This is an Ehrenfeucht–Fraïssé game
Is it possible that player $0$ or the attacker/competitor can win against the defender/player $1$/duplicator in $3$ rounds ?
I have thought about it but I could not find a winning strategy:
my attempt: (let A be the first set and B the second set)
0 chooses $0$ in first set 1 chooses some $a$ in B if $a\geq2$ then 0 wins by choosing $a-2$ in B in round 2. Player 1 cannot make a move and lost.
Hence if 1 chooses an $a<2$ then we have either $0$ or $1$ in B. What can 0 do in round 2 and round 3 to win the game? Is it possible? The competitor can win in 2 rounds therefore the attacker needs at least 3 rounds to win the game.
If the attacker plays $k$ in $A$, the defender can play $2k$ in $B$ to mimic the structure, so at some point the attacker needs to play in $B$ to win, and it seems plausible that it’s advantageous to play in $B$ right away.
Indeed, there is a winning strategy for the attacker that starts by playing $3$ in $B$. If the defender plays $a\ge2$ in $A$, the attacker then plays $a-1$ and $a-2$ in $A$ to win. On the other hand, if the defender plays $0$ in $A$, the attacker then plays $1$ in $B$ to win. So the defender has to play $1$ in $A$. But then the attacker plays $0$ in $B$. If the defender responds with $a\not\in\{0,2\}$ in $A$, the attacker then plays $a-1$ to win; if the defender responds with $0$ in $A$, the attacker then plays $2$ in $B$ to win; and if the defender responds with $2$ in $A$, the attacker then plays $5$ in $B$ to win.