Optimal strategies for a game on amenable semigroups

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.Comment: 17 pages. To appear in International Journal of Game Theor

Optimal strategies for a game on amenable semigroups

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.Comment: 17 pages. To appear in International Journal of Game Theor

Optimal strategies for a game on amenable semigroups

The semigroup game is a two-person zero-sum game defined on a semigroup $${(S,\cdot)}$$ as follows: Players 1 and 2 choose elements $${x \in S}$$ and $${y \in S}$$ , 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 gam

Existence of equilibria in countable games: an algebraic approach

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group and the payoffs are functions of the group operation. In order to obtain the existence of equilibria, finitely additive mixed strategies have to be allowed. This creates a problem of selection of a product measure of mixed strategies. We propose a family of such selections and prove existence of an equilibrium that does not depend on the selection. As a byproduct we show that if finitely additive mixed strategies are allowed, then Wald's game admits an equilibrium. We also prove existence of equilibria for nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we extend the main results to uncountable games

Adaptive high-order splitting schemes for large-scale differential Riccati equations

We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in a efficient way by utilizing a low-rank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used e.g. for time step adaptivity, and our numerical experiments in this direction show promising results.Comment: 23 pages, 7 figure