This is all joint work with Ali M. Khan (Johns Hopkins) and Paul Arthur Pedersen (CUNY).

Game theory as practiced by economists is often couched in a setting where players pick strategies, and then a utility function tells them who has which pay off (the so-called "normal form" of a game). For two person games, an important special case is the zero sum game: the case where pay offs always sum to zero. Aumann, the sixties, defined "strictly competitive games", two player games in which what is good for one player is bad for the other. Aumann frequently stated that this is the same class as the zero sum games—for an appropriate choice of utility function (and provided the players strategy spaces are closed under mixing).

We claim that Aumann must have known this because he knew the multidimensional theory of utility. But then in 2009, Adler, Daskalakis and Papadimitriou gave a non-trivial proof of the fact claimed by Aumann, for finite games, claiming that no such proof exists in the literature. This was generalized in 2023 by Raimondo to games where the set of strategies available to each player is an appropriate set of probability measures on \([0,1]\) (or if you're feeling fancy, on a standard Borel space).

In this talk, I shall show what Aumann and others must already have been aware of, but what has apparently been forgotten in the meantime: That these results, and more general ones, follow easily from the theory of mutlidimensional utility developed in the 60ies and early 70ies by Fishburn, Roberts, and others.