Chips and Cash: The need for an empirical formula for tournament equity

this was originaly published in Twoplustwo Magazine, May 5, 2005. since then i actually tried to do the analysis discussed in the bottom part of the article with a few other people. we made some progress, but eventually lost steam and gave up. rvg on 2p2 did some related work. at some point i might try to write up something more thorough on my experience and rvg's results, but not now.

Recently I played in a 10-player no-limit hold 'em tournament where an interesting hand came up. There were four players left, and I was in the big blind with 1,300 chips left (written T1,300) after posting the big blind of T200. The under the gun player (T6,000) folded, and the button (T1,200) raised all-in. The small blind (T1,200 after posting T100) folded, and it was my turn to act with

AK

Of course I called the additional T1,000, as ace-king suited is a big hand when you've only got about seven big blinds in your stack and you're heads up against a button raise. My opponent flipped up

44

As the board was dealt out, I began wondering whether or not I would have called if I had known what my opponent held.

I have 48.8 percent equity in this pot, so with the blind money in the pot, I knew that, on average, I could expect this call to win me chips. The total pot is

1,200 + 1,200 + 100 = 2,500 so my equity in it is

48.8% * 2,500 = 1,220. I only had to call T1,000, so my expected, or average, win is T220.

In a cash game, this would be an easy call, but this was a tournament, so there are other factors to consider. Only the top three finishers would be paid (50% - 30% - 20%), so going out in fourth was quite unappealing. The small blind was naturally thrilled to see the button and I tangle, since he and I must lose something by getting all-in. This effect is amplified because the under the gun player has a huge stack. Even doubling up leaves any of the three others as a big underdog to win the tournament, so there's more incentive to try to hold on and make it to second or third rather than gamble and play for first.

In order to determine whether I should call (knowing my opponent's hand), we need a formula that will calculate the expected win in real money for a particular chip distribution in a tournament. The most popular such formula is the independent chip model (ICM). I won't go into the details of how the formula works, but you can learn more about it (and use it) at: http://www.bol.ucla.edu/~sharnett/ICM/ICM.html

If I fold my ace-king, the chip distribution will look like this: T1,300 (me) - T6,000 - T1,500 - T1,200. According to the ICM, my expected win for this situation will be 19.6% of the prize pool. If I call, there is a 48.8 percent chance I win and the chips will be T2,800 (me) - T6,000 - T1,200, and my expected win is 33.0 percent of the prize pool. There is a 51.2% chance I lose and am left with T300, and my expected win is 6.2 percent. Taking a weighted average of those two shows that calling gives me an expected win of 19.3 percent of the total pool. This is lower than the 19.6 percent I have from folding, so, according to the ICM, I am better off folding.

This is useful analysis, and most people agree that the ICM is roughly correct. But there are a number of important factors that it fails to consider. One is the location of the button. With very large blinds, it can be very significant who has to pay the blinds next. Many people also feel the ICM short-changes the big stack, by not taking into account her freedom to steal blinds frequently.

Perhaps most importantly, the ICM is just a model. It has not been tested empirically. In the days before internet poker, it would have been incredibly difficult to do so, because one would need a very large data set. Now though, the technology exists to find an empirically-determined formula for converting tournament chips to real cash, one that can include factors like button position and a player's position to the right or left of the big stack.

If you open a single-table tournament on Party Poker, the hands are automatically saved to your hard drive. For a good programmer, it would be a fairly simple task to write a program to open four tournaments and then after 90 minutes close them and open four more. One computer could then accumulate complete histories for 64 tournaments per day, a number that could be multiplied by accounts on different Party skins.

Using a text parser, data on tournament results would then be culled from the hand histories. The chip stacks of the final four players could be recorded, and considered alongside the end result of the tournament. For example, one tournament's data could be something like (2,000, 4,000, 5,000, 1,000, 1, 3, 2, 4) to indicate chip stacks when the tournament got to 4-handed, and ending position. The math involved is outside the scope of this article, but it is then possible to compute a formula that estimates the likelihood of the different outcomes based on the actual observed tournaments. The likelihood of different outcomes could then easily be converted into expected cash winnings.

None of these steps are inherently very difficult, and the formula would allow us to answer some important questions. For example, it could be used to determine fair deals at the end of tournaments.

In addition, David Sklansky has written that chips in a tournament have decreasing marginal value. But Daniel Negreanu once wrote that he made a call that he knew would cost him chips (in expectation) because if he won the pot he would be "in the driver's seat to win the tournament," implying that although the call would (theoretically) cost him chips, it would gain him cash (again, in expectation). Sklansky's and Negreanu's ideas are directly contradictory: If chips do always have a decreasing marginal value then it is never correct to take a gamble that has negative expectation. An empirically-derived formula would help us settle this debate.

One limitation of this approach is that a formula would only truly apply to an "average" Party Poker player, and thus could not directly address Negreanu's claim. It could be the case that for most people, it is always a mistake to take a negative expectation gamble, but for a player of Negreanu's caliber it can be correct. Respected Two Plus Two poster Gigabet made an argument similar to Negreanu's in this thread.

Although it would not be perfect, an empirical formula would be a valuable step, and a marked improvement over using the untested ICM model.