Didn't understand that after reading it once, but I will read it again and ponder. Sounds a bit like a trick I've used in the past for speeding up the convergence of series. Thanks for the tip. And I like the added realism in your signature. Nice work! 
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OK. I think I get it. I've implemented this for all hands where exactly two players showdown, as this is easiest, and presumably (I haven't checked) you don't get three players all in very often. I haven't given any thought to your last suggestion. Let's see how well this works first. Thanks.
I'm going to let the simulations run for a bit longer, but they seem to be saying that there's nothing to choose between ICM and my suggestion for ICM+ to the accuracy at which I have computed so far. Hmmmm.
I also had an interesting conversation with nibbana last night where he pointed out the this thread on 2+2, which I will have to read carefully. In particular, he mentioned (and here I'm quoting from the thread, which is quoting from 'The Mathematics of Poker', which I've read, so D'oh! I should have remembered this!)
Malmuth-Weitzman -- is similar (to ICM) except that as you notionally work from the lowest prize (figure chance of a player busting next), and divide those chips equally among the remaining players. The probability of a player busting out next is inversely proportional to his stack size. This method is "probably near the edge of some players' ability to calculate at the table" and seems to be favored as most accurate.
The idea of dividing up the chances of finishing last in proportion to 1/(stack size) is nice. MW then divides up the chips of the player busting first equally amongst the remaining players, whilst I suggested dividing them up in proportion to stack sizes. Having seen this, it made me think that dividing up the chances of finishing last according to some power of the inverse stack size might be interesting. The higher the power, the more this would favour the big stack. Of course, it's no good for HU, as it then gives the wrong answer. But, in our very specific case, dividing up third place according to some power (to be determined by simulation?) of the inverse stack size seems promising.
But none of this addresses the problem of why ICM+ doesn't beat ICM, even though it estimates TEQ better. Maybe this chap would know? He looks like he's a top computer poker theorist. There's so much game theory I have to learn! 
The other interesting thing from the thread is the 2nd to last post - the guy has done simulations of M-W vs ICM and concluded that it has a small edge. I believe that he is the guy who runs Hold'Em resources as he mentions in the post updating the equity calculator to allow equity calculations using M-W
Payout Structure 0.65/0.35 - Stacks 15,10,5
Using ICM
Stack - Equity
15 - 44.75
10 - 35.6667
_5 - 19.5833
Using M-W
Stack - Equity
15 - 43.6364
10 - 35.4545
_5 - 20.9091
This is contrary to what jb and I discussed as for some reason, at first glance I thought that M-W favoured the big stacks more than ICM. It appears, at least in this example that it does the opposite. In fact both s1 and s2 have "lost" equity to s3.
EDIT - Comparison
Using ICM+
Stack - Equity
15 - 46.25
10 - 35.6667
_5 - 18.0833
Hm. This is interesting. I had some headache with interpreting the EVdiff results earlier, so this
Originally posted by muebarek
Just a hypothetical question to the extreme case (I don't think it's the case but imo considering even the weird options might help):
Can we safely exclude that it might be possible that ICM even gives a short stack too little equity which causes him to make bad calls in the ICM equilibrium (not realizing he's more risk averse than ICM says)? Pushing/Calling that bad makes us see a even lower TEQ than ICM predicts. Is there a valid argument for us to neglect this (vague) possibility?
might really be the case. But if this is true and the shortstacks' TEQ really is higher than ICM assumes, do we even have a method of measuring how good an idea is without just to try and error everything we come up with?
I mean M-W seems to have a (small but positive) edge over ICM in all types of payout structures...
EDIT:
So if we follow the idea of estimating a third place probability or in general the probability of busting first and proceeding from this, there are essentially two degrees of freedom:
- "Guessing" the third place probabilities. This is quite arbitrary and there are very few restrictions to it since you can't really know if an ICM+ type approach starting in 0th order with ICM or making it a function of inverse stack sizes (to the power of one seems reasonable since its HU-Limit is fine) or whatever model does better unless you test it. I don't really see much of chance to find arguments based on poker knowledge to say which one is closer to reality.
- The way how the chips of a busted player are split up. The advantage of thinking about this may be, that it might help to make some assumptions based on poker knowledge. The disadvantage of focussing on this point is that this (3handed) only influences the distribution of first and second place probabilities while the 3rd place probabilities stay constant. So in a 3handed satellite bubble, the split up rule doesn't matter at all in terms of TEQ. Also, adjusting 1st and 2nd place probabilities based on a bad 3rd place assumption doesn't make much sense either. But the split up will immensely gain importance for scenarios with more players left.
Originally posted by jbpatzer
But none of this addresses the problem of why ICM+ doesn't beat ICM, even though it estimates TEQ better. Maybe this chap would know? He looks like he's a top computer poker theorist. There's so much game theory I have to learn!
Imo, our problem is that we don't know the susceptibility of the nash ranges to deviations in TEQ predictions. If one can show that the ranges are quite stable to small changes in the TEQ distribution, then it might even be impossible to beat ICM with a significant ROI
Am currently at my parents' for the weekend, which involves a 4 hour drive, so I had time to do some thinking (needed something to mentally drown out the kids!). Two things.
i) My simulations with ICM and ICM+ suggest that there's really very little to choose between them. This is consistent with something I observed a while ago, which is that the Nash equilibrium is very shallow, by which I mean that the TEQ predicted changes very slowly as you vary the pushing and calling ranges close to equilibrium (I'll have to check this and post some graphs).
ii) When I get stuck in a hard maths problem, my usual reaction is to simplify the problem (doesn't matter if it stays realistic or not) to try to find the simplest problem that still retains enough features of the original that, when I understand the simpler problem, I can understand the harder problem. The obvious thing to do here is to reduce the number of possible hands. I've been using 25. How small can I make the number of hands and still have something interesting? My first thought was maybe 3, but on the drive down I realized that you can reduce it to just one hand! In other words, if you shove and somebody calls you coin flip for the pot. This is almost the same as the 'Game of Chicken' described in the 'The Maths of Poker'. If the stacks are large enough, with a satellite payout structure, if the button pushes, neither blind can call. So there are two questions. First of all, how small do the stack sizes have to be before, if the button pushed, the big blind should call (just did a back of the envelope calculation using MW, and came up with 2BB, but someone could check this), and how should the game, in general play out. Secondly, if the big blind is Reggie Kray, and says, 'If you push, I'm always calling, and then I'm going to nail your head to the wall! Grrr!' presumably the button should fold. (In the Game of Chicken, there are three equilibiria, always drive on, always swerve, and a mixed strategy, and I get the impression that it's generally accepted that multiplayer games are very different to two player games). From this I deduce that, however much we faff around with ICM and ICM+, it's reads on other players' intentions that really matter. I suspect a good poker player would say 'LDO!'.
Anyway, my last point notwithstanding, I think this game is worth analysing, as it can be simulated in no time at all, there are lots of pen and paper calculations you can do, and I think it still retains the stack size dynamics and (if we put it back in) position aspects of the full problem.
Over and out until I return from the seventh circle of hell.
Originally posted by muebarek
Originally posted by muebarek
Just a hypothetical question to the extreme case (I don't think it's the case but imo considering even the weird options might help):
Can we safely exclude that it might be possible that ICM even gives a short stack too little equitymight really be the case. But if this is true and the shortstacks' TEQ really is higher than ICM assumes, do we even have a method of measuring how good an idea is without just to try and error everything we come up with?
Originally posted by jbpatzer
but on the drive down I realized that you can reduce it to just one hand! In other words, if you shove and somebody calls you coin flip for the pot.
I've been thinking muchly about this. I started trying to work on something and explain it to jb but I think I got him and myself confused. It was after the post from jb about using ICM's 3rd place estimations as a starting point for ICM+. I really don't like this because my initial thoughts with the constraints of ICM were that it just blindly "divvies" up the chips into equity without realising that we're in the middle of a game of poker, so why should we use it's results for 3rd place when we don't like the way it reached the 1st and 2nd place equities.
So I thought I could find a way to express how often a player would be the next to go out in a tournament where every hand see's 2 players all-in on a coinflip.
If we say
1&2 is where s1 and s2 clash (are allin in a coinflip) and s1 wins
and
2&1 is where s1 and s2 clash and s2 wins
then we can come up with 6 possibilities of new stack sizes and see where we're at
_&_ s1 s2 s3
1&2 25 0 5
2&1 5 20 5
1&3 20 10 0
3&1 10 10 10
2&3 15 15 0
3&2 15 5 10
If you then take these stacks and say if one of them is 0 then the chance of that stack finishing 3rd is 1 and that tree ends. If all>0 repeat until either all stacks are equal or one becomes 0. If you keep doing this you can then add up all the 1's divide by the branches and have some %'s of each stack finishing last.
For some branches you would still need some way of calculating the equity though, for example if two stacks keep passing chips back and forth for 5 consecutive hands. I don't think using ICM or M-W would be so bad in this case as it would be watered down by the 7775 other branches, most of which should have been resolved. You could then use the ICM method for deciding 1st and 2nd place. I don't have the knowledge or software to program this into something manageable and have just been messing about with it in my head and on a spreadsheet but I think it could be a better way of estimating 3rd place.
Be interesting to hear your thoughts on this and the constraints/problems that this method has which ICM doesn't.
OK, so I finally managed to get my head around doing a few trials with the method I outlined. I'll been referring to this model as ICM- for the sake of self deprication. The results come from doing 4 rounds of "clashes" so a total of 1296 branches of possible ways the simplified game could end. Where the branches still had 3 active stacks I used M-W to determine the 3rd place %'s. Once all 3rd place %'s were gathered and calculated as a total, I used both the M-W method of dividing the 3rd place finishers chips equally between the 2 remaining players and the proportional method to calculate the 2nd and 1st place %'s.
Stacks 15,10,5
Payout 0.65/0.35
- - - - - - s1 - - s2 - - s3 - -
.ICM 0.448 - 0.357 - 0.196
M-W 0.436 - 0.355 - 0.209
ICM+0.463 - 0.357 - 0.181
ICM- 0.459 - 0.333 - 0.208 - Proportional 1st/2nd
ICM- 0.448 - 0.332 - 0.220 - M-W method 1st/2nd
s2 is going to be so mad 
@jbpatzer:
So you do actually know how stable the nash ranges are?! I’m looking forward to these results since they might give us an impression on how much there’s really to gain by trying to improve ICM. I’d suspect the equilibrium ranges of the bigstacks to be shallower than the ones of the short stacks.
Originally posted by jbpatzer
From this I deduce that, however much we faff around with ICM and ICM+, it's reads on other players' intentions that really matter. I suspect a good poker player would say 'LDO!'.
Ofc this is the case. But these abilities are easier to develop by playing (be coached by better players etc.) than by discussion here since this stuff always depends a lot on certain in-game situations which are mostly hard to explain in a forum. Nonetheless, imo, there’s nothing wrong doing this stuff (ICM+) for the sake of interest, while being aware that it probably won’t help your game a lot – especially for playing against fish when nash ranges are usually far off the holy grail anyway ;). But in SNGs with higher buy-ins where most of the players will shove pretty much in the ICM equilibrium, it might give you a small edge even against good regs in push or fold play.
@nibbana:
Hm. Your method seems to be kind of a hybrid method between M-W and the model I posted on the bottom on page 3. I like that you’re really focussing on busting probabilities which are based on probability paths of events that can happen and not only on stack ratios. This is the way poker is played out (a sequence of hands that change the stack distribution).
Imo, a little weakness in your model seems to be that all the “clash” cases are weighted with the same probability. I mean, on a bubble of 3 stacks, a clash between the two bigstacks will (at least if they play somewhat reasonably) be quite a rare event compared to clashes which involve the short stack. This is imo, why in your model the midstack seems to lose equity. You make him “go broke” more often than he usually does ( 1&2 has the same probability as 1&3 in the model !!!).
So I think if you weight your branches in a more realistic way (for example you could look up the ICM nash ranges for 15-10-5 to get a feel how often clashes 1&2 and 2&1 occur in real play compared to 1&3, 3&1, etc), your model may be a valid alternative.
Originally posted by muebarek
So I think if you weight your branches in a more realistic way (for example you could look up the ICM nash ranges for 15-10-5 to get a feel how often clashes 1&2 and 2&1 occur in real play compared to 1&3, 3&1, etc), your model may be a valid alternative.
Actually this would kill a few of problems all in one.
1. It would take into account the size of the blinds
2. It would take into account position
Then after each clash the new trees could be based on the new position and if necessary the new blind levels. So...
3. It would take into account future actions
Unfortunately this is not something I can continue to work on using a mere spreadsheet and as I mentioned I don't have the know how to do it any other way. I think what it lacks in simplicity compared with ICM, it could make up for in accuracy, though rather than an equity calculator it starts to look more like a game simulator and using the ICM Nash Ranges again uses ranges that I don't feel are accurate.
Use ICM to figure Equity --> Generate Nash Ranges --> Use ICM Nash Ranges to recalculate Equity using ICM- --> Generate New Nash Ranges
ICM is still in the loop !
Hm. It doesn't have to be ICM. I just see the necessity of taking into account that two bigstacks won't clash that often as two shortstacks would.
Maybe a very simple weighting rule might already do the trick to make it more realistic. What if you take the probability of two stacks s_i and s_j clashing (3handed) as
(g - s_i - s_j) / g = s_k/g with g = s1+s2+s3 (total number of chips)
so for example you'd have the stacks s1 and s2 clash with a probability of s3/g
In the 15-10-5 case, this would mean the 15-10 clash would only happen to 1/6 while 15-5 clash with probability 1/3, and 10-5 with 1/2.
Of course this simple approximation still lacks a payout dependence, but it's better than a sharp stick in the eye, i guess 
Originally posted by muebarek
I just see the necessity of taking into account that two bigstacks won't clash that often as two shortstacks would.
I like your idea, I'm just going to be devil's advocate for a moment.
Doesn't the assumption that the bigstacks won't clash as often come from the way ICM chops up the equity ?
If
We're unhappy with the way ICM calculates equity
and
ICM equity is used to calculate the ranges in the Nash Calculator's we use
then
We should also be unhappy with the ranges it produces ?
So we may not know how often the stacks "should" clash until we have a model that effectively calculates the equity ?
EDIT - Seems like a catch-22 infact. We need an equity model - to calculate ranges - to calculate % chance of clashing - to create an equity model
Originally posted by nibbana
Doesn't the assumption that the bigstacks won't clash as often come from the way ICM chops up the equity ?
No. Since it is quite a decent model, ICM predicts this as well, but he assumption above derives from general characteristics of tournament equity.
Take a 6max satellite with (1/2,1/2) structure. The fact that you (as the winner) have 5*your stack at the end of the tourney but only get 2,5*your buy-in back suggests that the real TEQ-curve flattens with increasing stacks. Each lost chip is worth more in $ than a won chip. This causes automatically a higher risk aversion for the midstack against the bigstack than the shortstack's risk av. This means that in the real TEQ nash equilibrium (remember, people play game theoretically optimal in the nash eq.) the midstack won't clash that often against the bigstack as the shortstack.
Sorry not to have been keeping up with the last few posts. I will read them carefully soon. Here are some results for ICM+.
For reference, ICM v ICM v ICM (ignore the legend!)
ICM+vICMvICM
ICMvICM+vICM+
To the accuracy I've calculated, they're not distinguishable.
Originally posted by muebarek
@jbpatzer:
So you do actually know how stable the nash ranges are?! I’m looking forward to these results since they might give us an impression on how much there’s really to gain by trying to improve ICM. I’d suspect the equilibrium ranges of the bigstacks to be shallower than the ones of the short stacks.
I think I may have misunderstood this. Not entirely sure what to plot against what to demonstrate this! (Although you might argue that the previous three plots demonstrate this?)
Originally posted by jbpatzer
From this I deduce that, however much we faff around with ICM and ICM+, it's reads on other players' intentions that really matter. I suspect a good poker player would say 'LDO!'.Ofc this is the case. But these abilities are easier to develop by playing (be coached by better players etc.) than by discussion here since this stuff always depends a lot on certain in-game situations which are mostly hard to explain in a forum. Nonetheless, imo, there’s nothing wrong doing this stuff (ICM+) for the sake of interest, while being aware that it probably won’t help your game a lot – especially for playing against fish when nash ranges are usually far off the holy grail anyway ;). But in SNGs with higher buy-ins where most of the players will shove pretty much in the ICM equilibrium, it might give you a small edge even against good regs in push or fold play.
I didn't really think doing this would help my game, although it could certainly do with some help!
My experience of applying mathematics to engineering problems is that a careful mathematical analysis often reveals that the engineers were doing a pretty good job before us boffins muscled in on the act. But, to immediately contradict myself, I find it hard to believe that that regs push the Nash equilibrium ranges all the time even at high buy in SnGs.
I am going to expend some effort on the three handed satellite coin flip game I described earlier, as I think it may be amenable to pencil and paper analysis.
Originally posted by jbpatzer
I am going to expend some effort on the three handed satellite coin flip game I described earlier, as I think it may be amenable to pencil and paper analysis.
I thought over the case of equal stacks on my way back home. The BB is never calling unless his stack is so small that his real TEQ after folding would be smaller than 1/4*pricepool. In this case, the BU always shoves and the SB always folds.
I think the BU is never shoving if the BB has to call since he's got always more chips after folding than the SB, so the SB would have to shove on the BB if folding leaves his TEQ under 1/4*pricepook and the BU is guaranteed second place.
Correct me if this is bullshit, bc I didn't write it down so I could easily be wrong
I think that sounds about right actually. Just needs to be quantified. And my analysis fizzled out at about the same point as yours!
And of course the stack size below which the BB will call and the button can't push depends on the model for the chance of finishing third. And so we continue in circles! ♦
Driving back tonight I realized that there's a fundamental problem with the way we're testing these models. The defining feature of the Nash equilibrium is that it's the unexploitable strategy. Consider the rock/paper/scissors game. The Nash equilibrium is to randomly choose each object 1/3 of the time. If I now come along and say 'I have a better strategy. I'm always going to choose the rock, because they're hard and tough and must be best.', how can you test whether I'm right? Well, what you don't do is play me off against the Nash equilibrium strategy. This just breaks even. The way to show that you have a better understanding of the game than me is to play 'paper' all the time and exploit my poor model to win.
What this means is that by playing Nash ICM off against Nash ICM+ we have really learnt nothing at all! D'oh! What we should be doing (I think) is playing the 'understanding' represented by the models off against each other by choosing the Nash equilibrium for each player based upon their model of the game. Each will then be trying to exploit the other.
Any thoughts before I try this?
Hmm. I'm not sure I if I understand your suggestion completely.
I get your point on why playing ICM vs ICM+ with the nash ranges doesn't make too much sense. Even if ICM+ was the perfect nash equilibrium, ICM could easily break even - as your (very well founded ) rock strategy does in the RPS game.
Where I'm struggling is the "understanding" part. I guess it's meant in a way that the ICM+ big-/midstack says: "Muhahaha, the ICM shortstack is such a wimp. He calls too tight since he thinks he has more TEQ than he really has. So I'm gonna shove the hell out of him blind vs blind to exploit him!"
While the ICM midstack will think: "This crappy ICM+ shortstack fish. Makes terrible calls on the bubble. I'm gonna exploit this by tightening up even further than my ICM-nash says against the bigstack since this idiot shorty will bust himself in a braindead way, anyway."
And so on... but the point that I don't get: How do you want to calculate this? What do you mean by nash eq. for each player based upon their model? I thought a deviation from the usual model nash ranges was needed.
Well, I currently calculate the Nash equilibrium iteratively. I cycle through the six ranges that have to be calculated and try to change them one hand at a time to increase the corresponding player's TEQ. If all players use the same model, this is the Nash equilibrium. All I have to do is use the appropriate model for the appropriate player, and each player thinks he is playing to maximise his TEQ; in other words to maximally exploit the other players. Not sure that there's guaranteed to be an equilibrium in this case (maybe we should email John Nash?). Not sure at all.