Graph value for cooperative games
This presentation is the property of its rightful owner.
Sponsored Links
1 / 26

Graph Value for Cooperative Games PowerPoint PPT Presentation


  • 111 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Graph Value for Cooperative Games

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Graph value for cooperative games

Graph Value for Cooperative Games

Ron Peretz (LSE)

Ziv Hellman (Bar Ilan)

Game Theory Seminar, Bar Ilan University, May 2014


Tu coalitional games

(TU) Coalitional Games

  • players

  • , .


Shapley value

Shapley value

  • ,where .

  • Additivity

  • Null player

  • Efficiency

  • Symmetry

  • Implied: dummy, linearity, Monotonicity.


Spectrum value alvarez hellman winter 2013

Spectrum value [Alvarez Hellman Winter (2013+)]

  • Admissible coalitions, intervals.


Graph value

Graph value

  • Admissible coalitions: connected sets.

  • Shapley axioms + monotonicity.


Highlights

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

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

Unrelated literature

  • Jackson (2005). Network games.

  • Forges & Serrano (2011). Incomplete information.


C oalitional games on a graph

Coalitional games on a graph

  • players.

  • , connected graph.

  • , admissible coalitions.

  • , (TU) coalitional games over .

  • are isomorphic if there is , such that , namely


Graph values

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

Existence [Edelman & Bilbao (2000)]

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

  • , all chains on .


Characterization

Characterization

  • acts on .

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


E xamples

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.


U niqueness

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

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

Proof outlines – Theorem 1

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

  • Linearity. determine .


Linearity null player

Linearity+ Null player

  • determine .


Linearity dummy

Linearity+ Dummy

  • Signed probabilistic value.

    Where


Linearity dummy monotonicity

Linearity+ Dummy + Monotonicity

  • Probabilistic value.

    Where , .


Linearity null player e fficiency

Linearity+ Null player + Efficiency

  • Implies Dummy.

  • Signed 1-flow


Linearity null player e fficiency1

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

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.


P roof of theorem 4 step 1

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

Proof of Theorem 4 – step 2

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

Max degree


Future research

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).


Graph value for cooperative games

Thank you!


  • Login