I was thinking about how to better answer your question concerning the sub-zero game statement I made. Let's remove ourselves from finance for a minute into som

I was thinking about how to better answer your question concerning the sub-zero game statement I made. Let's remove ourselves from finance for a minute into something that is more traditional in the game theory studies--namely poker. If the game is played where the players arrange transfers face to face such as in a home poker game, then the game is zero-sum. Your losses are my gains. No money is created. No money is destroyed. It stays inside the game. The minute we go to a casino to play the same game, the game becomes sub-zero due to the rake. You lose X to me and Y to the rake. I gain your X and lose Y to the rake. Your cash flows: -X - Y. My cash flows: X - Y. In a zero-sum game your cash flows are the negative of my cash flows. This implies -(-X - Y) = X + Y. This is a contradiction. Therefore, the game is not zero-sum.