Mixed Equilibrium Research Articles

Coevolutionary Algorithm for Building Robust Decision Trees under Minimax Regret

In recent years, there has been growing interest in developing robust machine learning (ML) models that can withstand adversarial attacks, including one of the most widely adopted, efficient, and interpretable ML algorithms—decision trees (DTs). This paper proposes a novel coevolutionary algorithm (CoEvoRDT) designed to create robust DTs capable of handling noisy high-dimensional data in adversarial contexts. Motivated by the limitations of traditional DT algorithms, we leverage adaptive coevolution to allow DTs to evolve and learn from interactions with perturbed input data. CoEvoRDT alternately evolves competing populations of DTs and perturbed features, enabling construction of DTs with desired properties. CoEvoRDT is easily adaptable to various target metrics, allowing the use of tailored robustness criteria such as minimax regret. Furthermore, CoEvoRDT has potential to improve the results of other state-of-the-art methods by incorporating their outcomes (DTs they produce) into the initial population and optimize them in the process of coevolution. Inspired by the game theory, CoEvoRDT utilizes mixed Nash equilibrium to enhance convergence. The method is tested on 20 popular datasets and shows superior performance compared to 4 state-of-the-art algorithms. It outperformed all competing methods on 13 datasets with adversarial accuracy metrics, and on all 20 considered datasets with minimax regret. Strong experimental results and flexibility in choosing the error measure make CoEvoRDT a promising approach for constructing robust DTs in real-world applications.

Efficient equilibria in common interest voting games

We develop a unified derivation of asymmetric pure strategy equilibria and their optimality in the canonical common interest voting model of Austen-Smith and Banks (Am Polit Sci Rev 90(1):34–45, 1996). We also study the relationship between the most efficient equilibria, which have a remarkably simple and intuitive structure, and the symmetric mixed strategy equilibrium that has been commonly studied in the literature. In particular, while the efficiency in the symmetric mixed strategy equilibrium under unanimity rule is known to be decreasing in the number of voters, the efficiency does not depend on the number of voters above a threshold in the most efficient equilibria.

Deep learning-based pore network generation: Numerical insights into pore geometry effects on microstructural fluid flow behaviors of unconventional resources

Pore-scale transport behaviors and mechanisms of rock reservoirs are still not well understood to increase unconventional resource production. This work mainly focuses on proposing a deep learning-based method to rapidly construct optimal pore network with different pore types, and deeply analyze its effects on pore-scale transport behaviors and mechanisms. The pore-scale variables reservoir evaluation indexes are defined to quantitatively evaluate pore geometry effects on the properties and production of rock reservoirs. The two-phase displacement simulations in pore network are conducted to study microstructural flow behaviors and transport mechanisms. Results suggest that the deep learning-based digital labeling algorithm (DL-DLA) has excellent abilities to rapidly construct pore network with errors less than 5%, compared with the classical algorithms. Square pores and circle throats are suggested as the optimal pore network assembly, considering the fluid phase drainage efficiency and production rate. The microstructural transport mechanisms are concluded as the pore-throat drainage, pore-filling, fluid phase mixing and fluid phase equilibrium processes. The novel theoretical relation between fluid phase drainage and microscopic production indexes provides effective tools to estimate rock reservoir production with errors all less than 10%, which are helpful for the technique developments to increase the production of unconventional resources in rock reservoirs.

Strong convergence of inertial shrinking projection method for split best proximity point problem and mixed equilibrium problem

Strong convergence of inertial shrinking projection method for split best proximity point problem and mixed equilibrium problem

Endogenous price discrimination with asymmetric firms

We explore a third-degree spatial price discrimination model within the context of asymmetric duopolies. We find that if the degree of asymmetry is small, both firms charge uniform price is the unique sub-game perfect Nash equilibrium. As the degree of asymmetry reaches intermediate, there exists no pure strategy sub-game perfect Nash equilibrium. In the mixed strategy equilibrium, both firms price discriminate with certain probability. When the degree of asymmetry is large, there exists one or two asymmetric equilibria: the advantaged firm price discriminating and the disadvantaged firm uniformly pricing IN and vice versa in the other NI. In both equilibria, the price-discriminating firm consistently benefits, and the uniform-pricing firm also experiences higher profits when the degree of asymmetry is large, despite being driven out of the smaller market. Moreover, when the asymmetry is large enough, social welfare can also be higher.

Routing of multi-modal autonomous vehicles for system optimal flows and average travel cost equilibrium over time

In the emerging era of autonomous vehicles (AVs), an important question is how to integrate AVs in a multi-modal framework. Recent studies on Transportation Management Centres (TMCs) for future AVs addressed the issue of providing system optimal (SO) solution. However, none of them considered the possibility of developing a multi-modal SO-based solution. In this regard, we propose a novel multi-modal Transportation Management Centre (MMTMC) for future AVs, which controls both the routing and mode use of each traveller and distributes the AVs in such a way that SO flows are maintained for the whole network every day, while travellers on the same origin-destination (OD) have the same Multi-modal Average Travel Time (MMAVT) over a cycle or period of time (in days). That is, travellers on the same OD may encounter different travel times or costs on different days, but over the planning cycle of many days, their travel times will all average to be the same MMAVT, hence preserving the MMAVT equilibrium. Travellers will travel on single occupancy vehicles (SOVs) or high occupancy vehicles (HOVs) on different routes for different days within the planning cycle, while maintaining the same MMAVT overall for the cycle. Travellers on SOVs will experience no crowdedness while on HOVs will experience crowdedness. We model both the cases of minimum system time and minimum system cost (including crowdedness cost), and provide link-based solution first for both cases. For minimum system time, we apply the conventional Frank Wolfe algorithm to generate feasible link flows, whereas for the minimum system cost problem, we incorporate a sensitivity-based equilibrium procedure to derive the SOV, HOV link flows and demand profiles. Afterwards, we utilize the most likely path method to compute unique SOV and HOV path flows and solve the MMAVT equilibrium. Then, we investigate the case of mixed-equilibrium (ME), where some travellers are given the option not to subscribe to the MMTMC and do their own private routing, i.e. user equilibrium (UE) travellers, who are penalized by a toll. The tolls imposed on these MMTMC non-subscribers are defined by the travel time/cost difference between the MMAVT of subscribers and UE users based on their value-of-time (VOT) distribution. This study investigates the properties of this MMTMC routing pattern, the relationship between MMAVT, tolling, and SOV and HOV market penetration, which offers insights to integrate AVs in a multi-modal framework.

Distributed strategies for mixed equilibrium problems: Continuous-time theoretical approaches

In this paper, the mixed equilibrium problem is studied by employing a continuous-time multi-agent system, where the objective of agents is to cooperatively find a point from the feasible set such that the sum of bifunctions with a free-variable is nonnegative. Different from existing works on mixed equilibrium problems with only convex set constraints, we consider a more general case where the feasible set is constrained by a set of convex inequalities. Moreover, each agent only has access to the information of its own bifunction and inequality constraint function, as well as the information of a local convex set. To handle this problem, the Karush–Kuhn–Tucker condition that is suitable for continuous-time theoretical approaches is provided, and a continuous-time distributed primal–dual algorithm is proposed. Under mild assumptions on the graph and bifunctions, we prove that the states of all agents asymptotically converge to the common solution of the mixed equilibrium problem. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.

Why cultural distance can promote - or impede - group-beneficial outcomes.

Quantifying the distance between cultural groups has received substantial recent interest. A key innovation, borrowed from population genetics, is the calculation of cultural FST (CFST) statistics on datasets of human culture. Measuring the variance between groups as a fraction of total variance, FST is theoretically important in additive models of cooperation. Consistent with this, recent empirical work has confirmed that high values of pairwise CFST (measuring cultural distance) strongly predict unwillingness to cooperate with strangers in coordination vignettes. As applications for CFST increase, however, there is greater need to understand its meaning in naturalistic situations beyond additive cooperation. Focusing on games with both positive and negative frequency dependence and high-diversity, mixed equilibria, we derive a simple relationship between FST and the evolution of group-beneficial traits across a broad spectrum of social interactions. Contrary to standard assumptions, this model shows why FST can have both positive and negative marginal effects on the spread of group-beneficial traits under certain realistic conditions. These results provide broader theoretical direction for empirical applications of CFST in the evolutionary study of culture.

Mixed strategy Nash equilibrium for scheduling games on batching-machines with activation cost

Mixed strategy Nash equilibrium for scheduling games on batching-machines with activation cost

A New Inertial-Projection Method for Solving Split Generalized Mixed Equilibrium and Hierarchical Fixed Point Problems

In this paper, we introduce a new iterative algorithm of inertial form for approximating the common solution of Split Generalized Mixed Equilibrium Problem (SGMEP) and Hierarchical Fixed Point Problem (HFPP) in real Hilbert spaces. Motivated by the subgradient extragradient method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and lower semicontinuity of the SGMEP and HFPP associated mappings, we establish the strong convergence of the iterative algorithm. Some numerical experiments are presented to illustrate the performance and behaviour of our method as well as comparing it with some related methods in the literature.

Modified monotone hybrid inertial algorithm for generalized nonexpansive maps, maximal monotone operators and generalized mixed equilibrium problems

Modified monotone hybrid inertial algorithm for generalized nonexpansive maps, maximal monotone operators and generalized mixed equilibrium problems

A bi-level framework for heterogeneous fleet sizing of ride-hailing services considering an approximated mixed equilibrium between automated and non-automated traffic

Ride-hailing companies will face the emergence and gradual expansion of AVs-only zones in urban areas where only automated vehicles (AVs) are allowed to circulate. When owning a mixed fleet (automated and conventional taxis), a ride-hailing company has to determine the optimal fleet size as a function of the gradually expanding coverage of AVs-only zones while taking into account interactions with privately-owned human-driven vehicles. To model this problem, we propose a bi-level framework in which the lower level captures the mixed routing behaviour of the vehicles and the endogenous traffic congestion, and the upper level determines fleet sizes to maximise profit. A parallel genetic algorithm is introduced to solve this bi-level framework, which is embedded with a tailored algorithm for solving the lower-level model. Numerical experiments are conducted on instances based on a small network and the network of the city of Delft, The Netherlands, to demonstrate the performance of the proposed solution method and investigate the impacts of AVs-only zones on traffic and ride-hailing operations. Results indicate that the fleet size of automated taxis increases nonlinearly with the expansion of the AVs-only zone while that of conventional taxis decreases as demand shifts from human-driven vehicles to automated taxis. The fleet size decision depends heavily on the fleet’s cost structure, the location and the distribution of parking depots. Furthermore, the existence of an AVs-only zone leads to detours for human-driven vehicles in the early stages, but it will bring major benefits by reducing congestion as its size increases.

Research on Exhibition Marketing Strategies in the Context of Network Era

Abstract In the network era, consumers’ behavior of shopping through the Internet can be seen everywhere in daily life. In this paper, starting from consumers’ online shopping behavior preference, the equilibrium mode of conventional advertising is studied, based on which the mass and targeted advertising strategies under decentralized and centralized decision-making modes are proposed, respectively. After comparing and studying the marketing channel structure in various modes, the corresponding advertising optimization model was established to derive four theorems. According to the deduced theorems, the enterprise revenue model is constructed and solved to maximize the benefits. Compare the mixed and separated equilibrium models of the two-stage model in the sales process, and at the same time, construct the XGBoost algorithm model to classify the characteristics of consumers’ shopping behaviors and collect the enterprise’s product marketing strategies. The results of the empirical experiments show that the reasons why consumers do not buy the exhibits of Company W are mainly due to “the price is too high”, “the sense of participation is insufficient” and “the attraction is too low”, with the values of 2.5, 2.5, 2.5, 2.5 and 2.5 respectively. The number of consumers who are satisfied and very satisfied with W brand exhibitions is the largest, and the number of people in all four aspects is around 50, so the marketing strategy of W company exhibitions adjusted according to consumer behavioral preferences is effective.

Effects of excess oxidizer coefficient on RBCC engine performance in ejector mode: A theoretical investigation

This study investigates the effects of the primary rocket’s excess oxidizer coefficient (EOC) on the performance of a kerosene-fueled rocket-based combined cycle (RBCC) engine operating in ejector mode using the diffuser and afterburning (DAB) cycle. Employing complete mixing and chemical equilibrium assumptions, a quasi-one-dimensional model is developed to simulate combustion processes in the primary rocket chamber, mixer, and afterburner. Experimental data and full three-dimensional simulation validate the model’s accuracy. Results indicate that increasing EOC mitigates thermal choking effects at the mixer exit, decreasing mixing and Rayleigh losses, enhancing the entrainment ratio, and providing more oxygen for secondary combustion. However, the gas mixture’s compression ratio and total temperature decrease, rendering secondary combustion more susceptible to thermal choking in a constant cross-sectional afterburner. As the flight Mach number increases from 0.8 to 2.5, the optimum EOC for maximum specific impulse shifts from 0.725 to 2.375. While the optimum EOC for maximum thrust remains consistently around 2.0. The findings underscore the necessity of a higher EOC to fully unlock the DAB cycle’s thrust augmentation potential.

Mixed nash equilibrium and its application in two-player zero-Sum games

The Colonel Blotto game represents a classic case in game theory, embodying the essence of two-player zero-sum games. A vital game theory theorem has demonstrated a mixed Nash equilibrium in all zero-sum games. In this paper, the author will start with this classic game to expound the principles of mixed Nash equilibrium. Furthermore, the paper will explore the practical applications of mixed Nash equilibrium in the context of two-player zero-sum games, illustrating its significance in addressing real-world issues and decision-making. By delving into the Colonel Blotto game and its associated theories, the paper will reveal the importance and broad applicability of mixed Nash equilibrium, showcasing its role in various real-world applications, including competitive strategies, resource allocation, risk management, and more. This research will assist readers in gaining a better understanding of the concept of mixed Nash equilibrium and how to apply it to the resolution of practical problems.

Strategic driver repositioning in ride-hailing networks with dual sourcing

In ride-hailing markets, the spatial mismatch between supply and demand characterizes one of the most fundamental operational bottlenecks. The voluntary customer search of drivers is oftentimes inadequate for narrowing the demand-supply gaps and falls short in responding timely to the dynamic market conditions. To facilitate better supply management, some ride-hailing platforms now adopt a dual-sourcing strategy by converting some freelance drivers as contractors, who would then follow platforms’ guidance on repositioning movements when idle. This paper thus aims to analyze the implications of such a dual-sourcing strategy on the overall operations of ride-hailing services. We consider a monopolistic platform that owns a blended workforce, with freelance drivers searching for customers following their own preferences whilst contracted drivers repositioning in a centralized manner as system actuators. A mixed network equilibrium model is devised to capture the interplay between the idle repositioning movements of both parties. The theoretical existence of the mixed equilibrium is proved, and real-world data is invited to quantify the effects of dual sourcing on various market sectors. Our numerical results confirm the effectiveness of dual sourcing in smoothing operations, increasing customer demand, and driving up the revenue of ride-hailing services. However, the benefit of dual sourcing to the platform remains sensitive to the proportion of contractors in the entire workforce as well as the stimulative commission beyond their regular service income.

Games on graphs with coalitions

This paper models games where the strategies are nodes of a graph G (we denote them as G-games) and in presence of coalition structures. The cases of one-shot and repeated games are presented. In the latter situation, coalitions are assumed to move from a strategy to another one under the constraint that they are adjacent in the graph. We introduce novel concepts of pure and mixed equilibria which are comparable with classical Nash and Berge equilibria. A Folk Theorem for G-games of repeated type is presented.

Pure strategy Nash equilibria for bargaining models of collective choice

This paper considers pure strategy Nash equilibria of non-cooperative legislative bargaining models. In contrast to existing legislative bargaining models, we derive legislators behavior from stochastic utility maximization. This approach allows us to prove the existence of a stationary Pure Local and Global Nash Equilibrium under rather general settings. The mathematical proof is based on a fixed point argument, which can also be used as a numerical method to determine an equilibrium. We characterize the equilibrium outcome as a lottery of legislators’ proposals and prove a Mean Voter Theorem, i.e., proposals result dimension-by-dimension as a weighted mean of legislators’ ideal points and are Pareto-optimal. Based on a simple example, we illustrate different logic of our model compared to mixed strategy equilibrium of the legislative bargaining model suggested by Banks and Duggan (Am Polit Sci Rev 94(1):73–88. https://doi.org/10.2307/2586381, 2000).

Study of Education and Punishment Based on Game Theory

Punishment as a means to regulate students behavior has reached a consensus in Chinese education for thousands of years, but it does not play a sufficient role in guiding the specif-ic problems in todays social practice. Based on the idea of game theory, this paper con-structs a mixed strategy game model in which schools, teachers and students participate at the same time, and through the analysis of the mixed strategy Nash equilibrium of the game model, puts forward methods to reduce students mistakes and improve students moral level. The results show that the probability of teachers implementation of educa-tional discipline is mainly related to the degree of punishment for teachers failure to im-plement educational discipline, the income and cost of paying attention to students moral cultivation; Students wrong behaviors are influenced by the spiritual benefits, costs, pun-ishments, rewards and costs obtained by teachers implementation of educational punish-ments, punishments obtained by teachers non implementation of educational punish-ments, and the benefits and costs of schools emphasis on students moral cultivation; It will be more helpful to improve the efficiency of the standardized training of students by improving the punishment for teachers failure to implement education discipline and stu-dents wrong behavior and reducing the cost of paying attention to students moral training.

Potential changes of regional natural gas market in China amidst liberalization: A mixed complementarity equilibrium simulation in 2030

Ensuring the continuous efficient use of natural gas is crucial for China to achieve its carbon peak goal by 2030. This study scrutinizes the prospective effects of an intensified liberalization of China’s natural gas sector in 2030. To accomplish this, a mixed complementarity equilibrium model is employed, capturing the interdependencies among relevant sectors within the industrial chain. Multiple scenarios of liberalization are simulated, encompassing variations in gas demand, pipeline capacity expansion, and the escalation of imported Liquefied Natural Gas (LNG), to identify potential modifications. Scenario results reveal that the insufficient capacity of long-distance pipeline network will significantly hinder the distribution of gas resources and cause a shortage of gas supplies in Eastern region. Furthermore, the expansion of pipeline capacity from Northwest to East is beneficial in low demand scenario, while increasing the receiving and transfer capacity of LNG in Northern and Eastern regions is crucial in face of strong demand growth. Moreover, a decline in LNG imports will put upward pressure on regional prices, and further affect the security of natural gas supply. The findings will offer critical data for policymakers and present a diversified view of natural gas liberalization market under China’s peak carbon emission target in 2030.