Suppose we have a two-player game which has some theory established about its solution concepts or other properties (anything from single simultaneous move games, to complex auctions, etc.).
Now suppose we construct a new, zero-sum game by, for each outcome independently, setting each player's new payoffs to the proportion of that player's old payoffs to the total old payoffs at that outcome. So if Player A previously got [imath]2[/imath] and Player B previously got [imath]7[/imath], then Player A now gets [imath]\frac{2}{9}[/imath] and Player B now gets [imath]\frac{7}{9}[/imath].
Does anything learned from the theory established for the former game remain applicable to the latter game? Is there a way to use equilibria from the former game to find equilibria for the latter game? Or must the latter game be analyzed completely from scratch?
Now suppose we construct a new, zero-sum game by, for each outcome independently, setting each player's new payoffs to the proportion of that player's old payoffs to the total old payoffs at that outcome. So if Player A previously got [imath]2[/imath] and Player B previously got [imath]7[/imath], then Player A now gets [imath]\frac{2}{9}[/imath] and Player B now gets [imath]\frac{7}{9}[/imath].
Does anything learned from the theory established for the former game remain applicable to the latter game? Is there a way to use equilibria from the former game to find equilibria for the latter game? Or must the latter game be analyzed completely from scratch?