Each agent ican evaluate the C-arguments and critical questions uttered in the dialogue that includei, by placing them in aValue-Based Argumentation Framework6(VAF) [19]. For a VAF, it is assumed that each agent/audienceihas a preference order over itssocial-values, of the form v1 ... vk(wherek =|Avi|), that ranksi’s known social-values into an order wherev1 is the most preferred andvk the least. This gives each agentian Audience specific Value-based Argumentation Framework (AVAF), denoted VAFi.
The classical VAF definitions ofobjective acceptanceandsubjective acceptance7, to find the acceptable arguments, are modified in this chapter to handle arguments for coalitions, as several issues were identified whilst developing a model to select the final recommended coalitions to form, from the communicated arguments in theC-pActdialogue. These issues were:
1. Coalition Membership- In this chapter it is assumed that each agentionly reasons over arguments for coalitions whereiis a member of that coalition. This condition is used so that an agent cannot try to manipulate other agents not in its coalition with false arguments over that coalition.
2. Incomplete Coalitions- There may be some arguments communicated in the dialogue, through the propose move, that have incomplete coalitions. These arguments should be discounted from consideration.
3. Agents Leaving- Arguments that include an agent who has left the dialogue before the dialogue completed should be removed from consideration.
4. Arguments Without Social-Values - Since several critical questions concern problem formulationissues [10], not all arguments have an associated social-value (i.e. those de- rived from CQ13, CQ14 and CQ15). Instead, these arguments are taken to automatically defeat any other argument that they attack in the AVAF, by being assigned the social- value ‘truth’, which always ranks higher than any other social-value [19], provided that these arguments areagreeable. Anagreeableargument for an agentiis one that matches the information in agenti’s VATS, i.e.: an agreeable argument can be formulated from VATSiif required. For example, CQ16 against the social-valuev7is onlyagreeabletoiif v7∈/ Avi. The same argument isnot agreeabletojifv7 ∈Avj. In future work, this pro- cess could be modeled withinuniform argumentation frameworks[11] where value-based and value independent arguments are modeled together.
5. Coalition to Coalition Attacks- Two different arguments in favour of two distinct coali- tions should attack each other if they share an agent, due to a traditional assumption of coalition formation that an agent can only be in one coalition at any one time [136]. Fur- thermore, given the model presented in this chapter, two different arguments A1 and A2 for two different coalitions should attack each other if EndState(A1) 6= EndState(A2),
6Described in background Section2.6.2. 7
which indicate that the two coalitions will attempt to bring about a state with at least one conflicting proposition. This is a logical contradiction.
After considering these issues, for an agentito construct an AVAF to reason over its accept- able coalition(s) given the arguments communicated during the dialogue, each agentishould: (1) find the arguments with coalitions includingi; (2) remove the arguments with incomplete coalitions; (3) remove the arguments that include an agentjwho has left the dialogue; (4) re- move the non-agreeable arguments that promote the value ’truth’; (5) and add the critical ques- tion attacks (which include the coalition to coalition attacks). Each agentican then evaluate the VAFiit has created, using its social-value ordering to find its preferred extension(s).
In this thesis, due to theses identified issues, model specific acceptability definitions are used:
Definition 75: Coalitionally Objectively-Stable: Given a VAF,hArgs,R,V,val,Ni, a coali- tionC⊆Nis coalitionally objectively-stable if and only if:∀i∈C,∃A∈Argsthat is inevery preferred extension ofV AFiwhere Coalition(A) =C.
Example 3.1. ConsiderN ={1,2,3}agents discussing what coalitions to form, where there are three possible coalitions that can form:
• Coalition{1,2}in argumentA1that promotes social-valuev3;
• Coalition{1,3}in argumentA2that promotes social-valuev1; and
• Coalition{2,3}in argumentA3that promotes social-valuev2.
In this example, all arguments attack every other argument. The social-value ordering of the agents are as follows:
• Agent 1 prefersv1overv2overv3;
• Agent 2 prefersv3overv2overv1; and
• Agent 3 prefersv1overv2overv3.
Given that the agents only reason over coalitions that involve itself, agent 1 has a preferred extension (PE) of PE11 ={A2}, agent 2 has PE21 ={A1}and agent 3 has PE31 ={A2}.
In this case, it can be seen that coalition{1,3}, in argumentA2is coalitionally objectively-
stable because the coalition exists in all the preferred extensions of all the agents that are in- cluded in it. For example,A2is in PE11and PE31while no more PEs exist for agents 1 or 3.
Definition 76: Coalitionally Subjectively-Stable:Given a VAF,hArgs,R,V,val,Ni, a coali- tion C ⊆ N is coalitionally subjectively-stable if and only if: ∀i∈C, ∃A∈Args that is in at least onepreferred extension ofV AFi where Coalition(A) = C; and¬∃A0 ∈Argswhere Coalition(A0) 6= Coalition(A), R(A0, A) and for all j ∈ Coalition(A0), A0 is inat least one preferred extension ofV AFj.
Example 3.2. ConsiderN ={1,2,3}agents discussing what coalitions to form, where there are four possible coalitions that can form:
• Coalition{1,2}in argumentA1that promotes social-valuev1;
• Coalition{1,3}in argumentA2that promotes social-valuev1;
• Coalition{2,3}in argumentA3that promotes social-valuev2; and
• Coalition{1,2,3}in argumentA4that promotes social-valuev3.
In this example, all arguments attack every other argument. The social-value ordering of the agents are as follows:
• Agent 1 prefersv1overv2overv3;
• Agent 2 also prefersv1overv2 overv3; and
• Agent 3 prefersv3overv2overv1.
Given that the agents only reason over coalitions that involve itself, agent 1 has two preferred extensions (PEs) of PE11 = {A1}or PE12 = {A2}, agent 2 has only PE21 ={A1}and agent 3
has only PE31 ={A4}.
In this case, it can be seen that the coalition{1,2}, in argumentA1is coalitionally subjective-
stable because even though it is not in an argument in all the preferred extensions of agent 1 (that is agent 1 thinks it has another option available), it is the only coalition option that is also in an argument in the other coalition member’s preferred extension. For example,A1 in PE11
and PE21while{1,3}(in argumentA2) is not in any PE of agent 3.
Definition 77: Coalitionally Subjectively-Unstable: Given a VAF, hArgs, R, V, val, Ni, a coalitionC ⊆N is coalitionally subjectively-unstable if and only if: ∀i∈C,∃A∈Argsthat is inat least onepreferred extension ofV AFiwhere Coalition(A) =C; and∃A0∈Argswhere Coalition(A0) 6= Coalition(A), R(A0, A) and for all j ∈ Coalition(A0), A0 is inat least one preferred extension ofV AFj.
Example 3.3. ConsiderN ={1,2,3}agents discussing what coalitions to form, where there are three possible coalitions that can form:
• Coalition{1,2}in argumentA1that promotes social-valuev1;
• Coalition{1,3}in argumentA2that promotes social-valuev1; and
• Coalition{2,3}in argumentA3that promotes social-valuev2.
In this example, all arguments attack every other argument. The social-value ordering of the agents are as follows:
• Agent 1 prefersv1overv2;
• Agent 2 prefersv1overv2; and
Given that the agents only reason over coalitions that involve itself, agent 1 has two preferred extensions (PEs) of PE11 = {A1}or PE12 = {A2}, agent 2 has only PE21 ={A1}and agent 3
has only PE31 ={A2}.
In this case, it can be seen that coalition {1,2} in argumentA1, and coalition{1,3} in
argumentA2, are coalitionally subjectively-unstable because both of the coalitions exists in at
least one argument of one of the preferred extensions of all their members. For example,A1 is
in PE11and PE21, whileA2is in PE12and PE31.
Any coalitionCthat is coalitionally objectively-stable should form, as all agents believe that Cis acceptable to them given their value order undereverypreferred extension of their AVAF, and they do not have another equally acceptable coalition (otherwiseCwould be either coali- tionally subjectively-stable or coalitionally subjectively-unstable due to the coalition to coalition attack issue).
Any coalition C that is coalitionally subjectively-stable should form, as all agents believe thatC is acceptable to them given their value order underat least onepreferred extension of their AVAF, and they do not have another equally acceptable coalition C0 to form that every agent ofC0 finds equally acceptable (otherwiseC would be coalitionally subjectively-unstable due to the coalition to coalition attack issue).
IfCSis the set of coalitions that should form after the VAF evaluation, then: for every coali- tionCo, that is coalitionally objectively-stable,Co ∈ CS should hold; and for every coalition Csthat is coalitionally subjective-stable,Cs ∈CSshould hold.
The remaining coalitions that should be considered for inclusion inCSare those coalitions that are coalitionally subjectively-unstable (denoted∆). That is, a coalitionCthat is acceptable to every agenti∈ Cunderat least onepreferred extension ofi, even though agentihas other coalitions of the same acceptability level. Firstly any subsetΛ ⊆∆chosen to be added toCS should have the following conditions so thatΛdoes not conflict with itself or the other coalitions inCS: (1) no coalitions inCS orΛ should share an agent; and (2) no coalitions inCS orΛ
should attempt to bring about a conflicting end state. That is,CSshould always remain conflict free.
The idea of partitioning the coalitions into three sets resembles the three set partition of Amgoud’s work in [4], where acceptable, abeyance and rejected coalitions were identified. Yet in [4], only one preference order for the coalitions was used. Having multiple preference orders gives rise to different degrees of acceptability (identified in this chapter as coalitionally objectively-stable, coalitionally subjectively-stable and coalitionally subjectively-unstable).
To resolve which coalitions Λ should be chosen from∆, inspiration could be taken from qualitative coalitional games (QCGs). In these games, the minimal coalition(s) that are success- ful are chosen. In this chapter, the minimal successful coalitions are the ones that satisfy as many agents as possible, i.e.¬∃Λ0⊆∆where|Λ0|>|Λ|. Yet this method of selection does not take into account the preferences of the agents.
Alternatively, to resolve which coalitions to choose from∆, inspiration could be taken from qualitative coalitional games with preferences. In these games, using the stability definition given in Section2.3.1, a stable subsetΛ ⊆ ∆could be identified where there does not exist a
coalitionS ∈∆,S /∈Λ, where every member ofSprefers coalitionScompared to its coalition inΛ.
To conclude, exactly which arguments of∆should be chosen to be in the final recommended coalition structureCSshould depend on what the dialogue designer wants to achieve, for ex- ample: are a minimal number of coalitions desired?; or are the most stable coalitions desired?