56 CHAPTER 2 Playing with Equilibria in Wireless Non-Cooperative Games Apply Rosen (1965) Yes CG? Apply Monderer and Shapley (1996) Yes PG? No No Not treated Is there a unique NE? Yes SG? No Apply Yates (1995) No BRs? Yes Prove BRs={} FIGURE 2.4 A (non-exhaustive) methodology for proving the uniqueness of a Nash equilibrium in strategic form games. The meaning of the used acronyms is as follows: CG? (Is the game concave?; Rosen, 1965), PG (potential games; Monderer and Shapley, 1996), SG (games with standard best responses; Yates, 1995), BRs? (Are the best-responses known?). 2.3.1 The Best Responses Do Not Need to be Explicated A natural question would be to ask whether the Debreu-Fan-Glicksberg theorem has a counterpart for uniqueness, that is, is there a general uniqueness theorem for quasi- concave K-player games. To the best of the author's knowledge, the answer is no. However, there is a powerful tool for proving the uniqueness of a pure NE when the players' utilities are concave: this tool is the uniqueness theorem derived by Rosen (1965). This theorem states that if a certain condition, called diagonally strict concavity (DSC), is met, then uniqueness is guaranteed. This theorem is as follows: ·························································································· · · · · · · · Assume that: i K: S i is a non-empty, compact, and convex set; u i (s) is a continuous · · function in s S and concave in s i . Let r = (r 1 , . . . , r K ) be an arbitrary vector of fixed · · · positive parameters. Define the pseudogradient of the function w r = r × u T by w (s) = · r Theorem 70: Rosen (1965)