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 15 B’s Payment 7 The core 11 15 A’s Payment • Questions: – How does approach generalize for more than two entities? – What are desirable properties of cost sharing protocols? 1
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 A pays 20 A+C=14 B=7 A = 11 The Core B+C=13 Equal share C=8 A+B=15 C pays 20 B pays 20 2
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 ( 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 � 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. • 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... 3
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: c ij ≥ 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., c 12 = c 21 = 5 – Edges not highlighted, e.g., (1 , 4) and (2 , 3) , have large/infinite costs – Minimum spanning tree (right) has total cost of 9 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 4
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 = i 0 , i 1 , . . . , i k = 0 such that all edges ( i 0 , i 1 ) , . . . , ( i k − 1 , i k ) 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 ) = c ij where ( i, j ) ∈ E ∗ N • 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 • Note that this is a valid cost sharing rule • Fact: Proposed allocation is in the core. 5
<latexit sha1_base64="tV5vPskwD1qjJrzqgTaDKG/LOQ=">AB7HicbVDLSgNBEOyNrxhfUY9eBoMgHsKuD/QYEQJIKbBJI1zE5mkyGzM8vMrBCWgH/gxYMiXv0gb/6Nk8dBEwsaiqpurvChDNtXPfbyS0sLi2v5FcLa+sbm1vF7Z2alqki1CeS9UIsacCeobZjhtJIriOS0HvYvR379kSrNpLg3g4QGMe4KFjGCjZX8q/btw1G7WHL7honhTUoIpqu3iV6sjSRpTYQjHWjc9NzFBhpVhNhoZVqmDSx13atFTgmOogGx87RAdW6aBIKlvCoLH6eyLDsdaDOLSdMTY9PeuNxP+8ZmqiyBjIkNFWSyKEo5MhKNPkcdpigxfGAJorZWxHpYWJsfkUbAje7MvzpHZc9k7KZ3enpcrN0ySOPOzBPhyCB+dQgWuog8EGDzDK7w5wnlx3p2PSWvOmUa4C3/gfP4ALy6Otg=</latexit> <latexit sha1_base64="aW8XgxCEh5CHIHWhkN7j7GyrodI=">AB63icbVBNSwMxEJ2tX7V+VT16CRbRU9m1ih4LIgiCVLAf0C4lm2b0CS7JFmhLAV/gRcPinj1D3nz35hte9DWBwOP92aYmRfEnGnjut9Obml5ZXUtv17Y2Nza3inu7jV0lChC6yTikWoFWFPOJK0bZjhtxYpiEXDaDIZXmd98pEqzSD6YUx9gfuShYxgk0nX3bvjbrHklt0J0CLxZqQEM9S6xa9OLyKJoNIQjrVue25s/BQrwin40In0TGZIj7tG2pxIJqP53cOkZHVumhMFK2pET9fdEioXWIxHYToHNQM97mfif105MeOmnTMaJoZJMF4UJRyZC2eOoxQlho8swUQxeysiA6wMTaeg3Bm395kTROy16lfH5/VqrePk3jyMBHMIJeHABVbiBGtSBwACe4RXeHOG8O/Ox7Q158wi3Ic/cD5/AHXhjks=</latexit> 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 � � CS ( i, N ) ≤ c ij i ∈ S ( i,j ) ∈ E ∗ S 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 � � CS ( i, N ) > c ij i ∈ S ( i,j ) ∈ E ∗ S • 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 � S c ij ( i,j ) ∈ E ∗ • Note that � N c ij > � N c ij by construction ( i,j ) ∈ E ∗ ( i,j ) ∈ E ′ • Previous example where S = { 3 } 0 0 0 5 6 1 2 1 2 1 2 5 4 2 2 4 2 3 3 3 3 3 3 3 4 3 4 3 4 1 1 E ∗ E 0 Minimum Spanning Tree Game N N 6
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