A few thoughts.
If we look at only the EW hands it is not at all clear that we want to be in any game. If we look at all four hands, we see that 4H can make, although everything has to be just so. On most NS layouts we will lose two hearts, we have to lose the spade ace, and we have to bring in the diamonds. Moreover, with most of the high cards in one hand it will not be all that easy to get to the E hand to run the diamond J. But it can all be done, so 4H is makable.
So, we have to first ask: looking only at teh EW cards, where would we like to play this hand?
Can we reach 3H passed out? Probably not. Some pairs have a way of signiing off in 3H after 1D-1H-2NT but imagine you had such an agreement. Looking only at the E hand would you then do it? W has two small spades and the K95 of hearts.. On the auction, he could just as well have two small hearts and the QJx of spades, in which case leaving the hand in 2NT sounds right.
So if the issue is to get to a part score, I expect the pair to end in 2NT. Yes there is a club fit as wel as a heart fit. but a club partial seems difficult to reach and, for that matter, it's not all that great if played by W.
I'll get to W in a moment, but let's first look at E's choices after 1D-1H-2NT. I think there is a good chance I would pass. That is only slightly better than 3NT passed out. If I were to decide to go on, I would check back for the heart fit. With many, perhaps most, pairs that is done by bidding 3C, new minor forcing. W then bids 3H and I bid, with fear in my voice, 4H.
Now about the 2NT bid from W. There was a time when opening 1NT with a worthless doubleton was a no no. But that time is long past. Sill, the 2NT after 1D-1H is scary. When I have am 18 count the chance that we will be in 3NT with partner having nothing in slades is substantial. When I open 1NT with xx in spades and partner takes me to 3NT then he has values somewhere and a decent chance some of those values are in spades. But here, after 1D-1D-2NT, partner does not need much to go on to 3NT and just who is it that will stoop spades? nonetheless, I might well have bid 2NT just as he did. No call is all that appealing.
So, after all this, I think my answer to "What is the best auction" is "I don't know". I believe that if I were playing with my clone the auction would begin 1D-1H-2NT, and then I would ask myself "Do I feel lucky?" If so, we end in 4H making. If not, we end in 2NT down a few.
I'll leave it at that, at least for now. Anyone else?
Oh, another thought. You will play 4H ow on a club lead? Yes, it works, but it really shows just how delicate 4H contract actually is.
