Argument Evaluation

This section evaluates the arguments presented in the exampleC-pAct dialogue using the ac- ceptability definitions introduced in Section3.5, which are used to determine which coalitions should be formed. To find either coalitionally objectively-stable, coalitionally subjectively- stable or coalitionally subjective unstable arguments, each agent i should generate their au- dience specific value-based argumentation framework, denoted VAFi, that contains arguments in theC-pActdialogue that: (i) haveiin the coalition; or (ii) includeagreeablearguments toi (i.e. arguments that include information that is within agenti’s VATS).

Looking initially at VAF1 in Figure3.18there are two main issues to notice: (a) Not all of the arguments from theC-pActdialogue are present because only some of the arguments in the dialogue included agent 1; and (b) the majority of the attack relationships are reciprocal, due to the nature of the critical questions between these arguments.

FIGURE3.18: Agent 1’s VAF taken from the exampleC-pActdialogue. Each node is labeled with the argument ID and the associated value. If there is no associated value then the label>

Agent 1 can evaluate these arguments using its value-ordering. As Agent 1 does not recognise the valuev4, arguments using this value are defeated in VAF1by argument A9. Assuming agent 1’s value ordering isv2 ≺ v3, then agent 1 has two preferred extensions: P E₁1 = {A1}and P E₂1 ={A4}. As Coalition(A1) = {1,2}and Coalition(A4) = {1,3,4}, agent 1 is currently undecided on which coalition it wants to form because there is no coalition that is coalitionally objectively-stable, due to there being no coalition in all of agent 1’s preferred extensions.

To find out if{1,2}is acceptable then VAF2 (Figure3.19) has to be evaluated. This eval- uation is very simple because there are only two arguments in the exampleC-pAct dialogue that involve a coalition that agent 2 is a member of. Both of these arguments promote different social-values, whilst suggesting that coalition{1,2} should form. Therefore, whatever value ordering agent 2 choses, there will be one argument with the coalition{1,2} in the preferred extension. It is now known that {1,2} is acceptable to its members. To find out if {1,2} is coalitionally subjectively-stable or coalitionally subjectively-unstable, agent 1 needs to know if the coalition{1,3,4}is also acceptable, and therefore a valid option.

FIGURE3.19: Agent 2’s VAF taken from the exampleC-pActdialogue. Each node is labeled with the argument ID and the associated value.

To find out agent 3’s acceptable coalitions (and to continue investigating if {1,3,4}is ac- ceptable) then VAF3 (Figure 3.20) has to be evaluated. In VAF3, argument A10 removes A6 and A7 from consideration due to these instantiations suggesting an end state that conflicts with VATS3. Therefore agent 3 is left with a choice between: coalition{1,3,4} supported by ar- guments A2 and A4; and coalition {3,4} supported by argument A5. In this case, because different social-values are promoted by the arguments; only one preferred extension is possible. The argument in the preferred extension depends on agent 3’s value ordering. Before discussing how the different value orderings will effect the system’s outcome, VAF4 will be introduced (Figure3.21).

It can be seen that VAF4is similar to agent 3’s apart from agent 4 has A8 (instead of A10) to eliminate arguments from consideration, where A8 is used to show that agent 4 does not recog- nisev2. Out of the remaining arguments, A5 and A7 support coalition{3,4}, while arguments A2 and A6 support coalition{1,3,4}. Assuming agent 1’s value ordering isv2 ≺v3, then to get thefinal recommended coalition structure to beCS = {{1,2},{3,4}}, agent 3’s value order should have v4 as the most preferred, whereas agent 4 can have any value order. This would make both coalitions ofCS coalitionally subjectively-stable as no other coalitions would have the same level of acceptability. The coalitions ofCS are not coalitionally objectively-stable because neither coalitions ofCSare present in all preferred extensions of agent 1 and agent 4.

FIGURE3.20: Agent 3’s VAF taken from the exampleC-pActdialogue. Each node is labeled with the argument ID and the associated value. If there is no associated value then the label>

is attached to the node to indicate the social-value ’truth’.

Alternatively, coalitions {1,2} and{1,3,4} would be coalitionally subjective-unstable if agent 1’s value order remained the same, agent 3 favouredv2orv3, and agent 4 had any value order. In this case, both{1,2}and{1,3,4} would exist in at least one (but not all) preferred extension(s) of every one of its members. As both coalitions would be coalitionally subjective- unstable and conflict with each other by sharing a member, then only one of them can form. This concludes the example.

FIGURE3.21: Agent 4’s VAF taken from the exampleC-pActdialogue. Each node is labeled with the argument ID and the associated value. If there is no associated value then the label>