computing least cores of supermodular …...computing least cores of supermodular cooperative games...
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![Page 1: Computing Least Cores of Supermodular …...Computing Least Cores of Supermodular Cooperative Games (AAAI’17) 波多野大督 (NII) ( Joint work with 𠮷𠮷田悠一(NII & PFI))](https://reader030.vdocuments.net/reader030/viewer/2022040800/5e3511736809302bab78e81e/html5/thumbnails/1.jpg)
Computing Least Cores of Supermodular Cooperative
Games (AAAI’17)
波多野大督(NII)(Joint work with 𠮷𠮷田悠一 (NII & PFI))
河原林ERATO感謝祭シーズンⅣ
2017年8月4日
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An example of the cooperative game• Given ( , )
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• = { }• = 15• = 25• = 30• = 30• = 50• = 50• = 80
Find a value division
• 𝑥𝑥 = 15• 𝑥𝑥 = 35• 𝑥𝑥 = 30
= 80
violate
leave from the cooperation
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An example of the cooperative game• Given ( , )
3
• = { }• = 15• = 25• = 30• = 30• = 50• = 50• = 80
Find a value division
• 𝑥𝑥 = 15• 𝑥𝑥 = 30• 𝑥𝑥 = 35
= 80
No violation!
No one leaves
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A goal of the cooperative game
Finding a core of a cooperative game is NP-complete
Finding a core can be described as the following program
• A value division• A value division is in the core if for all
Supermodular cooperative game4
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Supermodular cooperative game
• Supermodular cooperative game is a cooperative game where a characteristic function 𝜈𝜈 is supermodular
• Applications• Induced subgraph game [Deng and Papadimitriou 1994]• Airport game [Littlechild and Owen 1973]• Bidder collusion game [Graham, Marshall and Richard 1990]• Multicast tree game [Feigenbaum, Papadimitriou and
Shenker 2001]• Bankruptcy game [O’Neill 1982]
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An application of the supermodular cooperative gameInduced subgraph game
• Weighted hypergraph:• Characteristic function ∶ 2𝑉𝑉 → ℝ+ is the total weight of
hyperedges such that every vertex in belongs to
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e1
e2
1
1
Example:
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An example of induced subgraph game
In a certain K-project…• Professor K gives bonuses depending on the number of
accepted top conference-papers • Want to find a bonus division in the core
AAAI
ICML
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※この例はフィクションであり、実在する人物、団体等とは一切関係ありません7
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Motivation
• Property of supermodular cooperative game• A value division called Shapley value is always in the
core
• Focus on more general solution concepts than the core
• Want to find the stablest value division • Strong least core• Weak least core
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Goal of our work
• Goal is to find the strong and weak least core values of supermodular cooperative game
• The strong and weak least core values is obtained by solving the following LP
where is supermodular
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No computational analysis about the strong and weak least corevalues of supermodular cooperative game exists
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Idea of our approach (1)• Instead of solving the optimization problem,
fix and consider its feasibility• Define a function as
• Solve the following feasibility problem
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is a supermodular game
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Idea of our approach (2)
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• To check the feasibility…
• Define , where
• The above program is feasible iff• can be computed in polynomial time
(Frank and Tardos 1988; Naitoh and Fujishige 1992)
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Idea of our approach (3)
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is crossing submodular
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Idea of our approach (4)To compute …
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By the theorem, characterize the strong and weak least core values of the supermodular cooperative game
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Induced subgraph game
• The strong least core value
• A value division in the strong least core• Use the ellipsoid method to find the value division
e1
e2
1
1
How well the hypergraph is clustered by the partition
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Conclusion
• Goal:To find the strong least core and the weak least core of the supermodular cooperative game
• Contributions:• Provide theoretical characterizations of the strong and
the weak least core values• Derive explicit concise formulations of the induced
subgraph game and the airport game
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