user167: Edit: My mind played a trick on me after the fact. As Akavall notes in a comment, the Chadaev-Carlsen blitz game that I mention below actually proceeded 1. e4 g6 2. d4 Nf6 3. Nf3, a different gambit than the one my question asks about. So please ignore the references to Chadaev-Carlsen in reference to the Roscher Gambit here. (Incidentally, this means that Carlsen didn't buck the trend of highly-rated players avoiding 3. ... Nxe4 in the Roscher Gambit; see below.)

The Roscher Gambit goes as follows:
[FEN ""]

1. e4 d6 2. d4 Nf6 3. Nf3

I just saw Nikolai Chadaev convincingly defeat Magnus Carlsen at the World Blitz Championship using this line (where Carlsen accepted the gambit with 3. ... Nxe4). I realized I've basically never seen this gambit, and that I know nothing about it. (I even had to look up a name for it to ask this question.)
It is very seldom played, and especially so among highly-rated players: my database of 5.2M games contains a mere 99 games reaching the position after 3. Nf3, with the average White (respectively, Black) rating in the games being 1953 (1976). But there are 7 games in which 2400+ players tried the gambit as White against equally strong competition (and 2 games with 2500+ players). The most popular response over all games is indeed the obvious 3. ... Nxe4, being chosen in 46 of the 99 games. (Second most popular is continuing in Pirc Defense fashion with 3. ... g6; these two moves account for almost all the games, 83 of 99.)
All of that is about what I would expect. But, interestingly, in the games I have in this line with the highest-rated players, Black consistently avoids 3. ... Nxe4, preferring the 3. ... g6 Pirc approach, or instead aiming for the Philidor with 3. ... e5 (or 3. ... Nbd7 followed by 4. ... e5), Czech Defense with 3. ... c6, or even the Nimzowitsch Defense with 3. ... Nc6. The highest rated player I find playing 3. ... Nxe4 has Elo 2245 (until this Carlsen blitz game of course).
The sample size here is quite small, and maybe the highly-rated players have avoided 3. ... Nxe4 just because they are worried about stepping in to a well-prepared trap, instead preferring to head for more familiar ground. Nevertheless, the fact that the strongest players consistently avoid accepting the gambit - except for Carlsen, who got stomped, though it's just a blitz game - has made me curious:

What is the the theoretical status of the Roscher Gambit? Or is it too seldom played to even have a theoretical status?
And if it's the case that 3. ... Nxe4 isn't simply good for Black, then why doesn't the Roscher appear more often, or at least have the same popularity among amateurs as something like, say, the Blackmar-Diemer Gambit?

I had no luck finding any prose account of the gambit by a strong player, and would appreciate any pointers to such.

user12: I've taken a look at this gambit before as I play it frequently in blitz games, and I don't have very much faith in it for white. (Usually my games are from a different move order with c4 included for white).
The biggest problem with the gambit is that it can be quite simply declined with 3... g6 and what has white really accomplished?  There's nothing better than transposing into either a Pirc with 4. Nc3 or a King's Indian Defense with 4. c4 (incidentally allowing black another chance to accept the gambit in an even better way). 
Obviously declining the gambit isn't the critical path, however.  Black's most challenging move is 3... Nxe4 when white more or less must play 4. Bd3.  Any other move is inconsistent although there are a few move order tricks like 4. Bb5+ which I will return to later.
Now we see the biggest issue with the gambit - white doesn't get a "real" lead in development or force any concessions in the black position.  Black plays 4... Nf6 (entire line so far 1. e4 d6 2. Nf3 Nf6 3. d4 Nxe4 4. Bd3 Nf6) and now both players can finish development naturally.  Black will play g6, Bg7, O-O and Nc6 with a perfectly acceptable Dragon formation.  On the other hand, white cannot mount a standard kingside attack with g4 and h4 because the knight on f3 will allow the black bishop to come unmolested to g4.  
White doesn't have any targets to attack in the black position and will be completely unable to stop black from castling.  These are the two goals of a gambit - force a concession and attack with the lead in development.  While the pawn sac is probably not bad in and of itself because white keeps a normal edge as in so many Pirc and KID positions along with the open e file for future use, after black castles kingside, the e pawn is quite immune from capture due to moves like Re8, Kf8, and Nc6.
So at last we return to white's tricky attempt 4. Bb5+.  If black plays 4... c6 now white has something since the knight is deprived of the c6 square.  However, black simply plays 4... Bd7 and there are no problems because the d7 square is not reserved for other pieces in the Dragon formation.  Both sides "waste" a tempo so nothing is gainer or lost.
In fact, even after 4... c6, black can still claim barely equality, although this is a real concession in the position (Nb8 cannot develop naturally).  If I were to play white, I think that 4. Bb5+ is the best attempt simply because many players will play the natural looking 4... c6 to drive the bishop back.  
[FEN ""]
[White "Roscher Gambit"]
[Black "Analysis"]

1. e4 d6 2. d4 Nf6 3. Nf3 Nxe4 (3... g6!? {White has gained nothing over the usual Pirc move order}) 4. Bd3 (4. Bb5+!? c6?! {Black's knight is deprived of the c6 square} (4... Bd7!)) Nf6 {Both players will continue to develop normally and black doesn't have any weaknesses}


167: <p><strong>Edit:</strong> My mind played a trick on me after the fact. As Akavall notes in a comment, the Chadaev-Carlsen blitz game that I mention below actually proceeded <code>1. e4 g6 2. d4 Nf6 3. Nf3</code>, a different gambit than the one my question asks about. So please ignore the references to Chadaev-Carlsen in reference to the Roscher Gambit here. (Incidentally, this means that Carlsen didn't buck the trend of highly-rated players avoiding <code>3. ... Nxe4</code> in the Roscher Gambit; see below.)</p>
<hr />
<p>The Roscher Gambit goes as follows:</p>
<pre><code>[FEN &quot;&quot;]

1. e4 d6 2. d4 Nf6 3. Nf3
</code></pre>
<p>I just saw Nikolai Chadaev convincingly defeat Magnus Carlsen at the World Blitz Championship using this line (where Carlsen accepted the gambit with <code>3. ... Nxe4</code>). I realized I've basically never seen this gambit, and that I know nothing about it. (I even had to look up a name for it to ask this question.)</p>
<p>It is very seldom played, and especially so among highly-rated players: my database of 5.2M games contains a mere 99 games reaching the position after <code>3. Nf3</code>, with the average White (respectively, Black) rating in the games being 1953 (1976). But there are 7 games in which 2400+ players tried the gambit as White against equally strong competition (and 2 games with 2500+ players). The most popular response over all games is indeed the obvious <code>3. ... Nxe4</code>, being chosen in 46 of the 99 games. (Second most popular is continuing in Pirc Defense fashion with <code>3. ... g6</code>; these two moves account for almost all the games, 83 of 99.)</p>
<p>All of that is about what I would expect. But, interestingly, in the games I have in this line with the highest-rated players, Black consistently avoids <code>3. ... Nxe4</code>, preferring the <code>3. ... g6</code> Pirc approach, or instead aiming for the Philidor with <code>3. ... e5</code> (or <code>3. ... Nbd7</code> followed by <code>4. ... e5</code>), Czech Defense with <code>3. ... c6</code>, or even the Nimzowitsch Defense with <code>3. ... Nc6</code>. The highest rated player I find playing <code>3. ... Nxe4</code> has Elo 2245 (until this Carlsen blitz game of course).</p>
<p>The sample size here is quite small, and maybe the highly-rated players have avoided <code>3. ... Nxe4</code> just because they are worried about stepping in to a well-prepared trap, instead preferring to head for more familiar ground. Nevertheless, the fact that the strongest players consistently avoid accepting the gambit - except for Carlsen, who got stomped, though it's just a blitz game - has made me curious:</p>
<blockquote>
<p>What is the the theoretical status of the Roscher Gambit? Or is it too seldom played to even have a theoretical status?</p>
<p>And if it's the case that <code>3. ... Nxe4</code> isn't simply good for Black, then why doesn't the Roscher appear more often, or at least have the same popularity among amateurs as something like, say, the Blackmar-Diemer Gambit?</p>
</blockquote>
<p>I had no luck finding any prose account of the gambit by a strong player, and would appreciate any pointers to such.</p>
 12: <p>I've taken a look at this gambit before as I play it frequently in blitz games, and I don't have very much faith in it for white. (Usually my games are from a different move order with <code>c4</code> included for white).</p>

<p>The biggest problem with the gambit is that it can be quite simply declined with <code>3... g6</code> and what has white really accomplished?  There's nothing better than transposing into either a Pirc with <code>4. Nc3</code> or a King's Indian Defense with <code>4. c4</code> (incidentally allowing black another chance to accept the gambit in an even better way). </p>

<p>Obviously declining the gambit isn't the critical path, however.  Black's most challenging move is <code>3... Nxe4</code> when white more or less must play <code>4. Bd3</code>.  Any other move is inconsistent although there are a few move order tricks like <code>4. Bb5+</code> which I will return to later.</p>

<p>Now we see the biggest issue with the gambit - white doesn't get a "real" lead in development or force any concessions in the black position.  Black plays <code>4... Nf6</code> (entire line so far <code>1. e4 d6 2. Nf3 Nf6 3. d4 Nxe4 4. Bd3 Nf6</code>) and now both players can finish development naturally.  Black will play <code>g6</code>, <code>Bg7</code>, <code>O-O</code> and <code>Nc6</code> with a perfectly acceptable Dragon formation.  On the other hand, white cannot mount a standard kingside attack with <code>g4</code> and <code>h4</code> because the knight on f3 will allow the black bishop to come unmolested to g4.  </p>

<p>White doesn't have any targets to attack in the black position and will be completely unable to stop black from castling.  These are the two goals of a gambit - force a concession and attack with the lead in development.  While the pawn sac is probably not bad in and of itself because white keeps a normal edge as in so many Pirc and KID positions along with the open e file for future use, after black castles kingside, the e pawn is quite immune from capture due to moves like <code>Re8</code>, <code>Kf8</code>, and <code>Nc6</code>.</p>

<p>So at last we return to white's tricky attempt <code>4. Bb5+</code>.  If black plays <code>4... c6</code> now white has something since the knight is deprived of the c6 square.  However, black simply plays <code>4... Bd7</code> and there are no problems because the d7 square is not reserved for other pieces in the Dragon formation.  Both sides "waste" a tempo so nothing is gainer or lost.<br>
In fact, even after <code>4... c6</code>, black can still claim barely equality, although this is a real concession in the position (Nb8 cannot develop naturally).  If I were to play white, I think that <code>4. Bb5+</code> is the best attempt simply because many players will play the natural looking <code>4... c6</code> to drive the bishop back.  </p>

<pre><code>[FEN ""]
[White "Roscher Gambit"]
[Black "Analysis"]

1. e4 d6 2. d4 Nf6 3. Nf3 Nxe4 (3... g6!? {White has gained nothing over the usual Pirc move order}) 4. Bd3 (4. Bb5+!? c6?! {Black's knight is deprived of the c6 square} (4... Bd7!)) Nf6 {Both players will continue to develop normally and black doesn't have any weaknesses}
</code></pre>
