Alice and Bob play a game with $n$ cards. Alice writes the numbers $1,2,\ldots,n$ once each, and so does Bob (on the same set of cards). Then, they take turns choosing cards according to some specified sequence. In each turn, the player picks the card with his/her smallest number. At the end of the game, Alice notes the sum of her numbers on her cards.
Afterwards, the specified sequence is modified by moving some of Alice's turns earlier (no other kind of modification is allowed), and the game is repeated in the same way. Is it true that Alice necessarily gets a smaller or equal sum than before?
Example: Alice writes $1, 2, 3, 4$, Bob writes $1, 3, 4, 2$. Originally the sequence is Bob, Alice, Bob, Alice. Bob chooses the 1st card, Alice the 2nd, Bob the 4th (because Bob's number on the 4th card is smaller than on the 3rd), and Alice the 3rd, so Alice gets $2+3=5$. Afterwards, suppose the sequence is modified by moving Alice's first turn to the front, so it is now Alice, Bob, Bob, Alice. Alice chooses the 1st card, Bob the 4th, Bob the 2nd, and Alice the 3rd, so Alice gets $1+3=4$.
To prove that it is true, it would be enough to show that for every $k$, Alice gets a smaller or equal number in her $k$-th turn of the original game than in her $k$-th turn of the modified game. Can this be done possibly by induction?