Data-driven estimation in equilibrium using inverse optimization

TLDR: This work develops an efficient, data-driven technique for estimating the parameters of these models from observed equilibria, and supports both parametric and nonparametric estimation by leveraging ideas from statistical learning (kernel methods and regularization operators).

Abstract: Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and nonparametric estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.

Figures

Fig. 5 The left panel shows the histogram of of out-sample approximation errors induced by our nonparametric fit from Sect. 8.3. The right panel shows the norm of the difference of this flow from the observed flow, relative to the norm of the observed flow

Fig. 3 Bertrand–Nash example of Sect. 8.2. The left panel shows the out-of-sample approximation error. The right panel shows the out-of-sample prediction error

Fig. 2 The true marginal revenue function (dashed line), fitted parametric marginal revenue function (solid red line, circular markers), fitted non-parametric marginal revenue function (solid black line, square markers)and ambiguity sets (grey region) for each firm. We fix all variables except the firm’s own price to the median observation (color figure online)

Fig. 4 The left panel shows the true BPR function and fits based on polynomials of degrees 3, 4, 5, and 6. The right panel shows the true BPR function (dashed line), our degree 3 fit (solid black line with markers), and an ambiguity set around this function (grey region) (color figure online)

Fig. 1 An idealized scenario. The true marginal revenue function (dashed black line), our nonparametric fit (black line, square markers), and the ambiguity set (grey region) for both firms. Every function in the ambiguity set exactly reconciles all the data. A sample member (blue line, triangle markers) shown for comparison. All variables other than the firm’s own price have been fixed to the median observation (color figure online)

Full text

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Mathematical Programming
A Publication of the Mathematical
Optimization Society
ISSN 0025-5610
Volume 153
Number 2
Math. Program. (2015) 153:595-633
DOI 10.1007/s10107-014-0819-4
Data-driven estimation in equilibrium
using inverse optimization
Dimitris Bertsimas, Vishal Gupta &
Ioannis Ch.Paschalidis

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Math. Program., Ser. A (2015) 153:595–633
DOI 10.1007/s10107-014-0819-4
FULL LENGTH PAPER
Data-driven estimation in equilibrium using inverse
optimization
Dimitris Bertsimas · Vishal Gupta ·
Ioannis Ch. Paschalidis
Received: 23 November 2012 / Accepted: 12 September 2014 / Published online: 30 September 2014
© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014
Abstract Equilibrium modeling is common in a variety of fields such as game theory
and transportation science. The inputs for these models, however, are often diffi-
cult to estimate, while their outputs, i.e., the equilibria they are meant to describe,
are often directly observable. By combining ideas from inverse optimization with
the theory of variational inequalities, we develop an efficient, data-driven technique
for estimating the parameters of these models from observed equilibria. We use this
technique to estimate the utility functions of players in a game from their observed
actions and to estimate the congestion function on a road network from traffic count
data. A distinguishing feature of our approach is that it supports both parametric and
nonparametric estimation by leveraging ideas from statistical learning (kernel meth-
ods and regularization operators). In computational experiments involving Nash and
Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate
the unknown demand or congestion function, respectively, and b) our proposed reg-
ularization technique substantially improves the out-of-sample performance of our
estimators.
D. Bertsimas (
B
)
MIT, Sloan School of Management, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
e-mail: dbertsim@mit.edu
V. Gupta
Operations Research Center, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
e-mail: vgupta1@mit.edu
I. Ch. Paschalidis
Department of Electrical and Computer Engineering, Boston University,
Boston, MA 02215, USA
e-mail: yannisp@bu.edu
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596 D. Bertsimas et al.
Keywords Equilibrium · Nonparametric estimation · Utility estimation ·
Traffic assignment
Mathematics Subject Classification 74G75 Equilibrium: Inverse Problems ·
62G05 Nonparametric Inference: Estimation · 62P20 Applications to Economics ·
90B20 Operations Research and Management Science: Traffic Problems
1 Introduction
Modeling phenomena as equilibria is a common approach in a variety of fields. Exam-
ples include Nash equilibrium in game theory, traffic equilibrium in transportation sci-
ence and market equilibrium in economics. Often, however, the model primitives or
“inputs” needed to calculate equilibria are not directly observable and can be difficult
to estimate. Small errors in these estimates may have large impacts on the resulting
equilibrium. This problem is particularly serious in design applications, where one
seeks to (re)design a system so that the induced equilibrium satisfies some desirable
properties, such as maximizing social welfare. In this case, small errors in the estimates
may substantially affect the optimal design. Thus, developing accurate estimates of
the primitives is crucial.
In this work we propose a novel framework to estimate the unobservable model
primitives for systems in equilibrium. Our data-driven approach hinges on the fact
that although the model primitives may be unobservable, it is frequently possible to
observe equilibria experimentally. We use these observed equilibria to estimate the
original primitives.
We draw on an example from game theory to illustrate. Typically, one specifies
the utility functions for each player in a game and then calculates Nash equilibria. In
practice, however, it is essentially impossible to observe utilities directly. Worse, the
specific choice of utility function often makes a substantial difference in the resulting
equilibrium. Our approach amounts to estimating a player’s utility function from her
actions in previous games, assuming her actions were approximately equilibria with
respect to her opponents. In contrast to her utility function, her previous actions are
directly observable. This utility function can be used either to predict her actions in
future games, or as an input to subsequent mechanism design problems involving this
player in the future.
A second example comes from transportation science. Given a particular road net-
work, one typically specifies a cost function and then calculates the resulting flow
under user (Wardrop) equilibrium. However, measuring the cost function directly in a
large-scale network is challenging because of the interdependencies among arcs. Fur-
thermore, errors in estimates of cost functions can have severe and counterintuitive
effects; Braess paradox (see [13]) is one well-known example. Our approach amounts
to estimating cost functions using current traffic count data (flows) on the network,
assuming those flows are approximately in equilibrium. Again, in contrast to the cost
function, traffic count data are readily observable and frequently collected on many
real-life networks. Finally, our estimate can be used either to predict congestion on
the network in the future, or else to inform subsequent network design problems.
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Data-driven estimation in equilibrium 597
In general, we focus on equilibria that can be modeled as the solution to a variational
inequality (VI). VIs are a natural tool for describing equilibria with examples spanning
economics, transportation science, physics, differential equations, and optimization.
(See Sect. 2.1 or [26] for detailed examples.) Our model centers on solving an inverse
variational inequality problem: given data that we believe are equilibria, i.e., solutions
to some VI, estimate the function which describes this VI, i.e., the model primitives.
Our formulation and analysis is motivated in many ways by the inverse optimization
literature. In inverse optimization, one is given a candidate solution to an optimiza-
tion problem and seeks to characterize the cost function or other problem data that
would make that solution (approximately) optimal. See [27] for a survey of inverse
combinatorial optimization problems, [3] for the case of linear optimization and [28]
for the case of conic optimization. The critical difference, however, is that we seek
a cost function that would make t he observed data equilibria, not optimal solutions
to an optimization problem. In general, optimization problems can be reformulated
as variational inequalities (see Sect. 2.1), so that our inverse VI problem generalizes
inverse optimization, but this generalization allows us to address a variety of new
applications.
To the best of our knowledge, we are the first to consider inverse variational inequal-
ity problems. Previous work, however, has examined the problem of estimating para-
meters for systems assumed to be in equilibrium, most notably the structural estimation
literature in econometrics and operations management ([4,5,32,35]). Although there
are a myriad of techniques collectively referred to as structural estimation, roughly
speaking, they entail (1) assuming a parametric model for t he system including proba-
bilistic assumptions on random quantities, (2) deducing a set of necessary (structural)
equations for unknown parameters, and, finally, (3) solving a constrained optimization
problem corresponding to a generalized method of moments (GMM) estimate for the
parameters. The constraints of this optimization problem include the structural equa-
tions and possibly other application-specific constraints, e.g., orthogonality conditions
of instrumental variables. Moreover, this optimization problem is typically difficult to
solve numerically, as it can be non-convex with large flat regions and multiple local
optima (see [4] for some discussion).
Our approach differs from structural estimation and other specialized approaches in
a number of respects. From a philosophical point of view, the most critical difference is
in the objective of the methodology. Specifically, in the structural estimation paradigm,
one posits a “ground-truth” model of a system with a known parametric form. The
objective of the method is to learn the parameters in order to provide insight into
the system. By contrast, in our paradigm, we make no assumptions (parametric or
nonparametric) about the true mechanics of the system; we treat is as a “black-box.”
Our objective is to fit a model—in fact, a VI—that can be used to predict the behavior
of the system. We make no claim that this fitted model accurately reflects “reality,”
merely that it has good predictive power.
This distinction is subtle, mirroring the distinction between “data-modelling” in
classical statistics and “algorithmic modeling” in machine learning. (A famous, albeit
partisaned, account of this distinction is [15].) Our approach is kindred to the machine
learning point of view. For a more detailed discussion, please see Appendix 2.
This philosophical difference has a number of practical consequences:
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Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Data-driven estimation in equilibrium using inverse optimization" ?

The authors use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of their approach is that it supports both parametric and nonparametric estimation by leveraging ideas from statistical learning ( kernel methods and regularization operators ). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, the authors find that a ) they effectively estimate the unknown demand or congestion function, respectively, and b ) their proposed regularization technique substantially improves the out-of-sample performance of their estimators. 

Q2. What are the two tools the authors use to protect against faulty inference?

The authors see their ambiguity set technique and nonparametric analysis as important tools to protect against potentially faulty inference in these settings. 

Q3. How can the authors generate lower and upper bounds on the function f(x) quickly?

Using software for linear optimization, it is possible to generate lower and upper bounds on the function f(x̂) for various choices of x̂ quickly and efficiently. 

Q4. What is the common example of a cost function error?

errors in estimates of cost functions can have severe and counterintuitive effects; Braess paradox (see [13]) is one well-known example. 

Q5. What is the problem with measuring the cost function in a large-scale network?

measuring the cost function directly in a large-scale network is challenging because of the interdependencies among arcs. 

Q6. What is the cost of traveling a arc?

Note that because of interdependencies in the network, the cost of traveling arc a may depend not only on xa , but on the flows on other arcs as well. 

Q7. How can the authors use their approach to solve the inverse variational inequality problem?

3. Nonparametric estimation: Like existing methods in inverse optimization and structural estimation, their approach can be applied in a parametric setting. 

Q8. What is the purpose of their estimate?

their estimate can be used either to predict congestion on the network in the future, or else to inform subsequent network design problems. 

Q9. What is the way to estimate a function in equilibrium?

In this paper, the authors propose a computationally tractable technique for estimation in equilibrium based on an inverse variational inequality formulation.