Since the <why> part was already answered by <Once>, I will add a forced win against what I think would have been a stiffer defense:

<31.Be2> Rxf3 32.Rxf3 Rxf3 <33.Qxf3!?>

33...Bxf3 34.Bxf3 <e4!> 35.Bg4 (35.Bxe4 Qe3+, 35.Rxd4 Qe3+) Qxg4 36.Rxd4 Qe2 ... 0-1 >

Regarding 18.a4, Alekhine commented in, "My Best Games of Chess - 1924-1927", <...takes decidedly too much time and thus permits Black to build the ensuing instructive attack.>

Instead of 18.a4, Alekhine recommended 18.Qe5, with the threat of 19.Qc7. Alekhine indicated the game would then have continued with the moves: 18...f4 19.Qc7 Bxf3 20.Bxf3 fxe3 21.fxe3 Nf5, and White, according to Alekhine, would be able to protect his King.

Fritz indicates Alekhine's variation results in an equal evaluation: (.00) (22 ply) 22.Rd3 Qg6 23.Be4 Qg4 24.Qe5 g6, or (.00) (22 ply) 22.Qe5 Qg6 23 Rf1 Rc8 24.Rad1 d6.

While Fritz evaluates Alekhine's suggestion as equal, it also evaluates Stahlberg's 18.a4 as a better continuation, giving White the advantage.

Fritz indicates 18.a4 gives White an advantage, and provided the following analysis: (.70) (22 ply) 18...Rc8 19.Ne5 d5 20.c5 bxc5 21.bxc5 Nc6 22.Nf3, or (.88) (22 ply) 18...f4 19.exf4 Rxf4 20.a5 Bxf3 21.Bxf3 Rc8.

After his move 18...f4, Alekhine comments, <From now on, until the end, all Black's moves are very exactly timed. It is hardly possible to replace any one of them by a better one.>

AAA (via Kotov) states this:

<After 31.Qd2 (relatively best) comes Bxf3 32.Nxf3 Nxf3+ 33.Rxf3 Rxf3 34.Qxg5 Rxf1+ 35.Rxf1 Rxf1+ 36.Kxf1 hxg5 37.Ke2 Kf7 38.Kf3 Ke6 39.Ke4 b5! with a won pawn end-game.>

That makes me suspect that AAA did not properly consider the 31.Bd3 defense; or he ignored it in his comments because he did not see how to break it.

In this position:

click for larger view

Instead of playing the move 19.exf4!, White's advantage was less after: (.38) (22 ply) 19.a5 fxe3 20.Qxe3 Nf5 21.Qc3 d6 22.axb6 axb6.

Based on this comment, one may think 23.Ra7 was a weak move. On the contrary, it is White's best move, and the resulting position slightly favors White: (.38) (22 ply) 23.Ra7! Rd7 24.Nd4 Qg6 25.g3 Rdf7 26.Nxf5 Qxf5 27.f3.

In addition to 23.Ra7!, another reasonable choice for White was: (.30) (22 ply) 23.Nd4 Qg6.

Instead of one of these moves, or several other moves that give an approximately equal position, Stahlberg played: (-.26) (22 ply) 23.Ne1? e5! 24.Ra7. Even Alekhine may have been surprised by 23.Ne1?, as Fritz indicates White had more than 20 other moves that were superior to 23.Ne1?.

After 25...Rd7, Fritz prefers: (-.21) (22 ply) 26.Bd3 Rdf7 27.Rd2 Qc8, or (-.22) (22 ply) 26.Rd2 Rf4 27.Bd3 Rdf7 28.f3.

Why the analysis of Alekhine (or any other player) have to be perfect and withstand the test of computers a century later?

I understand that it is useful to learn how to improve the game, but not to judge as a court-martial

With his 27th move Stahlberg set the stage for the exciting conclusion of this game. Fritz agrees that 27.f3 was the best defense, but with so many Black pieces posed for the destruction of White's king side, and with those pieces commanded by Alekhine, who could have played 27.f3 with any real confidence of survival?

Alekhine remarked regarding 27.f3, <One would suppose that this Pawn, besides being protected by its neighbor, and easily supported by 3-4 pieces, cannot possibly form a welcome object for Black's attack.>

From here till the end of the game, all moves of Black are carefully calculated. It would be hard to replace any of them with a better one.>