1,624 research outputs found
On an unified framework for approachability in games with or without signals
We unify standard frameworks for approachability both in full or partial
monitoring by defining a new abstract game, called the "purely informative
game", where the outcome at each stage is the maximal information players can
obtain, represented as some probability measure. Objectives of players can be
rewritten as the convergence (to some given set) of sequences of averages of
these probability measures. We obtain new results extending the approachability
theory developed by Blackwell moreover this new abstract framework enables us
to characterize approachable sets with, as usual, a remarkably simple and clear
reformulation for convex sets. Translated into the original games, those
results become the first necessary and sufficient condition under which an
arbitrary set is approachable and they cover and extend previous known results
for convex sets. We also investigate a specific class of games where, thanks to
some unusual definition of averages and convexity, we again obtain a complete
characterization of approachable sets along with rates of convergence
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
Multipartite Causal Correlations: Polytopes and Inequalities
We consider the most general correlations that can be obtained by a group of
parties whose causal relations are well-defined, although possibly
probabilistic and dependent on past parties' operations. We show that, for any
fixed number of parties and inputs and outputs for each party, the set of such
correlations forms a convex polytope, whose vertices correspond to
deterministic strategies, and whose (nontrivial) facets define so-called causal
inequalities. We completely characterize the simplest tripartite polytope in
terms of its facet inequalities, propose generalizations of some inequalities
to scenarios with more parties, and show that our tripartite inequalities can
be violated within the process matrix formalism, where quantum mechanics is
locally valid but no global causal structure is assumed.Comment: 14 pages and 1 supplementary CDF fil
Quantum de Finetti Theorems under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in
quantum multipartite states. In this paper we prove two new quantum de Finetti
theorems, both showing that under tests formed by local measurements one can
get a much improved error dependence on the dimension of the subsystems. We
also obtain similar results for non-signaling probability distributions. We
give the following applications of the results:
We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the
exponential time hypothesis.
We show that the maximum winning probability of free games can be estimated
in polynomial time by linear programming. We also show that 3-SAT with m
variables can be reduced to obtaining a constant error approximation of the
maximum winning probability under entangled strategies of O(m^{1/2})-player
one-round non-local games, in which the players communicate O(m^{1/2}) bits all
together.
We show that the optimization of certain polynomials over the hypersphere can
be performed in quasipolynomial time in the number of variables n by
considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy
of semidefinite programs. As an application to entanglement theory, we find a
quasipolynomial-time algorithm for deciding multipartite separability.
We consider a result due to Aaronson -- showing that given an unknown n qubit
state one can perform tomography that works well for most observables by
measuring only O(n) independent and identically distributed (i.i.d.) copies of
the state -- and relax the assumption of having i.i.d copies of the state to
merely the ability to select subsystems at random from a quantum multipartite
state.
The proofs of the new quantum de Finetti theorems are based on information
theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor
improvements. v3: added some explanations, mostly about Theorem 1 and
Conjecture 5. STOC version. v4, v5. small improvements and fixe
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