Stable allocations and flows as Fleiner 1 Tam Summer School on - PowerPoint PPT Presentation

Stable allocations and flows as Fleiner 1 Tam´ Summer School on Matching Problems, Markets, and Mechanisms 26 June 2013, Budapest 1 Budapest University of Technology and Economics

Stable matchings Model:

Stable matchings Model: Boys

Stable matchings Model: Boys and girls

Stable matchings Model: Boys and girls with possible marriages are given.

Stable matchings Model: Boys and girls with possible marriages are given. Marriage scheme: matching .

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners.

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges .

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists.

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one:

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners,

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners, girls reject boys with no chance.

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners, girls reject boys with no chance. We iterate:

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners, girls reject boys with no chance. We iterate: rejected boys propose

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners, girls reject boys with no chance. We iterate: rejected boys propose and girls reject alternatingly.

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners, girls reject boys with no chance. We iterate: rejected boys propose and girls reject alternatingly. When no boy proposes

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners, girls reject boys with no chance. We iterate: rejected boys propose and girls reject alternatingly. When no boy proposes then we got a stable matching .

Stable matchings 1 3 2 1 2 1 2 1 2 1 3 2 2 2 1 1 1 3 1 2 3 4 1 5 4 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 Model: Boys and girls with possible marriages are given. Marriage scheme: matching . Preferences on possible partners. Instability may occur along blocking edges . A matching is stable if no blocking edge exists. The proposal algorithm of Gale and Shapley always finds one: Boys propose to best partners, girls reject boys with no chance. We iterate: rejected boys propose and girls reject alternatingly. When no boy proposes then we got a stable matching . Man-optimality: each boy gets the best stable partner.

Stable allocations and properties Extension of the model: capacities for vxs and edges (partnerships).

Stable allocations and properties 2 3 1 3 2 1 2 2 1 2 1 2 1 3 2 2 2 2 2 1 1 1 3 1 2 3 4 1 2 5 4 2 3 2 2 4 1 3 3 3 5 1 3 1 2 1 2 1 2 1 2 3 2 2 Extension of the model: capacities for vxs and edges (partnerships).

Stable allocations and properties 2 3 1 1 3 2 1 2 2 1 2 1 2 1 3 2 2 1 1 1 / 3 2 2 2 2 1 1 1 3 1 2 3 4 1 2 5 4 2 3 2 2 4 1 1 / 3 3 3 3 5 1 / 3 1 1 3 1 2 1 2 1 1 2 1 2 3 2 2 Extension of the model: capacities for vxs and edges (partnerships). An allocation an assignment of intensities to edges st capacities of edges and vxs are observed.

Stable allocations and flows as Fleiner 1 Tam Summer School on - PowerPoint PPT Presentation

Stable allocations and flows as Fleiner 1 Tam Summer School on Matching Problems, Markets, and Mechanisms 26 June 2013, Budapest 1 Budapest University of Technology and Economics Stable matchings Model: Stable matchings Model: Boys

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