Arbitrators in Overlapping Coalition Formation Games Yair Zick

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

Outline What are OCF Games ? Definition of an Arbitrator Our model: Arbitrated Cores We review our results for Arbitrated OCF Games

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

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 2 B 6 5 4 C Player 5 gives 75% to B and 25% to C

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 finite collection of coalitions so that: is a

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 Individual rationality An outcome is the pair then

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?

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

What is a stable outcome? Leslie Sue Florenc Social optimum is reached when agents make cake and lemonade. How should payoffs be divided? Suppose we propose this payoff division: C 0 0. 5 L 1 0. 5 1 0 Leslie Sue Florenc e Cake 0 15 30 Lemonade 32 8 0 Total 32 23 30

Is deviation worth it? Sue deviates and sells off the 1 kg of sugar promised to Leslie. What should Florence do? Florence lets denies Sue keep payoff her payoff Leslie Sue Florenc e Cake 0 15 30 Lemonade Sell off 20 32 8 10 0 Total 20 32 10 23 25 30

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).

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

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. Refined: Green gets payoff from , and. Green only gets payoff from and.

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.

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-

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.

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

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.

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

Classic Cooperative Games We normally assume that: • - 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)

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

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

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

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.

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).

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)

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.

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

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