e4 -- best by test.
What is the current theory or trend in theory behind this statement?
We have seen huge strides taken in chess AI in the past two years. How is this affecting opening theory regarding the "best first move"?
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It's still one of the best moves White can play. There's no clear consensus on whether 1.e4 or 1.d4 is better, but it's played frequently at the top level. Due to advancements in theory, I'd say 1.e4 isn't regarded quite as highly as it was in the past, but again it's most likely White's best/second best move.
Advancements in AI aren't really affecting theory on the first move. Games from AlphaZero/Leela are good for new ways of thinking in the middlegame or late opening, but they're not changing our fundamental opening understanding.
EDIT - also, technically 1.e4 isn't "best by test" currently. In my database it scores 52.8%, while 1.d4 scores 54.4%. But to be fair, 1.e4 has been played in roughly 1.5 million more games, and the types of players who favour 1.e4/1.d4 may differ.
Since 1.e4 and 1.d4 are considered equal by most experts, the value will really depend on your style: If you are a tactical player, you are more likely to get the types of positions that you favor. I am older, and a positional player, so I play 1.d4 for the same reason: I tend to get more positional games, and I am a pretty good positional player.
Whether an opening is truly equal, or even if you have a slight advantage, the real advantage is getting a position that you are comfortable with and have experience with, and that your opponent is less comfortable with. If you are in a tactical position, and the position is technically equal, but you stink at tactics, and are sure to make a mistake, it could be thought of as not equal for you, regardless of whether a computer likes it.
We see Carlsen all the time just trying to avoid book, and get "his" positions.
No. AI has not changed the evaluation of 1pe4. AI is too limited to be able to see far enough into the game to evaluate the first move that accurately. Best you can do is estimate it even with AI.
The best we can do ourselves is look at top GM results. At the highest levels it gives white 9% more wins but over 50% draws. Net scoring white to black is 55% to 45%.
For all players , meaning good players in tournaments not beginners and casual club players, e4 gives white under 7% advantage with just under 30% draws. Net is just under 53% for white to 47% for black.
And at the very highest levels white seems to do better than when used by players at lower levels.
What seems to matter more than the opening is the players themselves: How good they are, what type of position they prefer to play, and other intangible factors such as what moves followed e4 as all the other various openings would have their own statistics too.
The paucity of games by high rated players also colors the results. There are lots of games with e4, but only a handful starting with 1d3 even though ALL of those games were won by white. Does that make d3 the best first move? NO! It means that nakamura used it a few times and beat 4 other players with it. One was Carauna! Does that mean that d3 is the best first move? NO! It might mean that 1...Nf6 by carauna was not the best reply. Or it might mean that later on nakamura outplayed carauna and the previous moves were just fine after the position became a KID played by white with an extra tempo.