40,427 research outputs found
Quantum magic rectangles: Characterization and application to certified randomness expansion
We study a generalization of the Mermin-Peres magic square game to arbitrary
rectangular dimensions. After exhibiting some general properties, these
rectangular games are fully characterized in terms of their optimal win
probabilities for quantum strategies. We find that for rectangular
games of dimensions there are quantum strategies that win with
certainty, while for dimensions quantum strategies do not
outperform classical strategies. The final case of dimensions is
richer, and we give upper and lower bounds that both outperform the classical
strategies. Finally, we apply our findings to quantum certified randomness
expansion to find the noise tolerance and rates for all magic rectangle games.
To do this, we use our previous results to obtain the winning probability of
games with a distinguished input for which the devices give a deterministic
outcome, and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput.
46, 1304 (2017)].Comment: 23 pages, 3 figures; published version with minor correction
Square, Rectangular and Triangular Nim Games
Let p be an integer with p ≥ 2. We shall investigate the following two piles Nim games. Let S be the set of positive integers {1 ≤ i ≤ p − 1}. Each player can remove the number of tokens s1 ∈ {0} ∪ S from the first pile and s2 ∈ {0} ∪ S from the second pile with 0 < s1 + s2 at the same time. We shall identify (m, n) to a position of this Nim game, where m is the number of tokens in the first pile and n is the number of tokens in the second pile. We shall show the Sprague-Grundy sequence (or simply G-sequences) gs(m, n) satisfy the periodic relation gs(m+p, n+p) = gs(m, n) for any position (m, n). We will call this two piles Nim Square Nim. In case m and n are sufficiently large, we will show that G-sequences gs(m, n) are also periodic for each row and column with the same period p. Finally we shall introduce several related games, such as Rectangular Nim, Triangular Nim and Polytope Nim
A Note on Kuhn's Theorem with Ambiguity Averse Players
Kuhn's Theorem shows that extensive games with perfect recall can
equivalently be analyzed using mixed or behavioral strategies, as long as
players are expected utility maximizers. This note constructs an example that
illustrate the limits of Kuhn's Theorem in an environment with ambiguity averse
players who use maxmin decision rule and full Bayesian updating.Comment: 7 figure
Assume-Admissible Synthesis
In this paper, we introduce a novel rule for synthesis of reactive systems,
applicable to systems made of n components which have each their own
objectives. It is based on the notion of admissible strategies. We compare our
novel rule with previous rules defined in the literature, and we show that
contrary to the previous proposals, our rule defines sets of solutions which
are rectangular. This property leads to solutions which are robust and
resilient. We provide algorithms with optimal complexity and also an
abstraction framework.Comment: 31 page
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