5,345 research outputs found
Markovian assignment rules
We analyze dynamic assignment problems where agents successively receive different objects (positions, offices, etc.). A finite set of n vertically differentiated indivisible objects are assigned to n agents who live n periods. At each period, a new agent enters society, and the oldest agent retires, leaving his object to be reassigned. We define independent assignment rules (where the assignment of an object to an agent is independent of the way other objects are allocated to other agents), efficient assignment rules (where there does not exist another assignment rule with larger expected surplus), and fair assignment rules (where agents experiencing the same circumstances have identical histories in the long run). When agents are homogenous, we characterize efficient, independent and fair rules as generalizations of the seniority rule. When agents draw their types at random, we prove that independence and efficiency are incompatible, and that efficient and fair rules only exist when there are two types of agents. We characterize two simple rules (type-rank and type-seniority) which satisfy both efficiency and fairness criteria in dichotomous settings.dynamic assignment, finite Markov chains, seniority, promotion rules
Markovian assignment rules
We analyze dynamic assignment problems where agents successively receive different objects (positions, offices, etc.). A finite set of n vertically differentiated indivisible objects are assigned to n agents who live n periods. At each period, a new agent enters society, and the oldest agent retires, leaving his object to be reassigned. We define independent assignment rules (where the assignment of an object to an agent is independent of the way other objects are allocated to other agents), efficient assignment rules (where there does not exist another assignment rule with larger expected surplus), and fair assignment rules (where agents experiencing the same circumstances have identical histories in the long run). When agents are homogenous, we characterize efficient, independent and fair rules as generalizations of the seniority rule. When agents draw their types at random, we prove that independence and efficiency are incompatible, and that efficient and fair rules only exist when there are two types of agents. We characterize two simple rules (type-rank and type-seniority) which satisfy both efficiency and fairness criteria in dichotomous settings
Change Acceleration and Detection
A novel sequential change detection problem is proposed, in which the change
should be not only detected but also accelerated. Specifically, it is assumed
that the sequentially collected observations are responses to treatments
selected in real time. The assigned treatments not only determine the
pre-change and post-change distributions of the responses, but also influence
when the change happens. The problem is to find a treatment assignment rule and
a stopping rule that minimize the expected total number of observations subject
to a user-specified bound on the false alarm probability. The optimal solution
to this problem is obtained under a general Markovian change-point model.
Moreover, an alternative procedure is proposed, whose applicability is not
restricted to Markovian change-point models and whose design requires minimal
computation. For a large class of change-point models, the proposed procedure
is shown to achieve the optimal performance in an asymptotic sense. Finally,
its performance is found in two simulation studies to be close to the optimal,
uniformly with respect to the error probability
Optimal Assignment of Durable Objects to Successive Agents
This paper analyzes the assignment of durable objects to successive generations of agents who live for two periods. The optimal assignment rule is stationary, favors old agents and is determined by a selectivity function which satisfies an iterative functional differential equation. More patient social planners are more selective, as are social planners facing distributions of types with higher probabilities for higher types. The paper also characterizes optimal assignment rules when monetary transfers are allowed and agents face a recovery cost, when agents' types are private information and when agents can invest to improve their types.Dynamic Assignment ; Durable Objects ; Revenue Management ; Dynamic Mechanism Design ; Overlapping Generations ; Promotions and Intertemporal Assignments
The Complexity of Relating Quantum Channels to Master Equations
Completely positive, trace preserving (CPT) maps and Lindblad master
equations are both widely used to describe the dynamics of open quantum
systems. The connection between these two descriptions is a classic topic in
mathematical physics. One direction was solved by the now famous result due to
Lindblad, Kossakowski Gorini and Sudarshan, who gave a complete
characterisation of the master equations that generate completely positive
semi-groups. However, the other direction has remained open: given a CPT map,
is there a Lindblad master equation that generates it (and if so, can we find
it's form)? This is sometimes known as the Markovianity problem. Physically, it
is asking how one can deduce underlying physical processes from experimental
observations.
We give a complexity theoretic answer to this problem: it is NP-hard. We also
give an explicit algorithm that reduces the problem to integer semi-definite
programming, a well-known NP problem. Together, these results imply that
resolving the question of which CPT maps can be generated by master equations
is tantamount to solving P=NP: any efficiently computable criterion for
Markovianity would imply P=NP; whereas a proof that P=NP would imply that our
algorithm already gives an efficiently computable criterion. Thus, unless P
does equal NP, there cannot exist any simple criterion for determining when a
CPT map has a master equation description.
However, we also show that if the system dimension is fixed (relevant for
current quantum process tomography experiments), then our algorithm scales
efficiently in the required precision, allowing an underlying Lindblad master
equation to be determined efficiently from even a single snapshot in this case.
Our work also leads to similar complexity-theoretic answers to a related
long-standing open problem in probability theory.Comment: V1: 43 pages, single column, 8 figures. V2: titled changed; added
proof-overview and accompanying figure; 50 pages, single column, 9 figure
Gibbs fragmentation trees
We study fragmentation trees of Gibbs type. In the binary case, we identify
the most general Gibbs-type fragmentation tree with Aldous' beta-splitting
model, which has an extended parameter range with respect to the
probability distributions on which it is based.
In the multifurcating case, we show that Gibbs fragmentation trees are
associated with the two-parameter Poisson--Dirichlet models for exchangeable
random partitions of , with an extended parameter range
, and , , .Comment: Published in at http://dx.doi.org/10.3150/08-BEJ134 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Generalized master equations leading to completely positive dynamics
We provide a general construction of quantum generalized master equations
with memory kernel leading to well defined, that is completely positive and
trace preserving, time evolutions. The approach builds on an operator
generalization of memory kernels appearing in the description of non-Markovian
classical processes, and puts into evidence the non uniqueness of the
relationship arising due to the typical quantum issue of operator ordering. The
approach provides a physical interpretation of the structure of the kernels,
and its connection with the classical viewpoint allows for a trajectory
description of the dynamics. Previous apparently unrelated results are now
connected in a unified framework, which further allows to phenomenologically
construct a large class of non-Markovian evolutions taking as starting point
collections of time dependent maps and instantaneous transformations describing
the microscopic interaction dynamics.Comment: 8 pages, to appear on PR
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