80 research outputs found
Precompact noncompact reflexive abelian groups
We present a series of examples of precompact, noncompact, reflexive
topological Abelian groups. Some of them are pseudocompact or even countably
compact, but we show that there exist precompact non-pseudocompact reflexive
groups as well. It is also proved that every pseudocompact Abelian group is a
quotient of a reflexive pseudocompact group with respect to a closed reflexive
pseudocompact subgroup
Quantum strategies
We consider game theory from the perspective of quantum algorithms.
Strategies in classical game theory are either pure (deterministic) or mixed
(probabilistic). We introduce these basic ideas in the context of a simple
example, closely related to the traditional Matching Pennies game. While not
every two-person zero-sum finite game has an equilibrium in the set of pure
strategies, von Neumann showed that there is always an equilibrium at which
each player follows a mixed strategy. A mixed strategy deviating from the
equilibrium strategy cannot increase a player's expected payoff. We show,
however, that in our example a player who implements a quantum strategy can
increase his expected payoff, and explain the relation to efficient quantum
algorithms. We prove that in general a quantum strategy is always at least as
good as a classical one, and furthermore that when both players use quantum
strategies there need not be any equilibrium, but if both are allowed mixed
quantum strategies there must be.Comment: 8 pages, plain TeX, 1 figur
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