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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