  The semigroup game is a two-person zero-sum game defined on a semigroup S as
follows: Players 1 and 2 choose elements x and y in S, respectively, and player
1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the
semigroup is amenable in the sense of Day and von Neumann, one can extend the
set of classical strategies, namely countably additive probability measures on
S, to include some finitely additive measures in a natural way. This extended
game has a value and the players have optimal strategies. This theorem extends
previous results for the multiplication game on a compact group or on the
positive integers with a specific payoff. We also prove that the procedure of
extending the set of allowed strategies preserves classical solutions: if a
semigroup game has a classical solution, this solution solves also the extended
game.

Computer Science and Game Theory;Group Theory;Probability