Sunday, April 10, 2016

Any N-person Game Whether Zero-sum or Not

In Chapter XII, "Coalitions,"  of Fights, Games and Debates, Anatol Rapoport wrote:
Any N-person game whether zero-sum or not can always be pictured as a zero-sum game by adding a fictitious (N + 1)-th player, whose winnings (or losses) equal the summed losses (or winnings) of all the others. Since the (N + 1)-th player makes no moves, his inclusion makes no difference in the original game’s strategic structure; but it is advantageous to include him, because his presence lets us treat every game as a zero-sum game. Zero-sum games are easier to treat from a unified point of view than non-zero-sum games. 
Here it may occur to ask why the difficulties of the non zero-sum game were at all emphasized if they can be circumvented by adding another player. Note, however, that adding a player to a two-person game turns it into a three-person game, which is complicated by the possibility of forming coalitions. ...
To be fair, when Paul Samuelson, Thomas Friedman, Alicia Munnell, George Shultz or other mouthpieces cut-and-paste "zero-sum game" words into their lump-of-labor boilerplate, they don't intend to refer to actual game theory but only use game-theoryish sounding terms to impress and intimidate the rubes.