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Graph Value for Cooperative Games

Graph Value for Cooperative Games. Ron P eretz (LSE) Ziv Hellman (Bar I lan ). Game Theory Seminar, Bar Ilan University, May 2014. (TU) Coalitional Games. players ,. Shapley value. , where . Additivity Null player Efficiency Symmetry

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Graph Value for Cooperative Games

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  1. Graph Value for Cooperative Games Ron Peretz (LSE) Ziv Hellman (Bar Ilan) Game Theory Seminar, Bar Ilan University, May 2014

  2. (TU) Coalitional Games • players • , .

  3. Shapley value • , where . • Additivity • Null player • Efficiency • Symmetry • Implied: dummy, linearity, Monotonicity.

  4. Spectrum value [Alvarez Hellman Winter (2013+)] • Admissible coalitions, intervals.

  5. Graph value • Admissible coalitions: connected sets. • Shapley axioms + monotonicity.

  6. Highlights • Graph values always exist and are not unique (generally). • On some graphs the value is unique. • Complete graph – Shapley value. • Line – Spectrum value is one of the graph values. • Cycle – unique but different from the Shapley value. • Linearity& Monotonicity are implied iff the graph value is unique.

  7. Related literature • Myerson (77,80). Graphs and cooperation. • Dubey & Weber (77), Weber (88). Probabilistic values. • Bilbao & Edelman (2000). Convex geometries. • Alvarez Hellman Winter (2013+). Spectrum value.

  8. Unrelated literature • Jackson (2005). Network games. • Forges & Serrano (2011). Incomplete information.

  9. Coalitional games on a graph • players. • , connected graph. • , admissible coalitions. • , (TU) coalitional games over . • are isomorphic if there is , such that , namely

  10. Graph values • , an allocation rule (point solution concept) over . • Additivity • Null player: • Efficiency • Symmetry: isomorphic games yield isomorphic allocations. Formally, • Monotonicity: .

  11. Existence [Edelman & Bilbao (2000)] • A chain is a total ordering of the players , such that • , all chains on .

  12. Characterization • acts on . • Theorem 1. The set of graph values over is given by

  13. Examples • (the complete graph), there is a unique invariant measure, the uniform distribution; the unique value is the Shapley value. • (line), there are many values one of which is the Spectrum value [Alvarez et al. 2013+]. • (star), simple majority; centre gets 0.

  14. Uniqueness • , the action of on is transitive; hence there is a unique invariant measure; hence a unique value. • , the action of on is nottransitive; hence there are many invariant measures… Nevertheless, the value may be unique. • (-cycle).

  15. Uniqueness and implied properties • Theorem 2. If is a graph on which there is a unique value that satisfying the Shapley axioms + monotonicity, then there is a unique value satisfying the Shapley axioms (monotonicity is implied). • Theorem 3.If is a graph on which there is more than one value satisfying the Shapley axioms + monotonicity, then there is a value satisfying the Shapley axioms + linearity but not monotonicity,and a valuesatisfying the Shapley axioms but not linearity. • Theorem 4. The complete graph and the cycle are the only graphs on with the graph value is unique.

  16. Proof outlines – Theorem 1 • Following Weber (‘88), axioms are introduced gradually. • Linearity. determine .

  17. Linearity+ Null player • determine .

  18. Linearity+ Dummy • Signed probabilistic value. Where

  19. Linearity+ Dummy + Monotonicity • Probabilistic value. Where , .

  20. Linearity+ Null player + Efficiency • Implies Dummy. • Signed 1-flow

  21. Linearity+ Null player + Efficiency • Signed random-order value. where is a singed measure on , . • + Monotonicity: random-order value. I.e., • + symmetry: must be -invariant.

  22. Theorem 4 – proof outlines • If the value on is unique, then • is transitive, • the complement of any connected vertex set is connected (Property C). • The cycle and the complete graph are the only graphs satisfying Property C.

  23. Proof of Theorem 4 – step 1 • A transitive graph that does not satisfy Property C has multiple values. Path 1: Path 2: – value=?

  24. Proof of Theorem 4 – step 2 • Lemma. The cycle and the complete graph are the only graphs satisfying Property C. Max degree

  25. Future research • What are the graphs on which the value is unique? Solved! – complete graph and cycle. • Extensions to infinitely many players. • Infinite graphs. • Converging sequences of graphs, • Continuum of players (convex geometries).

  26. Thank you!

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