10 Player SNG. Standard 5-3-2 Payout. Everyone starts w/ 1500 chips.

On the first 3 hands, 3 different pairs of players go all-in. You fold each hand. So, we now have 3 players at 3,000 and 4 players (inc. you) at 1500.

Are your chances of finishing in the money better or worse than they were at the beginning of the tourney?

same.

Ok, so a simple model would say that if a player with 1500 stack's chance of making the money is x, then the chance of a 3000 player making the money is 2x. ICM is a little different, but this is an ok simplified model. Using this model we get the following equation:

(2x)^3 + 3(2x)^2*4*x + 3*2x*6*x^2 + 4*x*3 = 1
<-> 78 x^3 = 1 <-> x = (1/78)^(1/3) ~= 0.234.

Initially each player's probability of making the money is 0.3. So your chances are lower using this model unless I made an arithmetic mistake.

But this is also very misleading because the blinds will be going up very quickly and will be much more significant later. This is a limitation of ICM as well.

Standard theory says better.

A good tool to use for this type of problem is the Independent Chip Model. You can find many ICM calculators on the web.

With 4 stacks of 1500, and 3 stacks of 3000, the ICM says the finishing probabilities are

1500 1:10% 2:10.83% 3:11.90% Total: 32.73%, equity 10.63%
3000 1:20% 2:18.89% 3:17.46% Total: 56.35%, equity 19.16%

I'm not sure what model beserious was using, and there must have been some error in the calculation. The average probability to cash has to be 3/7, since 3/7 of the players will cash. If you assume that the players with 3000 chips will cash twice as often, then you cash 30%, but this was an unjustified and questionable assumption.

3 answers, all different.

Intuitively, I feel that there is less of a chance of making the money. Even though 3 opponents are out, you are now 1500 chips behind third place. It seems like if you are 20 games into the baseball season, you're 10-10 but someon in your division is off to a 20-0 start.

I know that most of the time I double up early, I make the money. I don't necessarily win, but if you play it smart it's pretty easy to hold on for third.

Of course, the thing that is disturbing about this (and for that matter any tournament) is your chances of winning have been hurt by events totally out of your control. Things get even worse, in my mind, if it is a multi-table tournament. Let's say your table is playing tight and the good hands seem to be spread out evenly, but at another table one player is cleaning up big time. Unless he blows up you're playing for 2nd place.

Your chances of making money improve.

To get an idea as to why, let's continue your example. Assume that all the players are of equal ability. Now let's say that the other players in the SNG continue to go allin against one another until there is just one player with about 90% of the chips and you have about 10% of the chips.

Guess what? You've locked up at least 2nd place money! You started the SNG with an expectation of 1/3rd as much. So you've tripled up by just sitting on your hands while all but one of the others got knocked out.

Moreover, you may still get 1st place money! It turns out that you have a 10% equity in the difference between 1st and 2nd place money.

For example, if the buyin is $10 making 1st place $50, 2nd place $30, 3rd place i$20, then you would have locked up $30 and have a 10% equity in $50 - $30 = $20. 10% of $20 is $2, so your position in the SNG is worth $32.

You can verify this using the ICM calculator linked by pzhon. Play around with it for awhile by testing a variety of situations to get a feel for what is going on.

When you have done this, the next time you play in a SNG and some other players go allin early on, you'll know to smile, because you'll know that you are making money with zero risk.

Yeah sorry messed it up, obviously they should average 3/7. In my model your prob of cashing ends up being 0.3 and the 3000 guys 0.6.

Anyway, if you look at archived posts in sttf, people have looked into the actual effect of doubling up early since obviously both ICM and my model really don't apply very well to this early a stage in the tournament. I think the consensus is that you are indifferent between taking a 60/40 for all your chips at lvl 1 or so.

In reality, in your example your probability of cashing is much higher than 30% (obviously still less than 3/7) for a variety of reasons. First off, the fact that they have double your stack at the first blind level is not hugely significant (definitely not as significant as ICM thinks). In a few levels blinds will be 75/150 and 100/200 and stealing once or twice will put you even with everyone else.

Also more importantly, the players are probably maniacs given that three of them got all-in at an early level. This means it is likely they'll continue to be crazy and go at it with each other or hand u chips when you hit a hand. So the ICM/game theory assumption of "perfect rationality" is unlikely to be very accurate in practice here.

R Gilbert, your main argument is kind of confusing. First you say "assume all the other players are of equal ability" and refer to using ICM, but then you claim all the other players will go all-in until just 1 remains with 90% of the chips. If the latter were the case, clearly the players are all horrible and ICM would grossly undervalue your equity.

I would have thought that the ICM would be more accurate in the first level. The ICM ignores your position relative to the blinds, for example, which is not important when the blinds are small, but which significantly distorts value of stacks with the blinds are high, and a short stack might be all-in next hand.

Do you have a reference to the discussion? I would have expected a much lower calling threshold, something like 56% with no dead money, and less as the dead money increases. It depends on the other assumptions, though, such as your skill advantage.

They can be horrible players or not. It doesn't matter. Take your pick. You're getting sidetracked by a detail at the expense of getting the point. I see now it would have been simpler to leave out my "assume all the other players are of equal ability."

The level of ability of the guy sitting on his hands with respect to the level of ability of everyone else isn't relevant. He could be a potted plant and still come out ahead

R Gilbert: I do get your point. I think this whole discussion is getting complicated by the fact that it's not clear whether we are discussing game theory/ICM or reality.

OP: I think the consensus is unanimous that your probability of cashing and prize equity increase. Just using ICM says it becomes 32.73% vs. 30% originally. Furthermore it is higher than this because it is likely you have a big skill edge over the other players given that 3 of them got all-in in the first three hands. I don't your probability of cashing is 100% as R Gilbert claims, because we are not sure that ALL of the other players are lunatics. And even if they are, they might tighten up around the bubble and still force you to double up.

pzhon: I don't have references, but you could maybe ask in the sttf or search the archives. Maybe ICM would suggest your equity needs to be 56% to stack off early, but if ICM were a perfect model then everyone would have expected -9% ROI (for rake) and it would be -EV to play in the first place. Also, the main reason I am arguing ICM is more applicable at higher blind levels is because of fold equity (also how it relates to skill advantage). For example, if two people have 2000 and 1000 stacks at 50/100, they both still have similar fold equity to the extent that if they shove people won't be committed to call out of pot odds. However, at 200/400 the 1000 stack is definitely getting called and will need to win a 50/50 on avg. just to get his stack up to 2000.

Btw, I recently got a paper accepted that presents an algorithm for computing an actual jam/fold equilibrium in a tournament (not using ICM). It had previously been done for 2 players (for which ICM actually applies very well), but I do it for 3 players with 300/600 blinds (13500 total chips) and show that in some cases ICM is way off (like you suspected). If you're interested I might add a link when camera-ready version is done.

Anyway, I prob. can't reply to responses for a few days b/c of a deadline, but PM me if you want to talk more about this topic.

Actually, the ICM suggests you need significantly less, about 54%, although the exact amount depends on the dead money. The 56% was an adjustment based on a simple model with a skill advantage assumption. The figure 60% does not agree with my intuition at all.

I just limped UTG on the first hand, then called the button's push. AKo>A2o.

I wonder if the "consensus" is actually based on logic, or whether it is tainted by clueless players who would argue for folding AA on the first hand of the WSOP main event, or the better but still bad idea of running away from bad players. A quick search didn't find anything, so I'd appreciate it if you could give more details that could help me to find relevant discussions.

That seems quite interesting. Did your results agree with the statstical analysis of empirically observed finishes better than the ICM?

Better. This is fundamental to understanding SNGs. They reward survival precisely because you gain some of the equity every player who is knocked out loses.

Pzhon, the consensus is based on experience, which is of course anecdotal, because I think most of STTF is aware that you can't be sure what your equity actually is and errs on the safe side.

Your equity is better

This is a maths question and not a poker question. How would knowledge of this improve your game?

In helping me to decide whether I should play SNG's. If my chances of winning were hurt because of either the other players playing like maniacs or even if it were AA vs. KK or set over set, I wouldn't want to play them. I don't think you would be as effected by other player's stack sizes as much in a cash game as you would in an SNG.

I understand what every is saying in their answers, but if we continue my example for a few more hands you could get to 4 players left:

Player 1 - 4,500
Player 2 - 4,500
Player 3 - 4,500
You - 1,500

If the other 3 tighten up, I think we're in trouble.

That's just a failure of imagination. It looks fine to me. If the others tighten up, you can steal a lot of small pots and build up your stack rapidly. If they tighten up and think the others don't give then credit for tightening up, then it is still quite possible for two to play a big pot against each other.

SNGs aren't all about survival. Proper play is a balance between accumulating chips and surviving. If your opponents are making it hard for you to limp into the money in the short run, they are often making it easy for you to accumulate chips, in which case you won't be the short stack for very long.

By the way, the ICM says that with 4500-4500-4500-1500 stacks, the short stack makes the money 42.15% of the time, and has an equity of 12.71% of the prize pool. If your equity is 12.71% with a rake of 10%, even if you had no skill advantage in the rest of the game, your ROI is 15.5%. If you can play, your ROI should be higher. Is that better or worse than normal for you?

I think what's being ignored, or stated clearly enough, is that the three players who doubled-up are loose enough that they are also willing to get into a lot more hands. This style dictates that there will be a certain fluctuation in their stack, and it's quite likely they'll give some back - at least one or two of them.

If you tighten up a bit, and only get involved with any of them when you are confident in your hand, it's very likely you'll benefit from both less players at the table, and their loose styles. You're still at the first level of blinds at this point, so you have plenty of time to pick your spots.

IMO

I would think that your chances are better than b4 given that 3 people are knocked and you are now 4 people knocked out away from getting into the money.