Constructive Proof Of Existence Research Articles

Equilibrium arrivals to a last-come first-served preemptive-resume queue

We consider a queueing system which opens at a given point in time and serves a finite number of users according to the last-come first-served discipline with preemptive-resume (LCFS-PR). Each user must decide individually when to join the queue. We allow for general classes of user preferences and service time distributions and show existence and uniqueness of a symmetric Nash equilibrium. Furthermore, we show that no continuous asymmetric equilibrium exists, if the population consists of only two users, or if arrival strategies satisfy a mild regularity condition. For an illustrative example, we implement a numerical procedure for computing the symmetric equilibrium strategy based on our constructive existence proof for the symmetric equilibrium. We then compare its social efficiency to that obtained if users are instead served on a first-come first-served (FCFS) basis.

A Complementary Pivot Algorithm for Competitive Allocation of a Mixed Manna

We study the fair division problem of allocating a mixed manna under additively separable piecewise linear concave (SPLC) utilities. A mixed manna contains goods that everyone likes and bads (chores) that everyone dislikes as well as items that some like and others dislike. The seminal work of Bogomolnaia et al. argues why allocating a mixed manna is genuinely more complicated than a good or a bad manna and why competitive equilibrium is the best mechanism. It also provides the existence of equilibrium and establishes its distinctive properties (e.g., nonconvex and disconnected set of equilibria even under linear utilities) but leaves the problem of computing an equilibrium open. Our main results are a linear complementarity problem formulation that captures all competitive equilibria of a mixed manna under SPLC utilities (a strict generalization of linear) and a complementary pivot algorithm based on Lemke’s scheme for finding one. Experimental results on randomly generated instances suggest that our algorithm is fast in practice. Given the [Formula: see text]-hardness of the problem, designing such an algorithm is the only non–brute force (nonenumerative) option known; for example, the classic Lemke–Howson algorithm for computing a Nash equilibrium in a two-player game is still one of the most widely used algorithms in practice. Our algorithm also yields several new structural properties as simple corollaries. We obtain a (constructive) proof of existence for a far more general setting, membership of the problem in [Formula: see text], a rational-valued solution, and an odd number of solutions property. The last property also settles the conjecture of Bogomolnaia et al. in the affirmative. Furthermore, we show that, if the number of either agents or items is a constant, then the number of pivots in our algorithm is strongly polynomial when the mixed manna contains all bads. Funding: Financial support from the Division of Computing and Communication Foundations, National Science Foundation (NSF) [Grants 1942321, 1750436] is gratefully acknowledged.

Computation of Dynamic Equilibria in Series-Parallel Networks

We consider dynamic equilibria for flows over time under the fluid queuing model. In this model, queues on the links of a network take care of flow propagation. Flow enters the network at a single source and leaves at a single sink. In a dynamic equilibrium, every infinitesimally small flow particle reaches the sink as early as possible given the pattern of the rest of the flow. Although this model has been examined for many decades, progress has been relatively recent. In particular, the derivatives of dynamic equilibria have been characterized as thin flows with resetting, which allows for more structural results. Our two main results are based on the formulation of thin flows with resetting as a linear complementarity problem and its analysis. We present a constructive proof of existence for dynamic equilibria if the inflow rate is right-monotone. The complexity of computing thin flows with resetting, which occurs as a subproblem in this method, is still open. We settle it for the class of two-terminal, series-parallel networks by giving a recursive algorithm that solves the problem for all flow values simultaneously in polynomial time.

Improper priors and improper posteriors

AbstractWhat is a good prior? Actual prior knowledge should be used, but for complex models this is often not easily available. The knowledge can be in the form of symmetry assumptions, and then the choice will typically be an improper prior. Also more generally, it is quite common to choose improper priors. Motivated by this we consider a theoretical framework for statistics that includes both improper priors and improper posteriors. Knowledge is then represented by a possibly unbounded measure with interpretation as explained by Rényi in 1955. The main mathematical result here is a constructive proof of existence of a transformation from prior to posterior knowledge. The posterior always exists and is uniquely defined by the prior, the observed data, and the statistical model. The transformation is, as it should be, an extension of conventional Bayesian inference as defined by the axioms of Kolmogorov. It is an extension since the novel construction is valid also when replacing the axioms of Kolmogorov by the axioms of Rényi for a conditional probability space. A concrete case based on Markov Chain Monte Carlo simulations and data for different species of tropical butterflies illustrate that an improper posterior may appear naturally and is useful. The theory is also exemplified by more elementary examples.

Spontaneous Periodic Orbits in the Navier–Stokes Flow

In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton–Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic estimates to verify the contractibility of the operator are presented in full generality and symmetries from the model are used to reduce the size of the problem to be solved. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier–Stokes equations with Taylor–Green forcing.

Nonlinearly bandlimited signals

In this paper, we study the inverse scattering problem for a class of signals that have a compactly supported reflection coefficient. The problem boils down to the solution of the Gelfand–Levitan–Marchenko (GLM) integral equations with a kernel that is bandlimited. By adopting a sampling theory approach to the associated Hankel operators in the Bernstein spaces, a constructive proof of existence of a solution of the GLM equations is obtained under various restrictions on the nonlinear impulse response (NIR). The formalism developed in this article also lends itself well to numerical computations yielding algorithms that are shown to have algebraic rates of convergence. In particular, the use Whittaker–Kotelnikov–Shannon sampling series yields an algorithm that converges as whereas the use of Helms and Thomas (HT) version of the sampling expansion yields an algorithm that converges as for any m > 0 provided the regularity conditions are fulfilled. The complexity of the algorithms depend on the linear solver used. The use of conjugate-gradient (CG) method yields an algorithm of complexity per sample of the signal where N is the number of sampling basis functions used and is the number of CG iterations involved. The HT version of the sampling expansions facilitates the development of algorithms of complexity (per sample of the signal) by exploiting the special structure as well as the (approximate) sparsity of the matrices involved. The algorithms are numerically validated using Schwartz class functions as NIRs that are either bandlimited or effectively bandlimited. The results suggest that the HT variant of our algorithm is spectrally convergent for an input of the aforementioned class.

From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory

In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zero-sum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of games.

Smooth and non-smooth regularizations of the nonlinear diffusion equation

We illustrate an alternative derivation of the viscous regularization of a nonlinear forward-backward diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. Trans. Amer. Math. Soc. , 324:331-351]. We propose and discuss a new ''non-smooth'' variant of the viscous regularization and we offer an heuristic argument that indicates that this variant should display interesting hysteretic effects. Finally, we offer a constructive proof of existence of solutions for the viscous regularization based on a suitable approximation scheme.

Competitive Equilibria in Two-Sided Matching Markets with General Utility Functions

We present an exact characterization of utilities in competitive equilibria of two-sided matching markets in which the utility of each agent depends on the choice of partner and the terms of the partnership, potentially including monetary transfer. Examples of such markets include sellers and buyers or jobs and workers. Demange and Gale showed that the set of competitive equilibria in this type of market forms a complete lattice with each extreme point of the lattice representing an equilibrium with the highest utilities for the agents on one side and the lowest utilities for the agents on the opposite side. Our characterization is based on establishing a connection between the competitive equilibria of a market and the competitive equilibria of certain strict subsets of that market—each obtained by removing exactly one agent. This characterization captures the effect of competition when agents are added to the market or removed from the market. It gives a precise procedure for constructing competitive equilibria and provides a constructive proof of existence of such equilibria; in contrast, previous proofs have been based on fixed point theorems.

A Rejoinder to Notes on a ‘Constructive Proof of the Existence of a Collateral Equilibrium’

This rejoinder contains a few remarks on my paper published in this journal with the title `A Constructive Proof of the Existence of Collateral Equilibrium for a Two-Period Exchange Economy Based on a Smooth Interior-Point Path' and Ragupathy and Velupillai's notes on a `Constructive Proof of the Existence of Collateral Equilibrium' published recently in the same journal.

Notes on a ‘Constructive Proof of the Existence of a Collateral Equilibrium’

In an interesting article in this journal, Ma (Comput Econ 45:1---30, 2015), claims that a constructive proof of existence of a Collateral Equilibrium (with a Leontief utility function) is provided in the paper. Moreover, there is also the additional claim that `on the basis of this proof, we can (and we shall) develop an algorithm for computing that equilibrium' (ibid, p. 1). Thirdly, the statements that `the algorithm is shown by simulation to be effective', and its effectiveness is demonstrated by `the accuracy of the equilibrium it provided'. These claims are, from a recursion theoretic (or computability theory) point of view, meaningless.

On Existence and Uniqueness of Equilibrium in a Class of Noisy Rational Expectations Models: Figure 1

I study a general class of noisy rational expectations models that nests the standard Grossman and Stiglitz (1980) and Hellwig (1980) models, but relaxes the usual assumption of joint normality of asset pay-offs and supply, and allows for more general signal structures. I provide a constructive proof of existence of equilibrium, characterize the price function, and provide sufficient conditions for uniqueness within the class of equilibria with continuous price functions, which are met by both the Grossman and Stiglitz (1980) models and the Hellwig (1980) models with a continuum of investors. My solution approach does not rely on the typical “conjecture and verify” method, and I exhibit a number of non-normal examples in which asset prices can be characterized explicitly and in a closed form. The results presented here open up a broad class of models for applied work. To illustrate the usefulness of generalizing the standard model, I show that in settings with non-normally distributed pay-offs, shocks to fundamentals may be amplified purely due to learning effects, price drifts can arise naturally, and the disagreement–return relation depends in a novel way on return skewness.

Dynamic Equilibria in Fluid Queueing Networks

Flows over time provide a natural and convenient description for the dynamics of a continuous stream of particles traveling from a source to a sink in a network, allowing to track the progress of each infinitesimal particle along time. A basic model for the propagation of flow is the so-called fluid queue model in which the time to traverse an edge is composed of a flow-dependent waiting time in a queue at the entrance of the edge plus a constant travel time after leaving the queue. In a dynamic network routing game each infinitesimal particle is interpreted as a player that seeks to complete its journey in the least possible time. Players are forward looking and anticipate the congestion and queuing delays induced by others upon arrival to any edge in the network. Equilibrium occurs when each particle travels along a shortest path. This paper is concerned with the study of equilibria in the fluid queue model and provides a constructive proof of existence and uniqueness of equilibria in single origin-destination networks with piecewise constant inflow rate. This is done through a detailed analysis of the underlying static flows obtained as derivatives of a dynamic equilibrium. Furthermore, for multicommodity networks, we give a general nonconstructive proof of existence of equilibria when the inflow rates belong to Lp.

Symplectic packings in dimension $4$ and singular curves

The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves, which can be derived from McDuff’s inflations on the blow-ups, is revisited through a new inflation technique that lives at the level of the manifold. As an application, we explain constructions of maximal symplectic packings of P by 6, 7 or 8 balls.

Tight lower bound for percolation threshold on an infinite graph.

We construct a tight lower bound for the site percolation threshold on an infinite graph, which becomes exact for an infinite tree. The bound is given by the inverse of the maximal eigenvalue of the Hashimoto matrix used to count nonbacktracking walks on the original graph. Our bound always exceeds the inverse spectral radius of the graph's adjacency matrix, and it is also generally tighter than the existing bound in terms of the maximum degree. We give a constructive proof for existence of such an eigenvalue in the case of a connected infinite quasitransitive graph, a graph-theoretic analog of a translationally invariant system.

Super-KMS Functionals for Graded-Local Conformal Nets

Motivated by a few preceding papers and a question of R. Longo, we introduce super-KMS functionals for graded translation-covariant nets over R with superderivations, roughly speaking as a certain supersymmetric modification of classical KMS states on translation-covariant nets over R, fundamental objects in chiral algebraic quantum field theory. Although we are able to make a few statements concerning their general structure, most properties will be studied in the setting of specific graded-local (super-) conformal models. In particular, we provide a constructive existence and partial uniqueness proof of super-KMS functionals for the supersymmetric free field, for certain subnets, and for the super-Virasoro net with central charge c>= 3/2. Moreover, as a separate result, we classify bounded super-KMS functionals for graded-local conformal nets over S^1 with respect to rotations.

Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

We characterize quasi-static rate-independent evolutions, by means of their graph parame- trization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals F of class C 1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals Fn and its Γ-limit F we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

An infeasible homotopy method for solving fixed point problems on a general unbounded set

In this paper, we consider an infeasible homotopy method to solve fixed point problems on a general unbounded set. The method enlarges the choice scope of initial points by a perturbation on equality constraint. Under a new and weaker condition for the defining mapping \(\Phi \), a constructive proof of the existence of fixed points and the global convergence are obtained. The computational result illustrates that our method improves greatly the computational efficiency than the existing interior point homotopy methods.

Uniform Price Auctions: Equilibria and Efficiency

We study the Uniform Price Auction, one of the standard sealed-bid multi-unit auction formats in Auction Theory, for selling multiple identical units of a single good to multi-demand bidders. Contrary to the truthful and efficient multi-unit Vickrey auction, the Uniform Price Auction encourages strategic bidding and is generally inefficient, due to a "Demand Reduction" effect; bidders tend to bid for fewer (identical) units, so as to receive them at a lower uniform price. All the same, the uniform pricing rule is popular by its appeal to the anticipation that identical items should be identically priced. Its applications include, among others, sales of U.S. Treasury notes to investors and trade exchanges over the Internet facilitated by popular online brokers. In this work, we characterize pure undominated bidding strategies and give an algorithm for computing pure Nash equilibria in such strategies. Subsequently we show that their Price of Anarchy is ee?1$\frac {e}{e-1}$. Finally, we show that the Price of Anarchy of mixed Bayes-Nash equilibria with undominated support is at most 4?2k$4-\frac {2}{k}$, where k is the number of auctioned items. To the best of our knowledge, our work provides the first (constructive) proof of existence of pure Nash equilibria in undominated strategies and the first performance evaluation (with respect to economic efficiency) of this popular auction format.

Naive configurations

We describe a new way to construct finite geometric objects. For every $$k$$ we obtain a symmetric configuration $$\mathcal{E }(k-1)$$ with $$k$$ points on a line. In particular, we have a constructive existence proof for such configurations. The method is very simple and purely geometric. It also produces interesting periodic matrices.