ICM with a skill factor

[eastbay]

Good ole' ICM is defined by:

P^m_i = prob. of mth place for ith player

P^m_i = sum_{k!=i} P^1_k*P^m-1_i(S_~k)

where S_~k is the stacks after removing the k'th stack.

The recursion is closed with:

P^1_i = S_i/sum_k S_k

That's the usual ICM deal. But a reasonable objection is always that this situation is symmetric for all players, so we are undervaluing our equity if we are a winning player.

So let's generalize the method to include a value factor on our chips that gives our chips more value due to our skill.

Rewrite the same equation for S' now, where S'_k = a_k*S_k.

Nominally we can choose a_k = 1, and a_i > 1 to represent our skill (we're player i).

Matching a_i to a nominal 25% ROI for a 10-handed calculation from even starting stacks, we find the finish probabilities are about 14,13,12% respectively. Hey, doesn't that seem familiar? It should. It's a pretty close approximation to most 25% ROI players finish distribution.

The value of a_i which generates 25% ROI is about 1.5. But then it becomes clear that a_i is really a function of handedness, since it's unlikely a 25% ROI players is 60/40 in a HU situation. But it makes sense that a_i is a function of handedness, as we have more opportunities to use our skill against the field early, and fewer opportunities later with big blinds. In the limit of course of blinds bigger than stacks nobody has a skill advantage, so a_i must converge to 1. A reasonable value HU is probably a_i = 1.2, giving a 55/45 advantage. You can probably linearly interpolate between to arrive at a complete 2-parameter (ROI and HU edge) skill-accounting equity distribution model.

The modified formula obeys the sanity check properties of summing to one over both placings and players. This is clear when you realize the formula is just the original ICM from adjusted stack sizes which assign value to each chip depending on who owns it.

We may have a winner. More investigation needed. An obvious next question is: for a typical stealing calculation, does it matter? Results to come.

eastbay

[jcm4ccc]

Well, I understood about 3% of that. But here is one thing I think I did understand:

[ QUOTE ]
A reasonable value HU is probably a_i = 1.2, giving a 55/45 advantage. You can probably linearly interpolate between to arrive at a complete 2-parameter (ROI and HU edge) skill-accounting equity distribution model.

[/ QUOTE ]

You are making two large assumptions here, neither of which I would make. You are assuming a skilled player has a 55/45 advantage in heads-up play. You don’t have any data to support that—it seems to be a number you just pulled out of your hat. The rest of ICM is based on hard data. I think you are treading into murky waters when you start introducing parameters like this.

Also, I believe your model assumes that a player’s skill advantage decreases linearly as the blinds increase. If I understand correctly, the skill advantage in your example starts at 1.5 and decreases linearly until it gets to 1.2. I don’t think this is a good assumption to make. I think that the skill advantage of a player who is experienced at the unique structure of SnGs actually increases as the blinds go up, at least for a while. Eventually, the blinds get to the point where it is basically a crap shoot. But personally, I feel that I have a greater skill advantage over the average fish when the blinds are 100/200 as opposed to 10/20. So I would see it more as a curvilinear function, rather than a linear function.

[pzhon]

[ QUOTE ]
the formula is just the original ICM from adjusted stack sizes which assign value to each chip depending on who owns it.

[/ QUOTE ]
While this does give you more parameters, it is farther from descriptive. Perhaps it allows you to predict the percentages of reaching each place more accurately, allowing you to study the risk/reward of blind steals and defenses (I don't think you will see much of a difference from the standard ICM), but so would other models.

Is there a way to extract data from actual tournaments? For example, take snapshots of the chip distribution at the start of the bubble. From this data and the actual finishes, estimate the probability that you hit each place. See if this matches the ICM, your adjusted ICM, or diverges badly. That is a better test of these models than an application at the start of the tournament with equal stack sizes.

[eastbay]

[ QUOTE ]
Well, I understood about 3% of that. But here is one thing I think I did understand:

[ QUOTE ]
A reasonable value HU is probably a_i = 1.2, giving a 55/45 advantage. You can probably linearly interpolate between to arrive at a complete 2-parameter (ROI and HU edge) skill-accounting equity distribution model.

[/ QUOTE ]

You are making two large assumptions here, neither of which I would make. You are assuming a skilled player has a 55/45 advantage in heads-up play. You don’t have any data to support that—it seems to be a number you just pulled out of your hat. The rest of ICM is based on hard data.

[/ QUOTE ]

Of course I pulled it out of a hat just like I pulled 25% ROI out of a hat - just for illustration. Certainly it's no different than using an empirical ROI figure. Those are both empirical numbers that you would choose based on your own results, aka "hard data."

ICM is not based on "hard data". It's based on logic, conditional probability, and reasonable assumptions, and no data at all.

One of those assumptions is of "equal skill." Many people revile at that assumption. I have replaced that assumption with something more general that does include "hard data." So I find the basis of your objections curious.

eastbay

[eastbay]

[ QUOTE ]
[ QUOTE ]
the formula is just the original ICM from adjusted stack sizes which assign value to each chip depending on who owns it.

[/ QUOTE ]
While this does give you more parameters, it is farther from descriptive. Perhaps it allows you to predict the percentages of reaching each place more accurately, allowing you to study the risk/reward of blind steals and defenses (I don't think you will see much of a difference from the standard ICM), but so would other models.


[/ QUOTE ]

I am unaware of any other nonsymmetric predictive model for placings as a function of stack sizes. Can you give an example?

I agree that we may see very little difference in a steal/defense application. However, this is would be a great utility of the method in and of itself, to eliminate or at least placate the objections associated with the symmetry of the original model.

[ QUOTE ]

Is there a way to extract data from actual tournaments?


[/ QUOTE ]

Clearly we have lots of data for 10-handed, even stacks. This is a player's actual finish distribution. But everything else is deeply problematic due to sample size.

Sample sizes are a severe problem otherwise because the size of the combinatoric space is so large. Even for short-handed play with a fairly coarse discretization of stack sizes, there is an explosion of possibilities that would require far more data than any one player has to arrive at any mean numbers as a function of stack distribution with any confidence.

eastbay

[rachelwxm]

Eastbay,
Thanks, this is interesting stuff. Not a lot of people dare to challange or even understand the foundations of ICM. While I kind of lost in the math fomular, wish we could have some Pdf or gif attachment capability for post, I find it interesting.

In my experience I think player's skill has more to do with stack size rather than number of players left (obvious input of ICM). In a crude way you can consider drop stack size to begin with since they have strong correlation anyway.

Also, I imagine you need to normalize the a_i first? Is it just sum(a_i)=1? ROI of 25% is not 25% anymore if everyone at the table is 25%.

[jcm4ccc]

[ QUOTE ]
[ QUOTE ]
Well, I understood about 3% of that. But here is one thing I think I did understand:

[ QUOTE ]
A reasonable value HU is probably a_i = 1.2, giving a 55/45 advantage. You can probably linearly interpolate between to arrive at a complete 2-parameter (ROI and HU edge) skill-accounting equity distribution model.

[/ QUOTE ]

You are making two large assumptions here, neither of which I would make. You are assuming a skilled player has a 55/45 advantage in heads-up play. You don’t have any data to support that—it seems to be a number you just pulled out of your hat. The rest of ICM is based on hard data.

[/ QUOTE ]

Of course I pulled it out of a hat just like I pulled 25% ROI out of a hat - just for illustration. Certainly it's no different than using an empirical ROI figure. Those are both empirical numbers that you would choose based on your own results, aka "hard data."

ICM is not based on "hard data". It's based on logic, conditional probability, and reasonable assumptions, and no data at all.

One of those assumptions is of "equal skill." Many people revile at that assumption. I have replaced that assumption with something more general that does include "hard data." So I find the basis of your objections curious.

eastbay

[/ QUOTE ]

I had two objections. You addressed one. You also assumed a linear relationship between skill level and blinds (impact of skill level decreases linearly as the blinds increase). That objection you didn't address, though to me that is the more important objection.

About the objection you did address--you will generate a skill value for HU. The example you gave above was 1.2. Two questions about that: Will you generate a unique skill value for HU for each individual person, and how will you generate that value? My objection above is that you will pull that number out of your hat; enlighten me as to how I am wrong.

[eastbay]

[ QUOTE ]
Eastbay,
Thanks, this is interesting stuff. Not a lot of people dare to challange or even understand the foundations of ICM. While I kind of lost in the math fomular, wish we could have some Pdf or gif attachment capability for post, I find it interesting.


[/ QUOTE ]

Agree it's hard to read in plaintext. One place a pencil and paper still blows away digital.

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In my experience I think player's skill has more to do with stack size rather than number of players left (obvious input of ICM). In a crude way you can consider drop stack size to begin with since they have strong correlation anyway.


[/ QUOTE ]

Not sure I follow.

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Also, I imagine you need to normalize the a_i first? Is it just sum(a_i)=1? ROI of 25% is not 25% anymore if everyone at the table is 25%.

[/ QUOTE ]

No need to normalize the a_i. It normalizes itself in the way I've written the recursion closing step.

When you choose a_k = 1, and a_i = 1.5 to generate ROI for player i of 25%, he is taking his money equally from the other players, so they are all adjusted downwards. It has the proper sum no matter how you choose the a_k.

eastbay

[jcm4ccc]

[ QUOTE ]

About the objection you did address--you will generate a skill value for HU. The example you gave above was 1.2. Two questions about that: Will you generate a unique skill value for HU for each individual person, and how will you generate that value? My objection above is that you will pull that number out of your hat; enlighten me as to how I am wrong.

[/ QUOTE ]

OK, now that I'm reading it over again, I understand. The person himself will enter in the parameter as to his/her heads-up skill (I think). I would say that that is soft data (I think I have 55/45 advantage in heads-up play) as opposed to hard data (I am in first place 14% of the time, second place 13% of the time, and third place 12% of the time).

The hard data in ICM is the chip count and the payout structure. You want to add to that a person's first/second/third place percentages. I think that is also good, hard data. You also want to add to that a person's advantage in heads-up play (55 vs. 45). I would classify that as soft, squishy data.

[eastbay]

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I had two objections. You addressed one. You also assumed a linear relationship between skill level and blinds (impact of skill level decreases linearly as the blinds increase). That objection you didn't address, though to me that is the more important objection.


[/ QUOTE ]

That is a relative detail. You can choose any relationship you like, maybe taking data for even or nearly even stack sizes at several points in the tournament. Linear is just the only reasonable way to do it when you use two numbers. Use more if you like.

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About the objection you did address--you will generate a skill value for HU. The example you gave above was 1.2. Two questions about that: Will you generate a unique skill value for HU for each individual person,


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Absolutely. That is the whole idea.

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and how will you generate that value? My objection above is that you will pull that number out of your hat; enlighten me as to how I am wrong.

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I will "generate" it by looking at tournament results, the same way a person "generates" his ROI. No hat required.

eastbay

[lorinda]

If someone has a 14%-13% 1st/2nd ratio they would only be 51.85% if they started heads up equal.

However, I suspect that most strong players are probably behind more often than in front when it reaches this stage, so 55% sounds reasonable.

(Feel free to put more useful numbers in there, I was just making a start on it)

Lori

[eastbay]

[ QUOTE ]

OK, now that I'm reading it over again, I understand. The person himself will enter in the parameter as to his/her heads-up skill (I think). I would say that that is soft data (I think I have 55/45 advantage in heads-up play) as opposed to hard data (I am in first place 14% of the time, second place 13% of the time, and third place 12% of the time).

The hard data in ICM is the chip count and the payout structure. You want to add to that a person's first/second/third place percentages. I think that is also good, hard data. You also want to add to that a person's advantage in heads-up play (55 vs. 45). I would classify that as soft, squishy data.

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I think it can be measured fairly well. I agree it is not quite as hard-edged as ROI, but I do think it's plenty quantifiable enough.

eastbay

[jcm4ccc]

[ QUOTE ]
[ QUOTE ]

OK, now that I'm reading it over again, I understand. The person himself will enter in the parameter as to his/her heads-up skill (I think). I would say that that is soft data (I think I have 55/45 advantage in heads-up play) as opposed to hard data (I am in first place 14% of the time, second place 13% of the time, and third place 12% of the time).

The hard data in ICM is the chip count and the payout structure. You want to add to that a person's first/second/third place percentages. I think that is also good, hard data. You also want to add to that a person's advantage in heads-up play (55 vs. 45). I would classify that as soft, squishy data.

[/ QUOTE ]

I think it can be measured fairly well. I agree it is not quite as hard-edged as ROI, but I do think it's plenty quantifiable enough.

eastbay

[/ QUOTE ]

That makes sense to me. A little caveat there. Just as with ICM, use it how it makes sense to you.

I will be interested in seeing what you come up with. I tried something crude on my own, and the problem I had was that adjusting for skill level had the result of sometimes predicting that the person would win more percentage of the prize pool than is possible (i.e., over 50%). Here is what I played around with, for what it's worth:

I have thought of a crude method to do what you are doing in a more sophisticated way, which is to account for skill differences. I think you can alter the payout structure in the ICM calculator to reflect a player’s 1st/2nd/3rd place ratio.

The payout structure of a normal SnG is .5/.3/.2. These numbers can also reflect the expected payout of the average person. Here’s how: There are 10 players. They are all average players. For player 1, what is his expected payout for 1st place, 2nd place, and 3rd place? It is the actual payout, divided by the number of players:

1st place: .5/10 = .05
2nd place: .3/10 = .03
3rd place: .2/10 = .02

Add these numbers together, and you get the total expected payout: 10%. If we are talking about a $10+1 SnG, the average player would earn, on average, $5 from first place, $3 from second place, and $2 from 3rd place. This adds up to $10. Of course, he is paying $11 to play, so on average he is losing $1 a game.

But a skilled player’s expected payout is different. Let’s take your example of 14% first place finishes, 13% second place finishes, and 12% third place finishes. Over the course of 100 SnGs, this player would get 1st place 14 times, 2nd place 13 times, and 3rd place 12 times. On average, over 100 SnGs, this player is earning $700 in first place money, $390 in second place money, and $240 in third place money. So, in actuality, when this player sits down at an SnG, his expected payout is:

1st place: 700/10000 = .07
2nd place: 390/10000 = .039
3rd place: 240/10000 = .024

The numerator is the amount of money won over 100 SnGs, and the denominator is the amount of prize pool money awarded over 100 SnGs.

If we multiply these numbers by the payout for 1 SnG, we’ll get that player’s expected earnings for every SnG:

1st place: .07 * 100 = $7
2nd place: .039 * 100 = $3.90
3rd place: .024 * 100 = $2.40

So this player earns, on average, $13.30 per SnG. His profit is $2.30, and his ROI is 20.9%

So, when this player sits down at an SnG with his 1000 chips, his expected payout is not .05/.03/.02, it is actually .07/.039/.024. Multiply these numbers by 10 (for the number of players), and you get .7/.39/.24. These are the numbers that you can put into the payout structure boxes in the ICM calculator.

When you plug these numbers into the ICM calculator and give everybody 1000 chips, your $EV is 0.133. Multiply this by the $100 prize pook, and the ICM calculator indicates that you will earn, on average, $13.30 for this SnG. Which is exactly the number we figured 2 paragraphs up.

The problem with doing this, of course, is that every player’s $EV becomes 0.133. But for the majority of things that we do, we are only interested in the change in our $EV. We’re not looking at the change in the other player’s $EV. So I think this method can work for some of the things we are interested in.

The other problem with doing this, is that the expected payout becomes greater than 1st place money when only looking at 2 or 3 players. So this is not something you would want to do when the number of players is small. I would be interested if your adjustments will see the same kind of problem (an expected value greater than 1st place money).

There are probably numerous other problems with this. No need to pick it apart. I was just playing around with numbers.

[eastbay]

[ QUOTE ]

The other problem with doing this, is that the expected payout becomes greater than 1st place money when only looking at 2 or 3 players. So this is not something you would want to do when the number of players is small. I would be interested if your adjustments will see the same kind of problem (an expected value greater than 1st place money).

[/ QUOTE ]

No, this can't happen with my method. This becomes clear when you see the method as simply adjusting the number of chips I have in my stack before you do the calculation.

eastbay

[jcm4ccc]

[ QUOTE ]
[ QUOTE ]

The other problem with doing this, is that the expected payout becomes greater than 1st place money when only looking at 2 or 3 players. So this is not something you would want to do when the number of players is small. I would be interested if your adjustments will see the same kind of problem (an expected value greater than 1st place money).

[/ QUOTE ]

No, this can't happen with my method. This becomes clear when you see the method as simply adjusting the number of chips I have in my stack before you do the calculation.

eastbay

[/ QUOTE ]

Very good. It's becoming clearer in my mind now. Should be interesting.

[rachelwxm]

[ QUOTE ]

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In my experience I think player's skill has more to do with stack size rather than number of players left (obvious input of ICM). In a crude way you can consider drop stack size to begin with since they have strong correlation anyway.


[/ QUOTE ]

Not sure I follow.


[/ QUOTE ]

Just another way of saying blinds increase when less people left. /images/graemlins/smile.gif Of course it is more accurate to account for blinds but it's much harder in pratice to calibrate this I imagine.

I think 400 gives you ROI estimate and 1000 gives you more details into different stages, eventually it will run into the problem of statistical significance.

[pzhon]

[ QUOTE ]
I am unaware of any other nonsymmetric predictive model for placings as a function of stack sizes. Can you give an example?

[/ QUOTE ]
Sure, take a random walk with drift. For many players, it is difficult to come up with a closed-form expression for the probability of each place as a function of the stack sizes, but it's not too hard with just two players. What you get is different from your modified ICM when the skill difference is extreme. Which do you believe more for 2 players?

I just played a 4 player freezeout by accident. A planned MTT on Bowmans had just 4 players, so it paid only the winner. The other players were absolutely terrible, and after a few hands I was heads-up. My opponent almost never raised preflop, didn't bluff, bet infrequently, and folded a lot. The blinds were very small, with a 100 big blind when the tournament ended. What do you think the probability of winning should be as a function of the stacks against such a horrible player?

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I agree that we may see very little difference in a steal/defense application. However, this is would be a great utility of the method in and of itself, to eliminate or at least placate the objections associated with the symmetry of the original model.

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I think most of the objections to the ICM were objections to mathematics. Making a more complicated model will not placate people who don't like mathematical analysis.

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Clearly we have lots of data for 10-handed, even stacks. This is a player's actual finish distribution. But everything else is deeply problematic due to sample size.

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It shouldn't be. Individuals play hundreds of SNGs. That should be enough to fit some parameters or even to reject a model, but if it isn't, there are many active players here. Why not pool the data?

[eastbay]

[ QUOTE ]

Sure, take a random walk with drift. For many players, it is difficult to come up with a closed-form expression for the probability of each place as a function of the stack sizes, but it's not too hard with just two players. What you get is different from your modified ICM when the skill difference is extreme. Which do you believe more for 2 players?


[/ QUOTE ]

I think it's not particularly relevant for my target application of SnG where there is no such thing as an extreme skill difference HU due to the blind size. Even a complete chook can win a good portion of the time.

I worked on biased random walk methods for awhile but found that it:

1 - didn't extrapolate well to higher dimensionality (can you solve your biased random walk methods 4-handed?), and
2 - required some disposable parameters that I didn't really understand how to choose.

[ QUOTE ]

I think most of the objections to the ICM were objections to mathematics. Making a more complicated model will not placate people who don't like mathematical analysis.



[/ QUOTE ]

You may have a point there. But I admit to be interested in how different, if at all, the results look.

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But everything else is deeply problematic due to sample size.

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It shouldn't be.

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It is. Think about how big the stack state space is for 4-handed play with increments of 100 chips. Even for tens of thousands of tournaments, you aren't going to traverse all of those states sufficiently many times to get good mean values.

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Individuals play hundreds of SNGs. That should be enough to fit some parameters or even to reject a model, but if it isn't, there are many active players here. Why not pool the data?

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Because I really want to work with the data from a particular skill level. If we pooled randomly, we'd be back to modeling the symmetric problem, and we already know how to solve that.

eastbay

[pzhon]

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But everything else is deeply problematic due to sample size.

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It shouldn't be.

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It is. Think about how big the stack state space is for 4-handed play with increments of 100 chips. Even for tens of thousands of tournaments, you aren't going to traverse all of those states sufficiently many times to get good mean values.

[/ QUOTE ]
That doesn't matter. Use a model with far fewer parameters than you have data points. Fit the parameters to the data. You will have enough data after hundreds of tournaments.

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If we pooled randomly, we'd be back to modeling the symmetric problem, and we already know how to solve that.

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Pooling data from 2+2 1TT regulars who claim to be winning players would be far from random. There would be a distinguished player who may be assumed to be stronger, so this would say something about skill differences.