The Spectrum of Equilibria for the Colonel Blotto and the Colonel Lotto Games

We study Nash equilibria of a symmetric Colonel Blotto and Colonel Lotto games. In these game two players, with \(N \ge 1\) units of resources each, distribute their resources simultaneously across \(K \ge 2\) battlefields. We introduce a characteristic of equilibria in this game called spectrum which stands for the fraction of battlefields receiving given numbers of units. We provide complete characterization of spectra of Nash equilibria in Colonel Lotto game as well as necessary conditions on spectra of Nash equilibria in Colonel Blotto game for the case of \({K}\! \mid \!{N}\).