poker paradox?

hold'em in a nutshell

Each player has 2 secret cards, and later 5 shared cards are revealed. Players bet on who has the best combination of cards both before and after the shared cards are revealed. Most secret cards are not worth betting on, good players will play between 10% and 30% of their hands.

three common principles

• Deception (fundamental theorem of poker): if your opponent has less (or wrong) information, you make more money than if they have more information
• Hand strength: all else equal, you make more money with higher cards than lower cards. (E.g. 9♣8♣ vs 8♣7♣)
• Style: a range of styles can be equally profitable, for example you may be able to maximise your winnings by playing anything from 10% to 30% of hands.

the problem

These principles seem axiomatic (except perhaps the third, but empirically it seems true). But they conflict:

• Take two optimal strategies: a tight 10%-of-hands game "A" and a loose 30% game "B".
• At the start of each hand, flip a coin, and play either strategy "A" or "B" accordingly. Call this compound strategy "C".
• A player who's played against either A or B for long enough knows they play 10% or 30% of hands respectively, and they're winning so it's probably the best 10% or 30% of hands. (Strictly, a best 10%. Hand strength is only a partial order, it doesn't compare 9♣8♣ to 2♢2♠).
• A player who's played against C knows that he plays 20% of hands. If the player is very observant he might know about the coin-flipping, but if the coin is hidden he doesn't know whether he's playing against A or B for a particular hand. In either case, he's playing against an optimal strategy with less information than last time, so when the coin comes up heads C makes more money than A, and when it comes up tails C makes more money than B. So C is more profitable than A and B (which were supposed to be equal and optimal).
• Strategy C involves sometimes playing hands in the 20-30th percentile, and sometimes not playing hands in the 10-20th percentile. According to hand strength, we'd make more money by replacing some worse hands we're playing with some better ones we're not playing. This gives a strategy "D" where we flip a coin play the top 20% of hands according to strategy A or B.

So D > C > A and B, but A and B were supposedly optimal.

resolutions

The first weakness I can see is the assumption that it's always incorrect to throw away a more profitable and play a less profitable one. A hand like 5♡7♡ that rarely wins may be profitable if played occasionally because when it does win your opponent will discount the possibility that you have it. However you have to fold it most of the time, even though playing it would be profitable for that individual hand, to maintain this value.

I'm surprised that the benefit from playing a probabilistic strategy is big enough to counteract the 'paradox'. Most analysis online tends to focus on opponent ranges - the range of cards they might have based on their actions in the hand - and it's assumed in calculations that the opponent would play their cards that way every time. Against good opponents, the edges of the ranges will be significantly fuzzy, and not accounting for this makes for flawed analysis that loses money (fundamental theorem again).

The second weakness (thanks Kyle!) is the assumption that C gives our opponent less information than A or B. This is intuitively true if A and B are deterministic, in the same way that a coin-flip is more uncertain than 'heads' or 'tails' individually; but if they A and B are probabilistic then C may be no more uncertain - the average of two random numbers is not 'more random'.