Stackelberg Problem Research Articles

Mean Field Games in a Stackelberg Problem with an Informed Major Player

Mean Field Games in a Stackelberg Problem with an Informed Major Player

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.

Competitive Location: New Models and Methods and Future Trends

Two new models for sitting a new facility in a competitive environment are introduced. Both the location and the quality of the new facility are to be found, so as to maximize the profit obtained by the locating firm. The patronizing behavior of customers is assumed to be probabilistic, i.e., they split their demand among all the existing facilities in the area, proportionally to the attraction they feel for them. The attraction is determined both by the distance between the demand point and the facility and by the quality of the facility. Contrarily to what is commonly done in literature, the demand is not fixed, but varies depending on the location of the facilities. The first model assumes a static scenario, whereas in the second one a competing chain reacts by location a single new facility too, leading to a Stackelberg (or leader-follower) problem. The new continuous location models lead to hard-to-solve global optimization problems. A new evolutionary algorithm called UEGO was used to deal with those problems. The computational results showed its usefulness and robustness. Parallel implementations of UEGO are also presented to cope with large instances. The efficiency and scalability of the parallel algorithms were shown through a computational study. Future trends which will allow the construction of an expert system for facility location are also discussed.

Competitive scheduling in a hybrid flow shop problem using multi-leader–multi-follower game - A case study from Iran

It is logical that organizations pay more attention to their major customers which have more impacts on their abilities or benefits. In many real cases, decision makers in scheduling system must prioritize the sequence of processing orders according to the agent of orders while the classic production scheduling do not have the ability to consider the relationships between agents. These problems can be modeled by the Stackelberg problems in the field of game theory. In this study, an agent-based hybrid flow shop problem is formulated by using the multi-leader multi-follower mathematical model. Each level is evaluated by Nash equilibrium and Nash bargaining with the new co-evolutionary and genetic algorithm. The proposed model is validated in a real case problem in the tire manufacturing industry. The results show that Nash equilibrium is superior to Nash bargaining for all agents.

Patrol Strategies to Maximize Pristine Forest Area

Illegal extraction of forest resources is fought, in many developing countries, by patrols that try to make this activity less profitable, using the threat of confiscation. With a limited budget, officials will try to distribute the patrols throughout the forest intelligently, in order to most effectively limit extraction. Prior work in forest economics has formalized this as a Stackelberg game, one very different in character from the discrete Stackelberg problem settings previously studied in the multiagent literature. Specifically, the leader wishes to minimize the distance by which a profit-maximizing extractor will trespass into the forest---or to maximize the radius of the remaining ``pristine'' forest area. The follower's cost-benefit analysis of potential trespass distances is affected by the likelihood of being caught and suffering confiscation. In this paper, we give a near-optimal patrol allocation algorithm and a 1/2-approximation algorithm, the latter of which is more efficient and yields simpler, more practical patrol allocations. Our simulations indicate that these algorithms substantially outperform existing heuristic allocations.

A two-stage stochastic Stackelberg model for microgrid operation with chance constraints for renewable energy generation uncertainty

In order to reduce greenhouse gas emissions, countries worldwide are transforming their energy systems with higher shares of renewable energy and smart technologies for demand response. Microgrids play an essential role in the transformation of electric grids to smart grids. However, renewable sources present new challenges, particularly those of high variability, which creates uncertainties in the supply side that can affect the security of electricity access at affordable prices. This paper proposes a novel Stackelberg stochastic model to account for different sources of uncertainty. The Stackelberg model considers microgrids as leaders (upper-level problem) with uncertainty regarding the availability of wind and solar sources and electricity prices. Availability of renewable sources is modeled via chance constraints, which allows assessing the risk of microgrids over-committing supply levels. Uncertainty in electricity prices is modeled via a set of demand scenarios with a given probability distribution. The lower-level problem of the Stackelberg problem considers an electricity dispatch problem for each demand scenario. The proposed model allows measuring the strategic actions of microgrids when facing different types of uncertainties and how the smart grid should adapt to guarantee that demand levels are supplied. The results show the effectiveness of the proposed method. We find that microgrids risk levels above 30% do not correlate with further benefits, such as reduced electricity prices. We also identified that in average, depending on the social cost of carbon and demand level, microgrids can cover their own demand and supply 15% of the electricity demand in the grid.

Safe Search for Stackelberg Equilibria in Extensive-Form Games

Stackelberg equilibrium is a solution concept in two-player games where the leader has commitment rights over the follower. In recent years, it has become a cornerstone of many security applications, including airport patrolling and wildlife poaching prevention. Even though many of these settings are sequential in nature, existing techniques pre-compute the entire solution ahead of time. In this paper, we present a theoretically sound and empirically effective way to apply search, which leverages extra online computation to improve a solution, to the computation of Stackelberg equilibria in general-sum games. Instead of the leader attempting to solve the full game upfront, an approximate "blueprint" solution is first computed offline and is then improved online for the particular subgames encountered in actual play. We prove that our search technique is guaranteed to perform no worse than the pre-computed blueprint strategy, and empirically demonstrate that it enables approximately solving significantly larger games compared to purely offline methods. We also show that our search operation may be cast as a smaller Stackelberg problem, making our method complementary to existing algorithms based on strategy generation.

Optimal coordination strategy for multiple distributed energy systems considering supply, demand, and price uncertainties

Facing a significant increase in connected distributed energy systems, the optimal coordination strategy among multiple distributed energy systems is vitally important to explore. In this paper, an energy management system is introduced and operated by an energy service company, which is responsible for managing the interaction of multiple distributed energy systems. To optimize the day-ahead scheduling of the distributed energy systems, a coordination scheme with a bilevel framework is proposed. The energy interaction between the energy management system and distributed energy systems contains electricity and heat, which is a Stackelberg problem. Two types of internal price schemes, namely, real-time pricing and time-of-use pricing, are discussed. Moreover, the uncertainties of renewable energy resources, energy demand, and energy prices are considered within both upper- and lower-level problems. The problem is formulated as a nonlinear bilevel robust optimization model and transformed into a single-level mixed-integer linear problem. Numerical cases illustrate how the energy management system coordinates with distributed energy systems and show the effectiveness of the coordination strategy such that all participators benefit from the proposed strategy and create a win-win situation. The model and results can serve as references for the business managers of companies that provide energy services for building clusters.

A Multi-Objective Coordinated Charging and Discharging Strategy for Electric Vehicles Based on Stackelberg Game

For the negative impact of large-scale electric vehicles (EVs) disorderly charging on the power grid, a multi-objective optimization strategy for coordinated charging and discharging of EVs based on Stackelberg game is proposed. As the leader, the grid company aims to stabilize load fluctuations and formulate a reasonable electricity price strategy to guide EVs to participate in vehicle-to-grid (V2G); As followers, EV users optimize their charging plans based on electricity price information with the objective of reducing costs and obtaining good comfort. This paper uses the MOPSO algorithm to solve the proposed multi-objective Stackelberg problem, and calculates the optimization results under various preferences, which proves the effectiveness of the proposed model and method.

Variational Principles for supinf Problems with Constraints

Some recent results on variational principles of constrained supinf problems of the form $\sup_{x \in X} \inf_{y \in Kx} \{f(x,y)\}$ are presented. Conditions that ensure validity of the presented results in case of Stackelberg problem are provided.

Solving Stackelberg equilibrium for multi objective aerodynamic shape optimization

• The Stackelberg game is introduced into multi-objective aerodynamic shape optimization . • Numerical methods coupled (EAs) and/or adjoint method with a Stackelberg game are Presented. • Virtual Nash and virtual Stackelberg problems are equivalent, and their solutions are the same. • Final equilibrium solutions are only related to what information is exchanged. • The equilibrium of the weak form Stackelberg problem is as the same as the Nash equilibrium. Numerical methods coupling evolutionary algorithms and/or adjoint methods with a two level hierarchical game between leader and follower, called the Stackelberg game, are presented for solving multi objective aerodynamic shape optimization problems in this paper, in which both leader and follower can be a Nash sub-game with multiple players. The decision making process of leader and follower is hierarchically and non-cooperatively in numerical implementation, more accurately they are alternate. The leader enounces a decision (a Nash sub-game solution), and the follower takes into account this decision and solves the other Nash sub-game optimization problem. The elitist information that follower passes to leader is the response function of follower to leader’s decision, instead of its response only. Then, the leader optimizes his objective based on elitist information accepted from follower depending on his behavior. The approximate response function is obtained by recalculating the follower’s Nash sub-game with a small perturbation on leader’s decision. Finally, a multi-objective shape optimization problem of hypersonic air-breathing vehicle is solved by using the optimization algorithm presented in this paper, and the converged result of the algorithm is proved to be a Stackelberg equilibrium.

Possibilistic Stackelberg solutions to bilevel linear programming problems with fuzzy parameters

This article considers bilevel linear programming problems where the coefficients of the objective functions and the constraints in the problem are given as fuzzy parameters. Stackelberg solutions under fuzziness are defined by incorporating the notions of possibility theory into the original concept of Stackelberg solutions. It is shown that Stackelberg problems under fuzziness are transformed into deterministic bilevel linear or nonlinear programming problems, and that the resulting problems are exactly solved by using conventional bilevel linear or nonlinear programming techniques.

Control of Stackelberg for Coupled Parabolic Equations

We consider the Stackelberg problem for coupled parabolic equations with a finite number of constraints on one of the states. This notion assumes that we have two controls to determine. The first control is supposed to bring the solution of the coupled system subjected to a finite number of constraints at rest at time zero while the second expresses that the states do not move too far from given states. The results are achieved by means of an observability inequality of Carleman adapted to the constraints.

Solving Bilevel Multicriterion Optimization Problems With Lower Level Decision Uncertainty

Bilevel optimization problems are characterized by a hierarchical leader-follower structure, in which the leader desires to optimize her own strategy taking the response of the follower into account. These problems are referred to as Stackelberg problems in the domain of game theory, and as bilevel problems in the domain of mathematical programming. In a number of practical scenarios, a bilevel problem is solved by a leader who needs to take multiple objectives into account and simultaneously deal with the decision uncertainty involved in modeling the follower's behavior. Such problems are often encountered in strategic product design, homeland security applications, and taxation policy. However, the hierarchical nature makes the problem difficult to solve and they are commonly simplified by assuming a deterministic setup with smooth objective functions. In this paper, we focus our attention on the development of a flexible evolutionary algorithm for solving multicriterion bilevel problems with lower level (follower) decision uncertainty. The performance of the algorithm is evaluated in a comparative study on a number of test problems. In addition to the numerical experiments, we consider two real-world examples from the field of environmental economics and management to illustrate how the framework can be used to obtain optimal strategies.

Solving a Huff-like Stackelberg location problem on networks

This work deals with a Huff-like Stackelberg problem where the leader wants to locate a facility so that its profit is maximal after the competitor (the follower) has built its facility. We assume that the follower makes a rational decision, maximizing its own profit. The inelastic demand is aggregated into the vertices of a graph, and facilities can be located along the edges. For this computationally hard problem we give a Branch and Bound algorithm using interval analysis and DC bounds. Our computational experience shows that the problem can be solved on medium sized networks in reasonable time.

Zero Lower Bounds in New Keynesian Models and a Constrained Ramsey Problem

There exist difficulties in escaping from stagnation and/or deflation if the economy hits zero lower bounds on short-term nominal interest rates because the central bank cannot stimulate the economy further using rate cuts. How can it escape from them? Answering the question, we extend a closed-looped solution of a Ramsey problem (also known as a Stackelberg problem) by introducing zero lower bounds. Herein, we formulate a constrained Ramsey problem and derive a solution for it. In our extension, we found that the discounted Lyapunov equation is necessary to obtain the shadow price of the economy that hits the zero lower bounds. Herein we apply our method to new Keynesian models with zero lower bounds. Our simulation shows that committing to mild inflation engenders positive effects on the economy and that managing inflation expectations is necessary to escape from the bounds.

Systematic evolutionary algorithm for general multilevel Stackelberg problems with bounded decision variables (SEAMSP)

Multilevel Stackelberg problems are nested optimization problems which reply optimally to hierarchical decisions of subproblems. These kind of problems are common in hierarchical decision making systems and are known to be NP-hard. In this paper, a systematic evolutionary algorithm has been proposed for such types of problems. A unique feature of the algorithm is that it is not affected by the nature of the objective and constraint functions involved in the problem as long as the problem has a solution. The convergence proof of the proposed algorithm is given for special problems containing non-convex and non-differentiable functions. Moreover, a new concept of $$(\varepsilon ,\delta )$$ -approximation for Stackelberg solutions is defined. Using this definition comparison of approximate Stackelberg solutions has been studied in this work. The numerical results on various problems demonstrated that the proposed algorithm is very much promising to multilevel Stackelberg problems with bounded constraints, and it can be used as a benchmark for a comparison of approximate results by other algorithms.

Random fuzzy bilevel linear programming through possibility-based value at risk model

This article considers bilevel linear programming problems where random fuzzy variables are contained in objective functions and constraints. In order to construct a new optimization criterion under fuzziness and randomness, the concept of value at risk and possibility theory are incorporated. The purpose of the proposed decision making model is to optimize possibility-based values at risk. It is shown that the original bilevel programming problems involving random fuzzy variables are transformed into deterministic problems. The characteristic of the proposed model is that the corresponding Stackelberg problem is exactly solved by using nonlinear bilevel programming techniques under some convexity properties. A simple numerical example is provided to show the applicability of the proposed methodology to real-world hierarchical problems.

Solution algorithm for an optimistic linear Stackelberg problem

The optimistic Stackelberg problem is a bilevel programming problem where the constraints in the lower level problem are parameter independent. For linear problems of that type, algorithms for computing local and global optimal solutions are suggested. Their convergence is shown. In the last part, problems with perturbed right-hand side of the lower level constraints are considered, and the behavior of optimal solutions and of the optimal function value is investigated.

Zero Lower Bounds and a Stackelberg Problem: A Stochastic Analysis of Unconventional Monetary Policy

There exist difficulties to escape from stagnation and/or deflation if the economy hits zero lower bounds on short-term nominal interest rates because the central bank cannot stimulate the economy using rate cuts. How to escape from them? Answering the question, we extend a closed-looped solution of a Stackelberg problem (also known as a Ramsey problem) by introducing zero lower bounds. In this paper, we formulate a constrained Stackelberg problem and derive a solution of it. In our extension, we found that the discounted Lyapunov equation is required to obtain the shadow price of the economy which hits the zero lower bounds. Additionally, we stress that our method is consistent with rational expectations hypothesis. In this paper, we apply our method to new Keynesian models with zero lower bounds. In the numerical analysis, we evaluate the quantitative effects of zero interest rate policies with committing mild or zero inflation. Our simulation indicates that committing mild inflation causes positive effects on the economy and managing inflation expectations is necessary to escape from the bounds.