846 research outputs found
Learning Economic Parameters from Revealed Preferences
A recent line of work, starting with Beigman and Vohra (2006) and
Zadimoghaddam and Roth (2012), has addressed the problem of {\em learning} a
utility function from revealed preference data. The goal here is to make use of
past data describing the purchases of a utility maximizing agent when faced
with certain prices and budget constraints in order to produce a hypothesis
function that can accurately forecast the {\em future} behavior of the agent.
In this work we advance this line of work by providing sample complexity
guarantees and efficient algorithms for a number of important classes. By
drawing a connection to recent advances in multi-class learning, we provide a
computationally efficient algorithm with tight sample complexity guarantees
( for the case of goods) for learning linear utility
functions under a linear price model. This solves an open question in
Zadimoghaddam and Roth (2012). Our technique yields numerous generalizations
including the ability to learn other well-studied classes of utility functions,
to deal with a misspecified model, and with non-linear prices
The Combinatorial World (of Auctions) According to GARP
Revealed preference techniques are used to test whether a data set is
compatible with rational behaviour. They are also incorporated as constraints
in mechanism design to encourage truthful behaviour in applications such as
combinatorial auctions. In the auction setting, we present an efficient
combinatorial algorithm to find a virtual valuation function with the optimal
(additive) rationality guarantee. Moreover, we show that there exists such a
valuation function that both is individually rational and is minimum (that is,
it is component-wise dominated by any other individually rational, virtual
valuation function that approximately fits the data). Similarly, given upper
bound constraints on the valuation function, we show how to fit the maximum
virtual valuation function with the optimal additive rationality guarantee. In
practice, revealed preference bidding constraints are very demanding. We
explain how approximate rationality can be used to create relaxed revealed
preference constraints in an auction. We then show how combinatorial methods
can be used to implement these relaxed constraints. Worst/best-case welfare
guarantees that result from the use of such mechanisms can be quantified via
the minimum/maximum virtual valuation function
Behavioral implications of shortlisting procedures
We consider two-stage “shortlisting procedures” in which the menu of alternatives is first pruned by some process or criterion and then a binary relation is maximized. Given a particular first-stage process, our main result supplies a necessary and sufficient condition for choice data to be consistent with a procedure in the designated class. This result applies to any class of procedures with a certain lattice structure, including the cases of “consideration filters,” “satisficing with salience effects,” and “rational shortlist methods.” The theory avoids background assumptions made for mathematical convenience; in this and other respects following Richter’s classical analysis of preference-maximizing choice in the absence of shortlisting
Testing Consumer Rationality using Perfect Graphs and Oriented Discs
Given a consumer data-set, the axioms of revealed preference proffer a binary
test for rational behaviour. A natural (non-binary) measure of the degree of
rationality exhibited by the consumer is the minimum number of data points
whose removal induces a rationalisable data-set.We study the computational
complexity of the resultant consumer rationality problem in this paper. This
problem is, in the worst case, equivalent (in terms of approximation) to the
directed feedback vertex set problem. Our main result is to obtain an exact
threshold on the number of commodities that separates easy cases and hard
cases. Specifically, for two-commodity markets the consumer rationality problem
is polynomial time solvable; we prove this via a reduction to the vertex cover
problem on perfect graphs. For three-commodity markets, however, the problem is
NP-complete; we prove thisusing a reduction from planar 3-SAT that is based
upon oriented-disc drawings
Social welfare and profit maximization from revealed preferences
Consider the seller's problem of finding optimal prices for her
(divisible) goods when faced with a set of consumers, given that she can
only observe their purchased bundles at posted prices, i.e., revealed
preferences. We study both social welfare and profit maximization with revealed
preferences. Although social welfare maximization is a seemingly non-convex
optimization problem in prices, we show that (i) it can be reduced to a dual
convex optimization problem in prices, and (ii) the revealed preferences can be
interpreted as supergradients of the concave conjugate of valuation, with which
subgradients of the dual function can be computed. We thereby obtain a simple
subgradient-based algorithm for strongly concave valuations and convex cost,
with query complexity , where is the additive
difference between the social welfare induced by our algorithm and the optimum
social welfare. We also study social welfare maximization under the online
setting, specifically the random permutation model, where consumers arrive
one-by-one in a random order. For the case where consumer valuations can be
arbitrary continuous functions, we propose a price posting mechanism that
achieves an expected social welfare up to an additive factor of
from the maximum social welfare. Finally, for profit maximization (which may be
non-convex in simple cases), we give nearly matching upper and lower bounds on
the query complexity for separable valuations and cost (i.e., each good can be
treated independently)
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