Hello! First time poster. I am having trouble understanding a step of the following proof that I have highlighted. It basically substitutes one players strategy vector for the other, as they are assumed equal. Apparently, through this expansion and the fact that a skew symmetric matrix is one such that a_ij = -a_ji you can show the payoff of the game to be zero. I can infer this easily, but the highlighted step seems to me wrong. The game is rock, paper, scissors btw. Am I correct? Thanks! edit: it was pointed out to me that the inner summation is changed from the first row to j=i through n as opposed to j=1. i am not familiar with this transformation...
asked
citrusvanilla |

The transformation is an attempt to collect symmetric terms. If (i < j), then the term in the second sum includes the (ij) entry and the (ji) entry in the first sum. You're right that it doesn't quite work as intended. The problem is that the diagonal elements, where (i=j), are accumulated twice in the second sum. So they should have been summed separately or they need to be subtracted off once.
answered
Matthew Salt... ♦ Paul Rubin ♦♦ |