There have been threads over the years which have discussed the strengths and weaknesses of the ICM model's valuation of chip stacks in a tournament given specific payouts.

First I'd like to explore the basis of the ICM model. Suppose there are N players left in the tournament. Suppose that P players will be paid (N>=P; truncate payouts as necessary).

The basis of the ICM model is that, abstracting from several factors (such as skill levels, position, blinds, etc.), the probability of finishing in first place out of M (M<=N) players is given by the percentage of chips each of the M players has.

Heads-Up Theory

Consider M=2 (heads-up). This assumption of the ICM model seems to rest upon a fairly well-known result in stochastic processes. The probability of a symmetric random walk with two absorbing states will reach each state is proportionate to the distance to the other state.

That is, beginning at the origin, and moving one unit either up or down with equal 1/2 chance, if the two absorbing states are at A and B (think of them as at positive A and negative B), then the prob of reaching A before B is given by B/(A+B).

Translating to the heads-up poker world, this means that if two players begin with X and Y chips, the prob that the player with X chips will win is X/(X+Y) and the prob that the player with Y chips will win is Y/(X+Y). [In poker, you win when your opponent runs out of chips, which corresponds to reaching the "other" absorbing state.]

Heads-Up Simulations

Over the weekend I programmed up a simulation of heads-up play where each player utilizes the set of Nash strategies presented in the Mathematics of Poker. The authors gave the optimal push-fold and call-fold ranges for each player given the effective number of big blinds at play.

The following table presents simulation results of optimal heads-up play where I varied the starting chips of the two players.

Starting Chips	Chip Leader Win Pct	Chip Trailer Win Pct
(6000,4000)	59.95%	40.05%
(7000,3000)	69.89%	30.11%
(8000,2000)	80.02%	19.98%
(9000,1000)	90.01%	9.99%
(15000,1000)	93.73%	6.27%

Each simulation consisted of 100,000 tournaments. The blinds were (250,500) throughout, with no antes. Regarding the last entry in the table, for those without a calculator handy, 15000/16000 = 93.75% and 1000/16000 = 6.25%.

As you can see, the above table is quite anti-climactic. All the Win Pcts correspond almost exactly with the Chip Stack Percentages. Which agrees with the stochastic processes result. Optimal heads-up strategies are tantamount to making heads-up poker a symmetrical random walk.

I guess this can be seen as confirmation that "ICM works" in the heads-up case. Of course, in heads-up there are no ICM considerations and maximizing chip EV is identical to maximizing $ EV. Players' theoretical win probs do indeed correspond to the players' respective percentages of chips in play when a tournament reaches heads-up play.

Questions which Arise

Several questions immediately spring to mind.

(1) Is there a corresponding theoretical stochastic processes result covering the case of 3+ players which parallels the case of two players?

That is, in a correlated multivariate random walk setting (each player has a 1/3 chance of winning the hand in which she would win two chips, and in the other 2/3 of the time the two losers would each lose one chip), is it true that the probs of "winning" correspond to the respective fractions of the starting distances.

I have run a few simple simulations, and this seems to be the case. Does anyone know if this theoretical result is indeed true?

(2) Unlike the heads-up case, in cases of 3+ players remaining in a tournament, the set of "Maximize Chip EV" strategies differ from the set of "Maximize $ EV" strategies. This is well-known, of course, and is at the heart of ICM.

Has anyone run simulations of 3-way tournaments using "Max Chip EV" strategies to see what the resulting Win Percents look like (viz a viz percentages of starting chip stacks)? I would assume that the two would be equal or very close, as they were in the heads-up world.

(3) Has anyone run simulations of 3-way tournaments using "Max $ EV" strategies to see what the resulting Win Percents look like (viz a viz percentages of starting chip stacks)?

I would imagine that these sets of strategies for button, small blind, and big blind, for all possible chip stacks and for all possible actions in front of each player, may not be readily available to the public.

To cite but one example, we all have heard that ICM valuations undervalue large chip stacks. Can we quantify this "disparity" between true values and ICM approximate values?

Are there any efforts (public or private) under way to look into these sorts of issues??

Very interesting op.
Depends on the payouts and how close we are to the bubble and the relative stacks. Having a big stack to my left makes me tighten up significantly near the bubble, for example folding hands that are likely slightly +chip ev, but -$ev as shoves.

I can't quantify the disparity, but here's what I think about your question:

Let's look at two extreme situations:

2 players left, 1st pays $100, 2nd pays $1, I have 10 big blinds, my opponent has 150 big blinds.

2 players left, 1st pays $51, 2nd pays $50, I have 10 big blinds and my opponent has 150 big blinds.

Chip ev in each situation is identical, but clearly my $ev is much better in the 2nd situation. I could simply sit out and earn $$ in the second situation and I'm losing <$1 in ev by doing so.

Really interesting thread. I can only comment in theoretical without simulations to back it up.

If every player plays perfect then the chip stack should perfectly predict the number of futures that include that player winning the tournament, versus the total chips in play.

A big stack should have advantages, but the small stacks also have advantages, and if every player plays to maximize the strength of their stack size then it is a wash.

So with regards to ICM, would not ICM awareness just become part of the ideal solution to every decision made by every player, with ICM going from zero at the beginning to highly important at the final table, and then back to zero heads up.

So as the chip EV varies as the later stages of the tournament progress, it should vary in a nonlinear fashion and be present in every chip stack.

So the chips at the bottom of a big stack are rarely at risk, the chips at the top of a big stack become closer to regular EV.

This, imho, is why the big stacks are not estimated correctly, if in fact solvers are not gauging big stacks correctly.

The small stacks are almost always at risk and easier to account for by ICM.

I would agree with this and perhaps go further but a warning this is quite a bit out of my knowledge comfort zone and so may be quite far off the mark.

If all players are playing the same perfect push-fold Nash frequencies then the prediction of the final results wouldn't change after one or many games is played forward (ie,if this starting setup was re-played out millions of times and the long run used).

After the first hand the players should still 'own' the same amount of final equity as they started with, in the long run that is. The BB should be able to call at the exact frequencies to recover it's starting hand equity overall and move into the Btn with this exact overall equity, sometimes doubled up sometimes out but on the average would still amount to the starting values going forward. I think this is why the shown simulation gets that result for the winner-takes-all HU case.

I also think this would hold if all players were playing exactly standard icm push-fold Nash frequencies. In this case the final result is predicted by the starting icm valuation and the hand frequencies chosen by the players would match this. After one hand the situation in the long run, ie, the multiple repeated situation, all players would still own the same overall equity. The ranges of hands played would be completely different to the winner-takes-all version but still all come out equal after one forward game or after many if averaged out over many trials.

I think the same thing would happen if all played using exactly the same future-game-strategy but again the ranges are now quite different and the result predicted by the algorithm still using the starting stacks.

Maybe ICM does undervalue large chip stacks, but if you did simulate the future games then after each subsequent hand each player, after many repeats, would still be on the same equity due to how the Nash frequencies are calculated using this icm algorithm.

I don't really know how the typical FGS strategy is calculated, I used to think each future hand was calculated using standard icm but now it seems I should re-think this as in my view playing 1 hand forward or 25 hands forward with all players using the same standard icm strategy doesn't seem to change things.

As I said I might be completely wrong and I look forward to future posts, I think it is an interesting thread.

I've looked into this in some detail a couple of years ago, you can find my ramblings about the topic here. Section 7.4 / page 44 should be relevant regarding winning probabilities.

Thanks much for the replies. I will cogitate on the replies above and respond later.

For now, here is a set of results of simple simulations of "random walks" with different number of players. The open issue (at least to me) was whether the probability of winning a "purely random walky" poker tournament is given by the respective percentage of chips each player has.

As stated in OP, we know this result holds for heads-up play due to a fairly well-known result in stochastic processes. If there are exactly two players the random walk is in one dimension and theoretical results are "easy" to prove.

Players	Starting Chip Stacks	Starting Chip Stack Pcts	Random Walk Winning Trials
2	[6,4]	[60%, 40%]	[600786, 399214]
3	[9,7,4]	[45%, 35%, 20%]	[450046, 350284, 199670]
3	[20,12,8]	[50%, 30%, 20%]	[500299, 300353, 199348]
4	[10,7,5,3]	[40%, 28%, 20%, 12%]	[400434, 279851, 200228, 119487]
5	[12,10,8,6,4]	[30%, 25%, 20%, 15%, 10%]	[299942, 249178, 200559, 150056, 100265]

The random walk simulations were described in OP. In each simulated deal, one of the players is randomly selected to win one chip from each of the other players (once a player's chips reach 0, she is no longer in the tournament). Play continues until a single player remains with all the chips.

Each simulation was performed 1,000,000 trials so I am pretty confident in the results. Obviously, the random walk winning percentages can be read directly from the number of winning trials (dividing by 1 million total trials).

As you can see, the random walk probabilities do indeed correspond to the starting chip percentages in all of the cases. While this is not iron-clad proof of the proposition, I am now convinced in its veracity. (If it was not true, differences would almost surely have shown up in the above table.)

Putting it poker terms, I think it is correct to conclude that if all players in a tournament play their optimal "Max Chip EV" strategies (question number two in OP above), then the expected win probabilities are given by the respective chip stack percentages.

If I have time, I will program this for the 3-player case since those sets of strategies are publicly available. (Possibly plexiq has already confirmed this in his paper linked above. I have not yet had time to review his paper.)

Cliffs regarding winning percentages: For 2 players the winning percentages are proportional to chip stacks, for 3-5 players the winning percentages are no longer proportional when players use "Max $ EV" strategies. Big-stacks win more than their proportional share there.

Could be that the strength of a big stack is non-linear. A big stack can be valued as multiple small stacks, or multiple entries in a small stack tourney. As I have proposed earlier, I believe the icm value of a stack varies within the stack itself, with the very top (first chip) being face value and the very bottom (last chip) is the most valuable, multiples of face value.

Cool thanks.

I think imagining extreme examples is helpful to figuring out this type of problem:

no limit holdem tourney in the money:

I have 1kbb on the button, sb has 2bb before posting, bb has 2 bb before posting.

I'm shoving 100% there no matter what; It's most likely -chipev vs all but the tightest strategies, but it wins lots of money.

Here's a 2008 paper by Sam Ganzfried and Tuomas Sandholm (both from Carnegie Mellon University) that looks into the difference between "Max Chip EV" strategies and "Max $ EV" strategies at 3-handed tables.

https://www.cs.cmu.edu/~sandholm/3-p...ld.AAMAS08.pdf

They report differences between the expected payouts using optimal tournament strategy (max $ EV) and the expected payouts given by the ICM model, but it is not the focus of the paper.

In the cited paper, the ICM deviations were attributed to the fact that ICM fails to account for the future forced bets of blinds – a well-known issue. For the example results cited, the largest ICM undervalue was the button who had the smallest stack while the SB having the largest stack was also undervalued by ICM.

I have now had a chance to read plexiq's fantastic paper linked above (see post 5). He has asked and answered virtually all of the questions posed in this thread.

As an aside, I do have a question regarding the numerical results plexiq reported. Over what set of starting stacks do these results emanate? Are these randomized? Is there a fixed canonical set of starting stacks used throughout? And how many tournaments are played for each set of starting stacks? (Sorry if this information was provided in the paper, but on my first reading I did not see this information.)

From a poker-player perspective, I think the results of the paper could be enhanced if the actual tournament solutions could be compared and contrasted to the standard ICM model (Malmuth-Harville) for a fixed set of "interesting" starting stacks.

For example, [30,25,20,15,10;50,30,20] or [50,20,15,10,5;50,30,20] for five-handed play (three spots paid). Or [40,30,20,10;50,30,10] or [50,30,15,5;50,30,20] for four-handed play (three spots paid). [Inside the brackets are the players' percentages of starting stacks; followed by the tournament payouts.]

I don't know if plexiq still has access to these computer programs or their results, but it would be cool if he could provide something like that here.

Of course, none of the above should be taken to be criticism of the research or the paper. Just asking for more pure poker information that can be more easily digested by the poker community.

Just like in the kelly thread where simple logic proves that EV is maximized by betting your whole bankroll on the good bets, simple logic gives an iron clad proof that the chip percentages are the chances of winning since each bet is zero EV.

Sorry for the short replies, I'm still out of town for a few more days. Will be more responsive when back.

Most of the results (e.g. the ones on winning percentages) are calculated over all the relevant tournament states of the abstract game. See section 5.5 for a summary of the game details: It's essentially a 5 player tournament with 50/30/20 payouts and 4.5BB starting stacks, constant blinds. There is a tiebreaker rule to always keep stacks at multiples of a small blind, that results in roughly 150k possible tournament states.

The only simulated results are in section 8.2 and 8.3: For the model-vs-model simulations, these start at equal stacks and used a fixed number of simulated tourneys for each of the 30 starting configurations (2^5 ways to assign the 5 players to models A/B, excluding the irrelevant AAAAA and BBBBB). Iirc it was 10^8 simulated tournaments per config, so a total of 3x10^9 simulated tourneys for each of the model comparisons.

I still have access to the program/results and actually just ran some new simulations on bounty models recently. Happy to dig out details of the old results if you are interested, the stacks need to add up to exactly 45 small blinds there though.

Thanks plexiq for the reply.

I think I now have a better understanding of your research/paper, but let me confirm that I am following you. Possibly others will find this of interest as well.

Looking at the very first number in Table 7.1, you report a Mean Absolute Percentage Deviation between the "true" player $ values and the ICM (Harville) player $ values of 7.20%.

I think this is a comparison result between true values and ICM values averaged over all 5 players and all 149,985 possible tournament states. And the 149,985 figure comes from there being 135,751 partitions of 45 into 5 parts (with additional states due to how you handle split pots). Meaning that each tournament state is the number (and position) of each of the five players, where their chips sum to a total of 45 small blinds. Right?

ICM values for each tournament state are straightforward to calculate (your paper even presents a more efficient way to calculate ICM values compared to brute force methods).

I think true values for each tournament state "come with" the optimal strategy 5-tuple your sophisticated algorithms derive. Is the optimal strategy 5-tuple for a specific tournament state "interconnected" with the strategy each player will employ in each other possible tournament state? It seems like a player cannot really determine how best to play a 20-blind stack without knowing how she will play a 10-blind stack and a 30-blind stack (say in the simple case of deciding in the big blind whether or not to call a shove from a 10 blind stack from the small blind).

If this is correct, the algorithm simultaneously derives the entire family of optimal 5-tuple strategies for every possible tournament state. And, the values "at" each possible tournament state are calculated as part of the algorithm (they are, of course, the optimization metrics).

I think that these strategies are derived over the entire space of possible deals (something like 10^21) consisting of two hole cards for each of five players and an additional five cards for the board. I believe match-up win pcts are pre-calculated for each possible match-up, abstracting suits in some cases, with the proviso that the number of players allowed to enter a pot (shove) on each deal is restricted to three.

Anyway, if the above is correct, then I think it is fair to state that no cards are actually (virtually) dealt to derive these figures that appear in Table 7.1 (other than in deriving the optimal strategies in the first place). Everything is done analytically, using win probabilities and values over all possible deals and all possible tournament states. If so, that is really neat.

Let us now return to something I mentioned in a previous post above. I think it might be illuminating if you present some canonical results of comparisons of your algorithmically derived true $ values vs. the well-known (but inaccurate) ICM $ values. Of course, the ICM values would be easy to find if you don't have them handy.

Here are some "starting" chip stack profiles (in SB) that I think may be interesting, but feel free to add in/replace others. From my perspective, these true values would best be presented as averaged over all the possible position orderings (as you do in Table 7.2).

3-handed: [30,10,5], [20,20,5], [25,15,5], [20,15,10]

4-handed: [30,5,5,5], [18,17,5,5], [20,12,8,5], [15,12,10,8]

5-handed: [25,5,5,5,5], [20,10,5,5,5], [15,15,5,5,5],[15,11,9,6,4]

From that we would be able to get a fairly good sense of how much ICM under-values large chip stacks (abstracting from position) and, perhaps, glean other valuable insights.

Later, other value profiles could be reported to help illuminate the value of specific positions (e.g., being to the left or right of a big stack, being a short stack immediately becoming the blinds, etc.).

Thanks much.

P.S. I should have asked above, are there antes in your tournaments?

Yup, above is essentially all correct - just some very minor notes:

All correct, except split pots are handled by a random tiebreaker, simply awarding the pot to one of the tied players. This doesn't result in additional states. The 135,751 states are for 5 players, the remaining ones are for 2-4 players.

Just wanted to add that card removal is also simplified (5.3.2):

I don't have access to the results from my laptop, but happy to post these later this week, once I'm back home.

No antes, that'd be a much more complex game because we can't keep the stacks at full multiples of a small blind.

You may know that some casinos are implementing button antes for both their cash games and tournaments. (The idea is that it significantly speeds up play, not having to wait for every player to ante, arguing over who did and did not ante, etc.)

The size of the ante is as follows (paid solely by the button):
- 1 big blind if 7-9 players at table
- 1 small blind if 3-6 players at table
- 0 (no antes) for heads-up play.

So you could introduce antes into your routines in this manner. Even in a game of pure shove/fold strategies, I imagine that having extra chips in the pot via the ante would affect player strategies. And, quite possibly, also affect the efficacy of short stacks since now an orbit at a 5-handed table would cost an extra small blind.

It may be of interest to see how player true $ values vary with and without antes. Of course, you may not be looking to perform a myriad of additional studies suggested herein.

Thanks again!

Ok, finally back. Having the entire ante posted by the BU would definitely simply things for the calculation, I may give that a try sometime.

Here are the position averaged $EV values:
[5, 5, 5, 5, 25]->[14.786, 14.786, 14.786, 14.786, 40.856]
[5, 5, 5, 10, 20]->[13.245, 13.245, 13.245, 23.983, 36.283]
[5, 5, 5, 15, 15]->[12.901, 12.901, 12.901, 30.649, 30.649]
[4, 6, 9, 11, 15]->[10.131, 14.551, 20.825, 24.378, 30.115]

[5, 5, 5, 30]->[18.583, 18.583, 18.583, 44.250]
[5, 5, 17, 18]->[15.567, 15.567, 33.937, 34.930]
[5, 8, 12, 20]->[13.878, 21.483, 28.017, 36.623]
[8, 10, 12, 15]->[19.392, 23.423, 26.636, 30.550]

[5, 10, 30]->[25.631, 31.113, 43.257]
[5, 20, 20]->[25.323, 37.339, 37.338]
[5, 15, 25]->[25.210, 34.331, 40.460]
[10, 15, 20]->[29.498, 33.519, 36.983]

ICM estimates for the same spots:
[5, 5, 5, 5, 25]->[15.377, 15.377, 15.377, 15.377, 38.492]
[5, 5, 5, 10, 20]->[13.960, 13.960, 13.960, 23.754, 34.365]
[5, 5, 5, 15, 15]->[13.579, 13.579, 13.579, 29.631, 29.631]
[4, 6, 9, 11, 15]->[10.582, 15.178, 21.019, 24.144, 29.078]

[5, 5, 5, 30]->[19.246, 19.246, 19.246, 42.262]
[5, 5, 17, 18]->[16.228, 16.228, 33.383, 34.161]
[5, 8, 12, 20]->[14.930, 21.953, 27.908, 35.209]
[8, 10, 12, 15]->[20.024, 23.580, 26.459, 29.936]

[5, 10, 30]->[25.873, 31.389, 42.738]
[5, 20, 20]->[25.111, 37.444, 37.444]
[5, 15, 25]->[25.278, 34.583, 40.139]
[10, 15, 20]->[29.556, 33.619, 36.825]

Lots of manual copy/pasting, hopefully I didn't mess anything up

Thanks plexiq for posting the positioned-averaged True $ values and the "baseline" ICM values.

Edge Model

I have attempted to use this data to estimate the "large stack edge" (over and above the values emanating from the ICM model) that everyone believes is part and parcel of tournament poker. From both common sense and looking at the data, the edge effect is surely non-linear. For much the same reason that the dollar value of tournament chips themselves are non-linear. For example, a chip leader having four times the chip average would not have twice as much "edge" as a chip leader with twice the chip average.

It appears to me that a log-log relationship is appropriate here. That is, I have posited a linear relationship between (i) the natural logarithm of each player's ratio of chips to the tournament chip average and (ii) the natural logarithm of each player's ratio of True $ value to the baseline ICM value.

One of the features of such a log-log predictive model is that the True $ value of any player with a chip stack equal to the chip average will be predicted to equal the ICM value. The True $ values of players with above average chip stacks will be predicted to be greater than the ICM value. And, of course, the True $ values of players with below average chip stacks will be predicted to be less than the ICM value.

Estimation and Analysis

Here are some preliminary findings of an analysis I have performed on the value data presented using the log-log model described above.

I grouped the 4-player and 5-player value data together as the True values (relative to ICM values) are comparable. Of course, in a tournament that pays only the top 3 finishers, when 4 or 5 players remain the tournament is in the "bubble zone". Accordingly, it could be believed that player chip stack edges are most relevant (and probably largest) in this bubble zone.

Here is the predictive model I have estimated:

LOG(True $ Value/ICM $ Value) = ALPHA * LOG(Chips/Avg Chips)

I estimated ALPHA by minimizing the sum of squared prediction errors. The best ALPHA that I found was 0.0635. Meaning that the impact (in logs) of chips, relative to the chip average, on True $ value, relative to the baseline ICM value, is around 6.35%.

Since the log-log relationship is difficult to get a handle on, below I present a table using the 6.35% ALPHA of the "direct" relationship between a player's chip stack (relative to chip average) and True $ value (relative to ICM $ value).

Edge Predictions

The table below presents the predictive "Chip Stack Edge" (True $ value relative to ICM $ value) based upon the player's chip stack relative to the tournament chip stack average.

Chip Stack / Avg Chip Stack	True $ Value / ICM $ Value
10.00	1.157
9.00	1.150
8.00	1.141
7.00	1.132
6.00	1.121
5.00	1.108
4.00	1.092
3.00	1.072
2.50	1.060
2.00	1.045
1.75	1.036
1.50	1.026
1.40	1.022
1.30	1.017
1.20	1.012
1.10	1.006
1.00	1.000
0.90	0.993
0.80	0.986
0.70	0.978
0.60	0.968
0.50	0.957
0.25	0.916

The first row of the table indicates that the True $ Value of a player with 10 times the tournament chip average is predicted to be 15.7% higher (a multiplier of 1.157) than the player's ICM $ value.

The True $ Value of a player with 2 times chip average is predicted to be 4.5% higher (a multiplier of 1.045) than the player's ICM $ value.

The True $ Value of a player with 1.5 times chip average is predicted to be 2.6% higher (a multiplier of 1.026) than the player's ICM $ value.

Of course, the edge for players with large stacks must come from somewhere, and it must come from players with small stacks.

The True $ Value of a player with 60% of the tournament chip average is predicted to be 3.2% lower (a multiplier of 0.968) than the player's ICM $ value.

To be honest, I was somewhat surprised that the large-stack "edges" were not larger than the factors that appear in the table. Possibly, my prior belief was based upon non-optimal tournament play in which many short stacks are very hesitant to enter pots before the money bubble bursts. Another possibility is that there is something special about the particular tournaments that plexiq studied that has caused the edges to be somewhat smaller than expected.

Possible Next Steps

Expand the analysis to encompass all of the tournament states covered by plexiq's study (the before/after the money bubble split probably should carry on). Above, we looked at only a handful of canonical states that I thought might prove illuminating. [This could be done within plexiq's existing framework.]

Expand the analysis to encompass antes. Posts above in the thread touched upon antes. I think most people believe that large stack edges would be greater in the presence of antes. But it would be interesting to do the comparison.

Expand the analysis to encompass additional tournament situations, such as more players than 5, different payout structures, etc. [Probably the most interesting stuff here could not be done within plexiq's existing framework.]

Successfully perform an analysis of In The Money values. From the value data plexiq presented above, the in-the-money True $ values (those pertaining to the tournaments when only 3 players remain) are much closer to the ICM values than I previously believed. So far I have been unable to successfully estimate a "large stack edge" model that adequately explains the True $ values in this case.

Comments welcome and encouraged.