Game Theory Lecture #6 Outline: Cost Sharing Problems The Core - - PDF document

Game Theory Lecture #6

Outline:

• Cost Sharing Problems
• The Core
• Minimum Spanning Tree Games

Example: A cost sharing problem among two towns

• Two nearby towns are considering building a joint water distribution system

– Town A can build its own facility for $11 million – Town B can build its own facility for $ 7 million – A joint facility serving Town A and B costs $15 million

• Fact: There is a financial incentive to build joint facility ($15 million vs. $18 million)
• Question: How should costs of a joint project be divided up?
• Solution #1: Divide costs equally

– Town A and B each pay $7.5 million – Town B would rather build its own facility for $ 7 million

• Solution #2: The core = All parties have an incentive to cooperate

B’s Payment A’s Payment 15 15 7 11 The core

• Questions:

– How does approach generalize for more than two entities? – What are desirable properties of cost sharing protocols?

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Example: A cost sharing problem among three towns

• Three nearby towns are considering building a joint water distribution system

– Town A can build its own facility for $11 million – Town B can build its own facility for $7 million – Town C can build its own facility for $8 million – Towns A+B can build a joint facility for $15 million – Towns A+C can build a joint facility for $14 million – Towns B+C can build a joint facility for $13 million – Towns A+B+C can build a joint facility for $20 million

• Question: Is there an incentive to cooperate?
• Question: How should costs of a joint project be divided up? Equal?
• Definition: The core is all cost division such that

– No town pays more than its individual (or opportunity) cost – No group of towns pays more than its opportunity cost

B pays 20 A pays 20 C pays 20

A+B=15 B+C=13 A+C=14 A = 11 C=8 B=7 The Core Equal share

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Cooperative Game Model

• Setup: Cost sharing game

– Players: N = {1, 2, ..., n} – Opportunity costs: c : 2N → R.

• Previous example:

– Players: N = {A, B, C} – Opportunity costs: c(A) = 11, c(B) = 7, c(C) = 8 c(AB) = 15, c(AC) = 13, c(BC) = 10 c(ABC) = 20

• Cost sharing rule: A function CS(·) that allocates the total cost of a venture among

the members of a group for every possible group of players S ⊆ N, i.e., for any set of players S ⊆ N the cost sharing rule satisfies

• i∈S

CS(i, S) = c(S) where CS(i, S) represents the cost share of player i in group S.

• Coalition: A given subgroup of players S ⊆ N. The full set N is commonly referred to

as the grand coalition.

• Allocation: The cost shares generated for a specific cost function c(·).
• Core: The set of allocations such that no participant, or group of participants, pays more

than its opportunity cost.

• Goal: Establish methodology to find an allocation in the core.

– Is the core always nonempty? – How do you find an allocation in the core provided it is non-empty?

• Previous example suggests finding an allocation in the core is challenging...

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Minimum Spanning Tree Game

• Fact: The core can be empty or non-empty in a given cost sharing game (N, c)
• Questions: How do you determine whether or not the core is non-empty? Are there

relevant classes of games where the core is guaranteed to be non-empty?

• Motivating problem: Build infrastructure to connect towns to common source (e.g.,

cable, phone, electricity, water, etc.)

• Setup: Minimum Spanning Tree Game

– Source: Denoted by {0} – Individuals: Denoted by N = {1, . . . , |N|} – Possible connections/edges: (i, j) where i, j ∈ {0, 1, . . . , |N|}. Convention: (i, j) is directed edge pointing from i to j – Edge costs: cij ≥ 0 for each possible edge (i, j)

• Goal: Find a collection of edges E∗

N that connects all individuals N (either directly or

indirectly) to the source with the least possible cumulative cost. Note that E∗

N will have

exactly |N| edges and is known as a minimum spanning tree.

• Example: N = {1, 2, 3, 4} with undirected edges and costs

– Undirected edges with highlighted costs, e.g., c12 = c21 = 5 – Edges not highlighted, e.g., (1, 4) and (2, 3), have large/infinite costs – Minimum spanning tree (right) has total cost of 9

1 2 4 3 6 4 2 3 5 5 1 3 1 2 4 3 2 3 1 3 Minimum Spanning Tree Game Minimum Spanning Tree

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Minimum Spanning Tree Game (2)

• Questions:

– How should you distribute the costs of a minimum spanning tree? – Is the core non-empty? What is an example of an allocation in the core?

• Theorem: The core is non-empty in any minimum spanning tree game.
• Proof approach: Specify allocation that is in the core
• Proposed allocation:

– Recall: E∗

N is the minimum spanning tree over the set N ∪ {0}

– Since E∗

N is a minimum spanning tree, then for each individual i ∈ N there ex-

ists a unique path to the source, i.e., i = i0, i1, . . . , ik = 0 such that all edges (i0, i1), . . . , (ik−1, ik) are in E∗

N

– Define the cost of individual i as the full cost of the outgoing edge in E∗

N, i.e., for

each individual i ∈ N we have CS(i, N) = cij where (i, j) ∈ E∗

N

• Previous example: CS(1, N) = 3, CS(2, N) = 3, CS(3, N) = 1, CS(4, N) = 2.

1 2 4 3 6 4 2 3 5 5 1 3 1 2 4 3 2 3 1 3 Minimum Spanning Tree Game Minimum Spanning Tree

• Note that this is a valid cost sharing rule
• Fact: Proposed allocation is in the core.

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Minimum Spanning Tree Game (3)

• How would you show that the proposed allocation is in the core?
• Answer: Show that for any coalition S ⊆ N
• i∈S

CS(i, N) ≤

• (i,j)∈E∗

S

cij where E∗

S is the minimum spanning tree over the set S ∪ {0}. Note edges in E∗ S are of

the form (i, j) where i, j ∈ S ∪ {0}, i.e., no nodes outside S can be used.

• Prove by contradiction: Suppose that there exists a coalition S such that
• i∈S

CS(i, N) >

• (i,j)∈E∗

S

cij

• Step 1: Inspect new collection of edges E′

N constructed as follows

– Start with the edges in the set E∗

N

– Remove all edges leaving nodes in S in set E∗

N, i.e., all edges of the form (i, j) where

i ∈ S (this is |S| edges). Total cost edges removed is

i∈S CS(i, N)

– Add edges from the set E∗

• S. Total cost edges added is

(i,j)∈E∗

S cij

• Note that

(i,j)∈E∗

N cij >

(i,j)∈E′

N cij by construction

• Previous example where S = {3}

1 2 4 3 6 4 2 3 5 5 1 3 Minimum Spanning Tree Game 1 2 4 3 2 3 1 3 1 2 4 3 4 2 3 3

E0

N

E∗

N

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Minimum Spanning Tree Game (4)

• Step 2: Explore whether or not E′

N forms a spanning tree. In particular, does there exists

a path from i to {0} in the graph E′

N?

– Consider any individual i. By definition, there exists a path from i → {0} in E∗

N

– Case #1: Nodes in path do not include any nodes in S (see nodes {2, 4} in example) ∗ Then there exists a path from i → {0} in E′

N (same path)

– Case #2: Nodes in path do include nodes in S (see nodes {1, 3} in example) ∗ Then there exists a path from i to some node s ∈ S in E′

N

∗ There exists a path from s to {0} in E∗

S and hence E′ N

∗ Hence, there exists a path from i to {0} in E′

N

1 2 4 3 6 4 2 3 5 5 1 3 Minimum Spanning Tree Game 1 2 4 3 2 3 1 3 1 2 4 3 4 2 3 3

E0

N

E∗

N

• Conclusion: E′

N is a spanning tree!

• However, since E′

N is a spanning tree then we must have (i,j)∈E∗

N cij ≤

(i,j)∈E′

N cij

since E∗

N is a minimal spanning tree. Hence we have a contradiction, meaning that there

does not exists a set S ⊆ N such that

• i∈S

CS(i, N) >

• (i,j)∈E∗

S

cij

• This implies that the proposed allocation is in core and we are done!

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