Backward induction algorithm for a class of closed-loop Stackelberg games

Kicsiny, Richárd; Varga, Zoltán; Scarelli, Antonino

doi:10.1016/j.ejor.2014.02.057

Cited by 23 publications

(7 citation statements)

References 31 publications

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“…Our single-level reformulation is based on the Best-Response (BR) of the follower, which has been used for the backward induction in Stackelberg game [204], [205]. Specifically, given the leader x, we define a unified BR mapping of the follower y (denoted as y * (x)) for two categories of BLOs as follows:…”

Section: Single-level Reformulationmentioning

confidence: 99%

Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond

Liu¹,

Gao²,

Zhang³

et al. 2021

Preprint

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Bi-Level Optimization (BLO) is originated from the area of economic game theory and then introduced into the optimization community. BLO is able to handle problems with a hierarchical structure, involving two levels of optimization tasks, where one task is nested inside the other. In machine learning and computer vision fields, despite the different motivations and mechanisms, a lot of complex problems, such as hyper-parameter optimization, multi-task and meta-learning, neural architecture search, adversarial learning and deep reinforcement learning, actually all contain a series of closely related subproblms. In this paper, we first uniformly express these complex learning and vision problems from the perspective of BLO. Then we construct a value-function-based single-level reformulation and establish a unified algorithmic framework to understand and formulate mainstream gradient-based BLO methodologies, covering aspects ranging from fundamental automatic differentiation schemes to various accelerations, simplifications, extensions and their convergence and complexity properties. Last but not least, we discuss the potentials of our unified BLO framework for designing new algorithms and point out some promising directions for future research.

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Section: Single-level Reformulationmentioning

confidence: 99%

Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond

Liu¹,

Gao²,

Zhang³

et al. 2021

Preprint

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“…Backward induction [41,42] is applied in calculating, which solves decision variables in the order of 'seller-remanufacturer-manufacturer'. To determine the sales price, we have the following Equation 11derived from the first order linear condition of Equation 9to p:…”

Section: Remanufacturer-remanufacturing Mode With Authorizationmentioning

confidence: 99%

Collection and Remanufacturing of Waste Products under Patent Protection and Government Regulation

Zhang

Shi

et al. 2018

Sustainability

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There is increasing academic and pragmatic interest in leveraging patent rights to invigorate remanufacturing for waste products under governmental interventions via regulations and reward-penalty instruments. In practice, many original manufacturers that are possessed with intellectual property rights allow third-party remanufacturers to implement reproducing operations through authorization and charging licensing fees. The general purpose of this paper is to explore favorable strategies for a closed-loop supply chain (CLSC) system of waste product collection and remanufacturing, in the context of either manufacturer-remanufacturing or remanufacturer-remanufacturing. To achieve such an objective, game theory is adopted to establish models of three collection and remanufacturing modes among channel members involving a manufacturer, a seller, and a remanufacturer. In so doing, the results show that a government's allocations of elementary remanufacturing ratio and the unit amount of reward-penalty count significantly in CLSC operations, especially for the manufacturer, who acts as the leader in the system and makes mode selections.

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“…Stackelberg game is widely used in two or more levels of supply chain operations, especially in fields of raw material procurements , wholesale price and market price determinations , production and inventory strategies , and product quality selections . Generally, backward induction method is applied to solve Stackelberg models .…”

Section: Literature Reviewmentioning

confidence: 99%

“…The manufacturer acts as the leader who decides the wholesale price w first, while the retailer follows to determine the order quantity q and sales price of the product p . Considering

q = a - b p

and using backward induction , we briefly get the optimal solutions as follows (see Appendix A),

q_{0}^{*} = true[a - b true(c_{M} + c_{R} true) true] / 4

p_{0}^{*} = true[3 a + b true(c_{M} + c_{R} true) true] / true(4 b true)

w_{0}^{*} = true[a - b true(c_{M} + c_{R} true) true] / true(2 b true)

π_{M, 0}^{*} = {true[a - b true(c_{M} + c_{R} true) true]}^{2} / true(8 b true)

π_{R, 0}^{*} = {true[a - b true(c_{M} + c_{R} true) true]}^{2} / true(16 b true)

π_{T, 0}^{*} = 3 {true[a - b true(c_{M} + c_{R} true) true]}^{2} /

…”

Section: Analytical Modelmentioning

confidence: 99%

Supply chain coordination with revenue‐sharing contracts considering carbon emissions and governmental policy making

Cao

Zhang

Zhou

2015

Env Prog and Sustain Energy

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Considering the opening up of carbon emission trading market, this article investigates the government's role in allocating the appropriate emission quota to maximize social members' (including enterprises and customers) utilities and analyzes how the emission-dependent enterprise improves revenues of both itself and the whole system through supply chain collaboration. Within this, a two-echelon supply chain consisting of one emission-dependent manufacturer and one retailer is discussed. Taking revenue-sharing contract into account and combining practical situations with Stackelberg game method, decision-makings of traditional supply chain (T-SC) without carbon emission limits and carbon-emissionconsidering supply chain (C-SC) are examined, respectively, along with the determination of the optimal range of revenue-sharing ratio. Furthermore, government's policy making in distributing the optimal emission quota is studied as well as supply chain members' efforts on fulfilling the revenue-sharing contract to accomplish Pareto improvements of social members' utilities. It is proved that profits of both manufacturer and retailer increase with the growth of the emission quota, proper governmental regulations not only help to achieve balance among social members but also improve overall utilities, and a reasonable revenue-sharing contract is essential to raise supply chain members' profits even under low-carbon conditions.

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Backward induction algorithm for a class of closed-loop Stackelberg games

Cited by 23 publications

References 31 publications

Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond

Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond

Collection and Remanufacturing of Waste Products under Patent Protection and Government Regulation

Supply chain coordination with revenue‐sharing contracts considering carbon emissions and governmental policy making

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