51,391 research outputs found
Non-Archimedean Preferences Over Countable Lotteries
We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces
Constructive Decision Theory
Contemporary approaches to decision making describe a decision problem by sets of states and outcomes, and a rich set of acts: functions from states to outcomes over which the decision maker (DM) has preferences. Real problems do not come so equipped. It is often unclear what the state and outcome spaces would be. We present an alternative foundation for decision making, in which the primitive objects of choice are syntactic programs. We show that if the DM's preference relation on objects of choice satisfies appropriate axioms, then we can find states, outcomes, and an embedding of the programs into Savage acts such that preferences can be represented by EU in the Savage framework. A modeler can test for SEU behavior without having access to the subjective states and outcomes. We illustrate the power of our approach by showing that it can represent DMs who are subject to framing effects.Decision theory, subjective expected utility, behavioral anomalies
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that appear in imprecise-probabilistic
decision making. We provide these choice functions with a clear interpretation
in terms of desirability, use this interpretation to derive a set of basic
coherence axioms, and show that this notion of coherence leads to a
representation in terms of sets of strict preference orders. By imposing
additional properties such as totality, the mixing property and Archimedeanity,
we obtain representation in terms of sets of strict total orders, lexicographic
probability systems, coherent lower previsions or linear previsions.Comment: arXiv admin note: text overlap with arXiv:1806.0104
Ordinal Games
We study strategic games where players' preferences are weak orders which need not admit utility representations. First of all, we ex- tend Voorneveld's concept of best-response potential from cardinal to ordi- nal games and derive the analogue of his characterization result: An ordi- nal game is a best-response potential game if and only if it does not have a best-response cycle. Further, Milgrom and Shannon's concept of quasi- supermodularity is extended from cardinal games to ordinal games. We ¯nd that under certain compactness and semicontinuity assumptions, the ordinal Nash equilibria of a quasi-supermodular game form a nonempty complete lattice. Finally, we extend several set-valued solution concepts from cardinal to ordinal games in our sense.Ordinal Games, Potential Games, Quasi-Supermodularity, Rationalizable Sets, Sets Closed under Behavior Correspondences
Common Mathematical Foundations of Expected Utility and Dual Utility Theories
We show that the main results of the expected utility and dual utility
theories can be derived in a unified way from two fundamental mathematical
ideas: the separation principle of convex analysis, and integral
representations of continuous linear functionals from functional analysis. Our
analysis reveals the dual character of utility functions. We also derive new
integral representations of dual utility models
When is there state independence?
It has been noticed that whether a preference relation can be represented by state-independent utilities as opposed to state-dependent utilities may depend on which acts count as constant acts [Schervish et al., 1990]. Indeed, this remark underlies an extension of Savage’s expected utility theory to the state-dependent case that was proposed by Edi Karni [Karni, 1993]. This paper contains a characterisation of the preference relations that permit a choice of acts which can play the role of constant acts, and relative to which there is a representation involving a state-independent utility function. This result applies both in the Savage and in the Anscombe & Aumann frameworks. It has as an immediate corollary an extension of Karni’s representation theorem. Finally, it is of methodological interest, insofar that it harnesses techniques from mathematical logic to prove a theorem of interest to decision theorists and economists.Subjective expected utility; State-dependent utility; Monotonicity axiom
Knowledge-aware Complementary Product Representation Learning
Learning product representations that reflect complementary relationship
plays a central role in e-commerce recommender system. In the absence of the
product relationships graph, which existing methods rely on, there is a need to
detect the complementary relationships directly from noisy and sparse customer
purchase activities. Furthermore, unlike simple relationships such as
similarity, complementariness is asymmetric and non-transitive. Standard usage
of representation learning emphasizes on only one set of embedding, which is
problematic for modelling such properties of complementariness. We propose
using knowledge-aware learning with dual product embedding to solve the above
challenges. We encode contextual knowledge into product representation by
multi-task learning, to alleviate the sparsity issue. By explicitly modelling
with user bias terms, we separate the noise of customer-specific preferences
from the complementariness. Furthermore, we adopt the dual embedding framework
to capture the intrinsic properties of complementariness and provide geometric
interpretation motivated by the classic separating hyperplane theory. Finally,
we propose a Bayesian network structure that unifies all the components, which
also concludes several popular models as special cases. The proposed method
compares favourably to state-of-art methods, in downstream classification and
recommendation tasks. We also develop an implementation that scales efficiently
to a dataset with millions of items and customers
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