Multiple Equilibria Research Articles

Statistical Investigation about the SDRE Optimality for Satellites with Nonlinearities

The Attitude Control System (ACS) of a Satellite can be designed with success by linear control theory if the satellite has slow angular motions. However, for fast maneuvers, the linearized models are not able to represent the effects of the nonlinear terms. One candidate technique to design the ACS of the satellite under fast maneuvers is the State-Dependent Riccati Equation (SDRE), which provides an effective algorithm for synthesizing nonlinear feedback control taking into account the nonlinearities. Nonetheless, much criticism has been leveled against the SDRE method since it does not assure global asymptotic stability (GAS), because there are situations in which global asymptotic stability cannot be achieved (e.g., systems with multiple equilibrium points). One way to study the GAS is by estimating the region of attraction (ROA) which in turn is fundamental to investigate the performance of the controller designed by the SDRE method The Brazilian National Institute for Space Research (INPE) is responsible to build a Remote-Sensing Satellites, called Amazonia-1. The ACS of the Amazonia-1 satellite must be stabilized in three axes so that the optical payload can point to the desired target. In this paper, one investigates the performance of the Amazonia-1 ACS using a statistical test (unpaired t-test) that compares the optimality inside the ROA controlled by LQR and SDRE. By several simulations running a full Monte Carlo perturbation satellite model one observes a significant difference between the optimality of the two controllers in favor of SDRE when nonlinearities are accounted for.

GENERAL THEORY FOR SIGNIFICANCE OF CULLING IN TWO-WAY DISEASE TRANSMISSION BETWEEN HUMANS AND ANIMALS

An epidemic model is proposed to comprehend the disease dynamics between humans and animals and back to humans with a culling intervention strategy. The proposed model is separated into two cases with two different culling rates: (1) at a per-capita constant rate and (2) constant population being culled. The global asymptotic stability of equilibria is determined in terms of the basic reproduction numbers. Further, we find that the culling rate (2) considered in the model could change the dynamics by having multiple positive equilibria. Sensitivity analysis recommends developing a strategy that promotes animals’ natural and disease-related death rates. By ranking the efficacies of various intervention strategies, we obtain that vaccination in the human population, isolation and public awareness are the largely effective control interventions. Our general theory raises concerns about both human and animal populations becoming reservoirs of the disease and affecting each other dynamically.

Stability Analysis of Ice Crystal Orientation

Abstract The stability of ice crystal orientation is studied by modeling the airflow around ice crystals at moderate Reynolds number, where an ice crystal is approximated by a cylinder with three parameters: diameter D, length L, and zenith angle of the axis θ. In this paper, the torque acting on ice crystals is simulated at different θ first, and then a special θ with zero horizontal torque, denoted as θe, is sought as an equilibrium of ice crystal orientation. The equilibrium is classified into two kinds: stable and unstable. Ice crystals rotate to θe of stable equilibriums while deviating from θe of unstable ones once they are released into quiet air. Multiple equilibriums of ice crystal orientation are found via numerical simulations. A cylinder with D/L close to one has three equilibriums, two of which are stable (i.e., θe = 0° and 90°). A cylinder with D/L away from one has only two equilibriums, one of which is stable (i.e., either θe = 0° or 90°). In addition, an asymmetric cylinder has two, three, or five equilibriums, and their θe is sensitive to the distance between its geometrical center and its center of gravity. The sensitivity of θe to crystal asymmetry suggests large symmetric ice crystals tend to become asymmetric (or irregular) and subsequently oriented randomly. Significance Statement Ice crystal orientation impacts high-cloud reflectance and satellite-based observations of high clouds significantly. However, its laboratory and field observations look dissimilar: the percentage of horizontally oriented ice crystals (HOICs) observed in the laboratory is quite high, while in the field it is often low and varies greatly in space and time. The motivation for this study is to elucidate what causes the difference between the laboratory and field observations. The torque acting on ice crystals are computed by modeling the airflow around ice crystals, revealing the conditions for nonhorizontal orientations of ice crystals. In quiet air, an ice crystal is oriented either horizontally or vertically when its shape is close to sphere. When its shape is elongated in one direction, its orientation depends on its asymmetry in density and shape. The sensitivity of ice crystal orientation to ice crystal asymmetry explains the low percentage of HOICs in the field, because asymmetric ice crystals are common in clouds. As an application, this sensitivity together with the observed percentage of HOICs can be used to infer the processes of ice crystal growth in clouds, providing clues to better representation of ice crystals in weather and climate models.

Region stability of switched DTL systems with all subsystems unstable and multi-equilibria

This paper addresses region stability issues of switched discrete-time linear (DTL) systems with all subsystems unstable and multiple equilibrium points (ASU-MEPs). The new concepts of region stability (RS), globally asymptotical region stability (GARS), and region instability (RIS) are defined for the switched DTL systems with ASU-MEPs. Based on these concepts introduced, this paper proposes several sufficient conditions of RS, GARS, and RIS for switched DTL systems with ASU-MEPs under the case as follows. Each subsystem of such kind of switched systems owns a single equilibrium point (SEP). There are at least two different subsystems' equilibrium points. The stability criteria obtained in this paper are with respect to a suitable region that contains all the subsystems' MEPs. Moreover, such kind of stability criteria is an extension of the classical eigenvalue stability criterion of non-switching DTL systems with SEP. By means of the GARS result, this paper designs an algorithm of globally asymptotical region stability for switched DTL systems with ASU-MEPs. The above results are applied to a commodity-selling market network.

Optimal pricing strategy for content products under competition: Pay-as-you-want or fixed-price?

PAYW (pay as you want) has become a popular pricing strategy for content products in practice and has received considerable attention in the literature. However, the previous literature mainly justifies this pricing strategy in the monopoly setting. By exploiting a duopoly setting, this paper analytically investigates the impact of competition on the value of PAYW by comparing it to the traditional fixed pricing (FP) strategy. We find that only symmetric equilibria exist, i.e., both sellers choose PAYW when the fairness ideal is sufficiently high in equilibrium; otherwise, both choose FP. In sharp contrast to the prior literature, we find that multiple equilibria can exist, and a prisoner’s dilemma situation can arise when the fairness ideal is at a medium level. We also find that the fairness ideal has a nonmonotonic impact on social welfare, and the presence of network externality makes PAYW preferable under competition. We also show that our results qualitative hold when consumers have heterogeneous product values.

Collateral Framework: Liquidity Premia and Multiple Equilibria

AbstractCentral banks normally accept debt of their own governments as collateral in liquidity operations without reservations. This gives rise to a valuable liquidity premium that reduces the cost of government finance. The ECB is an interesting exception in this respect. It relies on external assessments of the creditworthiness of its member states, such as credit ratings, to determine eligibility and the haircut it imposes on such debt. We show how such features in a central bank's collateral framework can give rise to cliff effects and multiple equilibria in bond yields and increase the vulnerability of governments to external shocks. This policy can potentially induce sovereign debt crises and defaults that would not otherwise occur. The success of the ECB's temporary suspension of these features of its collateral framework during the pandemic illustrates the practical relevance of this mechanism.

Public disclosure and private information acquisition: A global game approach

This paper studies public information disclosure in a model of dynamic financial markets with endogenous information acquisition. Due to an information complementarity, multiple equilibria may emerge, complicating comparative statics analysis. By adding noise to agents' information costs, we establish equilibrium uniqueness using global-game techniques. We show that while public information always crowds out private information in all underlying equilibria, it can crowd in private information acquisition in the unique global-game equilibrium. This result is driven by the strategic uncertainty introduced through the global-game refinement. The crowding-in effect is more pronounced when there is a high level of fundamental uncertainty, which supports the case for greater information disclosure during times of increased market volatility.

Quality Strategies in Network Markets

This paper studies network market problems in which firm(s)/platform(s) sets quality in addition to price. A well-established result in the network economics literature is that a profit-maximizing firm concerns only how quality is valued by the marginal consumer but not by inframarginal consumers, aka the Spence effect/distortion. For markets with strong network effects under which multiple market-tipping equilibria exist, I show that the validity of the previous result depends on the choice of the equilibrium selection criterion. Precisely, I show that all criteria commonly used in this literature give rise to the Spence effect, whereas the well-justified risk dominance criterion in game theory and its generalizations do not. Novel quality strategies are derived based on the latter criteria. This paper was accepted by Joshua Gans, business strategy.

A new institutional perspective on business associations: Filling a gap between firms and states in the dynamic analysis of Richard Day

In much of his work, Richard H. Day was concerned with the dynamics of growth among firms, households, industries and entire economies, and the fact that normal processes could end up at multiple equilibria or even out of equilibrium. To his great credit, he made use of optimizing models based on rational expectations to examine how firms and their managers would react in ways that would deal with the obstacles facing them, in some cases with the help of suitable policies of democratic governments. The purpose of this analysis is to point to a number of circumstances in which one type of institution omitted from his analysis, namely, business associations, might be able to serve as a very useful intermediary between the firms and government in mitigating the risks and inefficiencies that otherwise might lead to unfortunate equilibria and disequilibria in which firms and entire economies might find themselves.

A strongly convergent algorithm for solving multiple set split equality equilibrium and fixed point problems in Banach spaces

AbstractIn this article, using an Halpern extragradient method, we study a new iterative scheme for finding a common element of the set of solutions of multiple set split equality equilibrium problems consisting of pseudomonotone bifunctions and the set of fixed points for two finite families of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex Banach spaces, which are also uniformly smooth. For this purpose, we design an algorithm so that it does not depend on prior estimates of the Lipschitz-type constants for the pseudomonotone bifunctions. Furthermore, we present an application of our study for finding a common element of the set of solutions of multiple set split equality variational inequality problems and fixed point sets for two finite families of Bregman quasi-nonexpansive mappings. Finally, we conclude with two numerical experiments to support our proposed algorithm.

Equilibrium refinement in finite action evidence games

Evidence games study situations where a sender persuades a receiver by selectively disclosing hard evidence about an unknown state of the world. Evidence games often have multiple equilibria. Hart et al. (Am Econ Rev 107:690-713, 2017) propose to focus on truth-leaning equilibria, i.e., perfect Bayesian equilibria where the sender discloses truthfully when indifferent, and the receiver takes off-path disclosure at face value. They show that a truth-leaning equilibrium is an equilibrium of a perturbed game where the sender has an infinitesimal reward for truth-telling. We show that, when the receiver’s action space is finite, truth-leaning equilibrium may fail to exist, and it is not equivalent to equilibrium of the perturbed game. To restore existence, we introduce a disturbed game with a small uncertainty about the receiver’s payoff. A purifiable truthful equilibrium is the limit of a sequence of truth-leaning equilibria in the disturbed games as the disturbances converge to zero. It exists and features a simple characterization. A truth-leaning equilibrium that is also purifiable truthful is an equilibrium of the perturbed game. Moreover, purifiable truthful equilibria are receiver optimal and give the receiver the same payoff as the optimal deterministic mechanism.

Exploring Subtropical Stratocumulus Multiple Equilibria Using a Mixed-Layer Model

Abstract Stratocumulus clouds cover about a fifth of Earth’s surface, and due to their albedo and low-latitude location, they have a strong effect on Earth’s radiation budget. Previous studies using large-eddy simulations have shown that multiple equilibria (both stratocumulus-covered and cloud-free/scattered cumulus states) exist as a function of fixed SST, with relevance to equatorward advected air masses. Multiple equilibria have also been found as a function of atmospheric CO2, with a subtropical SST nearly 10 K higher in the cloud-free state and with suggested relevance to warm climate dynamics. In this study, we use a mixed-layer model with an added surface energy balance and the ability to simulate both the stratocumulus (coupled) and cloud-free/scattered cumulus (decoupled) states using a “stacked” mixed-layer approach to study both types of multiple equilibria and the corresponding hysteresis. The model’s simplicity and computational efficiency allow us to qualitatively explore the mechanisms critical to the stratocumulus cloud instability and hysteresis as well as isolate key processes that allow for multiple equilibria via mechanism-denial experiments not possible with a full-complexity model. For the hysteresis in fixed SST, we find that decoupling can occur due to either enhanced entrainment warming or a reduction in cloud-top longwave cooling. The critical SST at which decoupling occurs is highly sensitive to precipitation and entrainment parameterizations. In the CO2 hysteresis, decoupling occurs in the simple model used even without the inclusion of SST–cloud cover feedbacks, and the width of the hysteresis displays the same sensitivities as the fixed SST case. Overall, the simple model analysis and results motivate further studies using higher complexity models.

Role of Sn(II) in Mediating the Formation and Organization of a Langmuir Monolayer.

The influence of the Sn(II) ion on the formation and morphology of an arachidic acid (AA) monolayer was investigated using Langmuir film formation technology, pressure-area (Π-A) isotherm measurements, and Brewster angle microscopy (BAM). Our findings indicate that AA Langmuir monolayers exhibit organization that depends on subphase pH and Sn2+ concentration. There are multiple equilibria that are relevant to the complexation of AA monolayers, and the balance of Sn(OH)n equilibria and Sn(AA)n equilibria gives rise to unusual monolayer structural effects. With Sn2+ in the subphase, the AA monolayer exhibits an isotherm characterized by the absence of a collapse point and with a pH-dependent change in isotherm shape not consistent with the formation of an ordered solid phase. The amphiphile headgroup equilibrium mediates the absence of collapse seen experimentally and the ability of the monolayer to retain organization at a surface pressure of ca. 70 mN/m. BAM images show that the morphology of the monolayer depends on the Sn2+ concentration, consistent with several species of Sn(AA)n, where n = 1, 2, or 3, contributing to the overall monolayer order.

Multi-million-year cycles in modelled δ13C as a response to astronomical forcing of organic matter fluxes

Abstract. Along with 400 kyr periodicities, multi-million-year cycles have been found in δ13C records over different time periods. An ∼ 8–9 Myr periodicity is found throughout the Cenozoic and part of the Mesozoic. The robust presence of this periodicity in δ13C records suggests an astronomical origin. However, this periodicity is barely visible in the astronomical forcing. Due to the large fractionation factor of organic matter, its burial or oxidation produces large δ13C variations for moderate carbon variations. Therefore, astronomical forcing of organic matter fluxes is a plausible candidate to explain the oscillations observed in the δ13C records. So far, modelling studies forcing astronomically the organic matter burial have been able to produce 400 kyr and 2.4 Myr cycles in δ13C but were not able to produce longer cycles, such as 8–9 Myr cycles. Here, we propose a mathematical mechanism compatible with the biogeochemistry that could explain the presence of multi-million-year cycles in the δ13C records and their stability over time: a preferential phase locking to multiples of the 2.4 Myr eccentricity period. With a simple non-linear conceptual model for the carbon cycle that has multiple equilibria, we are able to extract longer periods than with a simple linear model – more specifically, multi-million-year periods.

A new integrated regulation strategy and modelling for wind turbine with battery energy storage system

Recent decades have seen a great penetration of renewable energy in electricity supply market. However, the intermittency of wind energy requires careful control of wind turbine (WT), for which a promising solution is to combine wind turbine with a battery energy storage system (BESS). This motivates our work on regulation strategy and modelling for the wind turbine with battery energy storage system (WT/BESS). We propose a new integrated regulation strategy (IRS) for WT/BESS which includes a dual-hysteresis charging/discharging mechanism for BESS and a cooperative output adjustment scheme for WT. The advantages of the proposed IRS are triple-folds: 1) the output fluctuation of WT/BESS is smoothed remarkably; 2) the status of charge (SOC) of BESS is kept in a suitable range for most of the time, which can improve the service life of BESS; 3) the phenomenon of chattering is escaped effectively. To describe the dynamics of the WT/BESS supervised by the proposed IRS, a model of composite switched system with multiple equilibria is developed, in which the switching strategy is related to three factors simultaneously, i.e., the variation of wind speed point, the working mode change of BESS and the control method of WT. The effectiveness of the proposed IRS is demonstrated by comparing several numerical simulations with other existing results.

SHARP TEST FOR EQUILIBRIUM UNIQUENESS IN DISCRETE GAMES WITH PRIVATE INFORMATION AND COMMON KNOWLEDGE UNOBSERVED HETEROGENEITY

This paper proposes a test of the single equilibrium in the data assumption commonly maintained when estimating static discrete games of incomplete information. By allowing for discrete common knowledge payoff-relevant unobserved heterogeneity, the test generalizes existing methods attributing all correlation between players’ decisions to multiple equilibria. It does not require the estimation of payoffs and is therefore useful in empirical applications leveraging multiple equilibria to identify the model’s primitives. The procedure boils down to testing the emptiness of the set of data generating processes that can rationalize the sample through a single equilibrium and a finite mixture over unobserved heterogeneity. Under verifiable conditions, this testable implication is generically sufficient for degenerate equilibrium selection. The main identifying assumption is the existence of an observable variable that plays the role of a proxy for the unobservable heterogeneity. Examples of such proxies are provided based on empirical applications from the existing literature.

Efficiency evaluation for decision making units with fixed-sum outputs using data envelopment analysis and stochastic multicriteria acceptability analysis

The generalized equilibrium efficient frontier data envelopment analysis (GEEFDEA) approach, an extension of the DEA method, has been widely used to solve the problem of evaluating decision making units (DMUs) producing fixed-sum outputs. It constructs a common equilibrium efficient frontier through a minimum reduction strategy for fixed-sum outputs and uses this frontier as a benchmark to achieve a complete ranking of DMUs. However, the existence of multiple feasible equilibrium efficient frontiers may lead to inconsistency in the evaluation criteria, and this possibility limits the method’s usefulness. In this paper, an integrated framework for solving this problem is proposed to rank DMUs by using stochastic multicriteria acceptability analysis (SMAA-2) method combined with the GEEFDEA approach. Instead of using a certain common equilibrium efficient frontier as in conventional GEEFDEA approaches, we explore all possible frontiers to answer various robustness questions by computing rank acceptability indices and pairwise winning indices. Furthermore, we derive the complete ranking from the dominance relationships among the DMUs. Two numerical examples are used to demonstrate the effectiveness and rationality of the proposed hybrid approach.

Pollution havens and agglomeration: The effect of globalization and technological spillover

ABSTRACT The statistical significance of the pollution haven hypothesis remains low. This suggests that some elements may have been overlooked in hypothesis testing. Here, pollution havens due to trade liberalization may depend on the degree of technological spillover. This spillover affects the industrial concentration of polluting firms. Using an agglomeration model with technological spillover, we analysed the effect of transport costs on pollution havens to clarify this relationship. Interactions between an international emission permit trading market and local spillover in innovation bring about the following multiple equilibria: the poverty trap equilibrium leads to similar results in the case of global spillover, where lower transport costs encourage economic growth in addition to relocation from the South to the North. By contrast, equilibrium with a higher concentration of firms leads to the pollution havens hypothesis because decreased transport costs prompt polluting firms to relocate to South with relatively small markets due to the home market effect, leading to lower economic growth. Our findings imply that the type of spillover in innovation (local or global) determines the pollution haven hypothesis. In a global economy, narrowing the North-South technology gap could contribute to eliminating pollution havens.

Dynamic Asset-Backed Security Design

Abstract Borrowers obtain liquidity by issuing securities backed by the current period payoff and resale price of a long-lived collateral asset, and they are privately informed about the payoff distribution. Asset price can be self-fulfilling: a higher asset price lowers adverse selection and allows borrowers to raise greater funding, which makes the asset more valuable, leading to multiple equilibria. Optimal security design eliminates multiple equilibria, improves welfare, and can be implemented as a repo contract. Persistent adverse selection lowers debt funding, generates volatility in asset prices, and exacerbates credit crunches. The theory demonstrates the role of asset-backed securities on stability of market-based financial systems.

On equilibrium uniqueness in generalized multi-prize nested lottery contests

The generalized multiple-prize nested lottery contest framework has been broadly applied to model noisy competitions that award prizes to multiple recipients. Equilibrium existence was not formally established until the recent contribution of Fu et al. (2022). This paper comprehensively examines the equilibrium uniqueness of this contest model. We first consider a multi-prize contest with identical players, a scenario that is commonly assumed in the literature. We verify that a symmetric equilibrium exists and that the equilibrium is unique, which lays a foundation for the numerous studies that adopt this model setting. We then proceed to an asymmetric setting in which players may differ in their prize valuations, impact functions, and/or cost functions. We show that the equilibrium uniqueness persists when players are weakly heterogeneous. However, equilibrium uniqueness may fail—and multiple equilibria may arise—when players are sufficiently heterogeneous.