Simple summation, difficult inference. (game theory)

[citrusvanilla]

hi guys, new user, long time lurker.

the following simple proof is proposed with the highlighted summations. you do not need to know what the proof is of to answer my question (it is that the payoff for a skew symmetric game, rock paper scissors, is zero). i need help understanding how you can substitute variables in summation signs. specifically, how the sum of the term in row 1 from 1 to n is somehow equivalent to the sum in row 2 from i to n. this doesn't make any sense to me. how can something be summed from i to n?

fyi assume x_i = y_i for all i

http://i.imgur.com/s9cdo.jpg

http://i.imgur.com/s9cdo.jpg

 

[lurflurf]

The limits of the inside sigma can depend upon the variable of the outside.
Like if we have
1+(1+1)+(1+1+1)
we could write
$$\Sigma_{i=1}^3\Sigma_{j=i}^3 1$$

if we have
x³+(x²+x³)+(x¹+x²+x³)
we could write
$$\Sigma_{i=1}^3\Sigma_{j=i}^3 x^j$$

to manipulate such we can focus on the possible value of each variable
if 0<i,j<n
if we prefer to work with i+j and i instead of j and i we have
0<i<n
i<i+j<n+i

 

[citrusvanilla]

thanks lurf i really appreciate it. after staring at your answer for 10 minutes i understood it, this is compared to the 3 hours i took to not understand the proof.