I gave two caveats:
All statistics need to be approached cautiously. In this case it is useful to remember that the vast majority of games in The Database are drawn from over a decade of games played online at FICS, regardless of outcome; so they should be somewhat representative of Jerome Gambit play among club players who contest online...
Of course, were I to subject each of these positions to the scrutiny of, say, Stockfish 8, the computer's evaluation would likely, in some cases, vary significant from the game outcomes in The Database!
Then I decided to give Stockfish 8 a chance, after all.
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5
If the game continues 3...Nd4 4.Bxf7+ instead, Stockfish 8 rates the position at 30 ply depth of search as -.18, about an even game.
4.Bxf7+
Stockfish 8 rates the position as -1.85 at 30 ply depth of search, meaning that Black is not quite 2 pawns better.
If Black plays 4...h6 instead, after 5.0-0 Bc5 6.Bxf7+ Stockfish 8 rates the position as -1.77 at 30 ply search depth. After 5.Nc3 Bc5 6.Bxf7+, it rates the position at 30 ply search depth as -1.45.
4...Kxf7
As for declining the Bishop, Stockfish 8 rates 4...Kf8 at 30 ply depth of search as +1.68, giving White more than a pawn and a half advantage. It rates 4...Ke7 at +2.01, giving White a two pawn advantage.
5.Nxe5+
Stockfish 8 rates this position as -2.41 at 30 ply depth of search.
If, instead, the game continues 5.Nc3 Nf6, the game reaches the Four Knights variation by transposition. Stockfish 8 rates the position as -2.85 at 30 ply depth of search.
5...Nxe5
Stockfish 8 rates this position as -2.32 at 30 ply depth of search.
If Black replies, instead, 5...Kf8, Stockfish 8 rates the position -.56; if Black replies 5...Ke8 Stockfish 8 rates the position +.22; if Black replies 5...Ke6, Stockfish 8 rates the position +.94; if Black replies 5...Ke7, Stockfish 8 rates the position +1.04; all at 30 ply depth of search.
6.Qh5+
Stockfish 8 rates this position as -2.27 at 30 ply of depth.
If White plays 6.d4 instead, Stockfish 8 rates the position as -2.46 at 30 ply depth.
6...g6
Stockfish 8 rates this position as -1.55 at 30 ply of depth.
It rates 6...Kf8 as -1.75 ; 6...Ke6 as -2.42; and 6...Ke7 as +2.71; all at 30 ply of depth.
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