is the amount of the prize pool you would expect to take using the ICM)
(v) your tEV at the end of the hand if you win
(vi) your tEV at the end of the hand if you lose
(vii) the amount of tEV you stand to gain by playing hands this situation
(viii) the amount you stand to lose by playing hands this situation
(ix) the Hand Certainty you need to play in this situation. This is how likely you need to be to win the hand given the gain you can make against the loss you may incur

Once we calculate the Hand Certainty required, we can calculate the hand ranges for opening the pot (ie. The hand certainty vs a random hand) and the hand range for a call (ie. the hand certainty vs an opening range)

Lets look at the first scenario. There are 10 players in the tournament and everyone has their initial starting stack of 1500. Everyone has a 0.100 starting tEV. If you manage to double up your tEV moves up to 0.156 (a gain of 0.056). However if you end up all in and lose your tEV is zero (you are out and lost tEV of 0.100).

This means you are risking 0.100 to win 0.056. According to the model you need a hand certainty of 0.643 to break even ( calculated by 0.643 x 0.056 = ( 1 - 0.643) x 0.100 ).

So your hand needs to win 64.3% of the time against a random hand. The range for this is the following 12 hands AA-77, AKs-AJs, AK. However, if someone before you makes a large raise you need to be 64.3% certain to win against the hand range they could open with. Assuming that it is the same range we would open with, the calling range would be AA-QQ only.

Looking through the table (shown below), we will see most of the situations have a Hand Certainty Requirement of about 65%. This means, as a default our opening hand range should be AA-77, AKs-AJs, AK and our calling range AA-QQ. Notice that this scenario is pretty much a generalization for playing whenever your opponent has a stack size around the starting stack or smaller.

From TopCat's blog:

"Even though these are turbo games, you have time to wait for good starting hands, DO NOT play speculative hands in this format, if you don't have a top 10 starting hand, there is no reason to see a flop unless you are in the big blind and the pot has not been raised !"

This is a very accurate observation... although I would increase the number of hands out to the top 12 starting hands.

Again, from TopCat's blog:
"Try to only play pots with people who have smaller stacks than yours if at all possible"

The reason for this is that the risk to reward required to play against the bigger stack pretty much limits your choice of starting hands to AA and in a couple of situations actually requires you to sit-out !!

So, looking at the remainder of the situations in the table we see that unless you have a very strong starting hand you should avoid playing big pots against the big stacks. Losing large chunks of your big stack to another big stack is a cardinal mistake.

TopCat's blog:
"Be sure to keep track of who has how many chips in every hand you play, because in this format, survival isthe key, it doesn't matter if you are 1st, or 5th in chips when the game is over, the message that appears says you are a winner !"


Bubble Play:
The situation at the bubble is really quite interesting. Unless you have a big stack (more than 3000) and get an opportunity to finish off a small stack (1500 or less), you are pretty much risking a substantial tEV for a very very small gain. In fact if you have a big stack and your opponent has a stack bigger than half of your stack ... you should fold everything but Aces...and in some situations it is correct to fold the Aces.

From TopCat's blog:
"When it gets down to 6 or 7 players left DON'T play any hand, unless it is a top 5 or 6 hand, and only against shorter stacks than yours. It is ok to fold pocket Queens to a pre flop all in, if losing puts you out on the bubble"

Again, TC is pretty much spot on. In a situation where both you and your opponent are short stacked, the top 5 or 6 hands is the exact range you can open with (AA-99), but only call with AA or KK (ie. fold those pocket Qs).

(the table)

Players	Your Stack	Opponent Stack	tEV Start	tEV Win	tEV Loss	tEV Gain	tEV Loss	Hand Certainty Required	Hand Range Push	Hand Range Call
10	1500	1500	0.100	0.156	0.000	0.056	0.100	0.643
9	1500	1500	0.106	0.162	0.000	0.056	0.106	0.652
	3000	1500	0.156	0.183	0.106	0.028	0.050	0.644
8	1500	1500	0.113	0.170	0.000	0.057	0.113	0.664
	3000	1500	0.162	0.188	0.113	0.026	0.049	0.653
	3000	3000	0.162	0.195	0.000	0.033	0.162	0.829	AA	AA
	4500	1500	0.183	0.195	0.162	0.012	0.021	0.643
7	1500	1500	0.123	0.184	0.000	0.061	0.123	0.667
	3000	1500	0.170	0.195	0.128	0.025	0.042	0.627
	3000	3000	0.170	0.198	0.000	0.028	0.170	0.858	AA	AA
	4500	1500	0.188	0.198	0.170	0.010	0.018	0.654
	4500	3000	0.188	0.199	0.134	0.011	0.054	0.829	AA	AA
	6000	1500	0.195	0.199	0.188	0.004	0.007	0.640
6	1500	1500	0.146	0.200	0.000	0.054	0.146	0.732	AA-99	AA-KK
	3000	1500	0.184	0.200	0.146	0.016	0.038	0.698		AA-KK
	3000	3000	0.184	0.200	0.000	0.016	0.184	0.919	NO HAND
	4500	1500	0.194	0.200	0.181	0.006	0.013	0.670
	4500	3000	0.194	0.200	0.154	0.006	0.040	0.861	AA	AA
	4500	4500	0.195	0.200	0.000	0.005	0.195	0.974	NO HAND
	6000	1500	0.198	0.200	0.194	0.002	0.004	0.656
	6000	3000	0.198	0.200	0.187	0.002	0.011	0.831	AA	AA

Summary:
So there you have it ... a mathematical proof for TopCat's blog that incorporates the ICM.

What we also have now is a general strategy for playing these tournaments.
(i) The correct range for opening against a small stack is AA-77, AKs-AJs, AK. The correct calling range is AA-QQ
(ii) The correct range for playing a big stack against a stack more than half the size of your stack is AA.
(iii) On the bubble, only play against the small stacks and then only if you have AA, unless you are a small stack, then you can extend your range to the top 6 hands AA-99.

My next task is to apply this in my quest to successfully multi-table 10+ of these at a time.

I'll let you know how I go, until then ... take care and good luck.

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ADDENDUM : Short Stack Play

Supertramp77 points out that you need to play much more aggressively when the blinds are big and your stack is small, to avoid blinding out. Generally I agree, but we need to show mathematically that this is correct and look at the situations we might find ourselves in.

The scenario is this: There are 8 players left, Blinds are 200/100 with a 20 ante. Somehow, we find ourselves short stacked with 500 chips only. What is the correct range to push in with ?

If we push and win we increase our stack to 1400 (double up +400 in blinds and antes) ... if we lose, well we get 0.

Our tEV to start the hand is 0.0478, if we win we gain 0.061 if we lose we have lost the 0.0478 we started the hand with. Note that what we stand to gain is alot more than what we could lose. Our Hand Certaintity in this situation only needs to be 43.8% . As you probably guessed this is a very big range of hands. In fact our pushing range become Any pair, or any hand with one card that is a Ten or better. Our calling range becomes AA-66, AK-AT,KQ.

What about if it was 7 or 6 handed ? From the table below we can see that a small stack (500) playing against a larger stack when the blinds are big can push with any hand with a winning rate of 50% against a random hand. The calling range extends out to AA-88, AK-AJ.

What if we have 1000 chips ? In this situation the Hand Range you need to have tightens up to include hands that have about a 60% chance of winning against a random hand. AA-66, AKs-A7s, KQs-KTs, KQs, AK-A9, KQ-KT. Your calling range would be AA-JJ, AK.

The table below illustrates the ranges depending on the stack size you have, players remaining and if another short stack is in play.

Bubble play when very short stacked

On the bubble the only situation that does not have extended hand ranges is where you are short, but someone else is much shorter than you are.

The example is where with 6 players left, blinds are 200/100 ante 20. You have 1000 and an opponent has 500. In this case you still need a top 6 hand to open (AA-99) and should only be calling this player with AA-KK (this situation has been described above). The point here is that it does not matter the size of the blinds on the bubble, IF there is a smaller stack than yours. So long as you have them (and the Blinds) covered, you still need to play tight.

(the table)

Players	Your Stack	Opponent Stack	tEV Start	tEV Win	tEV Loss	tEV Gain	tEV Loss	Hand Certainty Required	Hand Range Push	Hand Range Call
8	500	1500	0.048	0.109	0	0.061	0.048	0.438
	1000	1500	0.084	0.146	0	0.062	0.084	0.577
7	500	1500	0.057	0.122	0	0.065	0.057	0.464
	1000	1500	0.095	0.163	0	0.068	0.095	0.584
6	500	500	0.108	0.200	0	0.092	0.108	0.540
	500	1000	0.082	0.162	0	0.081	0.082	0.503
	500	1500	0.070	0.144	0	0.073	0.070	0.489
	1000	500	0.146	0.200	0	0.054	0.146	0.731	AA-99	AA-KK
	1000	1000	0.123	0.200	0	0.077	0.123	0.615
	1000	1500	0.112	0.182	0	0.070	0.112	0.614