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  • 1.
    Ait Ali, Abderrahman
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport Planning, Economics and Engineering.
    Lindberg, Per Olov
    KTH.
    Nilsson, Jan-Eric
    Eliasson, Jonas
    Aronsson, Martin
    Disaggregation in Bundle Methods: Application to the Train Timetabling Problem2017Conference paper (Refereed)
    Abstract [en]

    Bundle methods are often used to solve dual problems that arise from Lagrangian relaxations of large scale optimization problems. An example of such problems is the train timetabling problem. This paper focuses on solving a dual problem that arises from Lagrangian relaxation of a train timetabling optimization program. The dual problem is solved using bundle methods. We formulate and compare the performances of two different bundle methods: the aggregate method, which is a standard method, and a new, disaggregate, method which is proposed here. The two methods were tested on realistic train timetabling scenarios from the Iron Ore railway line. The numerical results show that the new disaggregate approach generally yields faster convergence than the standard aggregate approach.

  • 2.
    Engelson, Leonid
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Congestion Pricing of Road Networks with Users Having Different Time Values2006In: Mathematical and Computational Models for Congestion Charging / [ed] Lawphongpanich, Hearn, Smith, Springer Science+Business Media B.V., 2006, p. 81-104Chapter in book (Refereed)
    Abstract [en]

    We study congestion pricing of road networks with users differing onlyin their time values. In particular, we analyze the marginal social cost (MSC) pricing,a tolling scheme that charges each user a penalty corresponding to the value of thedelays inflicted on other users, as well as its implementation through fixed tolls. Weshow that the variational inequalities characterizing the corresponding equilibria canbe stated in symmetric or nonsymmetric forms. The symmetric forms correspondto optimization problems, convex in the fixed-toll case and nonconvex in the MSCcase, which hence may have multiple equilibria. The objective of the latter problemis the total value of travel time, which thus is minimized at the global optima of thatproblem. Implementing close-to-optimal MSC tolls as fixed tolls leads to equilibriawith possibly non-unique class specific flows, but with identical close-to-optimalvalues of the total value of travel time. Finally we give an adaptation, to the MSCsetting, of the Frank-Wolfe algorithm, which is further applied to some test cases,including Stockholm.

  • 3.
    Engelson, Leonid
    et al.
    Linköpings Universitet.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Convexification of the Traffic Equilibrium Problem with Social Marginal Cost Tolls2004In: Operations Research Proceedings 2003: Selected Papers of the International Conference on Operations Research (OR 2003) Heidelberg, September 3–5, 2003, Springer Berlin/Heidelberg, 2004, p. 141-148Conference paper (Other academic)
    Abstract [en]

    In an earlier paper, we have demonstrated that traffic equilibria under social marginal cost tolls can be computedas a local optima of a nonconvex optimization problem. The nonconvexity of this problem implies in particular that linearizations, e.g. the Frank-Wolfe method, do not give underestimates of the optimal value. In this paper we derive the convex hull of nonconvex arc cost functions of BPR type. These convexifications can be used to get underestimates of the optimal value, or get better search directions in the initial phase of the Frank-Wolfe method. Computational results for the Sioux Falls and Stockholm networks are reported.

  • 4.
    Engelson, Leonid
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS. KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Multi-Class User Equilibria under Social Marginal Cost Pricing 2003In: Operations Research Proceedings 2002, 2003Chapter in book (Other academic)
  • 5.
    Fosgerau, Mogens
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport Planning, Economics and Engineering. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Mattsson, Lars-Göran
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport Planning, Economics and Engineering. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Weibull, Jörgen
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Invariance of the distribution of the maximumManuscript (preprint) (Other academic)
    Abstract [en]

    Many models in economics involve probabilistic choices where each decision-maker selects the best alternative from a finite set. Viewing the value of each alternative as a random variable, the analyst is then interested in the choice probabilities, that is, the probability for an alternative to give the maximum value. Much analytical power can be gained, both for positive and normative analysis, if the maximum value is statistically independent of which alternative obtains the highest value. This note synthesizes and generalizes previous results on this invariance property. We provide characterizations of this property within a wide class of distributions that comprises the McFadden GEV class, show implications in several directions, and establish connections with copulas. We illustrate the usefulness of the invariance property by way of a few examples.

  • 6.
    Holmgren, Johan
    et al.
    Blekinge Tekniska Högskola.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Upright Stiff: subproblem updating in the FW method for traffic assignment2013In: EURO Journal on Transportation and Logistics, ISSN 2192-4376, E-ISSN 2192-4384Article in journal (Refereed)
    Abstract [en]

    We present improvements of the Frank–Wolfe (FW) method for static vehicular traffic and telecom routing. The FW method has been the dominating method for these problem types, but due to its slow asymptotic convergence it has been considered dead by methods oriented researchers. However, the recent introduction of conjugate FW methods has shown that it is still viable, and in fact the winner on multi-core computers. In this paper, we show how to speed up the FW iterations, by updating the subproblems in the FW method, instead of solving them from scratch. The subproblem updating is achieved by viewing the subproblems as network flow problems with a threaded representation of the shortest path trees. In addition, we introduce a new technique, thread following, implying that a single traversal of the thread is enough to find a new shortest path tree. Our computational tests show that very few nodes in practice are visited more than once when searching for improving arcs. Moreover, we update also the all-or-nothing solutions of the subproblems, resulting in significantly reduced loading times. For a set of standard test problems, we observe speedups in the region of 25–50 % for the subproblem updating FW method, compared to the traditional non-updating version. We typically achieve higher speedups for more difficult problems and converged solutions.

  • 7.
    Lindberg, Per
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Transport and Location Analysis (closed 20110301).
    Aspects of static multi-class traffic equilibria under congestion pricing2010Report (Other academic)
  • 8.
    Lindberg, Per
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    On methods for the convex multi-commodity flow problem2009In: International journal of pure and applied mathematics, ISSN 1311-8080, E-ISSN 1314-3395, Vol. 50, no 2, p. 157-164Article in journal (Refereed)
    Abstract [en]

    The Convex Multi-Commodity Flow Problem is a central routing problem in road and telecom traffic. We show that the traditional method for these problems, the Frank-Wolfemethod, still has a lot to give, through better search directions, and other amendments. In the end it can beat recent competitors, for accuracies of practical interest.

  • 9.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    A Note on Power Invariance in Random Utility Models2012Report (Other academic)
    Abstract [en]

    Invariance in Random Utility (RU) Models is the property that the distribution of achievedutility is invariant across the alternatives chosen. In this note we study a generalization termedPower Invariance (PI). It generalizes the property, showed by Mattsson and Weibull, thatinvariance holds for an independent RU Model, where the cdf’s of the random terms arepositive powers of a given base cdf . The power invariant copulas turn out to be the copulasof Multivariate Extreme Value distributions. We further show that, under natural regularityconditions, the copulas of the marginal distributions of a PI copula are power invariant.

  • 10.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    A Note on Two Papers by Dial2010Report (Other academic)
    Abstract [en]

    This note discusses some shortcomings of two papers by Dial. In these insightful but somewhat confused papers Dial studies the problem of determining “optimal” congestion tolls. The papers have several flaws, as well as outright errors. Moreover the presentation often is not very clear, often leaving the reader in doubt of what is proved about what. In this note we try to pinpoint the main flaws of the papers.

  • 11.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    A simple derivation of the Williams-Daly-Zachery theorem2012Report (Other academic)
    Abstract [en]

    In this note we give a simple elementary proof of the Williams-Daly-Zachery theorem for additive random utility (ARU) models. This theorem says that the choice probabilities in an ARU model are the partial derivatives of the expected achieved utility. This is proved under the condition that the choice probabilities are continuous. We further prove that the choice probabilities are continuous exactly when the probability of ties is zero. Finally we prove that the probability of ties is zero if the cdf of the random term (of the ARU model) is differentiable with locally bounded gradient.

  • 12.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Transport and Location Analysis (closed 20110301).
    Aspects of Static Multi-Class Traffic equilibria under Congestion Pricing2010Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    Congestion charging is a now accepted means of influencing traffic to behave in a more socio-economic optimal way, like e.g. in the Stockholm project. Already early work, in the 1920’s, showed that road use can be inefficient due externalities, i.e. that users don’t experience their own (negative) effect on other users: an extra car on a traffic link causes delays for other cars, but the driver himself does not experience this cost.In the 1950’s it was further shown - for a congested road network with homogeneous users – that if each user is charged a toll equal to the total value of time loss incurred on other users of the network, then -if we have fixed travel demand - this will induce an equilibrium that is system optimal in the sense that the total cost of network usage is minimal (assuming that all users have fixed and identical time values).  But toll charges need to be levied in monetary units, and different travelers have different values of time. Therefore, to account for the effects of tolls, and to be able to compute equilibria, one needs to introduce different user classes, differing in their time values.

    In this thesis, consisting of four papers, we study congestion pricing of road networks with users differing only in their time values. In particular, we analyze marginal social cost (MSC) pricing, a tolling scheme that charges each user a penalty corresponding to the value of the delays inflicted on other users, as well as its implementation through fixed tolls.

    Paper III contains the main theoretical work of the thesis. In that paper we show that the variational inequalities characterizing the equilibria in question can be stated in symmetric or non-symmetric forms. The symmetric forms correspond to optimization problems, convex in the fixed-toll case and non-convex in the MSC case, which hence may have multiple equilibria. The objective of the latter problem is the total value of travel time, which thus is minimized at the global optima of that problem. Implementing close-to-optimal MSC tolls as fixed tolls leads to equilibria with possibly non-unique class specific flows, but with identical close-to-optimal values of the total value of travel time. Finally we give an adaptation, to the MSC setting, of the Frank-Wolfe algorithm, which is further applied to some test cases, including Stockholm.

    Paper I is an early application using Frank-Wolfe, after having realized the possibility to symmetrize the problem.

    Paper II gives a convexification of non-convex equilibrium problem for MSC tolls. We have used these convexifications to compute lower bounds when computing equilibria.

    Paper IV is a short note commenting some flaws in two papers by Dial on MSC tolls.

  • 13.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Choice Probabilities in Random Utility Models: Necessary and Sufficient Conditions for a Much Used Formula, andImplications for the Conditional Distributions of Achieved Utility2012Report (Other academic)
    Abstract [en]

    Random Utility (RU) Models have reached a wide applicability in many areas of applied Economics. In all application areas, the formulas for the choice probabilities play a central role. For the formulas to be useful, the probabilities further need to sum up to 1. In this note, we give necessary and sufficient conditions for the validity of the classical choice probability formula (i.e. that the choice probabilities are the integrals of the partial derivatives of the cdf of the random term along the diagonal), and that the probabilities sum up to 1. Further, it is proved sufficient that the gradient of said cdf is continuous and piecewise differentiable with locally bounded gradient, along the diagonal. Along the way we prove many intermediate results relating different derivatives of the cdf. Finally the results are applied to derive formulas for the distribution of achieved utility conditioned on a given alternative being chosen.

  • 14.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Contributions to Probabilistic Discrete Choice2012Doctoral thesis, comprehensive summary (Other academic)
  • 15.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    On traffic equilibrium models with a nonlinear time/money relation 2002In: TRANSPORTATION PLANNING: STATE OF THE ART , 2002Chapter in book (Other academic)
  • 16.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Random Utility Invariance Revisited2012Report (Other academic)
  • 17.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Random Utility Models, IIA, Mattsson-Weibull-Li Distributionsand a Conjecture by Luce and Suppes2012Report (Other academic)
    Abstract [en]

    In recent papers, Mattsson and Weibull, and independently Li, have introduced a class of independent distributions, where the individual cdf’s are powers of each other. For this class they have showed that the choice probabilities are of “Luce form”, i.e. the form following from the Independence from Irrelevant Alternatives (IIA) assumption. We show a converse to this result. For a class of “independent” Random Utility (RU) Models, allowing non-uniform expansion, and where the cdf’s in question are well-behaved in a certain sense, the only possibility is that the cdf’s are of the above mentioned Mattsson-Weibull-Li (MWL) type. This result resolves an old conjecture by Luce and Suppes. The result demands unbounded choice sets. For finite models, we show on the contrary, that any finite independent RU model with Luce form choice probabilities, there is a version with random terms that are not MWL-distributed

  • 18.
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    The Choice Axiom Revisited2012Report (Other academic)
    Abstract [en]

    The famous Choice Axiom by Luce is now more than half a century old. We investigate the consequences of going back to Luce’s original formulation of the axiom, which had a more succinct formulation. Using this, what we term Strong Choice Axiom, we prove a sort of “lexicographic” version of the “Luce form” choice probabilities. Further, we study whether these extended choice probabilities are representable by Random Utility models.

  • 19.
    Lindberg, Per Olov
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science.
    Engelson, Leonid
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport planning, economics and engineering.
    Tolled multi-class traffic equilibria and toll sensitivities2015In: EURO Journal on Transportation and Logistics, ISSN 2192-4376, E-ISSN 2192-4384, Vol. 4, no 2, p. 197-222Article in journal (Refereed)
    Abstract [en]

    We review properties of tolled equilibria in road networks, with users differing in their time values, and study corresponding sensitivities of equilibrium link flows w.r.t. tolls. Possible applications include modeling of individual travellers that have different trip purposes (e.g. work, business, leisure, etc.) and therefore perceive the relation between travel time and monetary cost in dissimilar ways. The typical objective is to reduce the total value of travel time (TVT) over all users. For first best congestion pricing, where all links in the network can be tolled, the solution can be internalized through marginal social cost (MSC) pricing. The MSC equilibrium typically has to be implemented through fixed tolls. The MSC as well as the fixed-toll equilibrium problems can be stated as optimization problems, which in general are convex in the fixed-toll case and non-convex in the MSC case. Thus, there may be several MSC equilibria. Second-best congestion pricing, where one only tolls a subset of the links, is much more complex, and equilibrium flows, times and TVT are not in general differentiable w.r.t. tolls in sub-routes used by several classes. For generic tolls, where the sets of shortest paths are stable, we show how to compute Jacobians (w.r.t positive tolls) of link flows and times as well as of the TVT. This can be used in descent schemes to find tolls that minimize the TVT at least locally. We further show that a condition of independent equilibrium cycles, together with a natural extension of the single class regularity condition of strict complementarity, leads to genericity, and hence existence of said Jacobians.

  • 20.
    Lindberg, Per Olov
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Eriksson, E. Anders
    Mattsson, Lars-Göran
    KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS. KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Homothetic functions revisited2002In: Economic Theory, Vol. 19, no 2, p. 417-427Article in journal (Refereed)
  • 21.
    Lindberg, Per Olov
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis. KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Eriksson, E.A.
    Mattsson, Lars-Göran
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Invariance of achieved utility in random utility models1995In: Environment and planning A, ISSN 0308-518X, E-ISSN 1472-3409, Vol. 27, p. 121-142Article in journal (Refereed)
  • 22.
    Lindberg, Per Olov
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Holmgren, Johan
    Upright Stiff: Subproblem Updating in the FW-Method with Applications to Traffic Assignment2013In: EURO Journal on Transportation and Logistics, ISSN 2192-4376, E-ISSN 2192-4384Article in journal (Refereed)
  • 23.
    Lindberg, Per Olov
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis. KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Mitradjieva, Maria
    Linköpings Universitet.
    The Stiff is Moving - Conjugate Direction Frank-Wolfe Methods with Applications to Traffic Assignment2012In: Transportation Science, ISSN 0041-1655, E-ISSN 1526-5447, Vol. 47, no 2, p. 280-293Article in journal (Refereed)
    Abstract [en]

    We present versions of the Frank-Wolfe method for linearly constrained convex programs, in which consecutive search directions are made conjugate. Preliminary computational studies in a MATLAB environment applying pure Frank-Wolfe, Conjugate direction Frank-Wolfe (CFW), Bi-conjugate Frank-Wolfe (BFW) and ”PARTANized” Frank-Wolfe methods to some classical Traffic Assignment Problems show that CFW and BFW compare favorably to the other methods. This spurred a more detailed study, comparing our methods to Bar-Gera’s origin-based algorithm. This study indicates that our methods are competitive for accuracy requirements suggested by Boyce et al. We further show that CFW is globally convergent. We moreover point at independent studies by other researchers that show that our methods compare favourably with recent bush-based and gradient projection algorithms on computers with several cores.

  • 24.
    Lindberg, Per Olov
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science.
    Smith, Tony E.
    A note on a recent paper by Dagsvik on IIA and random utilities2017In: Theory and Decision, ISSN 0040-5833, E-ISSN 1573-7187, Vol. 82, no 2, p. 305-307Article in journal (Refereed)
    Abstract [en]

    In a recent paper in this journal, Dagsvik derives the class of independent random utility representations that are "equivalent" to the independence-from-irrelevant-alternatives (IIA) assumption by Luce (Individual choice behavior: a theoretical analysis. Wiley, New York, 1959). In this short note, we clarify the relations between this paper by Dagsvik, and a paper in Lindberg's 2012 thesis.

  • 25.
    Mattsson, Lars-Göran
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Weibull, Jörgen
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Extreme values, invariance and choice probabilities2011Conference paper (Refereed)
    Abstract [en]

    Consider a finite set of alternatives, and an associatedcollection of random variables representing some given relevant property of eachalternative, such as its utility, cost or reliability. Suppose that there is a processthat selects exactly one of these alternatives, an alternative with an extreme(maximal or minimal) value. In economics, discrete-choice analysis based onrandom utility theory is the most well-known modelling approach to such aprocess. Among practitioners, it is widely believed that in order to analyze suchsituations, one needs to resort to particular parametric forms of the underlyingprobability distributions of the random variables. We show that this is notso. Indeed, parametric forms impose unnecessary theoretical and empiricalconstraints and may hide more general properties. For the special case ofstatistical independence, we further provide a characterization of the invarianceproperty that all random variables, representing the relevant property of thealternatives, have the same distribution, conditional upon being chosen.

  • 26.
    Mattsson, Lars-Göran
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Weibull, Jörgen
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Extreme values, invariance and choice probabilities2011Conference paper (Refereed)
    Abstract [en]

    Consider a finite set of alternatives, and an associatedcollection of random variables representing some given relevant property of eachalternative, such as its utility, cost or reliability. Suppose that there is a processthat selects exactly one of these alternatives, an alternative with an extreme(maximal or minimal) value. In economics, discrete-choice analysis based onrandom utility theory is the most well-known modelling approach to such aprocess. Among practitioners, it is widely believed that in order to analyze suchsituations, one needs to resort to particular parametric forms of the underlyingprobability distributions of the random variables. We show that this is notso. Indeed, parametric forms impose unnecessary theoretical and empiricalconstraints and may hide more general properties. For the special case ofstatistical independence, we further provide a characterization of the invarianceproperty that all random variables, representing the relevant property of thealternatives, have the same distribution, conditional upon being chosen.

  • 27.
    Mattsson, Lars-Göran
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Weibull, Jörgen
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Extreme values, invariance and choice probabilities2011Conference paper (Refereed)
    Abstract [en]

    Consider a finite set of alternatives, and an associatedcollection of random variables representing some given relevant property of eachalternative, such as its utility, cost or reliability. Suppose that there is a processthat selects exactly one of these alternatives, an alternative with an extreme(maximal or minimal) value. In economics, discrete-choice analysis based onrandom utility theory is the most well-known modelling approach to such aprocess. Among practitioners, it is widely believed that in order to analyze suchsituations, one needs to resort to particular parametric forms of the underlyingprobability distributions of the random variables. We show that this is notso. Indeed, parametric forms impose unnecessary theoretical and empiricalconstraints and may hide more general properties. For the special case ofstatistical independence, we further provide a characterization of the invarianceproperty that all random variables, representing the relevant property of thealternatives, have the same distribution, conditional upon being chosen.

  • 28.
    Mattsson, Lars-Göran
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS. KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Weibull, Jörgen
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis.
    Extreme values, invariance and choice probabilities2011Report (Other academic)
    Abstract [en]

    Since the pioneering work of McFadden (1974), discretechoice random-utility models have become work horses in many areas in economics.In these models, the random variables enter additively or multiplicatively, and the noise distributions take a particular parametric form. We show that the same qualitative results, with closed-form choice probabilities, can be obtained for a wide class of distributions without such specifications. This class generalizes the statistically independent distributions where any two c.d.f.:s are powers of each others to a class that allows for statistical dependence, in a way analogous to how GEV distributions generalize Gumbel distributions. We show that this generalization is sufficient, and under statistical independence also necessary, for the following invariance property: all conditional random variables, when conditioning upon a certain alternative having been chosen, are identically distributed. In our general framework, proofs become simpler, more direct and transparent, well-known results are obtained as special cases, and one can characterize the Gumbel, Fréchet and Weibull distributions.

  • 29.
    Mattsson, Lars-Göran
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Traffic and Logistics.
    Weibull, Jörgen
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lindberg, Per Olov
    KTH, School of Architecture and the Built Environment (ABE), Transport Science, Transport and Location Analysis. KTH, School of Architecture and the Built Environment (ABE), Centres, Centre for Transport Studies, CTS.
    Extreme values, invariance and choice probabilities2014In: Transportation Research Part B: Methodological, ISSN 0191-2615, E-ISSN 1879-2367, Vol. 59, p. 81-95Article in journal (Refereed)
    Abstract [en]

    Since the pioneering work of McFadden (1974), discrete choice random-utility models have become work horses in many areas in transportation analysis and economics. In these models, the random variables enter additively or multiplicatively and the noise distributions take a particular parametric form. We show that the same qualitative results, with closed-form choice probabilities, can be obtained for a wide class of distributions without such specifications. This class generalizes the statistically independent distributions where any two c.d.f.:s are powers of each others to a class that allows for statistical dependence, in a way analogous to how the independent distributions in the MNL models were generalized into the subclass of MEV distributions that generates the GEV choice models. We show that this generalization is sufficient, and under statistical independence also necessary, for the following invariance property: all conditional random variables, when conditioning upon a certain alternative having been chosen, are identically distributed. While some of these results have been published earlier, we place them in a general unified framework that allows us to extend several of the results and to provide proofs that are simpler, more direct and transparent. Well-known results are obtained as special cases, and we characterize the Gumbel, Frechet and Weibull distributions.

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