Graph value for cooperative games
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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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Graph Value for Cooperative Games

Ron Peretz (LSE)

Ziv Hellman (Bar Ilan)

Game Theory Seminar, Bar Ilan University, May 2014


(TU) Coalitional Games

  • players

  • , .


Shapley value

  • ,where .

  • Additivity

  • Null player

  • Efficiency

  • Symmetry

  • Implied: dummy, linearity, Monotonicity.


Spectrum value [Alvarez Hellman Winter (2013+)]

  • Admissible coalitions, intervals.


Graph value

  • Admissible coalitions: connected sets.

  • Shapley axioms + monotonicity.


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.


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.


Unrelated literature

  • Jackson (2005). Network games.

  • Forges & Serrano (2011). Incomplete information.


Coalitional games on a graph

  • players.

  • , connected graph.

  • , admissible coalitions.

  • , (TU) coalitional games over .

  • are isomorphic if there is , such that , namely


Graph values

  • , an allocation rule (point solution concept) over .

  • Additivity

  • Null player:

  • Efficiency

  • Symmetry: isomorphic games yield isomorphic allocations. Formally,

  • Monotonicity: .


Existence [Edelman & Bilbao (2000)]

  • A chain is a total ordering of the players , such that

  • , all chains on .


Characterization

  • acts on .

  • Theorem 1. The set of graph values over is given by


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.


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).


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.


Proof outlines – Theorem 1

  • Following Weber (‘88), axioms are introduced gradually.

  • Linearity. determine .


Linearity+ Null player

  • determine .


Linearity+ Dummy

  • Signed probabilistic value.

    Where


Linearity+ Dummy + Monotonicity

  • Probabilistic value.

    Where , .


Linearity+ Null player + Efficiency

  • Implies Dummy.

  • Signed 1-flow


Linearity+ Null player + Efficiency

  • Signed random-order value.

    where is a singed measure on , .

  • + Monotonicity: random-order value. I.e.,

  • + symmetry: must be -invariant.


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.


Proof of Theorem 4 – step 1

  • A transitive graph that does not satisfy Property C has multiple values.

Path 1:

Path 2: –

value=?


Proof of Theorem 4 – step 2

  • Lemma. The cycle and the complete graph are the only graphs satisfying Property C.

Max degree


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).


Thank you!


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