Here is how I have begun to think about this question.
Let's stylize things a bit so that we can do some simple equity calculations in the two cases (one board vs. two boards). Let's use the following stylized example.
Hero has been dealt two spades (think of it as Ace-Ten if you want). He will win if and only if he makes a flush in spades (we will ignore complications dealing with higher ranked hands throughout). To simplify further, suppose that the turn and river are dealt together. Let's look at our equities and strategic options after all possible flops.
The following table summarizes this situation under the One Board game rules.
Spades on Flop | Prob of Flop | Prob of Flush given Flop | Combined Prob of Flush |
---|
0 | 46.628% | 0.000% | 0.000% |
1 | 41.587% | 4.163% | 1.731% |
2 | 10.944% | 34.968% | 3.827% |
3 | 0.842% | 100.000% | 0.842% |
SUM | 100.000% | -- | 6.400% |
Hopefully, everything in the One Board game table above is self-explanatory. The flop can come down with any number of spades between 0 and 3, each with a different probability of occurrence. And each case gives a different probability of making your flush by the river. The far right column is the product of the previous two columns and therefore reflects the "unconditional" probability for each case.
Taking everything into account, in the One Board game if you are dealt two spades and stay in the hand until the end, you will have a 6.400% of making a flush in spades. And, by our stylized assumptions, a 6.400% chance of winning the hand.
Now let's turn our attention to the Two Board game. Of course, now things are more complicated. There are two flops, two turns, and two rivers.
The following table summarizes the situation under the Two Board game rules. Here we need to keep track of
how many flushes we can expect to make in each of the various cases.
Spades on [Flop1,Flop2] | Prob of Flop[,] | Prob of 0 Flush given Flop | Prob of 1 Flush given Flop | Prob of 2 Flush given Flop | Combined Prob of 0 Flush | Combined Prob of 1 Flush | Combined Prob of 2 Flush | Aggregate Prob of a Full Win |
---|
[0,0] | 20.532% | 100.000% | 0.000% | 0.000% | 20.532% | 0.000% | 0.000% | 0.000% |
[0,1] | 39.856% | 95.243% | 4.757% | 0.000% | 37.960% | 1.896% | 0.000% | 0.948% |
[0,2] | 11.387% | 62.896% | 37.104% | 0.000% | 7.162% | 4.225% | 0.000% | 2.113% |
[0,3] | 0.949% | 0.000% | 100.000% | 0.000% | 0.000% | 0.949% | 0.000% | 0.474% |
[1,1] | 17.081% | 92.482% | 7.425% | 0.093% | 15.797% | 1.268% | 0.016% | 0.650% |
[1,2] | 8.540% | 64.430% | 34.775% | 0.794% | 5.503% | 2.970% | 0.068% | 1.553% |
[1,3] | 0.616% | 0.000% | 97.780% | 2.220% | 0.000% | 0.602% | 0.014% | 0.315% |
[2,2] | 0.923% | 48.652% | 43.500% | 7.848% | 0.449% | 0.402% | 0.072% | 0.273% |
[2,3] | 0.113% | 0.000% | 74.313% | 25.687% | 0.000% | 0.084% | 0.029% | 0.071% |
[3,3] | 0.003% | 0.000% | 0.000% | 100.000% | 0.000% | 0.000% | 0.003% | 0.003% |
SUM | 100.000% | -- | -- | -- | -- | -- | -- | 6.400% |
Again, hopefully everything in the table above is self-explanatory. All the columns are essentially parallel to columns in the previous One Board game table. The far right column in this Two Boards game table may need a little further explanation. Since the rules of Two Board NLHE require two "half-winners" (can be the same player) of each deal, the far right column credits making 0 flushes with 0 half-wins, credits making 1 flush with 1 half-win, and credits making 2 flushes with 2 half-wins. The far right column then expresses these figures in terms of "full wins" so as to be comparable to the One Board game table.
You will see that the overall probability of winning a deal (see above for what this means) in Two Board NLHE with two spades is 6.400% which is the exact probability of winning a deal in One Board NLHE with two spades. I imagine that this result is what everybody expected. Remember that this is the ex-ante pre-flop percentage. Similar to the fact that "running it twice" after all the decisions in a deal are made (such as all-in with the river to come) will not change the expected value of any hand, dealing more runouts will not change the ex-ante pre-flop expected value of any hand.
But game theory dictates that optimal strategy at an early stage of a complicated decision tree is affected by the optimal strategy of late (downstream) stages in the decision tree. In our stylized example, we must ask ourselves if the more "granularity" presented by the twin flops in the Two Board game (we could wind up in any of ten different granular situations on the flop rather than only four "coarse" cases) materially affects how we would play post-flop. And, if we change the way we would play post-flop, then would we have a different pre-flop optimal strategy?
I am going to think about this some more and look forward to comments.
Last edited by whosnext; 03-31-2018 at 10:28 PM.