A Value-based Alternating Transition System (VATS) [9] is an appropriate representation for reasoning over a qualitative coalition formation process as the underlying AATS [135] was ini- tially used to show how (already formed) coalitions, performing joint-actions, change the state of the world. The VATS will be used in this chapter to help the agents of the system find acceptable coalitions, given their beliefs and social-value preference order.
To help the agents form coalitions, the following new function is added to the VATS repre- sentation of agenti, given the full set of agents of the coalitional game (denotedN):
• ζi : N ×Aci → {>,⊥} is a representation function, which defines whether agent i believes if agentj ∈ N will perform the given action (therefore>is returned) or not (therefore⊥is returned).
The Practical Reasoning Argumentation Scheme for Coalition Formation (PRASCF), mod- ified from the AS1 scheme3of [21], and presented in [98] is:
PRASCF: In the current circumstances R, joint action J should be performed, by coalition C, which will result in the new circumstances S, which will realise goal G and promote/demote the social-value V.
Circumstances RandS are represented as tuples of propositions. Joint actionJ is a tuple µ of single actions, denoted µ = hacm, ..., acni. The coalitions are assumed to be able to coordinate these joint-actions themselves4. Coalition C = {i, ..., j} is a set of agents where a tupleξ matches agents to single actions, denotedξ = h(i, acp), ...,(j, acq)i. The intended interpretation forξ is that if this coalition is accepted, then each agent inC will perform the single action that it is paired to inξ. An agent can put forth PRASCF arguments, to persuade others that a coalition should be formed to undertake a joint-action, by instantiating this scheme. The goal to be achieved, and the social-value to be promoted, provides the justification for the coalition to undertake the joint-action.
Due to the nature of decentralised environments, the following conditions are enforced when an agentiis creating a new instantiation of the argumentation scheme:
• The ξ tuple cannot have been previously proposed in an argumentation scheme and re- jected unless the interpretation of the current state has changed, through an agenttalking by doing (i.e. an agent performing an action or a coalition performing a joint-action) or through an agent doing by talking (i.e. an agent asserting a new B-arguments that could influence another agent’s opinion of the current state). This stops instantiations of argumentation schemes being repeated indefinitely, yet allows for situations where new information is uncovered that indicates the current reasoning was flawed.
• Every agent of the coalition must be an agent in theξtuple. This condition requires each member of the coalition not to be adummy player[136], i.e. each agent of the coalition must contribute to the coalition it is in.
• Each agent must occur once and only once inξ. This condition ensures that an agent is not given multiple single actions to perform at the same time.
• The coalition argued for/against does not have to be complete, i.e. theξ tuple does not have to be fully instantiated initially. This condition is due to an agentipotentially having only partial knowledge of the system.
A formally instantiated version of the PRASCF scheme is defined as:
3The AS1 scheme is described in Section2.6.3. 4
With this assumption the joint actions operate in much the same manner as actions in the AS1 schema (the only different is that a certain number of agents are required to perform a joint-action).
Definition 55: A PRASCF is denotedC = hqx, µ, ξ, qy, p, v, si where qx is the current state, µis the joint action, ξ is the tuple that matches agents to a joint action, qy is the new state, p is the goal in the new state, v is the social-value associated with this state transition and s (where s = {+,−,=}) is the sign indicating whether the value is promoted/demoted/not affected respectively.
In the PRASCF scheme, the coalition C is implicit in theξ tuple, found by: ∀hi, aci ∈ ξ then i ∈ C. A C-argument will represent a proposal if there exist single actions of µ not matched to an agent inξ. This situation is a proposal because the coalition does not yet have a sufficient number of members to carry out the joint-action and so requires others to complete the coalition. AC-argument will represent anassertionif all single actions ofµhave been assigned to an agent, because the coalition now has enough members to carry out the joint-action and is therefore ready to form.
Additionally agents will be able to communicate critical questions attacking any proposal, assertion or another critical question. A formalised critical question is instantiated as a modified version of aC-argument intended to reflect the question it represents, in a logical form. These critical questions are modifications of the critical questions of the AS1 scheme, presented in Section2.6.3.