Partial Order Research Articles

Minimal solutions of master equations for extended mean field games

In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is well known that, without any monotonicity conditions, mean field games typically contain multiple mean field equilibria and the wellposedness of their corresponding master equations fails. In this paper, a partial order for the set of probability measure flows is proposed to compare different mean field equilibria. The minimal and maximal mean field equilibria under this partial order are constructed and satisfy the flow property. The corresponding value functions, however, are in general discontinuous. We thus introduce a notion of weak-viscosity solutions for the master equation and verify that the value functions are indeed weak-viscosity solutions. Moreover, a comparison principle for weak-viscosity semi-solutions is established and thus these two value functions serve as the minimal and maximal weak-viscosity solutions in appropriate sense. In particular, when these two value functions coincide, the value function becomes the unique weak-viscosity solution to the master equation. The novelties of the work persist even when restricted to the standard mean field games.

The Comparative Statics of Sorting

We create a general and tractable theory of increasing sorting in pairwise matching models with monetary transfers. The positive quadrant dependence partial order subsumes Becker (1973) as the extreme cases with most and least sorting and implies increasing regression coefficients. Our theory turns on synergy—the cross-partial difference or derivative of match production. This reflects basic economic forces: diminishing returns, technological convexity, insurance, and learning dynamics. We prove sorting increases if match synergy globally increases, and is cross-sectionally monotone or single crossing. We use our results to derive sorting predictions in major economics sorting papers and in new applications. (JEL C78, D21, D82, D86, J12)

Bayesian Nonparametric Bivariate Survival Regression for Current Status Data

We consider Bayesian nonparametric inference for event time distributions based on current status data. We show that under dependent censoring conventional mixture priors, including the popular Dirichlet process mixture prior, lead to biologically uninterpretable results as they unnaturally skew the probability mass for the event times toward the extremes of the observed data. Simple assumptions on dependent censoring can fix the problem. We then extend the discussion to bivariate current status data with partial ordering of the two outcomes. In addition to dependent censoring, we also exploit some minimal known structure relating the two event times. We design a Markov chain Monte Carlo algorithm for posterior simulation. Applied to a recurrent infection study, the method provides novel insights into how symptoms-related hospital visits are affected by covariates.

The Colley Method is an Extension of the Borda Count

Summary The Colley and Massey methods are two well-known methods to rate sports teams. We prove that when the Colley method is applied to election data, then it returns the same outcome as the Borda count, a well-known election procedure. Hence, the Colley method is an extension of the Borda count to partial orderings of candidates. A similar relationship holds between the Massey method and range voting. Real data from Major League Baseball is used.

On Protected Quasi-Metrics

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations.

A criterion for the properness of star-transposition as an involution on n × n matrices over any associative star-ring

On the ring M n ( C ) of all n × n complex matrices A, the conjugate transpose involution A ↦ A * has the “properness” property A * A = 0 ⇒ A = 0 , as required for the existence of the partial order known as the “ * -order” on M n ( C ) . More generally, related questions are asked and answered about the ring M n ( R ) of all n × n matrices over an arbitrary associative ring R with any given involution R → R (as a generalization of the conjugacy map C → C ), yielding a corresponding “ * -transposition” involution on M n ( R ) . A criterion is found for this involution of M n ( R ) to be proper, so that M n ( R ) has a corresponding * -order. The special case R = Z k is also considered.

Characterization of a Cournot–Nash Equilibrium for a Fishery Model with Fuzzy Utilities

The article deals with the extensions of discrete-time games with infinite time horizon and their application in a fuzzy context to fishery models. The criteria for these games are the total discounted utility and the average utility in a fishing problem. However, in the fuzzy case, game theory is not the best way to represent a real fishing problem because players do not always have enough information to accurately estimate their utility in the context of fishing. For this reason, in this paper, trapezoidal-type fuzzy utility values are considered for a fishing model, and the terms of the Nash equilibrium are given in the fuzzy context, i.e., this equilibrium is represented using the partial order of the α-cuts of the fuzzy numbers; to the best of the authors’ knowledge, there is no work with this type of treatment. To obtain each equilibrium, a suitable fully determined fuzzy game is used in combination with the dynamic programming technique applied to this game in the context of fishing. The main results are (i) the Nash equilibria of the fuzzy games coincide with the Nash equilibria of the nonfuzzy games and are explicitly determined in a fishery model and (ii) the values of the fuzzy games are of trapezoidal type and are also explicitly given in the fishery model.

Model selection over partially ordered sets

In problems such as variable selection and graph estimation, models are characterized by Boolean logical structure such as the presence or absence of a variable or an edge. Consequently, false-positive error or false-negative error can be specified as the number of variables/edges that are incorrectly included or excluded in an estimated model. However, there are several other problems such as ranking, clustering, and causal inference in which the associated model classes do not admit transparent notions of false-positive and false-negative errors due to the lack of an underlying Boolean logical structure. In this paper, we present a generic approach to endow a collection of models with partial order structure, which leads to a hierarchical organization of model classes as well as natural analogs of false-positive and false-negative errors. We describe model selection procedures that provide false-positive error control in our general setting, and we illustrate their utility with numerical experiments.

Description of Paralocks language semantics in TLA+

One of the basic aspects of information flow control in applications is security policy language. Such language should allow to define security policies for evaluation environment elements in coherence with higher level access control rules. So the language is expected to be flexible because there may be different access control paradigms implemented on the system level: mandatory, role-based etc. An application may also have its own specific restrictions. Finally it is also desirable that the language support declassification (controlled release of information) during the computation. One of such languages is Paralocks. The research is devoted to the logical semantics of modified version of Paralocks realized in TLA+. Paralocks represents a language basis for the perspective information flow control platform PLIF which is being developed for the analysis of PL/SQL program blocks with author’s participation. It includes the proofs of the partial order and lattice defined on the set of security policy expressions.

Directed partial orders on complex numbers and quaternions

Let [Formula: see text] be an integral domain that is algebraic over [Formula: see text]. It is shown that each directed maximal partial order on [Formula: see text] is an Archimedean total order. Let [Formula: see text] be a subfield of [Formula: see text] and [Formula: see text] be the complex field over [Formula: see text]. As a consequence of the above result, if [Formula: see text] is algebraic over [Formula: see text], then [Formula: see text] does not have a directed partial order making it a partially ordered ring. In particular, [Formula: see text] cannot be a lattice-ordered ring. The result is proved for certain partially ordered algebras of quaternions as well.

Inferring from an imprecise Plackett–Luce model: Application to label ranking

The Plackett–Luce model is a popular parametric probabilistic model to define distributions between rankings of objects, modelling for instance observed preferences of users or ranked performances of algorithms. Since such observations may be scarce (users may provide partial preferences, or not all algorithms are run for a given experiment), it may be useful to consider the case where the parameters of the Plackett–Luce model are imprecisely known. In this paper, we first introduce the imprecise Plackett–Luce model, induced by a set of parameters (for instance, parameters with a high relative likelihood). Given a set of possible parameters for the model, we then provide an efficient algorithm to make cautious inferences, returning sets of possible optimal rankings (for instance in the form of partial orders). We illustrate the use of our imprecise model on label ranking, a specific kind of supervised learning.

Implication as Inclusion and the Causal Asymmetry

Abstract How does causation in the physical world relate to implication in logic? This article presents implication as fundamentally a relation of inclusion between propositions. Given this, it is argued that an event cannot “causally imply” another, also given the laws of nature. Then, by applying the notion of inclusion to physical objects, a relation “within the possibilities of” is developed, which generates a partial order on sets of entities and is independent of time. Based on this, it is shown that changes of physical objects in time (at any rate, a great many of them) imply, and thus counterfactually depend on, what we call “causes” – an asymmetric dependence which is robust despite the perspectival nature of the concept of “cause”.

Pseudospheres: combinatorics, topology and distributed systems

Pseudospheres are simplicial complexes defined in the late 1990s to model some aspects of distributed systems. Since then, combinatorial properties of pseudospheres combined with topological properties have been very useful to derive distributed computability results. The goal of this paper is to study pseudospheres in more depth as mathematical objects and to give an overview of the properties that have been used in distributed computing. In this work we focus in combinatorial and topological aspects of pseudospheres. While doing so, the paper shows that these structures can be viewed from different perspectives, in addition to models of distributed computing. We show that the properties of pseudospheres that have been proved in distributed computing, as well as new ones, can be derived using combinatorial topology techniques and other combinatorial techniques taken from matroids and partial orders. A subclass of pseudospheres is related to universal bundles, and the Borsuk–Ulam theorem can be extended to apply to them.

Hamiltonian Intervals in the Lattice of Binary Paths

Let $\mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P \le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P \in \mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $\mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $\mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.

Coherence of inequality measures with respect to partial orderings of income distributions

This paper investigates the coherence of a new class of ratio-based inequality indices introduced in Chan (2022) with respect to certain partial orderings of the underlying income distributions. While coherence with respect to stochastic dominance has been extensively studied, the appropriate partial ordering for this new class of indices is quantile ratio dominance. This paper also establishes coherence with respect to quantile ratio dominance for some other classes of inequality indices which have been introduced by other authors. Finally, some connections between the notion of coherence and transfer principles are explored.

Teaching Partial Order Relations: A Programming Approach

Teaching Partial Order Relations: A Programming Approach

Multipermutations and Stirling Multipermutations

We consider multipermutations and a certain partial order, the weak Bruhat order, on this set. This generalizes the Bruhat order for permutations, and is defined in terms of containment of inversions. Different characterizations of this order are given. We also study special multipermutations called Stirling multipermutations and their properties.

A Pilot Study of the Combination of Entinostat with Capecitabine in Advanced Breast Cancer.

Breast cancer has an unacceptably high recurrence rate when any residual disease is found following neoadjuvant treatment of high-risk disease. Based on clinical data suggesting an adjuvant role for epigenetic modifying agents in breast cancer and preclinical data suggesting synergistic activity of entinostat combined with capecitabine, we conducted a phase I, open-label study of these agents in metastatic breast cancer (MBC). Both agents have published doses for use in combination therapy, but the agents had not previously been combined with each other in a human trial. A multisite phase I dose escalation study was performed at two academic centers. Patients with pretreated, HER2-negative MBC, and measurable disease were enrolled. Dual dose escalation was performed via a Bayesian partial order continual assessment method. Dose levels ranged from entinostat 3 mg to 5 mg and capecitabine 800 mg/m2 to 1000 mg/m2. Thirteen patients with MBC and a median of 4 lines of prior therapy were enrolled across four dose level combinations. The most common toxicities were neutropenia, thrombocytopenia, and palmar-plantar dysesthesia, which were expected toxicities. No new safety signals were observed. One dose-limiting toxicity was observed, which did not exceed a prespecified toxicity rate of 25%. The median treatment duration was 2.37 months. No partial nor complete responses were observed. The study was halted early prior to entering an expansion phase, due to drug supply limitations. The tested dosing combinations of entinostat and capecitabine are likely safe in heavily pretreated metastatic breast cancer. This study's clinical investigation of entinostat in breast cancer was halted, but drug development of this agent continues outside the US. There remains a need for postoperative adjuvant drug therapy for the subpopulation of breast cancer patients with high-risk residual cancer after curative therapy. This trial is registered with NCT03473639.

C-S and Strongly C-S Orthogonal Matrices

In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and BAS=0, where AS is the generalized core inverse of A. The characterizations of C-S orthogonal matrices and the C-S additivity are also provided. And, the connection between the C-S orthogonality and C-S partial order has been given using their canonical form. Moreover, the concept of the strongly C-S orthogonality is defined and characterized.

Time-Varying Resource Graph Based Processing on the Way for Space-Terrestrial Integrated Vehicle Networks

Desirable information processing in space-terrestrial integrated vehicle networks (STINs) handles data distributed in different satellites while transmitting, where efficient modeling time-varying resources is critical. Existing works are not applicable to STINs, however, because they lack the joint consideration of different movement patterns and fluctuating loads. In this paper, we propose the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Time-Varying Resource Graph (TVRG)</i> to model dynamic resources in STINs, by leveraging the advantages of software-defined networking in flexible resource management. Firstly, we propose the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">STIN mobility model</i> to uniformly model different movement patterns in STINs. Then, we propose a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">layered Resource Modeling and Abstraction (RMA)</i> approach, where evolutions of node resources are modeled as Markov processes, by encoding predictable topologies and influences of fluctuating loads as states. Besides, we propose the low-complexity domain resource abstraction algorithm by defining two mobility-based and load-aware partial orders on resource abilities. Finally, we formulate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">TVRG-based Processing on the Way (TPoW)</i> problem for data flows with processing requirements and multiple sources. We propose a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Multi-level Processing on the Way (MPoW)</i> approach with a bounded approximation ratio, realizing adaptive matching of resources and demands of processing and transmission. To evaluate the RMA approach, we propose a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">TVRG-based Routing (TR)</i> algorithm for time-sensitive and bandwidth-intensive data flows, with the multi-level on-demand scheduling ability. Comprehensive simulation results demonstrate that our RMA-TR and MPoW outperform most related schemes by decreasing nearly 40% bandwidth consumption with the shortest end-to-end delay.