Maximum entropy up to a given level is attained with a uniform distribution. I think that is why EHAA scores high - the entropy should be strongly correlated with using high opening bids relatively often, even conditional on the mean opening level.
If you guaranteed had the auction to yourself, you wouldn't go to far wrong by maximising information density (though there are tradeoffs with safety levels and information leakage - so this is by no means a safe assumption!). Traditional thinking has it that natural systems under this condition want to approximately reduce the frequency of each subsequent call by 50% compared to the one before that, e.g. half of all hands pass, a quarter opens 1
♣, an eight open 1
♦ etc., while relay systems have a theoretical (less information dense) limit of a factor 1.618.
Conversely, if I have it in enforcable legal writing that my LHO is about to bid 3
♣ over my opening, regardless of what I do, I maximise the information shared by picking a uniform frequency distribution from pass to 2NT inclusive (and some smidgen assigned to 3
♣ and up).
Put differently, if the opponents don't jump the auction, we have more space after cheaper bids, so we want more hands in it (to entangle later).
In practice, not only are the frequenty arguments too simple to be of much use for system design, also the lack of knowledge on which type of auction we are about to enter suggests something between these extremes. I am not convinced that entropy of the opening distribution conditional on the mean opening measures much other than level of aggression. Instead the uncertainty in partner's decisions conditional on our information is probably of more interest.
helene_t, on 2025-March-05, 13:09, said:
Also, with respect to Adam's comment about bidding game with 19 points: maybe, instead of defining safety as 100% safety, I could define it as e.g., 95% safety in uncontested auctions and 75% safety after an enemy preempt. This also has the advantage that the 25 percentile is statistically more stable than the minimum, so I would need fewer sims.
This is where entropy
is useful. Responder can look at their hand, at the preempt, at the opening, and reason something like "there is a 85% chance that we have game" (e.g. based on a double dummy simulation, or on a less expensive evaluation metric such as "either we have 25 HCP, or a major suit fit and something that re-evaluates to 25 HCP"). The entropy of that yes/no question is a good quantitative criterion of the difficulty of deciding whether or not to go to game.
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