In this chapter, topics within the fields of characteristic function games, cooperative game theory, coalition formation in multi-agent systems, agent communication and argumentation relevant to the research in this thesis have been presented. The remaining chapters focus on the creation of new methods for decentralised coalition formation, drawing on the research presented within this literature review. Work within decentralised environments suggest that methods for coalition formation should be presented that (among other properties): (i) make use of the distributed computational resources; (ii) minimise costly communication; (iii) deal with possibly conflicting information between the agents; and (iv) find solutions that a centralised arbiter with complete knowledge would prescribe.
The following chapters are organised as follows. Chapter 3 begins with an investigation into how argumentation and dialogue game tools (detailed in Section 2.6 and Section 2.5 re- spectively) can be used for the agents to reason over conflicting information in qualitatively described environments, to find a conclusion over what coalitions to form. Additionally Chap- ter3has parallels to the non-transferable utility game literature of Section2.3.1.
Chapter 4 shows how the distributed computational resources of agents can be exploited when calculating all of the coalitions’ values. It describes the problem as a characteristic func- tion game, as described in Section 2.1, and focuses in depth on the literature of distributed coalition value calculation algorithms, detailed in Section 2.4.1. Chapter 5, builds on Chap- ter4’s work to show how cooperative game theory stable solutions can be found with distributed knowledge and minimal communication. Chapter5uses the characteristic function game model of Section2.1, the cooperative game theory solution concepts of Section2.2, and draws on the literature of coalition structure generation and payoff distribution, as described in Sections2.4.2 and Sections2.4.3respectively.
Finally Chapter 6 details a new formal model of coalitional games when the agent differ on their predicted coalition values. Chapter 6focuses in detail on the valuation disagreement coalitional game literature, described in Section2.3.2.
For a discussion on each Chapter’s contributions compared to the related work, see Sec- tion1.4.
Forming Coalitions with
Argumentation Schemes and Critical
Questions
This chapter discusses an argumentation-based method that agents can use to form coalitions in a decentralised manner. In this method, the agents can: (i) engage in an inquirydialogue, where they can use arguments over beliefs to find the current state of the world; and (ii) engage in a persuasiondialogue, where they can use arguments over actions to persuade others over whether or not to form a coalition. Both of these dialogue types are multi-agent dialogues. This chapter focuses on situations where the agents represent their world in qualitative terms by a set of propositions, where coalitions can achieve qualitative changes in the environment through joint-actions between the agents of the coalition. The qualitative representation of the world is assumed to be in Value-based Alternating Transition System (VATS) form1, which itself was an extended version of AATS [135]. In the model of this chapter, the use of joint-actions (orginally embedded within the AATS themselves) allows the agents to reason over which coalitions may conflict with each others capability.
Due to the qualitative representation, inquiry and persuasion dialogues were chosen to allow the agents to form coalitions. The contributions of this chapter are aninquiry dialogue over beliefs that allows the agents (with possible heterogeneous knowledge bases) to reason over the current state of the world. Once a logical conclusion over the state of the world is found, the next contribution is apersuasion dialoguethat will allow the agents to complete the coalition formation process by reasoning over what coalitions to form and what joint-actions these coali- tions should undertake. Both dialogues are defined in the style of a dialogue game. Dialogue games usually consist of a set of communicative acts called moves, a set of rules detailing the moves that are legal to make at any time in a dialogue (the protocol), a set of rules stating the effect of making each move, and a set of rules that detail when a dialogue terminates (e.g. [72]). In this chapter, the contribution of the inquiry dialogue (which is an extension of the Black & Hunter model of [22]) includes: (a) clarification on how the agents compare their current
1
See Section2.6.4for the formal definition of a VATS.
states; and (b) the definition of a protocol that agents can use to find the set of legal moves in the dialogue. Both of these contributions were presented within [101]. Furthermore, the con- tribution of the persuasion dialogue (which is an extension of the Black & Atkinson model of [21]) includes: (i) an argumentation scheme to allow for coalitions to perform joint-actions; (ii) the formalisation of the critical questions associated with the scheme of (i); (iii) a full protocol that allows the agents to find the set of legal moves in the dialogue; and (iv) a model specific ar- gument evaluation method to find the acceptable coalitions to form. Contribution (i) is based on [98], influenced by the argumentation scheme of [10]. Contribution (ii) is based on [101]. Con- tribution (iii) is an extended and modified version of the protocol in [21]. Finally, contribution (iv) is a modified version of the acceptability definitions presented in [19].
This chapter is structured as follows: Section3.1details how the agents find their state of the world. Section3.2details the practical reasoning model used. Section3.3details the inquiry and persuasion dialogues, including the formalisation of all the critical questions. Section3.4 gives a protocol for the inquiry dialogue and a protocol for the persuasion dialogue, where both protocols find all the legal moves for an agent in their respective dialogues. Section3.5 details the method to evaluate the practical reasoning arguments communicated in the persuasion dialogue, to find the acceptable coalitions. Section3.6 gives a full example of both dialogues and the argument evaluation method. Finally Section3.7concludes.
3.1
Finding the State of the World
Before reasoning qualitatively over what coalition to join, it is reasonable to suggest that each agent will want to find the current state of the world. This requires each agent to inquire over the true value of the propositions it uses to represent the world that are likely to influence it’s preference over which coalition to join. The inquiry dialogue, formally detailed in Section3.3, allows the agents to communicate defeasible facts, defeasible rules and B-arguments2. The result of these inquiries will identifyB-arguments that each agent should use to find a conclusion over the current state of the world.
Reasoning over theseB-arguments may be complicated because scenarios could arise whereby B-arguments may claim logical contradictions (e.g. pand¬p). In a decentralised environment, it is possible that some agents would put forthB-arguments that are due to an erroneous knowl- edge base or a manipulation attempt. In this situation, the agents could be equipped with a weighting system based on trust to help resolve which agent’s arguments should be prioritised. The exact details of this system is out of the scope of this thesis.
To find a conclusion for each inquired proposition, the agents can look at the corresponding B-arguments for and against the proposition’s correct Boolean value being true to find the most likely Boolean valuation of each proposition. For the purposes of simplifying the discussion in the rest of the chapter, it is assumed that no contradictions on the Boolean value of a proposition occur.
2
3.1.1 Comparing States
One particular issue that arose when implementing the inquiry dialogue was the need for a method to clarify how agents using different propositions to represent the state of the world can accurately compare states (since agents’ VATS reflect only an individual’s representation of the world). The solution is described in this subsection.
Two agents,iandjcan compare their respective current states (qiandqj), using their respec- tive propositional representation of the world (ΦiandΦj), and assume they are equal (denoted qi ≈ qj) iffπ(qi)∩Φj = π(qj)∩Φi. Otherwise the states are different (denotedqi 6= qj). This equality test utilises an intersection to eliminate propositions that reside in only one of the agent’s propositional representations of the world. When the above approximation holds, the two statesqi andqj cannot reasonably be said to be different, as both states will agree for each shared proposition. However, these two states may not be identical since the same con- clusion can be reached irrespective of the Boolean assignments of the distinct propositions. If the comparison does not hold, then the states are definitely different due to both agents holding inconsistent Boolean values for their shared propositions. This comparison requires: either each agent to have an internal model of the other agent’s beliefs; or for the agents to make (at least some) of the Boolean values ofΦipublic. In this chapter the latter option is chosen due to the revelation of each agent’s current state being a requirement of the practical reasoning argumen- tation scheme (introduced in the next section). If any agentihas any privacy concerns over some propositions that it uses to represent its state, thenican choose to not make these propositions public. But privacy concerns should be balanced with knowledge accuracy concerns, because the fewer propositions communicated byi, the more likely other agents will believe they are in different states toi, yet will not be able to formB-arguments to indicate this, becauseidid not reveal enough information to trigger aB-argument construction.
The following example shows how the state comparison definition works, when every agent reveals all their propositions:
Example 28:Consider the following propositional representations of the world and the current states for agents i andj: Φi = {p, q, r, t}; Φj = {p, r, v}; q
i = [p,¬q,¬r, t]; and qj =
[p,¬r, v]. Given the state comparison definition π(qi) ∩Φj = π(qj) ∩Φi, the substitution {p, t} ∩ {p, r, v}={p, v} ∩ {p, q, r, t}gives{p}={p}. Therefore, asqi ≈qj, the conclusion is that there is no evidence to suggest the states are necessarily different.