Game Theory Lecture #7 Outline: Cost Sharing Problems - - PDF document

Game Theory Lecture #7

Outline:

• Cost Sharing Problems
• Decomposition Principle
• Marginal Contribution
• Shapley Value

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({A, B}) = 15, c({A, C}) = 13, c({B, C}) = 10 c({A, B, C}) = 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.

• Fact: The core may be empty or non-empty. Hence, finding an allocation in the core

may be impossible

• Question: What properties should a reasonable cost sharing rule satisfy?

Are there procedures for deriving desirable cost sharing rules?

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Cost Sharing Mechanism

• Question: Are there procedures for deriving desirable cost sharing rules?
• Cost Sharing Mechanism: A systematic procedure (i.e., an algorithm) for going from a

cost sharing problem (N, c) to a given cost sharing rule CS(·)

Cost Sharing Mechanism

N

c : 2N → R

CS(·)

• Notation: Given a cost sharing mechanism, we will sometime denote the cost sharing

rule as CS(i, S; c) to highlight the dependence on the opportunity costs c(·)

• Ultimate goal: Derive cost sharing mechanism that results in cost shares CS(·) that

provides allocation in the core whenever the core of (N, c) is non-empty

• Problem: Attaining this goal is too challenging
• Revised goal: Derive cost sharing mechanism that results in cost shares CS(·) that

provides reasonable and fair cost sharing rule

• 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

• Is this allocation reasonable and fair?

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The Decomposition Principle

• Special case: Decomposable costs
• Example: Power lines

O A B C D 500 300 200 400

• Features: A players uses only part of the systems

– A uses only line (OA) – B uses lines (OA) and (AB)

• Decomposition principle: If a cost function decomposes into distinct cost elements, divide

the cost equally among the users that use it. – Ex: A is charged 500/4 = 125 – Ex: B is charged 500/4 + 300/3 = 225

• Fact: If a cost function decomposes into distinct elements, the decomposition principle

yields a solution in the core.

• Three ideas behind decomposition principle:

(i) Those who do not use a cost element should not be charged for it (ii) Everyone who uses a given cost element should be charged equally for it. (iii) The results of different cost allocation should be additive.

• How do these ideas generalize to alternative cost functions?

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Generalization of the decomposition principle

• Consider the following three properties of a cost sharing rule CS(·):
• Property #1:

Dummy – Suppose there exists and individual i such that for any coalition S ⊆ N the opportunity costs satisfy c(S) = c(S ∪ {i}). Then for any coalition S such that i ∈ S the cost sharing rule satisfies CS(i, S) = 0.

• Property #2: Symmetry – Let i, j be any two individuals. If for any coalition S ⊆

N \ {i, j} the opportunity costs satisfy c(S ∪ {i}) = c(S ∪ {j}), then for any coalition T such that i, j ∈ T the cost sharing rule satisfies CS(i, T) = CS(j, T). (1)

• Property #3: Additivity – If the opportunity cost function c(·) decomposes into the

sum of two function c′(·) and c′′(·), i.e., c(S) = c′(S)+c′′(S) for every coalition S ⊆ N, then the cost allocation derived for c is precisely the sum of the cost allocations derived for c′ and c′′. That is, for any coalition S ⊆ N and individual i ∈ S we have CS(i, S; c) = CS(i, S; c′) + CS(i, S; c′′) (2)

• Example: Consider previous example

– We can express c(·) as cOA(·) + cAB(·) + cBC(·) + cBD(·) – cOA(S) = 0 if S = {∅}, cOA(S) = 500 otherwise – cAB(S) = 0 if S ∈ {∅, {A}}, cAB(S) = 300 otherwise – cBC(S) = 0 if S ∈ {∅, {A}, {B}, {A, B}}, cBC(S) = 200 otherwise

• Observation (for cAB): A is a dummy and {B, C, D} are symmetric
• Hence, if we have the set S = {A, B, C} the cost sharing rule must satisfy CS(A, S; cAB) =

0, CS(B, S; cAB) = 300/2, and CS(C, S; cAB) = 300/2

• What cost sharing mechanisms satisfy Properties #1-3? Equal share?

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Marginal Contribution

• Charge each player their marginal contribution to the cost

CSMC(i, S) = c(S) − c(S \ {i})

• Example revisited:

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

• Questions:

– CSMC(A, {A}) = ? – CSMC(A, {A, B}) = ? – CSMC(A, {A, C}) = ? – CSMC(A, {A, B, C}) = ? – CSMC(B, {A, B, C}) = ? – CSMC(C, {A, B, C}) = ?

• Problems?

CSMC(A, {A, B, C}) + CSMC(B, {A, B, C}) + CSMC(C, {A, B, C}) = c({A, B, C})?

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Shapley Value

• Definition: Shapley value – For any coalition S ⊆ N and any i ∈ S

CSSV(i, S) =

• T⊆S\{i}

|T|!(|S| − |T| − 1)! |S|! (c(T ∪ {i}) − c(T))

• Interpretations:

– Marginal contribution ⇒ Marginal contribution to full coalition – Shapley value ⇒ Average marginal contribution to all sub-coalitions

• Example revisited: Consider the coalition {A, B, C}

– Marginal contribution of player A: c({A, B, C}) − c({B, C}) – Shapley value of player A: wABC (c({A, B, C})) − c({B, C})) + wAB (c({A, B}) − c({B})) +wAC (c({A, C}) − c({C})) + wA (c({A}) − c({∅})) where wABC, wAB, wAC, wA ≥ 0 and wABC + wAB + wAC + wA = 1

• What are the meaning of the weights?

Look at possible orderings of set {A, B, C} and define incremental marginal cost of an agent as the marginal cost to the group of individuals in front of that agent relative to the ordering, i.e., A ← B ← C ⇒ c({A}) − c(∅) A ← C ← B ⇒ c({A}) − c(∅) B ← A ← C ⇒ c({A, B}) − c({B}) C ← A ← B ⇒ c({A, C}) − c({C}) B ← C ← A ⇒ c({A, B, C}) − c({B, C}) C ← B ← A ⇒ c({A, B, C}) − c({B, C})

• Weights of a sub-coalition = Proportion of orders where marginal contribution to sub-

coalition = incremental marginal cost to order – wABC = 2/6, wAB = 1/6, wAC = 1/6, wA = 2/6

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Shapley Value Example

• Example revisited:

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

• Order specific marginal contribution (computation for player A)

A ← B ← C ⇒ c({A}) − c(∅) = 11 A ← C ← B ⇒ c({A}) − c(∅) = 11 B ← A ← C ⇒ c({A, B}) − c({B}) = 15 − 7 = 8 C ← A ← B ⇒ c({A, C}) − c({C}) = 13 − 8 = 5 B ← C ← A ⇒ c({A, B, C}) − c({B, C}) = 20 − 10 = 10 C ← B ← A ⇒ c({A, B, C}) − c({B, C}) = 20 − 10 = 10

• Shapley value = Average marginal contribution over all ordering

1 6 (11 + 11 + 8 + 5 + 10 + 10) = 55 6

• Alternatively, using the definition of Shapley value we have

CSSV(A, {A, B, C}) =

• T⊆{ABC}\{A}

|T|!(3 − |T| − 1)! 3! (c(T ∪ {A}) − c(T)) which simplifies to 2 6(c(A) − c(∅)) + 1 6(c(AB) − c(B)) + 1 6(c(AC) − c(C)) + 2 6(c(ABC) − c(BC))

• Further, observe that

CSSV(A, {A, B, C}) + CSSV(A, {A, B, C}) + CSSV(A, {A, B, C}) = c({A, B, C}) which always holds true for the Shapley value, i.e., it is a valid cost sharing rule.

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Shapley value continued

• Recall: The goal is to find a cost sharing rule that satisfies the following properties

(i) Dummy: (ii) Symmetry: (iii) Additivity:

• The following theorem identifies all such cost sharing mechanisms
• Theorem (Shapley 1953): The Shapley value is the unique cost sharing rule that

satisfies Properties (i)–(iii)

• If our goal is to find a cost sharing mechanism that satisfies our three properties and is in

the core, we only need to check whether or not the Shapley value gives us an allocation in the core.

• Proposition: If the cost function decomposes into distinct cost elements (like the exam-

ple on electricity distribution networks), then the Shapley value produces an allocation in the core.

• Interestingly, the Shapley value does not necessarily provide an allocation in the core in

minimum spanning tree games. This implies that the proposed cost sharing mechanism for minimum spanning tree games must fail one of our three properties. Which one?

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