Pokerstars has introduced a variant of No-Limit Texas Holdem in which there are two separate boards on each deal. Two flops, two turns, and two rivers.

Essentially the entire pot is split into two halves. One half of the pot goes to the winning hand based upon the first board and the other half of the pot goes to the winning hand based upon the second board.

A recent thread in the Poker Theory forum has begun a discussion whether and, if so how, the optimal strategy under Split Holdem differs from the optimal stategy under Regular Holdem.

A recent thread in the NVG forum has begun a discussion whether or not this new variant is a "good thing" for players and/or for Pokerstars. Many people have guessed that, since many pots in Split Holdem will be split pots, the rake will dramatically eat away at players' "normal" ROI and the games will not be profitable for players.

Herein (and cross-posted to the NVG thread) I want to explore this last point. How often should we expect a pot in Split Holdem to be a split pot? I initially tried to answer that question using pure reasoning. Unfortunately I did not get very far using that tack, so I resorted to that old stand-by the simulation.

I just ran a series of simulations to estimate how often Split Holdem will result in a split pot based upon the number of players.

In these simulations, all players go to showdown on every deal. Each simulation set consists of 1 million random deals.

Number of Players	Split Pot Pct
2	49.4%
3	66.0%
4	74.4%
5	79.3%

Are these results what you would expect to see? What do you expect to see as more players are added to the table?

I played this game for a couple of orbits on play money just to see what it is. almost every pot that went to showdown was split. it shouldn't really be a surprise, it's pretty hard to hit both boards with just one hand.

from your table it looks like split percentage is approx equal to (1 - 1/N)*100%. if all players play exactly the same hands, that table should be 100% correct. so this game really is a huge raketrap.

I have done a couple more cases, so I'll add them to the table below.

Number of Players	Split Pot Pct
2	49.4%
3	66.0%
4	74.4%
5	79.3%
6	82.7%
7	85.0%

Yes, the above results do look a lot like 1-1/N.

To be perfectly honest, I was quite surprised by those results. I initially believed that "good hands" would win both boards a fair amount of the time. Say a high pocket pair or even AK.

I might be misinterpreting your comment, but it sounds like you are saying that if all players play the same range of hands, we would expect a result of nearly 1-1/N fraction of split pots.

I do not agree with that unless I am totally missing something. For example, to take an extreme case, if the rules of holdem were changed to rank hands solely according to cards' ranks (i.e., ignore flushes, straights, pairs, etc.), then I am quite confident that this variant of split holdem would see very few split pots. In such a game a high card in the hole would often win both boards.

Since these are simple to do, I just ran a simulation of this variant and got only 3% split pots in this heads-up variant (again, all deals going to showdown).

So there is something about how two hole cards interact with five board cards that is "causing" the very high prevalence of split pots.

Let me ask the audience, if I re-ran the simulations but restricted all players to play only top hands, would you expect the split pot percentages to change significantly?

In the 2-player simulations, I could restrict both players to play only Top 50% hands. Would we still expect to see around 49% of all pots that went to showdown be split?

that's what I meant, you didn't misinterpret it. that's my guess anyway.

For completeness, here are two more cases.

Number of Players	Split Pot Pct
2	49.4%
3	66.0%
4	74.4%
5	79.3%
6	82.7%
7	85.0%
8	86.8%
9	88.1%

Great work as always. Regarding your question, I don't have a clear feeling. If you tighten the ranges to the maximum such as players only play aces, you have obviously a lot of splits. On the other hands, if you allow a AA-KK range for both players, you have less splits (since it's much more common to have AA vs KK instead of KK vs KK or AA vs AA and in the former case you often have only one winner). So don't know what to expect as ranges tighten. I'd bet a very small amount that split percentage stays more or less at that level until you arrive at a ~15-10% range, after which I expect a slight drop in split % (because of domination).

Three additional things to report.

1. In heads-up Split Holdem, when I restrict players to only play any ace, any king, or any pair, the split pot pct is 48.9%.

2. In heads-up Split Holdem, when I restrict players to only play any ace or any king (including AA and KK), the split pot pct is 49.4%.

3. In heads-up Split Omaha (high only, of course), the split pot pct is 51.0%. To be honest, I am somewhat confused by this result. I thought that the split pot pct would be lower in Omaha than in Holdem.

Ranges composed of only top hands (pocket pairs and Ax) have limited coverage of the deck and make fewer straights and flushes, so less split pots.

Ranges composed of any two cards, and played to showdown always, will result in the most split pots.

Not sure if this is helpful, but this is my idea on your simulation results.