Showdown Win Percentages Revisited

by Daniel Kimberg | Published: Mar 26, 2004

In my last column ("Showdown Win Percentages Against Legitimate Callers"), I discussed the results of some computer simulations of the situation in which all five players ahead of you have called. That column has generated more e-mail than all of my previous columns put together (as well as a brief thread on the twoplustwo.com discussion group), and fortunately most of the comments have been intelligent and well-conceived, even if some were appropriately critical. They also have made me realize that the subject deserves a bit more discussion, in part to clarify some things I didn't have a chance to get into last time, and also to go into some more detail on what you can and can't learn from showdown poker simulations. (If you missed my last column, you can of course find it at cardplayer.com.)

How many hands do you need to deal to get reliable results from computer simulations?

Several people have written to ask whether 200,000 deals is really enough data. This is really two questions. First, did I have enough data to draw the conclusions I drew? And second, how much data do you need to draw conclusions from poker simulations?

I should first clarify that 200,000 is just the number of sets of starting hands (holecards for nine players) that matched the conditions I set. For each of these 200,000 deals, I (well, my program) dealt out all possible five-card boards (all 201,376 of them), to calculate an exact showdown win rate. So, while there were only 200,000 preflop deals (about 1,200 per starting hand), there were more than 42 billion showdowns.

That sounds like a lot of data, but it doesn't tell you much about how reliable the simulations are. Fortunately, we can check on that statistically. Each win rate (for example, 16.4 percent for 4-4) is actually the average of about 1,200 win rates, so it's easy to calculate a standard error. Standard error is a statistical estimate of the accuracy of a measurement. In this case, it tells us how accurate the win rates are for each hand (given the assumptions of the simulation). The win rates I reported are accurate to about plus or minus 0.3 percent (that's two standard errors; it varies a bit, some were quite a bit smaller). That is, there's a 95 percent certainty that the true showdown win rate for 4-4 in the situation I described is in the range of 16.4 percent, plus or minus 0.3 percent. What are the practical implications of this? First, it appears that 200,000 simulations was more than enough to make the points in my last column. However, it would be wrong to read too much into small differences in ordering; 6-6 clearly fares better than J-J, but its edge over 5-5 could just be a bit of noise in the data. With more data, we could find out for certain, although it's not clear why we'd want to. As well, the smallest differences are most vulnerable to small changes in the starting assumptions.

Of course, none of this tells you anything about whether or not these win rates will be remotely close to your actual win rates at a real poker table, which depends on other factors (for example, whether you or your opponents can fold if you miss the flop). It just means that in terms of what the simulations are intended to estimate – namely, showdown win percentages – they're more than accurate enough.

Regarding the second question, there's no magic number for how many simulations are needed to work out a poker situation. Although the numbers tend to be large, the only way to know when you have enough data is to keep tabs on the estimation error. It's worth keeping this in mind whenever you read about simulation data. Results reported without some index of reliability are often hard to interpret. Since I did that in my last column, I'm as guilty of this as anyone (I had the numbers, but I just didn't think it was worth putting them in the column).

How realistic is this situation?

No matter how accurate your numbers are, they're not much use unless they're an accurate measure of something useful. A number of folks expressed concern that the situation I described – five solid players who would flat-call with exactly those starting hands – was not particularly realistic. I certainly agree with this criticism, and in retrospect I would probably set things up differently. At some point I will try a few more scenarios, and let you know what I find. But it's worth bearing in mind that the simulation results don't depend on the action being exactly as I described it; they just depend on your knowing that five live opponents all have legitimate playable hands. There may be situations in your game that qualify, even though the action isn't how I described it. For example, maybe two tight players call, followed by raises from two average players, and finally a flat-call from a player on whom you have a very good read. Perhaps at your table, that sequence of events would be a more reliable sign that five players have solid hands, and in that case the simulation results would be equally applicable. Perhaps you might have even more specific information – for example, from an opponent who would raise only with Q-Q or better. The simulation results are due to the quality of the information you have, and in particular how closely they constrain your opponents' hands. So, they're more useful the better your ability to read players.

A related concern is that even if it's realistic, it's certainly rare. How often do five of the six players to act ahead of you all hold hands that are playable by Lou Krieger's starting-hand charts? I didn't actually check on that when I ran the original simulations, but I have in the meantime, and it turns out to be roughly every 1,200 hands. So, if you play 1,000 hours a year and get in 36 hands per hour, it'll come up 30 times. Maybe three or four of those, you'll be on the button. And given that the action won't always cooperate, or you may have a garbage hand, it's unlikely to make a noticeable difference in your bottom line. But the main idea in looking at simulation results like this isn't always to examine the precise circumstances of the simulation as much as it is to get a feel for general principles that might inform your decisions. With four callers, including one very tight raiser, you might make a similar adjustment. It's easy to think of situations in which the effects would be smaller, but still noteworthy. While the extreme case I described is certainly rare, and assumes your opponents adhere to the starting-hand tables rigidly, cases that resemble it do occur regularly, if not frequently, in full games. However, to get full value from the simulated data, it will eventually be important to run a range of simulations to cover a broader set of circumstances.

Showdown poker bears little resemblance to real poker.

The most problematic issue for simulations like this concerns the usefulness of showdown simulations. The simulations tell you how often your hand will be the best at the table if everyone reaches showdown. But not everyone does reach showdown. Your true win rate could be higher (if, for example, other players will fold weak draws) or lower (if you fold weak draws). Some folks think this patently false assumption makes the simulations worthless. And, certainly, if the intent was to provide real-world winning rates, I would have to agree.

But the purpose of calculating showdown rates isn't to claim that these percentages really reflect how often you will win the pot. Showdown win rates are just one dimension of a hand's value. If used at all, they should be used in combination with other knowledge about the hand: whether it needs to hit on the flop to continue; whether it can outdraw opponents when it's behind; whether it's likely to get paid off by second-best hands or vice versa; and so on. In my simulations, A-A and 8-8 were very close in win percentage, but in reality, the two hands play very differently. So, it would be foolish to draw any conclusions about which of those two hands is really more profitable. But some of the differences were quite large; 4-4 is more than four times more likely than A-K to end up best. While other differences between the hands may make up some of that gap, it's hard to imagine that A-K can be played profitably on those occasions when you know your opponents all have real hands. The situation is similar for many of the other unpaired high-card hands.

In my last column, I did put some emphasis on the absolute win rates, relative to the average. In particular, I noted that pocket pairs were somewhat better than average, small suited connectors were somewhat worse, and unpaired big cards were dramatically worse. I certainly don't think that small differences within these three groups are meaningful. But it would be wrong to conclude that just because the suited connectors won less than their fair share in showdown, they couldn't be played profitably. The "fair share" cutoff of 12.5 percent is really just a point of reference, not an absolute playability cutoff, and certainly much more than showdown win percentage figures into profitability.

Didn't you recently write about bunching, with completely different results?

I did recently run a simple simulation to examine how hand values change when all players ahead of you have folded. In that simulation I also used the Krieger starting-hand charts, and found that the bunching effect was basically negligible. Why do five calls have such a dramatic effect, while folds tell you almost nothing? Because the starting-hand charts are fairly tight, you learn a lot more about someone's hand when they call than you do when they fold. To put it a little differently, the amount of information you get from someone's action is related to how unusual it is. When tight players fold, you don't get much information about their cards. When tight players call, you do. I haven't run a simulation to check on the effects of five loose players all folding, but it would likely be more helpful than the bunching simulation, even though, again, it would reflect a fairly rare occurrence.

Conclusions

The simulations I reported in my last column are not the kind of results that will make an immediate and dramatic impact in your game. They haven't been fleshed out into detailed strategic advice, except in one very specific and unusual case. However, there are certainly other circumstances under which dramatic adjustments to hand-value estimates are appropriate. The most obvious cases are the ones in which you get a huge amount of information from one player (typically a player you know would reraise only with A-A or K-K). But there may be other cases in which you get useful information about your hand's value from an aggregate of four or five other players' actions. These simulations shouldn't be taken as the final word on anything, but as a starting point for working out how to integrate telling information from players ahead of you into preflop adjustments.