9,362 research outputs found
Diversification Preferences in the Theory of Choice
Diversification represents the idea of choosing variety over uniformity.
Within the theory of choice, desirability of diversification is axiomatized as
preference for a convex combination of choices that are equivalently ranked.
This corresponds to the notion of risk aversion when one assumes the
von-Neumann-Morgenstern expected utility model, but the equivalence fails to
hold in other models. This paper studies axiomatizations of the concept of
diversification and their relationship to the related notions of risk aversion
and convex preferences within different choice theoretic models. Implications
of these notions on portfolio choice are discussed. We cover model-independent
diversification preferences, preferences within models of choice under risk,
including expected utility theory and the more general rank-dependent expected
utility theory, as well as models of choice under uncertainty axiomatized via
Choquet expected utility theory. Remarks on interpretations of diversification
preferences within models of behavioral choice are given in the conclusion
A distributed adaptive steplength stochastic approximation method for monotone stochastic Nash Games
We consider a distributed stochastic approximation (SA) scheme for computing
an equilibrium of a stochastic Nash game. Standard SA schemes employ
diminishing steplength sequences that are square summable but not summable.
Such requirements provide a little or no guidance for how to leverage
Lipschitzian and monotonicity properties of the problem and naive choices
generally do not preform uniformly well on a breadth of problems. While a
centralized adaptive stepsize SA scheme is proposed in [1] for the optimization
framework, such a scheme provides no freedom for the agents in choosing their
own stepsizes. Thus, a direct application of centralized stepsize schemes is
impractical in solving Nash games. Furthermore, extensions to game-theoretic
regimes where players may independently choose steplength sequences are limited
to recent work by Koshal et al. [2]. Motivated by these shortcomings, we
present a distributed algorithm in which each player updates his steplength
based on the previous steplength and some problem parameters. The steplength
rules are derived from minimizing an upper bound of the errors associated with
players' decisions. It is shown that these rules generate sequences that
converge almost surely to an equilibrium of the stochastic Nash game.
Importantly, variants of this rule are suggested where players independently
select steplength sequences while abiding by an overall coordination
requirement. Preliminary numerical results are seen to be promising.Comment: 8 pages, Proceedings of the American Control Conference, Washington,
201
On Sharp Identification Regions for Regression Under Interval Data
The reliable analysis of interval data (coarsened data) is one of the
most promising applications of imprecise probabilities in statistics. If one
refrains from making untestable, and often materially unjustified, strong
assumptions on the coarsening process, then the empirical distribution
of the data is imprecise, and statistical models are, in Manskiās terms,
partially identified. We first elaborate some subtle differences between
two natural ways of handling interval data in the dependent variable of
regression models, distinguishing between two different types of identification
regions, called Sharp Marrow Region (SMR) and Sharp Collection
Region (SCR) here. Focusing on the case of linear regression analysis, we
then derive some fundamental geometrical properties of SMR and SCR,
allowing a comparison of the regions and providing some guidelines for
their canonical construction.
Relying on the algebraic framework of adjunctions of two mappings between
partially ordered sets, we characterize SMR as a right adjoint and
as the monotone kernel of a criterion function based mapping, while SCR
is indeed interpretable as the corresponding monotone hull. Finally we
sketch some ideas on a compromise between SMR and SCR based on a
set-domained loss function.
This paper is an extended version of a shorter paper with the same title,
that is conditionally accepted for publication in the Proceedings of
the Eighth International Symposium on Imprecise Probability: Theories
and Applications. In the present paper we added proofs and the seventh
chapter with a small Monte-Carlo-Illustration, that would have made the
original paper too long
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