The semigroup game is a two-person zero-sum game defined on a semigroup {(S,.) as follows: Players 1 and 2 choose elements x ∈ S and y ∈ S, respectively, and player 1 receives a payoff f (x y) defined by a function f: S → [-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.

Capraro, V., Morrison, K. (2013). Optimal strategies for a game on amenable semigroups. INTERNATIONAL JOURNAL OF GAME THEORY, 42(4), 917-929 [10.1007/s00182-012-0345-7].

### Optimal strategies for a game on amenable semigroups

#### Abstract

The semigroup game is a two-person zero-sum game defined on a semigroup {(S,.) as follows: Players 1 and 2 choose elements x ∈ S and y ∈ S, respectively, and player 1 receives a payoff f (x y) defined by a function f: S → [-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.
##### Scheda breve Scheda completa Scheda completa (DC)
Articolo in rivista - Articolo scientifico
Amenability; Intrinsically measurable; Loaded game; Minimax strategy; Multiplicative game; Nash equilibrium; Optimal strategies;
English
3-ago-2012
2013
42
4
917
929
none
Capraro, V., Morrison, K. (2013). Optimal strategies for a game on amenable semigroups. INTERNATIONAL JOURNAL OF GAME THEORY, 42(4), 917-929 [10.1007/s00182-012-0345-7].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/398051`
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