Cyclic Monotonicity Research Articles

Convex integrals of molecules in Lipschitz-free spaces

We introduce convex integrals of molecules in Lipschitz-free spaces F(M) as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of F(M) that are induced by Radon measures on M, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of F(M). In particular, we show that all elements of F(M) are convex series of molecules when M is uniformly discrete and identify all extreme points of the unit ball of F(M) in that case.

Sufficiency of -cyclical monotonicity in a class of multi-marginal optimal transport problems

Abstract $c$ -cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

Optimal Transport Approaches for Cloud Fusion in 3D Reconstruction

Cloud fusion plays a crucial role in achieving accurate 3D models. This paper introduces two optimal transport-based approaches, Gaussian Fusion and Cyclical Fusion, to showcase the effectiveness of optimal transport in establishing dense correspondence among incoming 3D point clouds. These approaches encompass several important components. Firstly, the fusion correspondences are derived from the optimal transport problem. The core principle of Gaussian Fusion and Cyclical Fusion involves manipulating points along geodesic curves to effectively leverage significant local geometric information. Secondly, this paper explores the fundamental concepts underlying Gaussian Fusion (displacement interpolation) and Cyclical Fusion (cyclical monotonicity), which greatly enhance the accuracy and completeness of the reconstruction process. And provides evidence for the uniqueness of displacement interpolation as geodesics on L2-Wasserstein space and the rationality of cyclical monotonicity in this context. Finally, the proposed approaches excel in capturing intricate surface details, particularly on small objects, when compared to the original fusion scheme which often introduces severe artifacts.

Detecting profitable deviations

Rochet’s Theorem characterizes implementable allocations in a mechanism design environment in terms of cyclic monotonicity, a concept from convex analysis. In this paper, I offer an alternative approach to both the proof and interpretation of this result, based on linear duality. This duality reveals a formal relationship between incentives and detection, much like that between prices and quantities. Moreover, it allows me to extend Rochet’s Theorem and present a subdifferential characterization of implementing payments, revenue equivalence as differentiability of a value function, as well as budget-balanced implementation in terms of attributing innocence after unilateral deviations, together with other ancillary results.

Matrix models for cyclic monotone and monotone independences

Matrix models for cyclic monotone and monotone independences

On Douglas-Rachford splitting that generally fails to be a proximal mapping: a degenerate proximal point analysis

Based on a degenerate proximal point analysis, we show that the Douglas-Rachford splitting can be reduced to a well-defined resolvent, but generally fails to be a proximal mapping. This extends the recent result of [Bauschke, Schaad and Wang. Math. Program. 2018;168:55–61] to more general setting. The related concepts and consequences are also discussed. In particular, the results regarding the maximal and cyclic monotonicity are instrumental for analysing many operator splitting algorithms.

Fenchel subdifferential operators: A characterization without cyclic monotonicity

Fenchel subdifferential operators: A characterization without cyclic monotonicity

A non‐linear monotonicity principle and applications to Schrödinger‐type problems

A basic idea in optimal transport is that optimizers can be characterized through a geometric property of their support sets called cyclical monotonicity. In recent years, similar monotonicity principles have found applications in other fields where infinite-dimensional linear optimization problems play an important role. In this note, we observe how this approach can be transferred to non-linear optimization problems. Specifically we establish a monotonicity principle is applicable to the Schrödinger problem and use it to characterize the structure of optimizers for target functionals beyond relative entropy. In contrast to classical convex duality approaches, a main novelty is that the monotonicity principle allows to deal also with non-convex functionals.

A condition for the identification of multivariate models with binary instruments

This article introduces an empirical condition for the nonparametric point-identification of multivariate instrumental variable models with continuous endogenous variables using binary instruments. Verifying this condition can confirm point-identification in settings in which traditional approaches are not applicable. In particular, it shows that nonlinear instrumental variable models with general heterogeneity can be point-identified with only a binary instrument. This generalizes existing identification results which either restrict the unobserved heterogeneity substantially or require the instrument to have a large support. The main assumption on the instrumental variable model is cyclic monotonicity of its first stage, a multivariate generalization of the classical rank-invariance assumption for univariate models. Asymptotic convergence results for the empirical observable distributions are derived that allow to check the condition in practice. The identification rests on a fixed-set convergence result of cyclically monotone maps between quasi-concave functions.

Maxmin implementation

This paper studies the implementation problem of a mechanism designer with ambiguity averse agents. The mechanism designer, desiring to implement a choice correspondence, can create ambiguity for agents by committing to multiple allocation rules and transfer schemes without revealing which one to use. By extending the cyclical monotonicity condition from choice functions to choice correspondences, we show that the condition can fully characterize implementable choice correspondences. We then study the implementability of choice correspondences in supermodular environments. As an application, we consider a mechanism designer who wants to allocate one object to one of her most desired agents and show that she can strictly benefit from concealing the tie-breaking rules. An intuitive and computationally tractable condition is provided to characterize when the mechanism designer's preference induces an implementable choice correspondence.

Semi-parametric estimation of multinomial choice model with unobserved heterogeneity

We propose a semi-parametric method for estimating the multinomial choice model using cyclical monotonicity moment inequalities. In our semiparametric approach, we do not specify the parametric distribution for the utility shocks, thus accommodating a wide variety of substitution patterns among alternatives. Compared with the conditional likelihood estimation method proposed by Chamberlain and the method proposed by Shi, Shum, and Song, simulations show that our method has some advantages in terms of mean bias, standard deviation and root mean squared error.

Polynomial with cyclic monotone elements with applications to Random Matrices with discrete spectrum

We provide a generalization and new proofs of the formulas of Collins et al. for the spectrum of polynomials in cyclic monotone elements. This is applied to Random Matrices with discrete spectrum.

A computationally fast estimator for semiparametric multinomial choice model

In this article, based on cyclical monotonicity moment inequalities implied by the random utility choice model, we propose a computationally fast estimator for semiparametric multinomial choice models. The term semiparametric refers to the fact that we do not specify a particular functional form for the error term in the random utility function. The proposed estimators are consistent. Comparing with the estimators developed by the cyclic monotonicity method, simulations show the estimators we proposed have great advantages on the running time.

Existence, duality, and cyclical monotonicity for weak transport costs

The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier–Strassen Theorem of Gozlan and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681, 2018) to general probability measures on mathbb {R}^d under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (‘adapted’) topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

TESTING THE QUANTAL RESPONSE HYPOTHESIS

AbstractWe develop a nonparametric test for consistency of player behavior with the quantal response equilibrium (QRE). The test exploits a characterization of the equilibrium choice probabilities in any structural QRE as the gradient of a convex function; thereby, QRE‐consistent choices satisfy the cyclic monotonicity inequalities. Our testing procedure utilizes recent econometric results for moment inequality models. We assess our test using lab experimental data from a series of generalized matching pennies games. We reject the QRE hypothesis in the pooled data but cannot reject individual‐level quantal response behavior for over half of the subjects.

$C$-cyclical monotonicity as a sufficient criterion for optimality in the multimarginal Monge–Kantorovich problem

$C$-cyclical monotonicity as a sufficient criterion for optimality in the multimarginal Monge–Kantorovich problem

Free probability for purely discrete eigenvalues of random matrices

In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.

Estimating Semi-Parametric Panel Multinomial Choice Models Using Cyclic Monotonicity

This paper proposes a new semi-parametric identification and estimation approach to multinomial choice models in a panel data setting with individual fixed effects. Our approach is based on cyclic monotonicity, which is a defining feature of the random utility framework underlying multinomial choice models. From the cyclic monotonicity property, we derive identifying inequalities without requiring any shape restrictions for the distribution of the random utility shocks. These inequalities point identify model parameters under straightforward assumptions on the covariates. We propose a consistent estimator based on these inequalities.

Complete duality for martingale optimal transport on the line

We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.

Counterfactual Estimation in Semiparametric Multinomial Choice Models

We show how to construct bounds on counterfactual choice probabilities in semiparametric discrete-choice models. Our procedure is based on cyclic monotonicity, a convex-analytic property of the random utility discrete-choice model. These bounds are useful for typical counterfactual exercises in aggregate discrete-choice demand models. In our semiparametric approach, we do not specify the parametric distribution for the utility shocks, thus accommodating a wide variety of substitution patterns among alternatives. Computation of the counterfactual bounds is a tractable linear programming problem. We illustrate our approach in a series of Monte Carlo simulations and an empirical application using scanner data.