Robust Equilibrium Research Articles

RCEMIP-II: mock-Walker simulations as phase II of the radiative–convective equilibrium model intercomparison project

Abstract. The radiative–convective equilibrium (RCE) model intercomparison project (RCEMIP) leveraged the simplicity of RCE to focus attention on moist convective processes and their interactions with radiation and circulation across a wide range of model types including cloud-resolving models (CRMs), general circulation models (GCMs), single-column models, global cloud-resolving models, and large-eddy simulations. While several robust results emerged across the spectrum of models that participated in the first phase of RCEMIP (RCEMIP-I), two points that stand out are (1) the strikingly large diversity in simulated climate states and (2) the strong imprint of convective self-aggregation on the climate state. However, the lack of consensus in the structure of self-aggregation and its response to warming is a barrier to understanding. Gaining a deeper understanding of convective aggregation and tropical climate will require reducing the degrees of freedom with which convection can vary. Therefore, we propose phase II of RCEMIP (RCEMIP-II) that utilizes a prescribed sinusoidal sea surface temperature (SST) pattern to provide a constraint on the structure of convection and move one critical step up the model hierarchy. This so-called “mock-Walker” configuration generates features that resemble observed tropical circulations. The specification of the mock-Walker protocol for RCEMIP-II is described, along with example results from one CRM and one GCM. RCEMIP-II will consist of five required simulations: three simulations with the same three mean SSTs as in RCEMIP-I but with an SST gradient and two additional simulations at one of the mean SSTs with different values of the SST gradients. We also test the sensitivity to the imposed SST gradient and the domain size. Under weak SST gradients, unforced self-aggregation emerges across the entire domain, similar to what was found in RCEMIP. As the SST gradient increases, the convective region narrows and is more confined to the warmest SSTs. At warmer mean SSTs and stronger SST gradients, low-frequency variability in the convective aggregation emerges, suggesting that simulations of at least 200 d may be needed to achieve robust equilibrium statistics in this configuration. Simulations with different domain sizes generally have similar mean statistics and convective structures, depending on the value of the SST gradient. The prescribed SST boundary condition is the only difference in the set-up between RCEMIP-II and RCEMIP-I, which enables comparison between the two; however, we also welcome participation in RCEMIP-II from models that did not participate in RCEMIP-I.

Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

Non-zero-sum investment-reinsurance game with delay and ambiguity aversion

In this paper, we investigate a investment-reinsurance game problem with delay and ambiguity aversion for two insurers under mean–variance preference. Each insurer can purchase proportional reinsurance whose surplus process is a diffusion approximation process, and invest oneself’s wealth in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a jump–diffusion process. Based on the introduction of delay, we obtain wealth dynamics for each insurer depicted by a stochastic delay differential equation. By the stochastic control theory in the framework of game theory, the robust time-consistent Nash equilibrium investment-reinsurance strategies and the corresponding robust equilibrium value functions for each insurer are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment-reinsurance strategy.

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

DR-Label: Label Deconstruction and Reconstruction of GNN Models for Catalysis Systems

Attaining the equilibrium geometry of a catalyst-adsorbate system is key to fundamentally assessing its effective properties, such as adsorption energy. While machine learning methods with advanced representation or supervision strategies have been applied to boost and guide the relaxation processes of catalysis systems, existing methods that produce linearly aggregated geometry predictions are susceptible to edge representations ambiguity, and are therefore vulnerable to graph variations. In this paper, we present a novel graph neural network (GNN) supervision and prediction strategy DR-Label. Our approach mitigates the multiplicity of solutions in edge representation and encourages model predictions that are independent of graph structural variations. DR-Label first Deconstructs finer-grained equilibrium state information to the model by projecting the node-level supervision signal to each edge. Reversely, the model Reconstructs a more robust equilibrium state prediction by converting edge-level predictions to node-level via a sphere-fitting algorithm. When applied to three fundamentally different models, DR-Label consistently enhanced performance. Leveraging the graph structure invariance of the DR-Label strategy, we further propose DRFormer, which applied explicit intermediate positional update and achieves a new state-of-the-art performance on the Open Catalyst 2020 (OC20) dataset and the Cu-based single-atom alloys CO adsorption (SAA) dataset. We expect our work to highlight vital principles for advancing geometric GNN models for catalysis systems and beyond. Our code is available at https://github.com/bowenwang77/DR-Label

Robust reinsurance contract and investment with delay under mean-variance framework

We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract.

An Extended McKean–Vlasov Dynamic Programming Approach to Robust Equilibrium Controls Under Ambiguous Covariance Matrix

An Extended McKean–Vlasov Dynamic Programming Approach to Robust Equilibrium Controls Under Ambiguous Covariance Matrix

Dynamic Pricing Provides Robust Equilibria in Stochastic Ridesharing Networks

Using prices induced by dual variables of a centralized optimization problem induces welfare-optimal equilibria among strategic drivers. We reveal a stark deficiency of such static pricing algorithms: it is possible for them to induce additional equilibria with arbitrarily low social welfare. Moreover, small perturbations to the marketplace, such as those caused by idiosyncratic randomness or model misspecification, can cause the welfare-optimal equilibrium to be Pareto-dominated (in terms of driver utility) by suboptimal equilibria. We show that dynamic pricing solves this problem. We describe a dynamic pricing algorithm that resolves the centralized optimization problem in each time period and show that it satisfies a new equilibrium robustness property, which guarantees that every induced (approximate) equilibrium is (approximately) welfare optimal. We also propose a novel two-level model of ridesharing networks with strategic drivers and spatiotemporal dynamics that lets us retain macroscopic uncertainty, such as correlated shocks caused by weather or other public events, when analyzing a large market limit in which idiosyncratic sources of uncertainty vanish. Funding: J. M. Cashore was supported by an NSERC PGS D Fellowship. P. Frazier was supported by AFOSR [Grant FA9550-19-1-0283]. É. Tardos was supported by AFOSR [Grant FA9550-19-1-0183] and [NSF Grants CCF-1408673 and CCF-1563714]. Supplemental Material: The online companion is available at https://doi.org/10.1287/moor.2022.0163 .

Robust equilibria in tournaments

We study robust equilibria in tournaments, where agents endowed with power form coalitions, and the coalition formed with the highest power prevails. We introduce the No-Threat Equilibrium (NTE), a stable partition where if a coalition deviates, then a new coalition could counter by forming an even stronger coalition. The NTE exists for any power function and preferences if and only if the set of feasible coalitions is a ‘Helly’ family.In contrast, the core is a partition in which no group of agents can profitably deviate by forming a feasible coalition, assuming that other agents do not react to such a deviation. The core is not empty for any power function and preferences if and only if the set of feasible coalitions has a ‘hierarchical-structure’.The paper also adapts and characterizes other stability concepts to tournaments, including the α-core, β-core, farsighted core and bargaining set.

Managers’ choice of disclosure complexity

Aghamolla and Smith (2023) make a significant contribution to enhancing our understanding of how managers choose financial reporting complexity. I outline the key assumptions and implications of the theory, and discuss two empirical implications: (1) a U-shaped relationship between complexity and returns, and (2) a negative association between complexity and investor sophistication. However, the robust equilibrium also implies a counterfactual positive market response to complexity. I develop a simplified approach in which simple disclosures indicate positive surprises, and show that this implies greater investor skepticism toward complexity and a positive association between investor sophistication and complexity. More work is needed to understand complexity as an interaction of reporting and economic transactions, rather than solely as a reporting phenomenon.

Strategic mistakes

To study the equilibrium implications of decision frictions, we introduce a new class of control costs in continuum-player, continuum-action games in which agents interact via an aggregate of the actions of others. The costs that we study accommodate a rich class of decision frictions, including ex post misoptimization, imperfect ex ante planning, cognitive constraints that depend endogenously on the behavior of others, and consideration sets. We provide primitive conditions such that equilibria exist, are unique, are efficient, and feature monotone comparative statics for action distributions, aggregates, and the size of agents' mistakes. We apply the model to make robust equilibrium predictions in a monetary business-cycle model of price-setting with planning frictions and a model of consumption and savings during a liquidity trap when endogenous stress worsens decisions.

Investigating the Effects of Automated Vehicles on Large Urban Road Networks: Some Evidence from Rome

This paper explores the possibility of applying simulation models formalized in the macrosimulation approach to predict the effects from the presence of automated vehicles in our cities. It is based on the use of a robust equilibrium assignment model allowing us to obtain multiclass traffic flows, including automated vehicles (AVs) and conventional ones (CVs) on large real-sized road networks. This modelling framework has been successfully applied to the road network of the metropolitan area of Rome, allowing us to assess the effects of AVs in future traffic at increasing penetration rates and the effects of possible transport policies involving AVs.

Modeling data-driven adaptive distributionally robust equilibrium last mile relief network under centrality metric

This paper studies a variant of the two-stage last mile relief network problem by introducing a centrality metric objective including the distance centrality metric, the demand centrality value-at-risk and the residual supplies centrality metric. Under the scenario-based design, the demand in each scenario is described as a random fuzzy variable whose probability distribution is ambiguous. In order to characterize its ambiguousness, this paper designs a Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm based on sample data to construct a data-driven ambiguity set containing the support and mean, which not only realizes the visualization of the ambiguous probability distribution, but also provides a finite set of scenarios for the uncertain demand. Given the constructed data-driven ambiguity set, this paper proposes a data-driven adaptive distributionally robust equilibrium programming (DDADREP) method for the last mile relief network problem (LMRNP). In view of its computational difficulty, the safe approximation, duality technique and credibility measure theory are used to approximate it as a mixed integer second-order cone programming with significant computational advantages. Finally, the proposed model is validated with two real-world cases, and its effectiveness as well as advantages is demonstrated by comparison with the classical robust optimization and the nominal equilibrium optimization, respectively.

Robust equilibrium strategies for time-inconsistent stochastic optimal control problems with applications

In this paper, a new robust time-inconsistent stochastic optimal control problem is investigated under a general jump-diffusion model. Such a problem can be described as a sup-sup-inf problem consisting of the optimal control and the optimal selection of a probability measure that reflects the optimistic and pessimistic sentiments of the agent. Under a game-theoretic framework, the definition of the equilibrium control-measure strategy is given for the robust time-inconsistent stochastic optimal control problem and then an extended Hamilton–Jacobi–Bellman–Isaacs (HJBI) system and a verification theorem are derived for characterizing the equilibrium control-measure strategy and the corresponding equilibrium value function. Moreover, some financial optimization problems and numerical experiments are provided to illustrate the applicability of our newly derived results.

Lower Bounds on Implementing Mediators in Asynchronous Systems with Rational and Malicious Agents

Abraham, Dolev, Geffner, and Halpern [ 1 ] proved that, in asynchronous systems, a (k, t)-robust equilibrium for n players and a trusted mediator can be implemented without the mediator as long as n > 4( k+t ), where an equilibrium is ( k, t )-robust if, roughly speaking, no coalition of t players can decrease the payoff of any of the other players, and no coalition of k players can increase their payoff by deviating. We prove that this bound is tight, in the sense that if n ≤ 4( k+t ) there exist ( k, t )-robust equilibria with a mediator that cannot be implemented by the players alone. Even though implementing ( k, t )-robust mediators seems closely related to implementing asynchronous multiparty ( k+t )-secure computation [ 6 ], to the best of our knowledge there is no known straightforward reduction from one problem to another. Nevertheless, we show that there is a non-trivial reduction from a slightly weaker notion of ( k+t )-secure computation, which we call ( k+t )-strict secure computation , to implementing ( k, t )-robust mediators. We prove the desired lower bound by showing that there are functions on n variables that cannot be ( k+t )-strictly securely computed if n ≤ 4( k+t ). This also provides a simple alternative proof for the well-known lower bound of 4 t +1 on asynchronous secure computation in the presence of up to t malicious agents [ 4 , 8 , 10 ].

Robust equilibrium investment-reinsurance strategy for n competitive insurers with square-root factor process

We investigate a robust equilibrium investment-reinsurance problem for n ambiguity-averse competitive insurers, . Each insurer is allowed to purchase proportional reinsurance and invest in a risk-free asset and a risky asset. Each insurer aims to maximize the expected utility of a weighted relative terminal wealth with respect to the other competitors. In this article, the risky asset is assumed to follow a general and flexible model: the square root factor process. Following the game theory approach, we derive the closed solutions of the robust equilibrium investment-reinsurance strategies. Moreover, the verification theorem is provided in this article. Finally, we demonstrate some numerical analyses and give the economic explanations as well.

Robust equilibrium reinsurance and investment strategy for the insurer and reinsurer under weighted mean-variance criterion

<abstract><p>This paper investigates the time-consistent robust optimal reinsurance problem for the insurer and reinsurer under weighted objective criteria. The joint objective criterion is obtained by weighting the mean-variance objectives of both the insurer and reinsurer. Specifically, we assume that the net claim process is approximated by a diffusion model, and the insurer can purchase proportional reinsurance from the reinsurer. The insurer adopts the loss-dependent premium principle considering historical claims, while the reinsurance contract still uses the expected premium principle due to information asymmetry. Both the insurer and reinsurer can invest in risk-free assets and risky assets, where the risky asset price is described by the constant elasticity of variance model. Additionally, the ambiguity-averse insurer and ambiguity-averse reinsurer worry about the uncertainty of parameter estimation in the model, therefore, we obtain a robust optimization objective through the robust control method. By solving the corresponding extended Hamilton-Jacobi-Bellman equation, we derive the time-consistent robust equilibrium reinsurance and investment strategy and corresponding value function. Finally, we examined the impact of various parameters on the robust equilibrium strategy through numerical examples.</p></abstract>

Robust Pricing for a Dual-Channel Green Supply Chain Under Fuzzy Demand Ambiguity

The pricing decision is a fundamental problem in a two-echelon dual-channel supply chain, and it is a challenging issue to get the optimal pricing strategy due to high demand ambiguity. Although this problem has been studied by many researchers, the market demand is usually assumed to be deterministic or described as a fuzzy variable with exact distribution information in most of the existing literature. In this article, the uncertain demand is assumed to possess fuzzy uncertainty instead of stochastic nature. In the case that the demand distribution information is partially available, this article proposes a new uncertainty distribution set to depict the demand distribution uncertainty based on the type-2 fuzzy theory. Then, a novel robust pricing game modeling framework is developed for the dual-channel green supply chain with the demand distribution varying in the proposed uncertainty distribution set. Finally, a new method is proposed for obtaining the robust equilibrium decisions of supply chain members. Numerical analysis and comparisons are conducted to demonstrate the impacts of the demand ambiguity on the manufacturer’s equilibrium channel choice strategy. We also demonstrate that our solution results are robust in the setting that the prices of the green products are not equal in the retail and the direct channels.

Distributionally robust equilibrious hybrid vehicle routing problem under twofold uncertainty

A novel nonlinear hybrid vehicle routing problem is examined, and its nonlinear component is linearised considering the transportation costs associated with electricity and traditional fuel-based driving. Due to the fact that the transportation cost and fuel consumption involve the twofold uncertainty of randomness and fuzziness, and only partial probability distribution information may be available. Therefore, these two parameters are considered as random fuzzy variables with ambiguous probability distributions. An ambiguous equilibrium risk value objective function and an ambiguous equilibrium chance constraint are formulated. Accordingly, a distributionally robust equilibrium approach is proposed, in which the ambiguity sets are used to characterize the ambiguous probability distributions of the random fuzzy variables. Specifically, this paper first applies the central limit theorem to construct the ambiguity sets. Subsequently, the inner ambiguous probability constraint and the outer credibility constraint are derived into their equivalent counterparts, respectively. In this manner, the proposed model is successfully converted into a mixed integer second-order cone programming model, where the conventional branch-and-cut algorithm is adopted to obtain the optimal routing. Finally, the performance of the proposed model and its price of distributional robustness are verified in the numerical experiments. Overall, the main achievements of this paper are summarized as (1) the proposal of a distributionally robust equilibrium optimization model for a nonlinear hybrid vehicle routing problem, (2) the definitions of an ambiguous equilibrium risk value objective function and an ambiguous equilibrium chance constraint under twofold uncertainty, and (3) the derivation of an equivalent second-order cone programming model for computationally solvable.

Robust equilibrium strategies in a defined benefit pension plan game

This paper investigates the robust non-zero-sum games in an aggregated overfunded defined benefit pension plan. The sponsoring firm is concerned with the investment performance of the fund surplus, while the participants act like a union to claim a share of the fund surplus. The financial market consists of one risk-free asset and n risky assets. The firm and the union are ambiguous about the financial market and care about the robust strategies under the worst-case scenario. The union's objective is to maximize the expected discounted utility of the additional benefits. The firm's two objectives are to maximize the expected discounted utility of the fund surplus and the probability of the fund surplus reaching an upper level before hitting a lower level in the worst-case scenario. We present a general robust non-zero-sum game with stopping times, which contains the two objectives as special cases. Hamilton-Jacobi-Bellman-Isaacs equations and verification theorem are presented for the robust optimization problem. We obtain explicit solutions in the related two robust non-zero-sum games for the firm and the union. Numerical results are illustrated to depict the economic behaviors of the robust equilibrium strategies in these two different games.