1. Arbitrators in Overlapping Coalition Formation Games
Yair Zick and Edith Elkind
Division of Mathematical Sciences
School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore
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2. Outline What are OCF Games? Definition of an Arbitrator
Our model: Arbitrated Cores
We review our results for Arbitrated OCF Games

3. Coalition Formation – Classical Setting Shapley (1953,1967,1971), Aumann & Dreze (1974)
Players divide into coalitions to perform tasks A 2 5 3 4 1 6 C
Each coalition can divide its profits between its members B
How are profits divided?
The core: no set can get more on its own.
The nucleolus, the Shapley value...
Coalition structure CS=(A,B,C)

4. Games with Overlapping Coalitions G. Chalkiadakis, E. Elkind, E
Games with Overlapping Coalitions G. Chalkiadakis, E. Elkind, E. Markakis, M. Polukarov and N. R. Jennings "Cooperative Games with Overlapping Coalitions“ (JAIR 2010)
Each player has some resource that she may divide between tasks to gain profits. Players are allowed to participate in more than one task A
Player 2 Gives 40% to A and 60% to B 1 3 B 2
Player 5 gives 75% to B and 25% to C C 4 5 6

5. OCF Games - Definitions
In an Overlapping Coalition Formation (OCF) Game, players can choose to contribute only some of their weight to a task.
Each partial coalition represents participation levels of the n players.
The value of a coalition is determined by the characteristic function
A coalition structure is a finite collection of coalitions so that:

6. Outcomes in OCF Games
An imputation over CS is a division of the payoff of each coalition between its members:
Efficiency:
No payoff to outsiders! If then
Individual rationality
An outcome is the pair

7. What are Core Outcomes?
A set of agents does not want to deviate if they cannot improve their individual payoffs.
How do we define this?

8. A Short Example
Leslie has 2kg of lemons; Sue has 2kg of sugar; Florence has 2kg of flour.
All agents can sell their goods for 10$ a kg.
However: 2kg lemons + 1kg sugar = lemonade = 40$ 2kg flour + 1 kg sugar = cake = 45$

9. What is a stable outcome?
Social optimum is reached when agents make cake and lemonade.
How should payoffs be divided?
Suppose we propose this payoff division: C L
Leslie 1
Sue
0.5
Florence
Cake
Lemonade
Total
Leslie 32
Sue 15 8 23
Florence 30

10. Is deviation worth it?
Sue deviates and sells off the 1kg of sugar promised to Leslie.
What should Florence do?
Florence lets Sue keep her payoff
Florence denies Sue payoff
Cake
Total
Leslie
Sue 15
Florence 30
Lemonade
Sell off 20 32 20 32 10 8 10 23 25

11. Different Reactions make for Different Cores
Agents need to know what is the group reaction to deviation!
[Chalkiadakis et. al. 2010] define three notions of deviation:
Conservative
Refined
Optimistic
The OCF core is the set of all outcomes (CS, x) that no set of agents can profitably deviate from (depends on group reaction).

12. Our Proposed Model: Arbitrated OCF Games
There can be many types of deviation.
We introduce the notion of The Arbitrator

13. Different Arbitrators – Different cores
An arbitrator assigns a value to each coalition from which a deviating set withdrew resources.
Conservative: Green only gets payoff from
Green gets all the payoff from but only half from
Green only gets payoff from and
Refined: Green gets payoff from , and

14. Some of Our Main Results
Core Characterization Result: extends the characterization result in [Chalkiadakis et. al. 2010] to general arbitrators.
An outcome is in the arbitrated core if and only if all sets receive higher payoff than what the arbitrator will give them for any proposed deviation.

15. Our Results - Continued
Alternative Solution Concepts: the arbitrated nucleolus, the OCF Shapley value, the arbitrated bargaining set.
Solution Concepts exhibit similar properties to their non-OCF counterparts:
The arbitrated nucleolus is non-empty
It is always in the core if it is non-empty.
The core is in the bargaining set, etc.

16. Our Results - Continued
The OCF Shapley value is derived from a set of desirable axioms; different axioms yield two (unique) values.
Bounds on the generosity of arbitrators: if an arbitrator is too generous then the core is always empty.

17. Summary We extend the basic framework for cores of OCF games
We define other solution concepts for the OCF setting

18. Future Work Arbitrated convex games.
Application of OCF framework to new models (market models, networks...).
Utilizing arbitrators for coalition formation processes.
Computational Aspects of arbitration.

19. Thank You! We would like to thank:
The Theoretical Computer Science Group at Nanyang Technological University for listening to a preliminary version of this talk.
Thank You!

21. Classic Cooperative Games - Definitions
We normally assume that:
Classic Cooperative Games - Definitions
A cooperative game over n players is defined by a characteristic function:
An imputation represents payoff to agents. An imputation must be:
Feasible:
Coalitionally rational – no payoffs to outsiders!
Individually rational:
The set of all imputations: I(CS)

22. The Core – Stable Profit Division
Each Set of agents asks itself – Can we leave the current coalition structure and collaborate in order to get more? 2 5 3 4 1 6 A B C
Can players 2 and 5 get more on their own than what they’re getting now?
The core is the set of all such that for any set of agents J:

23. The Arbitration Function - Definitions
Given a coalition structure and a set of players , we first define a deviation of J from (CS, x).
Given the nature of the deviation, the arbitration function assigns a non- negative value to all coalitions that consist not only of members of J

24. The Arbitration Function – Definitions, Continued
The arbitration value is the payoff that J receives if it chooses to deviate; it consists of:
the most that J can gain by using its newly available resources; this is available regardless of the arbitrator.
the payoff to J from coalitions, decided by the arbitrator.
a freely distributable value.
Given a coalition c that J deviates from, only members of J that contribute to c receive the arbitrated payoff from c.

25. The Arbitration Function – Desired Properties
Autonomy
Increasing Severity: the more coalitions J hurts, the less J gets.
If J cannot ensure that all non-J members in a coalition c are given the same payoff as before, it will not get payoff from c.

26. The Arbitrated Nucleolus
Given an arbitrator , a subset J and an outcome (CS, x), we define as the most J can gain by deviating from (CS, x).
The difference between and the payoff to J under (CS, x) is called the excess of J in (CS, x) and is denoted e(CS, x, J).

27. Definitions: Continued
We can now order the subsets of players according to their excess:
This can be thought of as a vector: which is called the excess vector of the outcome (CS, x)

28. The Arbitrated Nucleolus
The arbitrated nucleolus is the set of all outcomes whose excess vectors are minimal (using the lexicographic order).
This definition of the arbitrated nucleolus extends the definition of the nucleolus given in Schmeidler (1969) for non-OCF cooperative games.
The definition requires some mild assumptions on the arbitrator.

29. Our Results: The arbitrated nucleolus is nonempty.
Unlike the nucleolus in the classical setting, it is not always unique (per coalition structure).
If the arbitrator is convex, then all sets have the same excess in all nucleolus outcomes.
The core contains the nucleolus if it is nonempty.

30. The OCF Shapley value No arbitrators involved.
Measures the relative contribution of each player.
Relies on an axiomatic approach.
Two natural definitions of the Shapley value for OCF games: one measures the general influence of players, and one the influence of players in a given coalition structure.