Playing holdem or omaha with two boards, (limit, pot limit, or no limit) is the GTO preflop strategy ever different than if there was only one board?

I would assume yes because the game becomes split pot more frequently.

To elaborate more I would expect that:

Hands that are slightly +EV or 0 EV in a single flop game would become unplayable

and

Hands with mixed strategies would significantly or completely shift to one strategy.

Two boards wth the same deck?

If so, then yeah - suited connectors should become stronger right?

^^ makes sense due to removal effects.

Suited / connected cards go up
high cards unsuited go down
Big pairs still ok
Small pairs way worse

The GTO frequency of bets/raises would not change, and the order of hands would not change either, since suited hands are already higher than offsuit hands. Sets, trips, full houses and quads are all twice as likely as well. Lastly, the pot gets split evenly between both boards. I am not seeing any GTO changes here, what am I missing?

Raw preflop equity becomes much more important. All pairs would increase in value. High card value is more important and hands that are considered trash now would become playable, like offsuit high cards/offsuit aces. Lower suited cards become trash.

The more boards you play the more hands values will correspond to preflop equity.

Another way to ask the question, would be:

Can I alter my preflop game because my table likes to “run it twice”?

No.

That’s actually a different question.

What if we play 10^100 different boards. Would our preflop strategy change? The answer is clearly yes as postflop is completely meaningless and is effectively just another 3 preflop betting rounds as postflop equities will have converged to preflop equities.

This is completely different from running it twice.

Lets stick to the two boards in the question.

So AA is more likely to hold up on at least one board out of the two, according to preflop equity. But it only wins half the pot for that board. So it gets twice the chance to realize half its preflop equity. What has changed here versus one board?

Analyzing the limit is a very good way of understanding problems, which is why I brought up what happens with a greater number of boards.

AA has the same equity regardless of how many boards we play, but it retains that equity better as ranges narrow across multiple boards. There is far less variation of equity depending on the flops, which means postflop is less significant. Hands which have low preflop equity but play well postflop become worse. Hands which have bad postflop playability but have good preflop equity become better.

Well there are not enough cards in a deck for 10^100 boards.

If you want to discuss boards that block other boards and suited hands that can scoop both boards, maybe that would have merit.

But unless you can boil this equity argument into a logic statement that can then be expressed mathematically, then I just disagree.

+1, your strategy would clearly change. As you rely less on the board with 2 of them, I would expect hands with more innate showdown value to gain value (in Hold'em, that would be pocket pairs and Ace high).

On the flip side, hands like 98s would lose value.

If there is any change to GTO preflop strategy, then there will be a set of two starting hands x,y such that:

Single board: EVx is less than EVy
Multiple board: EVx is greater than EVy

I suggest there is no set of such hands. This math is not hard, I will be happy to admit I am wrong if anyone can name these two hands and calculate the EV.

Remember that robust equity and raw equity are post flop concepts, and that with perfect play, are just equity.

Why do you think the math is not hard? I am not sure how to do it, which is why I haven’t answered your question yet.

The EVs for single boards could be estimated with solvers. I’m not aware of any solvers that could work with multiple boards and it wouldn’t be trivial to make one or to solve without one.

Point taken, the math should not be described pejoratively as "not hard".

When we put hands into an equity calculator, the calculator looks at hundreds of thousands of boards to calculate the EV of each hand. This does all the hard math for us, thankfully.

So, let me restate it (no math required) thus:

Take the hands Ad5d versus 2s2d

Both players because of complicated strategy, go all in preflop (just for example)...

On one board these are almost exactly the same raw equity of ~49.9/50.1 EV.

Now, the dealer allows as many boards as can be dealt.

Can either player win more or less money by agreeing to more or less than one board?

If so, then they would need to alter their ranges preflop to account for this.

Now, the question becomes scooping and having extra combos of potential scoopers in your range preflop (whatever those may be).

If a GTO strategy alters for a hand to have better scooping power (this is not proven possible) then that hand would have to be ranked differently in the order of the hands that can be dealt when playing a game of multiple boards.

This is why I used two different hands of identical EV in the example.

Neither player profits or loses if multiple boards are agreed upon.

Now, there is skill involved on multiple boards post flop. GTO will be especially helpful with card removal and boards blocking each other, etc.

But that is all post-flop, and the question here is about pre-flop.

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.

Certainly, as a pure exploit, we could rearrange our preflop hands to take advantage of weaker players. Just saying that as a given, even though we are talking GTO here.

Also, I suggest for now, that we keep the discussion among heads-up play and disregard multi-way.

With regards to GTO, I must admit that there exists the possibility of the slightest change, such that it would be very small and very very complicated.

GTO organizes ranges logically, with regard to mistakes. Sometimes, multiple GTO solutions are both optimal and admissible. When this occurs, the strategy more likely to encounter a "mistake" is chosen, even when no mistakes are expected. All GTO participants agree to this and choosing the other solution does not change the value of the game to any player.

However, what if one admissible strategy is likely to encounter mistakes more frequently, and that those mistakes might result in a scoop?

It is entirely possible, I must admit, that GTO post-flop ranges should be altered by one or more admissible strategies becoming the new choice in the Two Board game. But these would not change the outcome, with perfect play. Any player can pick any of the admissible strategies and not lose or gain.

So, I fall back on the order of hands dealt, and say that the order will not change among GTO players, but there will be a slight shuffle in the order of holdings post-flop.

Is it possible, for a pre-flop range to be altered so that the possible flops will contain more possibility for these "new mistakes" that involve two boards?

Possibly, but the new mistakes will make other "old mistakes" less likely since the new preferred boards will contain less of those opportunities, and a GTO player is free to play either way with zero gain or loss.

I'm surprised this is up for such debate. When you change the mechanics of postflop rules, you clearly change preflop strategy.

It could be an interesting task to mathematically prove this is the case, but even without that proof I am effectively certain the answer to OP is "yes".

I'm far from an expert in all this GTO concept, but just to get it right for myself.
You are saying that since by definition Nash equilibrium only occurs when the other players also play Nash, so in a multi board game GTO to stand unbeatable while the other players also switch to GTO is a self-propelling state of process.
But, the preflop ranges as which cards run good multiway or heads up / on multi boards are deviated. Am I right in this? Not sure if I picked up these things correctly.

Without turning the thread into a strategy session for nlhe split, let’s at least run through the basics and then we can theorize about preflop adjustments.

For starters, equity on one of the boards will be informed by any blockers on the other board and visa-versa.

So, a draw might be worth more or less, if outs to that draw are present or absent on the other board. Similarly, a made hand that may need to fade a draw might be weaker or stronger if outs that promote that made hand to a better hand are similarly present or absent on the other board.

So your equity in one board is more precisely available due to the extra information on the other board.

Also, the information you have about your opponents hand is increased, since you know he/she does not hold the cards on the extra board.

What are other considerations for strategy?

Why the two month delay to address this question.

I was away from the forum for a while, and seeing what all I had missed. I was surprised this thread had so few answers and were all in agreement with regard to pre-flop GTO.

Look at it this way:

Imagine we take it a step further and run out infinite boards with infinite split pots. Every board will now come out "average" and you will simply get your preflop equity vs your opponent at showdown (e.g. AKo will get 45% of the pot vs 66 at showdown every time). You would no longer care about actually reading the boards and runouts. You'd essentially be playing a game where there are 4 rounds of preflop betting, and then you turn over your cards (if no one folds) and get your equity.

Clearly the consequence for this we already know: playability stops mattering and only raw pre-flop equity matters. There would be no sense speculating with 32s "preflop" anymore since you lose to every other hand on every "runout".

It's easy to see with infinite (or say millions) of boards, GTO preflop strategy is going to change drastically. So intuitively, if going from 1 to 100 million boards results in a change, then going from 1 to 2 boards will also result in at least some change in strategy.

Sounds fine to me, I don’t really care, but the thread was interesting. If it seems so obvious to every other participant itt, then so be it.