Optimal strategies for equal-sum dice games

In this paper, we consider a non-cooperative two-person zero-sum matrix game, called dice game. In an (n,σ)(n,σ) dice game, two players can independently choose a dice from a collection of hypothetical dice having n   faces and with a total of σσ eyes distributed over these faces. They independently roll their dice and the player showing the highest number of eyes wins (in case of a tie, none of the players wins). The problem at hand in this paper is the characterization of all optimal strategies for these games. More precisely, we determine the (n,σ)(n,σ) dice games for which optimal strategies exist and derive for these games the number of optimal strategies as well as their explicit form.