535 research outputs found
The Robustness of ‘Enemy-of-My-Enemy-is-My-Friend’ Alliances
This paper examines the robustness of alliance formation in a three-player, two-stage game in which each of two players compete against a third player in disjoint sets of contests. Although the players with the common opponent share no common interests, we find that under the lottery contest success function (CSF) there exists a range of parameter configurations in which the players with the common opponent have incentive to form an alliance involving a pre-conflict transfer of resources. Models that utilize the lottery CSF typically yield qualitatively different results from those arising in models with the auction CSF (Fang 2002). However, under the lottery and the auction CSFs, the parameter configurations within which players with a common opponent form an alliance are closely related. Our results, thus, provide a partial robustness result for ‘enemy-of-my-enemy-is-my-friend’ alliances.Alliance Formation, Contests, Economics of Alliances, Conflict
Nonlocal Games and Quantum Permutation Groups
We present a strong connection between quantum information and quantum
permutation groups. Specifically, we define a notion of quantum isomorphisms of
graphs based on quantum automorphisms from the theory of quantum groups, and
then show that this is equivalent to the previously defined notion of quantum
isomorphism corresponding to perfect quantum strategies to the isomorphism
game. Moreover, we show that two connected graphs and are quantum
isomorphic if and only if there exists and that are
in the same orbit of the quantum automorphism group of the disjoint union of
and . This connection links quantum groups to the more concrete notion
of nonlocal games and physically observable quantum behaviours. We exploit this
link by using ideas and results from quantum information in order to prove new
results about quantum automorphism groups, and about quantum permutation groups
more generally. In particular, we show that asymptotically almost surely all
graphs have trivial quantum automorphism group. Furthermore, we use examples of
quantum isomorphic graphs from previous work to construct an infinite family of
graphs which are quantum vertex transitive but fail to be vertex transitive,
answering a question from the quantum group literature.
Our main tool for proving these results is the introduction of orbits and
orbitals (orbits on ordered pairs) of quantum permutation groups. We show that
the orbitals of a quantum permutation group form a coherent
configuration/algebra, a notion from the field of algebraic graph theory. We
then prove that the elements of this quantum orbital algebra are exactly the
matrices that commute with the magic unitary defining the quantum group. We
furthermore show that quantum isomorphic graphs admit an isomorphism of their
quantum orbital algebras which maps the adjacency matrix of one graph to that
of the other.Comment: 39 page
Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number
We investigate vector chromatic number, Lovasz theta of the complement, and
quantum chromatic number from the perspective of graph homomorphisms. We prove
an analog of Sabidussi's theorem for each of these parameters, i.e. that for
each of the parameters, the value on the Cartesian product of graphs is equal
to the maximum of the values on the factors. We also prove an analog of
Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value
on the categorical product of graphs is equal to the minimum of its values on
the factors. We conjecture that the analogous results hold for vector and
quantum chromatic number, and we prove that this is the case for some special
classes of graphs.Comment: 18 page
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