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 : 2 N → 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 � CS ( i, S ) = c ( S ) i ∈ 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? 1

<latexit sha1_base64="ZGcRD4PDP97p18ALFbl8couVwXE=">ACBXicbVC7TsMwFHXKq5RXgBEGiwqJqUoKCMRUwcKECqIPqQmV47qtVceObAdURV1Y+BUWBhBi5R/Y+BucNgO0HOlKR+fcq3vCSJGlXacbys3N7+wuJRfLqysrq1v2JtbdSViUkNCyZkM0CKMpJTVPNSDOSBIUBI41gcJH6jXsiFRX8Vg8j4oeox2mXYqSN1LZ38Vn57gp6kvb6GkpHqAXIt0PguRm1LaLTskZA84SNyNFkKHatr+8jsBxSLjGDCnVcp1I+wmSmJGRgUvViRCeIB6pGUoRyFRfjL+YgT3jdKBXSFNcQ3H6u+JBIVKDcPAdKYXqmkvFf/zWrHunvoJ5VGsCceTRd2YQS1gGgnsUEmwZkNDEJbU3ApxH0mEtQmuYEJwp1+eJfVyT0sHV8fFSvnWRx5sAP2wAFwQmogEtQBTWAwSN4Bq/gzXqyXqx362PSmrOymW3wB9bnD/BgmD4=</latexit> <latexit sha1_base64="BFD7WfVz/ua1zpfM6j4Ocdvx2xk=">AB8HicbVDJSgNBEO2JW4xb1KOXwSDES5hxQY/BXDxGNIskQ+jp6SRNehm6a4Qw5Cu8eFDEq5/jzb+xk8xBEx8UPN6roqpeGHNmwPO+ndzK6tr6Rn6zsLW9s7tX3D9oGpVoQhtEcaXbITaUM0kbwIDTdqwpFiGnrXBUm/qtJ6oNU/IBxjENB5I1mcEg5Uea/flLokUnPaKJa/izeAuEz8jJZSh3it+dSNFEkElEI6N6fheDEGKNTDC6aTQTQyNMRnhAe1YKrGgJkhnB0/cE6tEbl9pWxLcmfp7IsXCmLEIbafAMDSL3lT8z+sk0L8OUibjBKgk80X9hLug3On3bsQ0JcDHlmCimb3VJUOsMQGbUcG4C+vEyaZxX/vHJ5d1Gq3mRx5NEROkZl5KMrVEW3qI4aiCBntErenO08+K8Ox/z1pyTzRyiP3A+fwDUl4/I</latexit> <latexit sha1_base64="Kh1EhlhIADXmxIbhTUgfc4+baU=">AB6HicbVDLSgNBEOyNrxhfUY9eBoPgKez6QI9BL54kAfOAZAmzk04yZnZ2mZkVwpIv8OJBEa9+kjf/xkmyB0saCiqunuCmLBtXHdbye3srq2vpHfLGxt7+zuFfcPGjpKFM6i0SkWgHVKLjEuFGYCtWSMNAYDMY3U795hMqzSP5YMYx+iEdSN7njBor1e67xZJbdmcgy8TLSAkyVLvFr04vYkmI0jBtW57bmz8lCrDmcBJoZNojCkb0QG2LZU0RO2ns0Mn5MQqPdKPlC1pyEz9PZHSUOtxGNjOkJqhXvSm4n9eOzH9az/lMk4MSjZf1E8EMRGZfk16XCEzYmwJZYrbWwkbUkWZsdkUbAje4svLpHFW9s7Ll7WLUuUmiyMPR3AMp+DBFVTgDqpQBwYIz/AKb86j8+K8Ox/z1pyTzRzCHzifP6jVjNk=</latexit> 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 ( · ) N CS ( · ) Cost Sharing Mechanism c : 2 N → R • 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 . 0 0 5 6 1 2 1 2 5 2 4 2 3 3 3 3 3 4 3 4 1 1 Minimum Spanning Tree Game Minimum Spanning Tree • Is this allocation reasonable and fair? 2

The Decomposition Principle • Special case: Decomposable costs • Example: Power lines C 200 500 300 A B 400 D O • 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? 3

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 c OA ( · ) + c AB ( · ) + c BC ( · ) + c BD ( · ) – c OA ( S ) = 0 if S = {∅} , c OA ( S ) = 500 otherwise – c AB ( S ) = 0 if S ∈ {∅ , { A }} , c AB ( S ) = 300 otherwise – c BC ( S ) = 0 if S ∈ {∅ , { A } , { B } , { A, B }} , c BC ( S ) = 200 otherwise • Observation (for c AB ): 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 ; c AB ) = 0 , CS ( B, S ; c AB ) = 300 / 2 , and CS ( C, S ; c AB ) = 300 / 2 • What cost sharing mechanisms satisfy Properties #1-3? Equal share? 4

Marginal Contribution • Charge each player their marginal contribution to the cost CS MC ( 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: – CS MC ( A, { A } ) = ? – CS MC ( A, { A, B } ) = ? – CS MC ( A, { A, C } ) = ? – CS MC ( A, { A, B, C } ) = ? – CS MC ( B, { A, B, C } ) = ? – CS MC ( C, { A, B, C } ) = ? • Problems? CS MC ( A, { A, B, C } ) + CS MC ( B, { A, B, C } ) + CS MC ( C, { A, B, C } ) = c ( { A, B, C } )? 5

Shapley Value • Definition: Shapley value – For any coalition S ⊆ N and any i ∈ S | T | !( | S | − | T | − 1)! CS SV ( i, S ) = � ( c ( T ∪ { i } ) − c ( T )) | S | ! T ⊆ S \{ i } • 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 : w ABC ( c ( { A, B, C } )) − c ( { B, C } )) + w AB ( c ( { A, B } ) − c ( { B } )) + w AC ( c ( { A, C } ) − c ( { C } )) + w A ( c ( { A } ) − c ( {∅} )) where w ABC , w AB , w AC , w A ≥ 0 and w ABC + w AB + w AC + w A = 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 – w ABC = 2 / 6 , w AB = 1 / 6 , w AC = 1 / 6 , w A = 2 / 6 6