Here is my interpretation of the question:
There are n objects in the pile.
Some number p is chosen and each player in turn
can remove any number of objects from 1 up to and including p.
And it is decided whether taking the last piece is a win or a loss.
Then someone decides who goes first and you play the game under those conditions.
Let's say p = 3 and you can remove 1, 2, or 3 objects from the pile.
The rest of the analysis generalizes to any p quite readily.
Just substitute (p+1) for 4.
Let's consider all four cases:
1. n is a multiple of 4, objective is to take the last one
2. n is a multiple of 4, objective is to not take the last one.
3. n is not multiple of 4, take the last
4. n is not multiple of 4, not take the last.
1. Multiple of 4, objective is to take the last one.
This is a losing position:
Player 1 takes some number of objects (1,2,3),
player 2 takes (4 - that number), again leaving a multiple of 4.
That continues until there are 4 left.
Player 1 takes (1,2, or 3) and player 2 takes the rest, and wins.
Player 2 wins.
2. Multiple of 4, objective is to not take the last one.
This is a winning position:
Player 1 takes 3, leaving a multiple of 4 + 1.
Player 2 takes some, and Player 1 follows the strategy above,
always leaving a multiple of 4 + 1 for Player 2.
Eventually Player 2 is left with 5, and player 1 can leave the last one for him.
Player 1 wins.
3. Not multiple of 4, take the last
Player 1 leaves Player 2 with a multiple of 4,
and now we are in the same scenario as #1, but with the players reversed.
Player 1 wins.
4. Not multiple of 4, not take the last.
Now it depends on what the actual remainder is.
If the number is a multiple of 4 + 1, it's a losing position.
No matter what Player 1 does, Player 2 does the complementary move totaling 4,
and again leaves him with a multiple of 4 + 1.
Eventually it gets down to 5, and Player 1 takes some,
Player 2 leaves 1, and Player 2 wins.
If the number is a multiple of 4+2, Player 1 takes 1,
and leaves player 2 with a multiple of 4 + 1, and Player 1 wins,
as just described.