Weak Monotonicity Research Articles

Influence in Private-Goods Allocation

We reinterpret the `bossiness' of a private-goods allocation rule (Satterthwaite and Sonneschein, 1981) as the ability of an agent to `influence' another's welfare with no change to her own welfare. We propose simple conditions on (1) which agents may have influence (`acyclicity' and `preservation'), and (2) the welfare consequences of influence (`positivity' and `oppositeness'). We apply these conditions to three well-known bossy rules: the `Vickrey rule' in single-object auctions (Vickrey, 1961) (acyclic, positive), the `doctor-optimal stable rule' in matching with contracts (Hatfield and Milgrom, 2005) (acyclic, positive, preserving) and `generalised absorbing top-trading cycles (GATTC) rules' in housing markets with indifferences in preferences (Aziz and Keijzer, 2011) (acyclic, opposite, preserving). Under mild restrictions, we show how the nature of influence under a strategy-proof rule determines whether or not it satisfies `weak group-strategy-proofness' (requires acyclicity and either positivity or preservation), `weak Maskin monotonicity' (acyclicity and positivity) and `Pareto-efficiency' (acyclicity and oppositeness). In addition, we propose an influence-related generalisation of the`efficiency-adjusted deferred acceptance mechanism' in school choice (Kesten, 2010), and characterise influence for strategy-proof GATTC rules in housing markets.

Interim Rationalizable Implementation of Functions

This paper investigates rationalizable implementation of social choice functions (SCFs) in incomplete information environments. We identify weak interim rationalizable monotonicity (weak IRM) as a novel condition and show that weak IRM is a necessary and almost sufficient condition for rationalizable implementation. We show by means of an example that interim rationalizable monotonicity (IRM), found in the literature, is strictly stronger than weak IRM as its name suggests, and that IRM is not necessary for rationalizable implementation, as had been previously claimed. The same example also demonstrates that Bayesian monotonicity, the key condition for full Bayesian implementation, is not necessary for rationalizable implementation. This implies that rationalizable implementation can be more permissive than Bayesian implementation: one can exploit the fact that there are no mixed Bayesian equilibria in the implementing mechanism.

Join Idle Queue with Service Elasticity: Large-Scale Asymptotics of a Nonmonotone System

We consider the model of a token-based joint autoscaling and load-balancing strategy, proposed in a recent paper by Mukherjee et al. [Mukherjee D, Dhara S, Borst SC, Van Leeuwaarden JSH (2017) Optimal service elasticity in large-scale distributed systems. Proc. ACM Measurement Anal. Comput. Systems 1(1):25:1–25:28.], which offers an efficient scalable implementation and yet achieves asymptotically optimal steady-state delay performance and energy consumption as the number of servers N → ∞. In the aforementioned work, the asymptotic results are obtained under the assumption that the queues have fixed-size finite buffers, and therefore, the fundamental question of stability of the proposed scheme with infinite buffers was left open. In this paper, we address this fundamental stability question. The system stability under the usual subcritical load assumption is not automatic. Moreover, the stability may not even hold for all N. The key challenge stems from the fact that the process lacks monotonicity, which has been the powerful primary tool for establishing stability in load-balancing models. We develop a novel method to prove that the subcritically loaded system is stable for large enough N and establish convergence of steady-state distributions to the optimal one as N → ∞. The method goes beyond the state-of-the-art techniques; it uses an induction-based idea and a “weak monotonicity” property of the model. This technique is of independent interest and may have broader applicability.

Weak Monotone Comparative Statics

We develop a theory of monotone comparative statics based on weak set order -- in short, weak monotone comparative statics -- and identify the enabling conditions in the context of individual choices, Pareto optimal choices% for a coalition of agents, Nash equilibria of games, and matching theory. Compared with the existing theory based on strong set order, the conditions for weak monotone comparative statics are weaker, sometimes considerably, in terms of the structure of the choice environments and underlying preferences of agents. We apply the theory to establish existence and monotone comparative statics of Nash equilibria in games with strategic complementarities and of stable many-to-one matchings in two-sided matching problems, allowing for general preferences that accommodate indifferences and incompleteness.

Global asymptotic stability of nonconvex sweeping processes

Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes the conditions for global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraints. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result.

Existence of Weak Solutions to an Elliptic-Parabolic Equation with Variable Order of Nonlinearity

We consider an equation with variable nonlinearity of the form |u|p(x), in which the parabolic term can vanish, i.e., in the corresponding domain the parabolic equation becomes “elliptic.” Under the weak monotonicity conditions (nonstrict inequality) we prove the existence of a solution to the first mixed problem in a cylinder with a bounded base.

Existence and uniqueness of solutions for BDSDEs with weak monotonicity coefficients

In this paper, we study backward doubly stochastic differential equations (BDSDEs for short) with weak monotonicity coefficients. With the help of choosing the suitable approximation sequence, we derive the existence and uniqueness of solutions to BDSDEs. The comparison theorem is also established in one-dimensional situation.

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

The aim of this paper is to investigate strong convergence of modified truncated Euler–Maruyama method for neutral stochastic differential delay equations introduced in Lan (2018). Strong convergence rates of the given numerical scheme to the exact solutions at fixed time T are obtained under local Lipschitz and Khasminskii-type conditions. Moreover, convergence rates over a time interval [0,T] are also obtained under additional polynomial growth condition on g without the weak monotonicity condition (which is usually the standard assumption to obtain the convergence rate). Two examples are presented to interpret our conclusions.

Quasilinear elliptic systems with nonstandard growth and weak monotonicity

We prove the existence of solutions for a quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}\,\sigma (x,u,Du)&{}=f(x,u,Du)\quad \text {in}\;\varOmega ,\\ u&{}=0\quad \text {on}\;\partial \varOmega . \end{array} \right. \end{aligned}$$ The results are obtained in Orlicz–Sobolev spaces by means of the Young measures.

Investigating Treatment Effects of Participating Jointly in SNAP and WIC when the Treatment Is Validated Only for SNAP

The U.S. Department of Agriculture operates several food assistance programs aimed at alleviating food insecurity. We study whether participation in both participation in both SNAP and WIC alleviates food insecurity compared with participation in SNAP alone. We bound underlying causal effects by applying nonparametric treatment effect methods that allow for endogenous selection and underreported program participation when validation data are available for one program (treatment) but not the other. We estimate average treatment effects using data from the National Household Food Acquisition and Purchase Survey (FoodAPS). FoodAPS includes administrative data to validate SNAP participation. Information on local food prices allows us to construct a food expenditure‐based monotone instrumental variable that does not require a typical instrumental variable exclusion restriction. Under relatively weak monotonicity assumptions, we identify that the impact of participating in both programs relative to SNAP alone is strictly positive, suggesting that the programs are nonredundant. This evidence can support improved design and targeting of food programs.

Experimental study on transition characteristics of thermo-solutocapillary convection under buoyancy

An experimental study on the spatial and temporal evolution of thermo-solutocapillary convection in liquid bridges under buoyancy effect is conducted by Particle Image Velocimetry (PIV). In this experiment, 5cst silicone oil and the n-hexane are used as the solvent and solute, respectively. The velocity field for the steady thermocapillary-buoyancy convection is investigated firstly, and then the transitional characteristic of thermo-solutocapillary convection under buoyancy effect is analyzed as well. Experimental results show that, in the steady thermocapillary-buoyancy convection of liquid bridge with volume ratio Vr = 0.9 or Vr = 0.8, the vortex center of thermocapillary convection moves gradually toward the free surface with the increasing temperature difference, and the vortex evolves from a semilune “Ͻ” at a small temperature difference to a sickle-shaped “ر” at a large temperature difference. When the volume ratio is relatively large (Vr = 0.9), as the temperature difference increases, the connection line for velocity values at monitoring particles at the intermediate height of liquid bridge R(X, 1.5) shows a weak monotonicity. However, when the volume ratio is decreased (Vr = 0.8), the monotonicity is significantly better than that at large volume ratio. When Vr is 0.6, the temperature difference affects weakly the vortex position of thermocapillary convection, and the shape of the vortex is always the semilunar—“Ͻ”. The oscillatory mechanism and flow pattern of thermo-solutocapillary convection under gravity are different at different volume ratios. When the volume ratio is large (Vr = 0.9), vortices of buoyancy convection and thermo-solutocapillary convection appear an oscillatory pattern of mutual wane-wax—a “synchronous” oscillatory type. When Vr is 0.6, vortices of thermo-solutocapillary convection show an asynchronous feature of wane-wax—an “asynchronous” oscillatory type.

Weakly differentially monotonic solutions for cooperative games

The principle of differential monotonicity for cooperative games states that the differential of two players’ payoffs weakly increases whenever the differential of these players’ marginal contributions to coalitions containing neither of them weakly increases. Together with the standard efficiency property and a relaxation of the null player property, differential monotonicity characterizes the egalitarian Shapley values, i.e., the convex mixtures of the Shapley value and the equal division value for games with more than two players. For games that contain more than three players, we show that, cum grano salis, this characterization can be improved by using a substantially weaker property than differential monotonicity. Weak differential monotonicity refers to two players in situations where one player’s change of marginal contributions to coalitions containing neither of them is weakly greater than the other player’s change of these marginal contributions. If, in such situations, the latter player’s payoff weakly/strictly increases, then the former player’s payoff also weakly/strictly increases.

Curve-based monotonicity: a generalization of directional monotonicity

ABSTRACTIn this work we propose a generalization of the notion of directional monotonicity. Instead of considering increasingness or decreasingness along rays, we allow more general paths defined by curves in the n-dimensional space. These considerations lead us to the notion of α-monotonicity, where α is the corresponding curve. We study several theoretical properties of α-monotonicity and relate it to other notions of monotonicity, such as weak monotonicity and directional monotonicity.

Weak and directional monotonicity of functions on Riesz spaces to fuse uncertain data

In the theory of aggregation, there is a trend towards the relaxation of the axiom of monotonicity and also towards the extension of the definition to other domains besides real numbers. In this work, we join both approaches by defining the concept of directional monotonicity for functions that take values in Riesz spaces. Additionally, we adapt this notion in order to work in certain convex sublattices of a Riesz space, which makes it possible to define the concept of directional monotonicity for functions whose purpose is to fuse uncertain data coming from type-2 fuzzy sets, fuzzy multisets, n-dimensional fuzzy sets, Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets, among others. Focusing on the latter, we characterize directional monotonicity of interval-valued representable functions in terms of standard directional monotonicity.

Join-Idle-Queue with Service Elasticity

We consider the model of a token-based joint auto-scaling and load balancing strategy, proposed in a recent paper by Mukherjee, Dhara, Borst, and van Leeuwaarden [4] (SIGMETRICS '17), which offers an efficient scalable implementation and yet achieves asymptotically optimal steady-state delay performance and energy consumption as the number of servers N ! 1. In the above work, the asymptotic results are obtained under the assumption that the queues have fixed-size finite buffers, and therefore the fundamental question of stability of the proposed scheme with infinite buffers was left open. In this paper, we address this fundamental stability question. The system stability under the usual subcritical load assumption is not automatic. Moreover, the stability may not even hold for all N. The key challenge stems from the fact that the process lacks monotonicity, which has been the powerful primary tool for establishing stability in load balancing models. We develop a novel method to prove that the subcritically loaded system is stable for large enough N, and establish convergence of steady-state distributions to the optimal one, as N ! 1. The method goes beyond the state of the art techniques " it uses an induction-based idea and a "weak monotonicity" property of the model; this technique is of independent interest and may have broader applicability.

Mixture functions and their monotonicity

We consider mixture functions, which are a type of weighted averages for which the corresponding weights are calculated by means of appropriate continuous functions of their inputs. In general, these mixture function need not be monotone increasing. For this reason we study sufficient conditions to ensure standard, weak and directional monotonicity for specific types of weighting functions. We also analyze directional monotonicity when differentiability is assumed.

Weak Monotone Comparative Statics

We develop a theory of monotone comparative statics based on weak set order, or in short weak monotone comparative statics, and identify the enabling conditions in the context of individual choices, Pareto optimal choices for a coalition of agents, and Nash equilibria of games. Compared with the existing theory based on strong set order, the conditions for weak monotone comparative statics are weaker, sometimes considerably, in terms of the structure of the choice environment and underlying preferences of agents. We apply the theory to establish existence and monotone comparative statics of Nash equilibria in games with strategic complementarities and of stable many-to-one matchings in two-sided matching problems, allowing for general preferences that accommodate indifferences and incomplete preferences.

Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term

In this paper we are concerned with the fractional Schrodinger equation \begin{document}$ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $\end{document} , \begin{document}$ x\in {{\mathbb{R}}^{N}} $\end{document} , where \begin{document}$ f $\end{document} is superlinear, subcritical growth and \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} is nondecreasing. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are periodic in \begin{document}$ x_{1},\ldots, x_{N} $\end{document} , we show the existence of ground states and the infinitely many solutions if \begin{document}$ f $\end{document} is odd in \begin{document}$ u $\end{document} . When \begin{document}$ V $\end{document} is coercive or \begin{document}$ V $\end{document} has a bounded potential well and \begin{document}$ f(x, u) = f(u) $\end{document} , the ground states are obtained. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are asymptotically periodic in \begin{document}$ x $\end{document} , we also obtain the ground states solutions. In the previous research, \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.

Weakly Monotone Sets and Continuous Selection from a Near-Best Approximation Operator

A new notion of weak monotonicity of sets is introduced, and it is shown that an approximatively compact and weakly monotone connected (weakly Menger-connected) set in a Banach space admits a continuous additive (multiplicative) e-selection for any e < 0. Then a notion of weak monotone connectedness (weak Menger connectedness) of sets with respect to a set of d-defining functionals is introduced. For such sets, continuous (d−1, e)-selections are constructed on arbitrary compact sets.

On Wold’s approach to representation of preferences

This paper presents easily verifiable sufficient conditions on sequence spaces that guarantee representation of preference orders. Our approach involves identifying a suitable subset of the set of alternatives, such that (a) the preference order is representable on this subset, and (b) the subset has the property that for each alternative, there is some element in this subset which is indifferent to it. We follow Wold in choosing this subset to be the diagonal. Our first result uses a weak monotonicity condition (on the diagonal), and a substitution condition, and may be identified as the essence of Wold’s contribution. In the second result, we show that one can obtain a Wold-type representation result when weak monotonicity is replaced by a weak continuity condition. We use the countable order-dense characterization of representability in the proofs of both results, thereby integrating the contributions of Wold (1943) and Debreu (1954). Through a series of examples we show that our representation results are robust; they cannot be improved upon by dropping any of our conditions. An example is also presented to show that existence of degenerate indifference classes is compatible with the representation of monotone preferences. Our study thereby indicates that while the presence of substitution possibilities can be useful in representing preferences, they are not necessary for such results to hold.