Decided to sort out my code this lunchtime. You can download it here. Comments and criticisms welcome. Happy bug hunting! ♠_biggrin:
Examples:
To get plots of ICM ranges exploiting a fish in the BB
[x,E] = ICMranges([0 0 0],[0.65 0.35 0],s,169,[0 0 1 floor(169*[0.3 0.15 0.3 0.15 0.05 0.3])],1);
To simulate 5000 STTs with two ICM players trying to exploit a maniac
[EV, EVout] = STTsim([1 0 0],[1 0 0 N N N N N N],[0.65 0.35 0],[10 10 10],25,5000,0,'test','Maniac','ICM','ICM');
Originally posted by jbpatzer
I tried ICM v ICM v fish (fish = push 30%, call 15%, overcall 5%), and found that whether ICM was straight Nash ICM, or adjusting to exploit the fish, ICM won about 34.5% each and the fish 31%. This seems a bit surprising, but having seen the results for maniacs and wimps, perhaps it isn't. I think the problem is that the two ICM players have to respect each other, which limits the extent to which they can exploit the fish [...]
I thought a bit about this "problem" of having fish on the table (not that I don't love to have a lot of fish ingame ♠_biggrin: ) and I had to realize that this makes it really complicated to approach this mathematically. The reason for this is that you're breaking the "symmetry" of equal skilled players. But this unfortunately leads to a deviation from the basic assumption of all equity models we know.
Perfecting shoving/calling ranges highly depends on your TEQ predictions. Take calling for example where the minimum equity you need is
E = (TEQ(fold) - TEQ(call,lose))/(TEQ(call,win) - TEQ(call,lose)). This equity determines which hands are profitable to call. But without having a clue about the TEQ function this formula is worthless. Determining exploitative ranges based on equity models that assume equal skilled players could make your ROI even worse than just using the models' nash ranges. This seems to show up in your simulations! Adjusting might be easy if you are BvB (btw my fav football club 
) vs a wimp for example, but for not so extreme fish types, attempts to adjust seem to lean towards pure speculation.
So, imo we have to ask ourselves, if it really should be the symmetry of skill to break first or if it might be easier to start with breaking positional symmetry since we have more of a legitimation of using TEQ models like ICM, ICM+, MW etc. there .
Originally posted by jbpatzer
I'm going to tidy up and comment my MATLAB code and make it available here. Then other people can try it.
Cool. That's nice of you. I already downloaded it.
Originally posted by jbpatzer
And by other people, I probably mean muebarek!
After having a first look on the code, I have to admit it will probably take me quite a while to get used to it since I've never worked with Matlab before (and I only have Matlab available on university computers). But I see what I can do to run some simulations as well. 
Originally posted by muebarek
I thought a bit about this "problem" of having fish on the table (not that I don't love to have a lot of fish ingame ♠_biggrin: ) and I had to realize that this makes it really complicated to approach this mathematically. The reason for this is that you're breaking the "symmetry" of equal skilled players. But this unfortunately leads to a deviation from the basic assumption of all equity models we know.
Maybe, but remember that I've been trying to optimize the ranges for ICM players who know exactly what the fish's pushing and calling ranges are. There isn't therefore an assumption of equal skill. In fact the ICM players have perfect information about the fish's 'skill' or lack of it. The problem is more subtle I think. In fact, I'm surprised that the ICM players can't seem to exploit the fish. When it's BvB, the ICM player must have an advantage because he knows the fish's (exploitable) pushing and calling ranges, and can act appropriately.
Originally posted by jbpatzer
There isn't therefore an assumption of equal skill. In fact the ICM players have perfect information about the fish's 'skill' or lack of it.
That's true. But my point is: they know the fish's exact shoving/calling ranges but can't figure out how to change their own ranges perfectly to adjust to this because they need a TEQ prediction to do so.
To stay in the example I used above, they don't know how much equity they need to call a shove in the BB since they can't estimate their TEQ correctly for the cases which can come up.
For example the stacks are S_BU = 13bb, S_SB(Fish) = 10bb, S_BB = 7bb. How can the BB know how risk averse he is against a shove from the BU if he has no idea about his equity in case he wins [ S_BU = 6bb, S_SB(Fish) = 9,5bb, S_BB = 14,5bb ]
( because TEQ_ICM ( 6bb, 9,5bb, 14,5bb) != TEQ( 6bb, 9,5bb(FISH!), 14,5bb) )
Concerning BvB against the fish, I agree, at least you'll never make a mistake to shove lighter against an opponent who calls too tight and shove tighter against wide-calling opponents. But it'll be still hard to get the exact ranges right!
Hm. I guess I didn’t explain that too well, so I’m going to do a little calculation to show off the effect I was talking about:
Take a tournament of N equal skilled players with random positions and assume the case of equal stacks S_i = s for all i. This means, the real TEQ of player j at the moment is:
TEQ_j = 1/N
The preflop equity, a player j needs to go broke against another player m (ignoring the blinds for the moment) is
E = TEQ_j(fold) / TEQ_j(call,win) = 1 / (N*TEQ_j(call,win)) ~= 1 / (N * TEQ_ICM ) (*)
where TEQ_ICM = ICM_j( S_j = 2*s, S_m = 0, S_k = s for all k != j,m )
The last identity of (*) is an approximation assuming that for random positions and equal skill, it is acceptable to replace the real TEQ by ICM TEQ.
But now let’s assume the player m is a fish who therefore has less TEQ than 1/N. Furthermore, let’s assume we know exactly that his bad play loses him an amount e of TEQ in the long run. The TEQ distribution for equal stacks and random positions is then
TEQ_m = 1/N – e, TEQ_k = 1/N + e/(N-1) for all k != m
If we now calculate the equity, player j (!= m) needs to go broke against the fish, we get
E = TEQ_j(fold) / TEQ_j(call,win) ~= [ 1/N + e/(N-1) ] / TEQ_ICM
Note, that the approximation TEQ_ICM in the final state is ok, since after the fish busts, there are only equal skilled players left.
One sees: Having a fish on the table makes you more risk averse when calling all-in against hím (since there was a positive quantity added in the nominator). The other way round this means, that you’re less risk averse if you are an average player on an average table and one genius is there too, to call all-in against the genius (just change the sign in front of e).
The funny thing about this formula is that you can actually calculate the equities if you know e (which could be put into relation with the ROI in the early stages). But the problem is, that you can’t calculate anything which leaves final states where the fish/genius is still around since you need a skill dependent TEQ model to calculate e as a function of the stack distribution.
And this is the problem of adjusting your play when having non-equal skill.
So for the three handed bubble you can estimate e (Terrible choice of symbol btw. I am well-known for being a notation fascist!)) using my simulations for ICM v ICM v fish. It's about 2%, which is about 6% of the fish's TEQ. I could assume that his TEQ is always 6% of what ICM says it is, and that the missing TEQ is distributed to the other two player in proportion to stack sizes, and see if that helps the ICM players exploit the fish any better. What do you think?
Originally posted by jbpatzer
(Terrible choice of symbol btw. I am well-known for being a notation fascist!))
Didn't think it was so bad ♠_frown:. e for edge (I initially calculated it for a winning player, and on my paper it was an epsilon but in this text format writing epsilon isn't that great imo ). I mean, ok, we already have an E for equity but i don't think we'll ever have an Euler e in our context here! So what's wrong with it? Explanation plz!! 
But of course I don't mind us using another symbol.
Originally posted by jbpatzer
So for the three handed bubble you can estimate e (...)
using my simulations for ICM v ICM v fish. It's about 2%, which is about 6% of the fish's TEQ. I could assume that his TEQ is always 6% of what ICM says it is, and that the missing TEQ is distributed to the other two player in proportion to stack sizes, and see if that helps the ICM players exploit the fish any better. What do you think?
Hmm. Might be worth a try for sure. The problem is that we again have two "guessing games" in there which are to hope that the procentual loss of the fish will be at least approximately equal for all stack distributions and once again the split up rule. But ok, since we don't have more information for now, it'll probably be our most natural next step.
Originally posted by muebarek
Originally posted by jbpatzer
(Terrible choice of symbol btw. I am well-known for being a notation fascist!))Didn't think it was so bad ♠_frown:. e for edge (I initially calculated it for a winning player, and on my paper it was an epsilon but in this text format writing epsilon isn't that great imo ). I mean, ok, we already have an E for equity but i don't think we'll ever have an Euler e in our context here! So what's wrong with it? Explanation plz!!
But of course I don't mind us using another symbol.
Look, I'm a mathematician. e is Euler e whatever I'm working on!
Originally posted by jbpatzer
So for the three handed bubble you can estimate e (...)
using my simulations for ICM v ICM v fish. It's about 2%, which is about 6% of the fish's TEQ. I could assume that his TEQ is always 6% of what ICM says it is, and that the missing TEQ is distributed to the other two player in proportion to stack sizes, and see if that helps the ICM players exploit the fish any better. What do you think?Hmm. Might be worth a try for sure. The problem is that we again have two "guessing games" in there which are to hope that the procentual loss of the fish will be at least approximately equal for all stack distributions and once again the split up rule. But ok, since we don't have more information for now, it'll probably be our most natural next step.
I'll give it a try.
Originally posted by jbpatzer
Look, I'm a mathematician. e is Euler e whatever I'm working on!
... How you would hate my cashgame calculations where the potsize always is a small pi (since p is reserved for probabilities)!
Originally posted by jbpatzer
So for the three handed bubble you can estimate e (Terrible choice of symbol btw. I am well-known for being a notation fascist!)) using my simulations for ICM v ICM v fish. It's about 2%, which is about 6% of the fish's TEQ. I could assume that his TEQ is always 6% of what ICM says it is, and that the missing TEQ is distributed to the other two player in proportion to stack sizes, and see if that helps the ICM players exploit the fish any better. What do you think?
Tried this. Makes not a scrap of difference! Baffling! 
Hmm. I’m starting to believe we have to rethink how to approach this. So far, our procedure was like:
Try to study a certain scenario --> Come up with (hopefully) improved TEQ estimations
--> Let them play out and see whether they win more $
Our results seem to end up being: “Well, it makes no difference” though we are quite sure to have a better TEQ. Think of your nice ICM+ graph (jb 3rd place iteration method) which showed that your TEQ was even nearly a martingale and therefore had to be quite close to real TEQ. Or the case now where we can’t be sure whether our TEQ is better since we had two crucial assumptions to make for the simulation but my calculation above (which didn’t assume anything besides random position, really) showed that there has to be an effect of edge on risk aversion.
So maybe we should study another question: “How much can model TEQ predictions differ from real TEQ without having impact on the perfect ranges in a N hand model?”
Perhaps the results of this will show that we have to have at least N=45 hands for example to see an effect.
What do you think about this? I’d like to know your opinion on this before I blindly start suggesting how this could be realized 
Originally posted by muebarek
So maybe we should study another question: “How much can model TEQ predictions differ from real TEQ without having impact on the perfect ranges in a N hand model?”
Perhaps the results of this will show that we have to have at least N=45 hands for example to see an effect.
What do you think about this? I’d like to know your opinion on this before I blindly start suggesting how this could be realized
Seems reasonable. And I have been wondering whether N=25 is enough. The pictures below are from page 2 of this thread, and shows the Nash ranges for N = 100, 50 and 25. They're all pretty ragged, particularly N=25. This is probably an indication that the EV doesn't change much close to the Nash equilibrium (look how smoothly the EV varies with stack size).
I could try some calculations with N=50. Going to take four time as long though!
Another random thought. Do you think trying fixed positions would give us any insight?
Originally posted by jbpatzer
Seems reasonable. And I have been wondering whether N=25 is enough. The pictures below are from page 2 of this thread, and shows the Nash ranges for N = 100, 50 and 25. They're all pretty ragged, particularly N=25. This is probably an indication that the EV doesn't change much close to the Nash equilibrium (look how smoothly the EV varies with stack size).
Hmm. Yes. The flat lines do look suspicious to me and I was shocked at first, thinking "How could a model be susceptible to small TEQ changes if it doesn't matter if you add like 3-4bb to the stack?!".
But fortunately it isn't all that bad! Part of the effect we're seeing here isn't caused by stack size dynamics (and therefore TEQ model dependent risk aversion) but by plain pot odds, because you had equal stacks for every stack size here.
Imagine, you are in a cashgame. The equity you need to call an openshove of 20bb won't be much different to calling a 24bb openshove since they are both very large in comparison to the pot so you'll be calling a pretty similar range there.
In one of pzhon's video, he stated, that the equity needed to go broke is like
E = E(classic Potodds) + risk aversion
and the risk aversion is more or less independent of the absolute amount of bb you have in your stack but more dependent of the relative stack dynamics and the payouts (and therefore the TEQ model!!). And if you have a nearly constant risk aversion due to equal stacks and worse potodds which make E(classic Potodds) converge towards 50% this is a possible explanation for what we see in the graphs.
Nontheless, I'm pretty sure we'll find that N=25 isn't sufficient (as there are still differences between N=25 and N=50,100 - but that seems to be more significant towards the premium ranges since H(h) changes faster there).
Originally posted by jbpatzer
I could try some calculations with N=50. Going to take four time as long though!
Imo, we should really think about this and try to estimate the N we need before you spend 3-4 days running N=50 to find out it didn't help much.
Originally posted by jbpatzer
Another random thought. Do you think trying fixed positions would give us any insight?
Definitely! As we saw in your simulations with fixed positions, relative position corrections seemed to be at least as important as the TEQ differences due to stacksize. But then again, we don't know if N=25 will be sufficient to make us "see" anything ♠_cry:
So, imo, we should really find an answer to the question I put up above. The reason for this is, that we don't even care if positions are randomized or if we have equal skilled players but we just can find out in general how big TEQ differences (whereever they come from [position,stacks,edge,... ]) have to be to see differences in the ranges dependent on N (under the assumption of using your H(h) ). Sorry for being so stubborn on this ♠_cool:
Originally posted by jbpatzer
So precisely what simulation are you suggesting I do?
This is the crux
I'm still thinking about it. But I already have some ideas where we could start:
We're trying to improve our ranges to take advantage of the fact that we know the TEQ function better than our opponent. The easiest case to study this is a BvB situation.
Let's assume our opponent's strategy as static. This could be him always playing ICM nash ranges or any other (fish) strategy.
So if we know his static ranges exactly in a certain BvB spot, we are able to calculate the preflop equity E we need against his shoving/calling range R as a function of our TEQ predictions T which is a vector with entries of TEQ for the cases which can come up (for example call-win , call-lose, shove-opponent folds, ...).
Since we don't know the real T, we have to use a model giving us a T'. This will in general yield a different minimum equity E. Now when does this have an effect on your strategy?
Be h0 the last hand you could call/shove with using T' (and I really mean "hand" not range up to this hand as your h in the simulations) and E'(h0,R) the preflop equity of h0 vs the range R
If deltaE = E(T,R) - E(T',R) > 0 then we have to take out h0 of our range if
E'(h0,R) - E(T',R) < deltaE.
And if deltaE < 0, we have to take the hand h0+1 in our range if
E'(h0+1) - E(T',R) >= deltaE
We could now try to find out how big deltaE will be by inserting several T in the magnitude of the TEQ differences you got out of your simulations so far and calculate all
|E'(h0,R) - E'(h0+1,R)| for N=25, 50 and 100. (again, we have to be careful here since h0 is a single hand, not a range)
If we see for a certain N that |E'(h0,R) - E'(h0+1,R)| >> |deltaE| there probably won't be much of a strategy change to expect.
Now this is more of a spontaneous idea and it could be easily bullshit! But anyway... I'm looking forward to hear your opinion on this!
edit: if I keep editing like this, I'll eventually overtake pzhon's signature
EDIT:
To make my suggestion a bit more concrete, I'm going to start the calculation of E'(h0,R) - E'(h0+1,R).
Be h the range of hands in which h0 is the worst hand, then the preflop equity of h against villains range R according to your model is
E'(h,R) = H(h)/[ H(h) + H(R) ] where H is the hand strength function.
To get the equity of the single hand h0 against R, we can use the identity
E'(h,R) = (1/h) * [ sum_[k in h\{h0}] E'(k,R) + E'(h0,R) ]
where k is running over all single hands except h0 and every hand is weighted with the factor 1/h since every hand as an equal amount of combinations in your model. The sum can be expressed by E'(h-1,R), so:
E'(h,R) = (h-1)/h * E'(h-1,R) + 1/h * E'(h0,R)
<==> E'(h0,R) = h*E'(h,R) - (h-1)*E'(h-1,R)
==> E'(h0,R) - E'(h0+1,R) = 2h*E'(h,R) - (h-1)*E'(h-1,R) - (h+1)*E'(h+1,R)
Now all we have to do is plug in the numbers for N=25,50,100 (and maybe 169 ^^) which I can do tomorrow or friday if you think it makes sense to study this.
I hope I didn't screw up in the calculation ♠_ugly:
Thought I should post here. I don't think I'm going to be able to find the time to continue with this project atm. I think we have some evidence that:
i) In the absence of increasing blinds and with randomized positions, ICM is a very good model of tournament equity
ii) This simple discrete approach is a worthwhile way of modelling full NLHE three man bubbles.
I hope nibbana, muebarek and pzhon will consider carrying on with this if they can unravel my MATLAB code. There are lots of interesting ideas in this thread.
Also, I'm going to start offering 'The Mathematics of Poker' as a possible undergraduate project from next academic year, so I may be able to get a student to carry on with this too.
Anyway, good luck to anyone who picks up the ideas here.
Nice work jb, the stuff that I could decipher with my level 1 maths thinking has been enlightening. I don't think I'll be able to contribute anything meaningful to this venture but I am going to write to sngWiz and ask if they're looking at having functionality in future releases to be able to choose between different equity models (notably M/W) - it can't be so tough seen as they have $ev and chipEV already as options and seen as the latest version has a link to the Nash calculator at Hold'Em Resources I would think it's perhaps a natural step to be looking at this soon.
Take it easy.
