Several studies have combined argumentation with coalition formation (e.g. [4,24,25,28,29, 50]); two of the papers that are most relevant to the work presented in this thesis are discussed here.
Firstly, in Dung’s seminal paper that introduced the notion of Argumentation Frameworks (AFs) [50], he used characteristic function games to demonstrate the correctness of AFs. He showed that AFs can be used to represent core and von Neumann-Morgenstein stable solution concepts (where the grand coalition is assumed to form). The following theorem was given:
Theorem 2.1. LetImp be the set of imputations of a coalitional gameGand letattacksbe the corresponding domination relation between them4. Given the argumentation framework AF = (Imp, attacks), then the core of G coincides with all acceptable imputations defended by the empty set.
This Theorem details how all the core solutions can be found for a characteristic function game with transferable utility. Amgoud expanded on Dung’s work to show how argumenta- tion frameworks can be used to reason over whatcoalition structuresto form [4]. The formal framework was as follows:
Definition 52:ACoalition Structure Framework(CSF): is a framework for generating coali- tion structures denotedhC,R,i. In this framework: C is a set of possible coalitions;Ris a binary relationR ⊆ C × C representing the attack relationship between coalitions (the rea- soning for the attack is left undefined); and is a (partial or complete) pre-ordering onC, representing the preferences of the coalitions, where a coalitionCxthat attacksCy only defeats CyifCy Cxdoesnothold.
Due to the nature of argumentation frameworks, Amgoud outlined three classes of coali- tions:
1. The classAR,of acceptable coalitions.
2. The classRR,of rejected coalitions.
3. The classBR,of coalitions neither accepted or rejected, the so-called class of coalitions in abeyance, whereBR,=C\(AR,∪ RR,).
These three classes can be described as follows: theacceptablecoalitions are coalitions that are not attackedorcoalitions defended by other acceptable coalitions; the set of coalitions in abeyance are not defeated by the acceptable coalitions but may conflict with each other; and therejected coalitions are the set defeated by the acceptable coalitions. A two agent dialogue game was also detailed to determine whether a coalition is acceptable or not, which allowed one coalition’s acceptability to be checked without having to know the full set of acceptable coalitions.
Sometimes the set of acceptable coalitions may be empty. Additionally choosing a subset of coalitions, from the coalitions in abeyance, to join with the acceptable coalitions to create a
FIGURE2.8: An example coalition structure framework.
coalition structure may not be an obvious task. To help with these issues, Amgoud introduced semantics for the coalition structure framework, as well as a modified definition of how a set of arguments are conflict free:
Definition 53: For a CSF, a set of coalitionsS ⊆ C isconflict free, iff¬∃Cx, Cy ∈ Swhere R(Cx, Cy)and it is not the case thatCy Cx.
Definition 54:For a CSF, a set of coalitionsS ⊆ Cis apreferred extension, iff: • Sis conflict free
• Sdefends all its elements • Sis maximal (wrt set inclusion).
Each CSF has at least one preferred extension:
Example 27: Consider the following coalition structure framework, represented in Figure2.8: hC,R,i where C = {C1, C2, C3, C4, C5}, R = {(C2, C1),(C2, C3),(C2, C4),(C3, C2),
(C4, C5)}, C2 C1, C4 C2 andC4 C5 (as therelationship is partial, all pairs of coalitions not given a preference order are assumed to have equal preference). The following set of coalitions are conflict free: ∅, {C1}, {C2}, {C3}, {C4}, {C5}, {C1, C3}, {C1, C4}, {C1, C5},{C2, C4},{C2, C5},{C3, C4},{C3, C5},{C1, C3, C4}and{C1, C3, C5}. There exists two preferred extensions: {C1, C3, C4}(because the three coalitions do not attack each other and defend themselves from attacks) and{C2, C4}(because even thoughC2 does attack C4, this attack does not succeed becauseC4 C2).
The class of acceptable coalitions is therefore AR, = {C4} because C4 can never be defeated. The class of rejected coalitions is RR, = {C5} because C5 is defeated by an acceptable coalition. This leaves the coalitions in abeyance BR, = {C1, C2, C3} because these coalitions may or may not be part of the chosen coalition structure, depending on what other coalitions of the abeyance class are chosen.
Coalition Structure Frameworks are based onpreference-based argumentation theory (PBAT) [48]. The key computational issues from PBAT that relate to Coalition Structure Frameworks are:
• Theorem 4 of [48] states that deciding whether an argumentais accepted by at least one preference order in a PBAT is NP-hard. In the context of CSFs, this means that finding out if a coalition is accepted given at least one preference order is NP-Hard.
• Proposition 3 of [48] states that deciding whether an argumentais accepted by all prefer- ence orders in a PBAT is coNP-hard. In the context of CSFs, this means that finding out if a coalition is accepted given all the possible preference orders is coNP-Hard.
• In Section 5 of [48] it is shown that a stable extension of arguments can be found in a PBAT, given a single preference order, in polynomial time. In the context of CSFs, this means that finding a set of coalitions that are stable can be found in polynomial time.
In summary, there are issues that still remain in argumentation methods for coalition forma- tion:
• Detailing an individual imputation as one argument, as done in [50], can create a vastly complex argumentation framework, as a real value utility of a coalition can be distributed in a theoretically infinite number of ways (which, due to rounding, becomes a computa- tionally very large but finite number of ways).
• Additionally, [50] does not detail how agents can collaboratively choose one payoff vector imputation out of all the possible core (or von-Neuman Morgenstein) stable solutions (in a possibly decentralised manner).
• Amgoud [4] does not detail what happens if the agents have their own individual prefer- ences over which coalition to join, only one static system wide preference order is used.
• In [4], how the single static system wide preference order is found is not detailed.
• Finally, [4] does not detail what happens if the payoff is transferable, thus preferences between the agents over which coalition to join can change, depending on their received payoff.