Set Of Equilibria Research Articles

Existence of optimal strategies in bimatrix game and applications

This paper delves into interval-valued bimatrix games, where precise payoffs remain elusive, but lower and upper bounds on payoffs can be determined. The study explores several key questions in this context. Firstly, it addresses the issue of the existence of a universally applicable equilibrium across all instances of interval values. The paper establishes a fundamental equivalence by demonstrating that this property hinges on the solvability of a specific system of interval linear inequalities. Secondly, the research endeavors to characterize the comprehensive set of weak and strong equilibrium using a system of interval linear inequalities. The findings in this paper shed light on the complexities and intricacies of interval-valued bimatrix games, offering valuable insights into their equilibrium properties and computational aspects. Through illustrative examples, we underscore the practical utility of these approaches and compare them with previously developed state-of-the-art methods, demonstrating their ability to generate conservative solutions in the face of interval uncertainty. The findings of this research not only offer valuable insights into the equilibrium properties and computational aspects of interval-valued bimatrix games but extend their practical implications. In particular, the paper delves into real-life applications, exemplifying the significance of these findings for crude oil trading decision-making.

Hidden multi-scroll and coexisting self-excited attractors in optical injection semiconductor laser system: Its electronic control.

In this paper, we investigate hidden and coexisting self-excited multi-scroll attractors by using a modified rate equations model of semiconductor lasers (REM-SCLs) subjected to optical injection by exploring various quantifying analytical and numerical methods. The multi-leveled dynamics sticks out the existence of several sets of equilibria that asymptotically attract trajectories originating outside of them. Chaos topology based on the impact of equilibria allows the describing of the so-called stable or unstable multi-scroll chaotic attractors. Shaping of the new coexisting self-excited multi-scroll attractor, whose source is from coupling of equilibria, is analyzed, as well as its structural dynamics along with the dynamical emergence of the hidden multi-scroll attractor in the restricted interval, defined by an additional decisive parameter. Additionally, specific 3D plots with embedded contour plots obtained by harnessing two-parameter bifurcation analysis clarify structural dynamics of such a multi-scroll attractor and accurately circumscribe stretching of its fractal-like basin of attraction. Strange metamorphoses undergone by the fractal-like basin of attraction of the studied multi-scroll attractor are stepwisely parsed in the map of two-codimension bifurcation as its scroll number evolves. At last, an electronic circuit of equivalent REM-SCLs is designed and simulated in the PSpice environment alongside a tailored electronic controller. The achieved results align with the ones of numerical analysis; besides, temporal controlling of optical waves pertaining thereto is also fulfilled.

Biomolecular Condensates are Characterized by Interphase Electric Potentials.

Biomolecular condensates form via processes that combine phase separation and reversible associations of multivalent macromolecules. Condensates can be two- or multiphase systems defined by coexisting dense and dilute phases. Here, we show that solution ions partition asymmetrically across coexisting phases defined by condensates formed by intrinsically disordered proteins or homopolymeric RNA molecules. Our findings were enabled by direct measurements of the activities of cations and anions within coexisting phases of protein and RNA condensates. Asymmetries in ion partitioning between coexisting phases vary with protein sequence, macromolecular composition, salt concentration, and ion type. The Donnan equilibrium set up by the asymmetrical partitioning of solution ions generates interphase electric potentials known as Donnan and Nernst potentials. Our measurements show that the interphase potentials of condensates are of the same order of magnitude as membrane potentials of membrane-bound organelles. Interphase potentials quantify the degree to which microenvironments of coexisting phases are different from one another. Importantly, and based on condensate-specific interphase electric potentials, we reason that condensates are akin to capacitors that store charge. Interphase potentials should lead to electric double layers at condensate interfaces, thereby explaining recent observations of condensate interfaces being electrochemically active.

Stability of open-loop Nash equilibria for [formula omitted]-person nonzero-sum bounded rationality generalized differential games

The problem of determining the existence of Nash equilibria in n-person nonzero-sum generalized differential games is highly intricate and constrained by the advancement of partial differential equations theory. There is limited existing research literature on this subject. This paper presents an existence theorem for open-loop Nash equilibria employing the Fan-Glicksberg fixed point theorem. The n-person nonzero-sum bounded rationality generalized differential game model is formulated by introducing a bounded rationality function, and its structural stability and robustness are studied. The conclusions indicate that in the sense of Baire classification, most n-person nonzero-sum bounded rationality generalized differential games are structurally stable and robust in the set of ɛ-open-loop Nash equilibria, and we can approximate the equilibrium set obtained with full rationality generalized differential games by the ɛk-open-loop Nash equilibria set obtained with bounded rationality generalized differential games.

Biomolecular condensates are characterized by interphase electric potentials.

Biomolecular condensates form via processes that combine phase separation and reversible associations of multivalent macromolecules. Condensates can be two- or multi-phase systems defined by coexisting dense and dilute phases. Here, we show that solution ions can partition asymmetrically across coexisting phases defined by condensates formed by intrinsically disordered proteins or homopolymeric RNA molecules. Our findings were enabled by direct measurements of the activities of cations and anions within coexisting phases of protein and RNA condensates. Asymmetries in ion partitioning between coexisting phases vary with protein sequence, condensate type, salt concentration, and ion type. The Donnan equilibrium set up by asymmetrical partitioning of solution ions generates interphase electric potentials known as Donnan and Nernst potentials. Our measurements show that the interphase potentials of condensates are of the same order of magnitude as membrane potentials of membrane-bound organelles. Interphase potentials quantify the degree to which microenvironments of coexisting phases are different from one another. Importantly, and based on condensate-specific interphase electric potentials, which are membrane-like potentials of membraneless bodies, we reason that condensates are mesoscale capacitors that store charge. Interphase potentials lead to electric double layers at condensate interfaces. This helps explain recent observations of condensate interfaces being electrochemically active.

Robust Two-Layer Control of DC Microgrids With Fluctuating Constant Power Load Demand

This paper proposes a cascaded control structure for the regulation of DC Microgrids (MG). It is well-known that the negative impedance characteristics of a constant power load (CPL) adversely affect the stability of the network, and can cause problems such as voltage collapse or damage the electronic components. To mitigate this, we propose a two-layer control structure, where at the inner layer, the proposed controller achieves fast tracking of the supplied reference points and ultimate boundedness of the trajectories in a desired set. The outer layer generates the inner layer reference points, accounts for system constraints, and introduces robustness of the voltage dynamics to unknown perturbations of the CPL demand. For the first time, an investigation of the nonlinear geometric behaviour of the CPL is carried out to derive necessary conditions that ensure boundedness of the network dynamics and feasible regulation to a desired equilibrium set. Finally, Control Lyapunov functions are formulated to prove the stability and estimate the region of attraction of the closed loop dynamics. A simulated scenario of a meshed MG network is presented to confirm the validity of the results.

Excitatory action of low frequency depolarizing GABA/glycine synaptic inputs is prevalent in prenatal spinal SOD1G93A motoneurons.

Amyotrophic lateral sclerosis (ALS) is a fatal adult-onsetneurodegenerative disease characterized by progressive motor neuron degeneration and muscle paralysis. Recent evidence suggests the dysfunction of inhibitory signalling in ALS motor neurons. We have shown that embryonic day (E)17.5spinal motoneurons (MNs) of the SOD1G93A mouse model of ALS exhibit an altered chloride homeostasis. At this prenatal stage, inhibition of spinal motoneurons (MNs) is mediated by depolarizing GABAergic/glycinergic postsynaptic potentials (dGPSPs). Here, using an ex vivo preparation and patch clamp recording from MNs with a chloride equilibrium set below spike threshold, we report that low input resistance (Rin ) E17.5 MNs from the SOD1G93A ALS mouse model do not correctly integrate dGPSPs evoked by electrical stimulations of GABA/glycine inputs at different frequencies. Indeed, firing activity of most wild-type (WT) MNs with low Rin was inhibited by incoming dGPSPs, whereas low Rin SOD1G93A MNs were excited or exhibited a dual response (excited by low frequency dGPSPs and inhibited by high frequency dGPSPs). Simulation highlighted the importance of the GABA/glycine input density and showed that pure excitation could be obtained in SOD-like MNs by moving GABA/glycine input away from the cell body to dendrites. This was in agreement with confocal imaging showing a lack of peri-somatic inhibitory terminals in SOD1G93A MNs compared to WT littermates. Putative fast ALS-vulnerable MNs with low Rin are therefore lacking functional inhibition at the near-term prenatal stage. KEY POINTS: We analysed the integration of GABAergic/glycinergic synaptic events by embryonic spinal motoneurons (MNs) in a mouse model of the amyotrophic lateral sclerosis (ALS) neurodegenerative disease. We found that GABAergic/glycinergic synaptic events do not properly inhibit ALS MNs with low input resistance, most probably corresponding to future vulnerable MNs. We used a neuron model to highlight the importance of the GABA/glycine terminal location and density in the integration of the GABAergic/glycinergic synaptic events. Confocal imaging showed a lack of GABA/glycine terminals on the cell body of ALS MNs. The present study suggests that putative ALS vulnerable MNs with low Rin lack functional inhibition at the near-term stage.

A Stochastic Hybrid Framework for Event-Triggered Nonlinear Control Systems With Stochastic Denial of Service Attacks

In this paper, the stability problem for event-triggered nonlinear control systems is addressed in the presence of stochastic denial of service (DoS) attacks on the communication network. Dynamic event-triggered control (DETC) and periodic event-triggered control (PETC) are considered under the stochastic attacks. The stochastic attacks are modeled by a stochastic hybrid timer and a logic variable which is used to identify whether the network is attacked or not. Then the entire closed-loop system can be modeled into the framework of stochastic hybrid systems. Uniform global asymptotic stability in probability is concluded for the resulting stochastic system by proving that the equilibrium set is uniformly globally stable in probability and each neighborhood of the equilibrium set is uniformly globally recurrent. A nonlinear system example is provided to illustrate the effectiveness of the proposed results.

Submodularity and supermodularity in contest games

AbstractThis paper presents various examples of two‐player submodular or supermodular contest games. Emphasizing the three main elements of a contest model, our examples revolve around situations where (i) contest success function allows for a draw, (ii) winning prize is not exogenously given but rather jointly produced, or (iii) individual effort cost also depends on the rival's effort. We then illustrate how submodularity and supermodularity can be used to study the existence of equilibrium, the order structure of the equilibrium set, and monotone comparative statics in such contest game examples.

Some notes on possibilistic randomisation with t-norm based joint distributions in strategic-form games

This article continues the investigation started in [18] on the role of possibilistic mixed strategies in strategic-form games. In this earlier work we assumed, as standard in possibility theory, that joint possibility distributions were computed by combining possibilistic mixed strategies with the minimum t-norm. In this paper, we investigate the consequences of defining joint possibility distributions by using any continuous t-norm, with players' expected utilities based on the Choquet integral. We characterise under which conditions a pair of possibilistic mixed strategies is an equilibrium, generalising the results first presented in [18], and also show that the set of equilibria in possibilistic mixed strategies depends on the set of idempotent elements of a t-norm and not just on the chosen t-norm.

Adaptive Regulation With Global KL Guarantees

In the absence of persistency of excitation (PE), referring to adaptive control systems as “uniformly asymptotically stable” typically indicates insufficient understanding of stability concepts. While the state is indeed regulated to zero and the parameter estimate has some limit, namely, the overall state converges to some equilibrium, the equilibrium reached is not unique (and not even necessarily stable) but is dependent on the initial condition. The equilibrium set in the absence of PE is not uniformly attractive (from an open set containing the equilibrium set); hence, uniform asymptotic stability does not hold and KL-estimates are unavailable for the full state. In this paper we pursue adaptive control design with KL guarantees on the regulated state, something that is possible but previously unachieved with smooth, time-invariant and non-hybrid adaptive controllers. This property is referred to as Uniform Global Asymptotic Output Stability, where the regulated state is thought of as a system output. We provide designs for (i) systems with a matched uncertainty and (ii) systems in the parametric strict feedback form. We assume no a priori known bound for the parameters. To guarantee KL-estimates in the absence of PE, our designs employ time-invariant nonlinear damping terms, which depend both on the state and the parameter estimate. With an example, we illustrate the theory.

Scroll-growing/controlling chaotic attractors in cyclic Hopfield neural networks via memristive bridging

In this paper, a memristor-based neural network is proposed, which is implemented by two tri-neuron resistive-cyclic Hopfield neural networks (RC-HNNs) via memristive bridging. The memristor-bridged network has a line equilibrium set composed of infinitely many index-2 saddle-foci, but it can produce multi-scroll chaotic attractors contrary to Shil’nikov’s criterion. Complex bifurcation behaviors, scroll-growing chaotic attractors over time, and homogeneous coexisting attractors are revealed by numerical methods. Further, a scroll-control scheme is designed and scroll-controlling chaotic attractors are demonstrated numerically. The results show that the memristor-bridged network can not only generate scroll-growing chaotic attractors over time, but also produce scroll-controlling chaotic attractors by limiting the dynamic range of the internal state of the bridging memristor. Finally, an analog electronic circuit is designed for the memristor-bridged network, and PSIM circuit simulations are used to verify the numerical simulations.

Auxetic polymer networks: The role of crosslinking, density, and disorder

Low-crosslinked polymer networks have recently been found to behave auxetically when subjected to small tensions, that is, their Poisson’s ratio ν becomes negative. In addition, for specific state points, numerical simulations revealed that diamond-like networks reach the limit of mechanical stability, exhibiting values of ν = −1, a condition that we define as hyper-auxeticity. This behavior is interesting per se for its consequences in materials science but is also appealing for fundamental physics because the mechanical instability is accompanied by evidence of criticality. In this work, we deepen our understanding of this phenomenon by performing a large set of equilibrium and stress–strain simulations in combination with phenomenological elasticity theory. The two approaches are found to be in good agreement, confirming the above results. We also extend our investigations to disordered polymer networks and find that the hyper-auxetic behavior also holds in this case, still manifesting a similar critical-like behavior as in the diamond one. Finally, we highlight the role of the number density, which is found to be a relevant control parameter determining the elastic properties of the system. The validity of the results under disordered conditions paves the way for an experimental investigation of this phenomenon in real systems, such as hydrogels.

Experimental investigation and thermodynamic assessment of phase equilibria in the GaTe–AgGa5Te8–Te system below 600 K

Equilibrium T–x space of the Ag–Ga–Te system in the GaTe–AgGa5Te8–Te part was divided below 600 K into three-phase regions Ga2Te5–AgGa5Te8–Te, Ga2Te3–AgGa5Te8–Ga2Te5, Ga7Te10–AgGa5Te8–Ga2Te3, Ga3Te4–AgGa5Te8–Ga7Te10, and GaTe–AgGa5Te8–Ga3Te4 by the electromotive force (EMF) method. To accomplish accurate experimental data, the following electrochemical cells (ECs) were assembled: (−)IE|NE|SSЕ|R{Ag+}|PЕ|IE(+), where IE is the inert electrode (graphite powder), NE is the negative electrode (silver powder), SSE is the solid-state electrolyte (glassy Ag3GeS3Br), PE is the positive electrode, R{Ag+} is the region of PE that contact with SSE. At the stage of cell preparation, PE is a nonequilibrium phase mixture of the well-mixed powdered compounds Ag2Te, GaTe, Ga2Te3, and tellurium, taken in ratios corresponding to two or three different points in each of the mentioned regions. The equilibrium set of phases was formed in the R{Ag+} region at 600 K for 48 h with the participation of the Ag+ ions. Silver cations, displaced for thermodynamic reasons from the NE to the PE of the ECs, acted as catalyst, i.e., small nucleation centers of equilibrium phases. The spatial position of the established three-phase regions relative to the silver point was used to assign the overall potential-determining reactions of synthesis of the ternary AgGa5Te8 and binary Ga2Te5, Ga7Te10, Ga3Te4 compounds. For the first time, the values of the standard thermodynamic functions (Gibbs energies, enthalpies, and entropies) of these compounds were determined based on the temperature dependences of the EMF of the ECs.

The n-player Hirshleifer contest

While the game-theoretic analysis of conflict is often based on the assumption of multiplicative noise, additive noise such as considered by Hirshleifer (1989) may be equally plausible depending on the application. In this paper, we examine the equilibrium set of the n-player difference-form contest with heterogeneous valuations. For high and intermediate levels of noise, the equilibrium is in pure strategies, with at most one player being active. For small levels of noise, however, we find a variety of equilibria in which some but not necessarily all players randomize. In the case of homogeneous valuations, we obtain a partial uniqueness result for symmetric equilibria. As the contest becomes increasingly decisive, at least two contestants bid up to the valuation of the second-ranked contestant, while any others ultimately drop out. Thus, in the limit, equilibria of the Hirshleifer contest share important properties of equilibria of the corresponding all-pay auction.

Robustness Against Indirect Invasions in Asymmetric Games with Continuous Strategy Spaces

Strategies which are robust against indirect invasions (RAII) have been studied in the literature for symmetric games. In this paper, we introduce the notion of RAII profiles for asymmetric games with continuous strategy spaces. For these games, we prove that the set of all indirect neutral mutants of a RAII profile is equivalent to a minimal evolutionarily stable set. Furthermore, globally strong uninvadable profiles are shown to be RAII profiles. We also discuss connections between evolutionarily stable sets and strict equilibrium sets of profiles.

Isochoric bubble point data for binary systems of 1 (ethane) in 2 (decane, undecane, dodecane and tridecane)

In this communication, a new set of vapour liquid equilibrium (bubble point) data were obtained for ethane-decane, ethane-undecane, ethane-dodecane and ethane-tridecane binary systems. The datasets were obtained using constant-volume (isochoric) method. The datasets cover temperatures from 260 to 373 K and pressures from 0.6 to 10 MPa for all binary systems. A total of five compositions were studied for each of the binary systems ranging from ethane rich compositions to compositions rich in decane, undecane, dodecane and tridecane respectively. Datasets obtained were used to adjust the binary interaction parameters (BIP) for Peng-Robinson (PR), Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT) and the Multi-Fluid Helmholtz Energy Approximation (MFHEA) equations of state (EoS). Quadratic temperature dependent BIP functions were obtained for PR-EoS using a combination of datasets from this work and those published in the literature for the ethane-decane and ethane-dodecane binary systems. Correlation of the datapoints from this work with the PR-EoS, PC-SAFT-EoS and MFHEA EoS yielded very low average absolute deviations (AADs) of less than 5 % for all the datasets. Expanded uncertainties associated with each of the bubble point temperatures and pressures obtained are also reported at 95 % confidence interval. Expanded uncertainties in all the compositions studied were found to be less than 0.002 at 95 % confidence interval.

Offset-Control Plane Coexisting Behaviors in Two-Memristor-Based Hopfield Neural Network

Memristor synapse with activated synaptic plasticity can be taken as an adaptive connection synaptic weight. To demonstrate its kinetic effects, this article presents an improved Hopfield neural network with two memristive self-connection synaptic weights. The two-memristor-based Hopfield neural network (TM-HNN) has a plane equilibrium set related to two memristor initial conditions and its stability distributions are analyzed by two non-zero roots of the eigenvalue polynomial. Afterwards, the parameter-related bifurcation behaviors are investigated using bifurcation plots and phase portraits. Emphatically, the kinetic effects of memristor synapses are demonstrated by taking the memristor initial conditions as two invariant measures. The theoretical and numerical results show that the TM-HNN can exhibit the wondrous offset-control plane coexisting behaviors and its plane coexisting attractors can be controlled by switching the two memristor initial conditions. Besides, a digital hardware device is developed and the offset-control plane coexisting attractors are experimentally reproduced to verify the numerical ones.

Gyro orbit simulations of neutral beam injection in Wendelstein 7-X

Simulations exploring neutral beam operation in Wendelstein 7-X (W7-X) at reduced magnetic field are performed using a newly implemented gyro orbit model in the BEAMS3D code. Operation at field strengths below the nominal 2.5 T are seen as a path to explore both high beta plasmas and as a means to access magnetic configurations not possible at 2.5 T. As the field strength becomes smaller, the gyro radius for 55 keV fast protons grows from at 2.5 T to at 0.75 T in a device with minor radius bringing into question the applicability of the gyro center approximation. To address this a gyro orbit model was implemented in the BEAMS3D code. Agreement is found between the gyro center and gyro orbit models in a circular cross section tokamak equilibrium at high field. A set of W7-X equilibria are assessed with fixed density and temperature profiles but decreasing magnetic field strength (increasing plasma beta). Neutral beam deposition is found to be mostly unaffected with changes in the core of the plasma associated with the Shafranov-shift. In general good agreement is found between gyro orbit and gyro center simulations at 2.5 T. Both models indicate increasing losses with decreasing magnetic field strength with the gyro orbit losses being higher at all field strengths. Gyro orbit simulations to the first wall of W7-X show a change in loss pattern with decreasing magnetic field strength. A preliminary assessment of losses to fast ion loss detectors are made.

Location games with references

We study a class of location games where players want to attract as many resources as possible and pay a cost when deviating from an exogenous reference location. This class of games includes political competitions between policy-interested parties and firms' costly horizontal differentiation. We find that the introduction of reference locations simplifies the set of pure-strategy equilibrium to a unique candidate which has a strong property: at most four players, the two most-left and two most-right, deviate from their reference locations. We provide necessary and sufficient conditions for the candidate to be an equilibrium. We illustrate our results in particular cases including the duopoly competition where we moderate the principle of minimal differentiation.