Cooperative Games on Combinatorial Structures Basic Concepts

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Cooperative Games on Combinatorial
Structures----Basic Concepts Several Examples
in OR
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Outline of this lecture

• 1. Basic Concepts
• 2. Examples in OR
• 3. Problems Methodology
• 4. Some of my own ideas

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Basic Concepts(1/9)

• Def. 1 cooperative games vs. noncooperative games

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1. Basic Concepts(2/9)

• Def. 2 Transferable utility game
• Players elements of N1,2,, n
• Coalitions subsets of N
• Worth the value of S, v(S), is called the worth
  of S
• Characteristic function V
• Profit games cost games

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1. Basic Concepts(3/9)

• Subgame
• Efficient
• Pre-imputation set
• Imputation set
• Polyhedron associated to a game

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1. Basic Concepts(4/9)

• Def. 3 core
• Balanced games
• Totally balanced games
• Dual game
• Pro. 1 (N,v) is a profit game and (N,v) is a
  cost game, then

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1. Basic Concepts(5/9)

• Excess of S at x e(S,x)v(S)-x(S)
• Stronge-core
• Least core

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1. Basic Concepts(6/9)

• Multiplicative core
• Def. 4 The nucleolus of Y is the set of
  pre-imputations in Y that minimize T in the
  lexicographic ordering

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1. Basic Concepts(7/9)

• Shapley value
• Banzhaf value

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1. Basic Concepts(8/9)

• Unanimity game
• Every game is a unique linear combination of
  unanimity games

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1. Basic Concepts(9/9)

• Tijs value
• Upper value
• Utopia payoff
• Lower value
• Def. 5 quasi-balanced
• (QB1)
• (QB2)

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2. Examples

• 1. Assignment and matching games (Shapley and
  Shubik,1972)
• 2. Min-cost spanning tree games (Granot and
  Huberman,1981)

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2. Examples

• 3. Linear production games (Owen,1975)
• 4. Tax games (Tijs and Driessen,1986)

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2. Examples

• 5. Bin packing games (Faigle and Kern, 1993)
• 6. Simple flow games (Kalai and
  Zemel, 1982 Reijnierse,Maschler,Potters and
  Tijs,1996)
• 7. Permutation games
• (Tijs, Parthasarathy,Potters and Prasad,1984)

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2. Examples

• 8. Travelling salesman games (Potters, Curiel and
  Tijs, 1992)
• 9. Delivery games (Hamers, 1995)
• 10. Sequencing games (Curiel,Pederzoli and
  Tijs,1989)
• 11. Restricted games (Myerson, 1977 Owen, 1986)

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2. Examples
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3. Problems Methodology

• Problems
• 1. Is a cooperative game balanced\totally
  balanced?
• 1.1 cooperative games determined by combinatorial
  optimization problems
• 1.2 restricted games
• 2. How to allocate the profit among the grand
  coalition?
• 2.1 axiomatic approach efficiency, linearity,
  dummy axiom, chain axiom
• 2.2 computing of Shapley values
• 3. Complexity analysis
• 3.1 testing nonemptiness
• 3.2 checking membership
• 3.3 finding a core member

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3. Problems Methodology

• Methodology
• 1. to characterize the structure
• Graph theory
• Algebra poset, lattice, closure space, convex
  geometry, matroid, greedoid, antimatroid,
  partition system, unionstable system

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3. Problems Methodology

• 2. to characterize and analyze v
• Optimization theory dual theory, integer and
  combinatorial optimization
• Discrete convex analysis supermodular functions,
  subdifferentials, indirect functions, least
  increment functions
• Basic ideas (relaxing, transforming, extending)
• Bicooperative games, Fenchel conjugation, Lovasz
  extension
• 3. to analyze the complexity
• Computational complexity theory

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4. Some of my own ideas

• 1. A more realistic definition of v
• We call a situation, the worth of a
  coalition is situation depended.
• More realistic (at least in certain situations)
• It embodies more of the essence of game theory.

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4. Some of my own ideas

• 2. Equilibrium problems
• balancedness a kind of Nash Equilibrium
• Allocation rule before equilibrium
• Under this assumption, cooperative games can be
  converted to non-cooperative ones. Every one
  makes his decision to prefer a situation to
  maximize his profit.

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4. Some of my own ideas

• 3. New allocation rules
• To allocate profits among players according to
  their powers
• To allocate according to indispensability

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4. Some of my own ideas

• 4. dynamic approaches to distribute profits

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4. Some of my own ideas

• 5. Stability of the coalition
• The bigger the coalition is, the less stable it
  is.
• 6. Asymmetric information
• 7.Cost of forming the coalition

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• THANKS!