The game:
Given $S = \{ a_1,..., a_n \}$ of positive integers ($n \ge 2$). The game is played by two people. At each of their turns, the player chooses two different non-zero numbers and subtracts $1$ from each of them. The winner is the one, for the last time, being able to do the task.
The problem:
Suppose that the game is played by $\text{A}$ and herself.
$\text{a)}$ Find the necessary and sufficient conditions of $S$ (called $\mathbb{W}$), if there are any, in which $\text{A}$ always clear the set regardless of how she plays.
$\text{b)}$ Also, find the necessary and sufficient conditions of $S$ (called $\mathbb{L}$) in which $\text{A}$ is always unable to clear the set regardless of how she plays.
$\text{c)}$ Then, find the strategies/algorithm by which $\text{A}$ can clear the set with $S$ that doesn't satisfy $\mathbb{L} \vee \mathbb{W}$.
Next, suppose that the game is played by $\text{A}$ and $\text{B}$ respectively and $S$ that doesn't satisfy $\mathbb{W}$.
$\text{d)}$ Is there any of them having the strategies/algorithm to win the game? If so, who is her and what is her winning way? (It's possible to suppose that $\text{A}$ and $\text{B}$ play the game optimally)
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Note:
$\text{1)}$ This is not an assignment. I have just create this out of a familiar thing in my life. So, I haven't known whether there is an official research or even names for the game. If so, I'd be very appreciated if you shared those.
$\text{2)}$ The case of $n = 2$ is so obvious that we can eliminate that from consideration. We can do the same thing to an obvious condition in $\mathbb{W}$ (if $\mathbb{W} \neq \varnothing$): $\left ( \sum_{i \in S} i \right ) \; \vdots \; 2$.
Thanks in advance.
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Update 1: To clear many people's misunderstanding and to avoid it for new ones, I emphasize the word "different" above. And by "different", I mean different indices of numbers, not their values. If this is still not clear, I think we should consider $S$ as a finite natural sequence ($a_1$ to $a_n$) and not delete any of them once they become $0$.
Update 2: (d) has been renewed a little, thank to Greg Martin.