This presentation is the property of its rightful owner.
1 / 26

# Graph Value for Cooperative Games PowerPoint PPT Presentation

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

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 - - - - - - - - - - - - - - - - - - - - - - - - - -

## Graph Value for Cooperative Games

Ron Peretz (LSE)

Ziv Hellman (Bar Ilan)

Game Theory Seminar, Bar Ilan University, May 2014

• players

• , .

### Shapley value

• ,where .

• 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 .

• 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 .

• 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!