1.d3 is Mieses opening. Similar to 1.e3, white doesn’t commit to a center pawn advance on the first move but gives a hint at either playing e4 or c4 later on.
The idea of playing 1.d3 can be to transpose into a familiar opening with a different move order. We will see various possible structures arising after 1.d3
1.d3 e5 2.c4 Nf6 3.Nc3 Nc6 4.g3 g6 5.Bg2 Bg7 6.Nf3 d6 7.Rb1 transposes to English opening.
White will play on the queenside with b4-b5 ideas. Black can try to grab more center with 0-0- Ne8-f5.
1.e4 e5 sidelines
1.d3 e5 2.Nf3 Nc6 3.e4 leads to sideline of King’s pawn opening. One can reach the same position after 1.e4 e5 2.Nf3 Nc6 3.d3. But the move d3 is quite passive here. White gives away the initiative to black.
White’s idea can be to play g3 and fianchetto the bishop, transposing to structure similar to the King’s Indian attack. The game can go 3…Nf6 4.g3 d5 5.exd5 Nxd5 6.Bg2 Be7 (Bd6) 7.0-0 0-0 with an equal game.
King’s Indian attack
One good reason to play 1.d3 can be to transpose to the King’s Indian attack. For example, 1.d3 d5 2.Nf3 c5 3.g3 Nf6 4.Bg2 Nc6 5.Nbd2 Be7 6.e4
This is a typical structure of the King’s Indian attack. White will play Re1 and try to go Nf1-e5 Bf4 and strengthen the e5 square.
And finally, white can play a very funny set-up with 1.d3 e5 2.e3 d5 3.b3 c5 4.Bb2 Nc6 5.Ne2 Nf6 6.g3 Bd6 7.Bg2 Be6 8.Nd2 0-0 9.h3 Qd7 10.h3
But with this set-up, white is inviting black to initiate the attack. White has not advanced a single piece on the 4th rank. It is like trying to test your opponent’s abilities rather than playing a serious game of chess. So it can be played for fun games.
Game played with Mieses Opening
Here is a game played between Carlsen, Magnus vs Yatzenko, Anton
Mieses opening or 1.d3 has no practical advantage other than using a different move order to reach a familiar opening. It can be helpful against an opponent who is used to memorizing moves very rigidly. But against experienced players changing move orders may not be useful.
By playing 1.d3, you can only transpose to the king’s pawn-related structures mainly the King’s Indian attack. Other than that move 1.d3 has no real value.
You can try practice games against different opponents to see how black’s responses vary against 1.d3.