Stochastic Equilibrium Research Articles

Research on a collaborative evolution model of multi-airport route network considering subsidy strategies.

To efficiently utilize subsidy strategies for optimizing multi-airport route networks and promoting collaborative development among multiple airports, we delve into the tripartite strategic interactions between passengers, airlines and airports. A dual-layer game-theoretic model is constructed to optimize subsidy strategies, facilitating a synergistic alignment between multi-airport positioning and route networks. In the upper-layer game-theoretic model, Fermi rules are employed to analyze the interplay between pricing strategies of distinct airline brands and passenger travel preferences, aiding in determining optimal pricing strategies for airlines. The lower-layer game-theoretic model introduces an asymmetric stochastic best response equilibrium (QRE) model, drawing insights from optimal airline pricing and the impact of airport subsidies on airline route adjustments to formulate effective multi-airport subsidy strategies. The results reveal that: (ⅰ) Airline revenues display varying peaks based on travel distances, with optimal fare discount intervals clustering between 0.6 and 0.9, contingent upon travel distances and passenger rationality; (ⅱ) dynamic monopolistic intervals and inefficient ranges characterize airport subsidy strategies due to diverse competitive strategies employed by rivals; (ⅲ) targeted airport subsidy strategies can enhance inter-airport route coordination in alignment with their functional positioning. This research provides decision-making insights into collaborative airport group development, encompassing airport subsidy strategies and considerations for airline pricing.

Carbon spot prices in equilibrium frameworks associated with climate change

<p style='text-indent:20px;'>At present, it is believed that the best approach to mitigate global warming is the market-based formulation of carbon emission pricing. Thus, in this paper, we work on determining the carbon spot prices in a stochastic equilibrium framework associated with climate change. Two circumstances, differentiated by whether taking carbon trading in the market, are considered. We construct optimization problems and solve them by using dynamic programming principle. The Fourier transform and its properties are fully made use of to return the explicit formulas of carbon prices. In addition, some surprising but interesting properties of the carbon prices are also found. First, the carbon prices happen jumps at the end of the abatement period. Second, the return rates of carbon prices are completely dependent on the climate elements. Finally, we present some numeric results in response to our theoretical results.</p>

Emergent scale invariance in neural networks

We demonstrate, both analytically and numerically, that learning dynamics of neural networks is generically attracted towards a scale-invariant state. The effect can be modeled with quartic interactions between non-trainable variables (e.g. states of neurons) and trainable variables (e.g. weight matrix). Non-trainable variables are rapidly driven towards stochastic equilibrium and trainable variables are slowly driven towards learning equilibrium described by a scale-invariant distribution on a wide range of scales.

The role of demand response in mitigating market power: a quantitative analysis using a stochastic market equilibrium model

Market power is a dominant feature of many modern electricity markets with an oligopolistic structure, resulting in increased consumer cost. This work investigates how consumers, through demand response (DR), can mitigate against market power. Within DR, our analysis particularly focusses on the impacts of load shifting and self-generation. A stochastic mixed complementarity problem is presented to model an electricity market characterised by an oligopoly with a competitive fringe. It incorporates both energy and capacity markets, multiple generating firms and different consumer types. The model is applied to a case study based on data for the Irish power system. The results demonstrate how DR can help consumers mitigate against the negative effects of market power and that load shifting and self-generation are competing technologies, whose effectiveness against market power is similar for most consumers. We also find that DR does not necessarily reduce emissions in the presence of market power.

A Risk-Averse Energy Sharing Market Game for Renewable Energy Microgrid Aggregators

Energy sharing has the potential to decrease operation costs by exploiting the complementarity of different entities. The further increase of microgrids’ generation capacity in the distribution network could trigger the emergence of MW-level microgrids aggregator (MGA) to participate in the regional energy sharing market (ESM) and sell excess energy. However, the uncertainty of renewable energy generation will cause the failure of energy sharing and an economic loss for the energy sharing market. Therefore, a novel two-sided market with the clearing price and risk management is proposed in this paper. In order to optimize the energy sharing strategy of MGAs, a network-constrained risk-averse two-stage stochastic game model based on DC optimal power flow (OPF) is established. The objective of the MGA is to maximize the revenue in the ESM while the subtransmission network operator (SNO) aims to achieve regional energy self-sufficiency. The sample average approximation (SAA) method is employed to approximate the stochastic Cournot-Nash equilibrium. A distributed market clearing algorithm based on the alternating direction method of multipliers (ADMM) and best response seeking is developed to solve the optimal bidding problem of MGAs and the optimal dispatch problem of the SNO. Furthermore, the existence of the SAA Nash equilibrium is also investigated. Numerical simulations prove that the proposed game model can effectively control the overbidding risk and promote regional energy self-sufficiency.

Stochastic Nash Equilibrium Problems: Models, Analysis, and Algorithms

Decision making under uncertainty has been studied extensively over the last 70 years, if not earlier. In the field of optimization, models for two-stage, stochastic, linear programming, presented by Dantzig <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> and Beale <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> , are often viewed as the basis for the subsequent development of the field of stochastic optimization. This subfield of optimization now encompasses a breadth of models that can accommodate both convexity and nonconvexity, probabilistic constraints, risk-aversion, discreteness, and multistage decision-making (compare <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> , <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> ). Similarly, stochastic control <xref ref-type="bibr" rid="ref5" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[5]</xref> has proven to be an enormously impactful subarea of control theory. When one extends the decision-making paradigm to multiple self-interested decision makers, then the resulting problem can be viewed as a noncooperative game that is rooted in the groundbreaking text by Von Neumann and Morgenstern <xref ref-type="bibr" rid="ref6" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[6]</xref> .

ENDOGENOUS PROCYCLICAL LIQUIDITY, CAPITAL REALLOCATION, AND q

AbstractCapital reallocation is procyclical and an economic boom has a cleansing effect by shifting the distribution of firms from low quality to high quality. We explain these facts by modeling search frictions for used capital in the business cycle. The article characterizes the stochastic equilibrium analytically to prove that the liquidity and the price of reallocated capital are procyclical endogenously. We calibrate the model and construct proxies in the data for the unemployment rate of capital and the time on the market. These two variables have a strong positive relationship in both the model and the data.

Equilibrium pricing for carbon emission in response to the target of carbon emission peaking

The climate warming effect on global economic output and social sustainability should be paid adequate attention. Nowadays, the best way is thought as the market-based pricing policy for carbon emission, which can reflect the emission cost by directly considering the economic efficiency. In response to the target of peak carbon emission before 2030 by China, this paper formulates an equilibrium carbon price by setting up a stochastic equilibrium model, depicting the interplay between the national economic output, carbon emission and climate change. An explicit pricing formula for carbon emission is presented, which clearly discloses the intrinsic link of carbon price to other considered variables and parameters. In addition to this, some interesting results are achieved. First, we calculate the drift and the approximate volatility of the carbon price. It is found that the drift is completely determined by the climate elements, while the approximate volatility shows a three-dimensional surface structure, not only time- and price-dependent, but also dependent on the carbon emission. Second, we find that the carbon price happens a jump at the deadline of the abatement period, in response to the failure of carbon abatement. Third, via sensitivity analysis, we capture the bifurcation phenomena in the carbon price induced by the economic growth rate and negative effect of climate warming. In conclusion, our equilibrium carbon price plays the role of benchmark in the national carbon emission trading market, leading to the realization of the abatement target.

Coordinating Followers to Reach Better Equilibria: End-to-End Gradient Descent for Stackelberg Games

A growing body of work in game theory extends the traditional Stackelberg game to settings with one leader and multiple followers who play a Nash equilibrium. Standard approaches for computing equilibria in these games reformulate the followers' best response as constraints in the leader's optimization problem. These reformulation approaches can sometimes be effective, but make limiting assumptions on the followers' objectives and the equilibrium reached by followers, e.g., uniqueness, optimism, or pessimism. To overcome these limitations, we run gradient descent to update the leader's strategy by differentiating through the equilibrium reached by followers. Our approach generalizes to any stochastic equilibrium selection procedure that chooses from multiple equilibria, where we compute the stochastic gradient by back-propagating through a sampled Nash equilibrium using the solution to a partial differential equation to establish the unbiasedness of the stochastic gradient. Using the unbiased gradient estimate, we implement the gradient-based approach to solve three Stackelberg problems with multiple followers. Our approach consistently outperforms existing baselines to achieve higher utility for the leader.

General stochastic ridesharing user equilibrium problem with elastic demand

In this study, we propose a general stochastic ridesharing user equilibrium (SRUE) problem with elastic demand (ED) for urban transportation network analysis with ridesharing activities by extending the deterministic ridesharing user equilibrium (DRUE) problem, which is referred to as the general SRUE-ED problem. We develop a novel variational inequality (VI) model in terms of the Lagrange multipliers corresponding to the ride-matching constraints for the general SRUE-ED problem. The existence and uniqueness of the SRUE-ED solution are demonstrated under mild conditions. We proceed to put up a global convergent method to solve the developed VI model by incorporating the weighted cost averaging (WCA) method. The purpose of WCA method is to solve a parametric fixed-point model formulated for the general SRUE-ED problem without the ride-matching constraints. Finally, we use three numerical examples to provide practical insights into the operation and management of the ridesharing systems and to examine the effectiveness of the proposed model and solution method.

The effects of couple stress lubricants and surface roughness on squeeze EHL motion between porous medium layer and elastic ball

This paper investigated the effects of porous media layer (PML) and couple stress (CS) lubricants at pure squeeze action in an isothermal elastohydrodynamic lubrication (EHL) point contact with surface roughness (SRN) under constant load condition. The modified transient stochastic Reynolds equation was derived in polar coordinates by means of the Christensen’s stochastic theory and the Stokes’s CS fluid theory. The Gauss-Seidel method (GSM) and the finite difference method (FDM) were both used to solve simultaneously modified transient stochastic Reynolds, force equilibrium, elasticity deformation, and rheology equations. The simulation results revealed that the greater permeability ( K) and porous layer thickness (Φ) are, the smaller central pressures ( Pc) are, and the smaller film thicknesses ( H) are. The times required for achieving maximum central pressures (Pcmax) decrease with increasing K and Φ. Due to squeeze and elastic deformation effects, the slope values of central velocity ( Vc) change from negative to positive. The magnitude of Vc and Vmin decreased with decreasing K and Φ. The Hcmin of circular type RN are greater than that of radial type RN. The Pcmax of radial type RN are slightly greater than that of circular type RN.

Learning and firm dynamics in a stochastic equilibrium

New firms have unknown profitability types, but they learn in the market. With imperfect information, the learning mechanism matters at the firm level and in the aggregate. This paper studies the effect of learning on the distribution of firms, aggregate dynamics, and welfare when firm growth and the entry and exit dynamics are belief-driven. Aggregate uncertainty affects heterogeneous firm beliefs about own types. Faster learning raises firms' continuation values, lowers the selection threshold belief, and thickens the belief distribution's right tail. Aggregate uncertainty lowers firm value by reducing firms' learning speed. Transitory aggregate shocks have persistent impacts on the economy by shifting existing firms' belief updating paths, affecting firm decisions along both intensive and extensive margins. Faster learning improves social welfare.

Control Analysis of Stochastic Lagging Discrete Ecosystems

In this paper, control analysis of a stochastic lagging discrete ecosystem is investigated. Two-dimensional stochastic hysteresis discrete ecosystem equilibrium points with symmetry are discussed, and the dynamical behavior of equilibrium points with symmetry and their control analysis is discussed. Using the orthogonal polynomial approximation theory, the stochastic lagged discrete ecosystems are approximately transformed as its equivalent deterministic ecosystem. Based on the stability and bifurcation theory of deterministic discrete systems, through mathematical analysis, asymptotic stability and Hopf bifurcation are existent in the ecosystem, constructing control functions, controlling the behavior of the system dynamics. Finally, the effects of different random strengths on the bifurcation control and asymptotic stability control are verified by numerical simulations, which validate the correctness and effectiveness of the main results of this paper.

Affordable Housing and City Welfare

Abstract Housing affordability is the main policy challenge for most large cities in the world. Zoning changes, rent control, housing vouchers, and tax credits are the main levers employed by policymakers. How effective are they at combatting the affordability crisis? We build a dynamic stochastic spatial equilibrium model to evaluate the effect of these policies on the well-being of its citizens. The model endogenizes house prices, rents, construction, labour supply, output, income, and wealth inequality, the location decisions of households within the city as well as inter-city migration. Its main novel features are risk, risk aversion, and incomplete risk-sharing. We calibrate the model to the New York metropolitan statistical area. Housing affordability policies carry substantial insurance value but affect aggregate housing and labour supply and cause misallocation in labour and housing markets. Housing affordability policies that enhance access to this insurance especially for the neediest households create substantial net welfare gains.

Distributed Variable Sample-Size Gradient-Response and Best-Response Schemes for Stochastic Nash Equilibrium Problems

This paper considers an $n$-player stochastic Nash equilibrium problem (NEP) in which the $i$th player minimizes a composite objective $f_i( \bullet ,x_{-i}) + r_i( \bullet )$, where $f_i$ is an expectation-valued smooth function, $x_{-i}$ is a tuple of rival decisions, and $r_i$ is a nonsmooth convex function with an efficient prox-evaluation. In this context, we make the following contributions. (I) Under suitable monotonicity assumptions on the \redpseudogradient map, we derive optimal rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate with degree $v > 0$, the mean-squared error of the iterates decays at a corresponding polynomial rate; in particular, we prove that the iteration and oracle complexities to obtain an $\epsilon$-Nash equilibrium ($\epsilon$-NE) are $\mathcal{O}(1/\epsilon^{1/v})$ and $\mathcal{O}(1/\epsilon^{1+1/v})$, respectively. \redWhen the sample-size is held constant, the iterates converge geometrically to a neighborhood of the Nash equilibrium in an expected-value sense. (II) We then overlay \bf VS-PGR with a consensus phase with a view towards developing distributed protocols for aggregative stochastic NEPs. In the resulting \bf d-VS-PGR scheme, when the sample-size at each iteration grows at a geometric rate while the communication rounds per iteration grow at the rate of $ k+1 $, computing an $\epsilon$-NE requires similar iteration and oracle complexities to \bf VS-PGR with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$. Notably, (I) and (II) rely on weaker oracle assumptions in that the conditionally unbiasedness assumption is relaxed while the bound on the conditional second moment may be state-dependent. (III) Under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where each player solves a sample-average BR problem. When the sample-size increases at a suitable geometric rate, the resulting iterates converge at a geometric rate while the iteration and oracle complexity are, respectively, $\mathcal{O}(\ln(1/\epsilon))$ and $\mathcal{O}(1/\epsilon)$. If the sample-size increases at a polynomial rate with degree $v$, the mean-squared error decays at a corresponding polynomial rate while the iteration and oracle complexities are $\mathcal{O}(1/\epsilon^{1/v})$ and $\mathcal{O}(1/\epsilon^{1+1/v})$, respectively. (IV) Akin to (II), the distributed variant \bf d-VS-PBR achieves similar iteration and oracle complexities to the centralized VS-PBR with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$ when the communication rounds per iteration increase at the rate of $ k+1 $. Finally, we present preliminary numerics to provide empirical support for the rate and complexity statements.

A Stochastic Nash Equilibrium Problem for Medical Supply Competition

In this paper, we study the competition of healthcare institutions for medical supplies in emergencies caused by natural disasters. In particular, we develop a two-stage procurement planning model in a random environment. We consider a pre-event policy, in which each healthcare institution seeks to minimize the purchasing cost of medical items and the transportation time from the first stage, and a recourse decision process to optimize the expected overall costs and the penalty for the prior plan, in response to each disaster scenario. Thus, each institution deals with a two-stage stochastic programming model that takes into account the unmet demand at the first stage, and the consequent penalty. Then, the institutions simultaneously solve their own stochastic optimization problems and reach a stable state governed by the stochastic Nash equilibrium concept. Moreover, we formulate the problem as a variational inequality; both the discrete and the general probability distribution cases are described. We also present an alternative formulation using infinite-dimensional duality tools. Finally, we discuss some numerical illustrations applying the progressive hedging algorithm.

A novel deep-learning based surrogate modeling of stochastic electric vehicle traffic user equilibrium in low-carbon electricity–transportation nexus

The increasing penetration of electric vehicles (EV) and fast charging stations (FCS) is tightly coupling the operation of power and transportation systems. In this context, the characterization of the EV flows and charging demand in response to varying traffic conditions and coordinated optimization strategies play a vital role. Previous work on the computation of stochastic traffic user equilibrium (TUE) involve non-linearities in the traffic link and FCS congestion representations and are generally inefficient in dealing with the multi-source uncertainties associated with the operating conditions of the traffic network (TN) and power distribution network (PDN). To address this, this paper proposed a novel deep learning (DL) based surrogate modeling method, leveraging the strength of edge-conditioned convolutional network (ECCN) and deep belief network (DBN). ECCN enables automatic extraction of spatial dependencies, taking into account both node and edge features characterizing the operation of TN. DBN leverages the value of the extracted features and achieves an accurate mapping between the latter to the EV charging demand and EV flows in the TUE, while adaptively generalizing to the multi-dimensional uncertainties. Case studies on three test systems of different scales (including a real-world case involving the matched TN and PDN of Nanjing city) demonstrate that the proposed surrogate model achieves a higher solution accuracy with respect to the state-of-the-art DL-based methods, and exhibits favorable computational performance. Quantitative results also corroborate the benefits brought by the proposed coordinated spatial optimization of EV flows and charging demand on the operation of both TN and PDN.

Data envelopment analysis and stochastic equilibrium analysis for market power investigation in a bi-level market

This study investigates how the exercise of market power affects the economic efficiency and revenue sharing in a bi-level market, which integrates upstream contract market and downstream wholesale market. We consider price volatility and identify Nash equilibrium restricted by the data envelopment analysis (DEA)-estimated capacity constraint. This study develops stochastic mixed complementarity problems (MiCP) models embedded with stochastic programming (SP) and robust optimization (RO) to address price uncertainty in a bi-level market. We investigate the exercise of market power: (1) wholesalers as leaders or (2) bilateral market power between manufacturers and wholesalers. A numerical study of the power generation market is conducted to validate the proposed stochastic MiCP models. The results show that price uncertainties in the upstream contract market have a significant effect in a bi-level market, and thus controlling price volatility in the upstream contract market will improve productivity and support policies. In addition, we suggest the wholesalers should be market leaders and the manufacturers should negotiate profit-sharing contracts with the wholesalers in a deregulated system.

A macro-financial perspective to analyse maturity mismatch and default

The Basel Committee proposed the Net Stable Funding Ratio (NSFR) to curb excessive maturity mismatch within the banking sector. However, it remains to be ascertained as to what are the financial and real effects of the NSFR on bank credit quality, investment, and the pass-through of monetary policy. This paper develops a nominal dynamic general equilibrium model featuring bank maturity mismatch and the moral hazard due to costly monitoring. First, I show that a tightening of the NSFR to move loan maturity towards the long-run capital investment cycle would only increase real investment if it sufficiently improved bank credit quality. Then in the numerical example calibrated with the US data, I show that such tightening of the NSFR can indeed increase real investment and also reduce the aggregate fluctuation of the economy. In the steady states, a 10% tightening in the NSFR can decrease net charge-offs of non-performing loans by about 0.06 pp annually, despite squeezing bank interest margins. Moreover, the moral hazard stemming from banks’ unobserved monitoring efforts impairs the pass-through of monetary policy. However, a 10% tightening in the NSFR improves the pass-through of a 20-bp policy rate reduction by around 17% annually. Finally, the model simulates the stochastic dynamic equilibrium path to study the propagation of shocks, demonstrating that the NSFR complements monetary policy in reducing financial frictions.

A game-theoretic perspective to study a nonlinear stochastic parabolic model of population competition

In this paper, a game of competition between two populations is designed. Populations are considered as two players competing against each other to increase their sizes. Players have been provided with costly two-component strategies. Population densities are described by two coupled nonlinear stochastic second-order parabolic equations. We have added stochastic terms to the deterministic model to consider random perturbations. First, we have considered the stochastic payoff functions and proved the game has a unique stochastic Nash equilibrium. Then, we have shown the stochastic Nash equilibrium is also a Nash equilibrium for the game with deterministic payoff functions defined by expectation. Cooperative game is also studied and stochastic Pareto efficient strategies are obtained. Then, the existence and uniqueness analyses of the deterministic Nash equilibrium and deterministic Pareto efficient strategies for the game with deterministic payoff functions are provided. The explicit forms of Nash equilibriums and Pareto efficient strategies are presented. Results about the losses the players experience when they don't cooperate are given. Finally, some numerical experiments are included to illustrate what we have claimed in the statements of theorems and show the effects of the Nash equilibrium and Pareto efficient strategy on the payoffs of the players.