21 research outputs found
Efficiently Learning from Revealed Preference
In this paper, we consider the revealed preferences problem from a learning
perspective. Every day, a price vector and a budget is drawn from an unknown
distribution, and a rational agent buys his most preferred bundle according to
some unknown utility function, subject to the given prices and budget
constraint. We wish not only to find a utility function which rationalizes a
finite set of observations, but to produce a hypothesis valuation function
which accurately predicts the behavior of the agent in the future. We give
efficient algorithms with polynomial sample-complexity for agents with linear
valuation functions, as well as for agents with linearly separable, concave
valuation functions with bounded second derivative.Comment: Extended abstract appears in WINE 201
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)
Online Learning and Profit Maximization from Revealed Preferences
We consider the problem of learning from revealed preferences in an online
setting. In our framework, each period a consumer buys an optimal bundle of
goods from a merchant according to her (linear) utility function and current
prices, subject to a budget constraint. The merchant observes only the
purchased goods, and seeks to adapt prices to optimize his profits. We give an
efficient algorithm for the merchant's problem that consists of a learning
phase in which the consumer's utility function is (perhaps partially) inferred,
followed by a price optimization step. We also consider an alternative online
learning algorithm for the setting where prices are set exogenously, but the
merchant would still like to predict the bundle that will be bought by the
consumer for purposes of inventory or supply chain management. In contrast with
most prior work on the revealed preferences problem, we demonstrate that by
making stronger assumptions on the form of utility functions, efficient
algorithms for both learning and profit maximization are possible, even in
adaptive, online settings
Incentive Compatible Active Learning
We consider active learning under incentive compatibility constraints. The
main application of our results is to economic experiments, in which a learner
seeks to infer the parameters of a subject's preferences: for example their
attitudes towards risk, or their beliefs over uncertain events. By cleverly
adapting the experimental design, one can save on the time spent by subjects in
the laboratory, or maximize the information obtained from each subject in a
given laboratory session; but the resulting adaptive design raises
complications due to incentive compatibility. A subject in the lab may answer
questions strategically, and not truthfully, so as to steer subsequent
questions in a profitable direction.
We analyze two standard economic problems: inference of preferences over risk
from multiple price lists, and belief elicitation in experiments on choice over
uncertainty. In the first setting, we tune a simple and fast learning algorithm
to retain certain incentive compatibility properties. In the second setting, we
provide an incentive compatible learning algorithm based on scoring rules with
query complexity that differs from obvious methods of achieving fast learning
rates only by subpolynomial factors. Thus, for these areas of application,
incentive compatibility may be achieved without paying a large sample
complexity price.Comment: 22 page
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
On the Existence of Low-Rank Explanations for Mixed Strategy Behavior
Nash equilibrium is used as a model to explain the observed behavior of
players in strategic settings. For example, in many empirical applications we
observe player behavior, and the problem is to determine if there exist payoffs
for the players for which the equilibrium corresponds to observed player
behavior. Computational complexity of Nash equilibria is an important
consideration in this framework. If the instance of the model that explains
observed player behavior requires players to have solved a computationally hard
problem, then the explanation provided is questionable. In this paper we
provide conditions under which Nash equilibrium is a reasonable explanation for
strategic behavior, i.e., conditions under which observed behavior of players
can be explained by games in which Nash equilibria are easy to compute. We
identify three structural conditions and show that if the data set of observed
behavior satisfies any of these conditions, then it is consistent with payoff
matrices for which the observed Nash equilibria could have been computed
efficiently. Our conditions admit large and structurally complex data sets of
observed behavior, showing that even with complexity considerations, Nash
equilibrium is often a reasonable model.Comment: Updated writeup. 19 page
Learning Time Dependent Choice
We explore questions dealing with the learnability of models of choice over time. We present a large class of preference models defined by a structural criterion for which we are able to obtain an exponential improvement over previously known learning bounds for more general preference models. This in particular implies that the three most important discounted utility models of intertemporal choice - exponential, hyperbolic, and quasi-hyperbolic discounting - are learnable in the PAC setting with VC dimension that grows logarithmically in the number of time periods. We also examine these models in the framework of active learning. We find that the commonly studied stream-based setting is in general difficult to analyze for preference models, but we provide a redeeming situation in which the learner can indeed improve upon the guarantees provided by PAC learning. In contrast to the stream-based setting, we show that if the learner is given full power over the data he learns from - in the form of learning via membership queries - even very naive algorithms significantly outperform the guarantees provided by higher level active learning algorithms