I'm not sure this answer works, but I think that

Alice always wins.

We first determine a list of losing numbers. We draw a table with 2 columns and start with a number 3.

Losing    Previous
3

Now we write the greatest number less than half this number on the right hand side.

Losing    Previous
3         1

Now we add both of them and find the next number to use in the new row. (3+1=4, next number is 5)

Losing    Previous
3         1
5

Now we use the same process repeatedly.

Losing    Previous
3         1
5         2

Losing    Previous
3         1
5         2
8

and so on.....

Losing    Previous
3         1
5         2
8         3
12        5
18        8
27        13
41        20
62

Now on every turn, Alice will reduce the number of objects to the greatest losing number (ignore the column on the right now). The moment Alice gets an opportunity to win, she should take it (obviously).

I'm not sure this answer works, but I think that

Alice wins if $N$ is not a 'losing number'. (Converse may/ may not be true)

We first determine a list of losing numbers. We draw a table with 2 columns and start with a number 3.

Losing    Previous
3

Now we write the greatest number less than half this number on the right hand side.

Losing    Previous
3         1

Now we add both of them and find the next number to use in the new row. (3+1=4, next number is 5)

Losing    Previous
3         1
5

Now we use the same process repeatedly.

Losing    Previous
3         1
5         2

Losing    Previous
3         1
5         2
8

and so on.....

Losing    Previous
3         1
5         2
8         3
12        5
18        8
27        13
41        20
62

Now on every turn, Alice will reduce the number of objects to the greatest losing number (ignore the column on the right now). The moment Alice gets an opportunity to win, she should take it (obviously).

However, if the game itself starts with Alice facing a 'losing number', then she cannot use this strategy. I am yet to solve this case.