Game with two players and 120 points in total Assume the following game:

The game has two players $P_{1}$ and $P_{2}$ and 15 rounds in which they play against each other. Each round gives an amount of points equal to its number, i.e. the winner of the first game gets 1 point, the winner of the second game gets 2 points, and so on.
This means there is a total of $\sum_{i=1}^{15} i = 120$ points.
The winner of the game is the one with most points, so sometimes the game will not be played till game 15. For example, if $P_{1}$ wins the first 11 games he wins the game as he has 66 points.
In the case of a draw (each player having 60 points), a final game decides who wins.

I want to formally check in which scenarios the outcome of the first game (awarding 1 point) is important for the outcome of the entire game. How does one calculate this without having to check every possible combination of points by hand?
 A: You are basically asking for a subset of $ \{ 2, 3, \ldots, 14, 15 \}$ to sum up to 59 or 60. 
There isn't a really easy of doing so. You can either do it by hand, or consider the generating function
$$ (1+a_2 ^2) (1+a_3^3)  \times \ldots \times (1+ a_{14}^{14}) ( 1 + a_{15}^{15} ) . $$
A: We can consider the following three categories of outcomes of the game: end after the 15th round, arrive at a draw after the 15th round, end early.  
OUTCOME: "End after the 15th round"
If the game ends after the 15th round, is it possible to have ended with just one-point lead of the winner? No, because, in such a case the sum total of points of both players would be an odd number, while, the 15 rounds award in total 120 points, which is even.
So, tentative conclusion 1: The 1st round is not important for all scenaria in which the game ends after the 15th round.
OUTCOME: "Arrive at a draw after the 15th round"
Consider the following variant of the game, call it "game $B$": 14 rounds, with the first round awarding 2 points, the second 3 points etc. Then game $B$ distributes in all 119 points. So it is not possible for game $B$ to end in a draw. Therefore, tentative conclusion #2: The 1st round in game $A$ "is important" in that it permits the case of a draw after the 15th round. Obviously the same could be said for any round that awards an odd number of points, but we are focusing on round 1 here. 
I designate $P_1(m)$ to be the points of the leader after round $m$, and with lowercase $p_i$ the points of its round. Then the possibility of a draw is described by the following relation, after the 14th round has been played:
$$P_1(14) - P_2(14) = 15$$
while we also have $$ P_1(14) + P_2(14) = 105$$
Inserting the second into the first we have
$$P_1(14) - 105 + P_1(14) = 15 \Rightarrow P_1(14) = 60,\; P_2(14) = 45$$
So the "importance condition" here is reduced to: round 1 is important in all scenaria where the follower ends up with 45 points after the 14th round. 
But this covers and qualifies tentative conclusion 1, so we should keep only this last one selection criterion, since it automatically excludes all cases where the game ends after the 15th round but having "lost" the possibility of a draw. 
OUTCOME: "End early"
The game will end early if a player has 61 or more points after any given round. It is easy to deduce that this means that at least 11 round must be played for the possibility of the game ending really to be feasible (including the possibility of ending exactly after the 11th round). In this family of outcomes, round 1 is important for all scenaria in which win-lose combinations in all subsequent rounds give 60 points to the winner excluding the outcome of round 1: so if the 60-points holder counting from round 2 onwards has also won the first round, the game ends early, while if she hasn't won round 1, the game does not end at the specific stage. Hence the outcome of round 1 becomes critical.   
Let's see : after round 11 and not counting round 1, 65 points have been awarded in all. So round 1 is critical if the leader has won 60 out of these 65 points. This could be achieved by her winning all rounds from round 2 onwards, except round 5, or except rounds 2 & 3 - in no other case.
Now after round 12 (and not counting round 1), 77 points have been awarded in all. For her to have 60, she must have lost all combinations of rounds whose sum of points equals 17. Since the largest is 12, while the lowest is 2, we have the combinations of winning all rounds except rounds (12, 5), or (12, 2, 3) or (11,2,4), (11, 6), (10,2,5), (10, 3, 4), etc
The general selection criterion here is already stated : "round 1 is important in all scenaria in which a player acquires 60 points from rounds 2 onwards, before the playing of the 15th round."
