La ciudad y el Instituto General y Técnico.

arguments between one another to convince each other of the acceptability or not of a given claim (or negotiation stance as the case may be).

2.5

Summary

We have presented in this chapter background knowledge on the key subject areas underpinning this thesis. The introduction of resources, allocations, agent preferences and social welfare in Section 2.1, as well the choices we make in this thesis regarding these concepts, will be useful in framing the resource reallocation problem of interest in this thesis (see Chapter 3). The introduction, also in Section 2.1, of distributed allocation procedures and deals will be useful in capturing the kinds of deals that are necessary and sufficient for solving our resource reallocation problem of interest in this thesis (see Chapter 4). The discussion in Section 2.2 of issues involved with agent communication and interaction for solving multi-agent problems such as the resource reallocation problem by means of agent dialogues (negotiation), as well as the discussion in Section 2.3 of issues surrounding argumentation-based negotiation, will be useful in presenting an argumentation-based negotiation protocol and policy for solving our resource reallocation problem of interest in this thesis (see Chapter 5). Lastly, the presentation of assumption-based argumentation in Section 2.4 will provide a basis for the general-purpose argumentation framework which we define for multi-agent settings (see Chapter 6) and which we use for grounding our argumentative negotiation policy (see Chapter 7).

Agent Systems for the Resource

Reallocation Problem

We define in this chapter the key components of our problem domain, amounting to the notion of agent system (Section 3.1), resource reallocation problem (Section 3.3), and individual and social welfare (Section 3.4). Informally, the problem domain is as follows: An agent is initially allocated a set of resources, possibly none, where resources are single-unit (distinguishable by name), indivisible (an agent either has a particular resource or it does not) and unshareable (no two agents have the same resource). Each agent has a goal which it desires to fulfil. A certain goal may be fulfilled by a choice of different resources. Also, a certain resource may fulfil a choice of different goals. We demonstrate in Section 3.2 an example application domain.

Throughout the thesis we will adopt the following notation: ¬ stands for (classical) negation; terms beginning with a capital letter are variables; terms beginning in small-case are constants;

stands for an anonymous variable (as in Prolog). 50

3.1. Agent System 51

3.1

Agent System

We consider agent systems where (i) each (resource reallocation) agent may have in its posses- sion some (or no) resources; and (ii) each agent has a goal, which is fulfilled if the agent has a particular resource which the agent believes to fulfil its goal. In addition to beliefs about resources fulfilling goals, agents may hold beliefs about other agents existing in the system, holding resources, and having goals.

Definition 3.1 (Agent System) An agent system is a triple (G, R, A) such that

• G is a (finite) set of constants, representing goals of agents in the system

• R is a (finite) set of constants, representing resources (owned by agents) in the system • A is a (finite) set of (resource reallocation) agents, where an agent is a quadruple hx,

Res(x), G(x), B(x)i such that

– x, the name, is a constant, uniquely identifying the agent in A – Res(x) ⊆ R is the set of resources allocated to x,

– G(x) ∈ G is the goal of x,

– B(x), the belief set of x, is a consistent set of ground literals including (some) instances (for Y 6= x, hY, , , i ∈ A) of

(i) isAgent(Y ), representing the information that Y is another agent in the system; (ii) has(Y ,R), for R ∈ R, indicating x’s belief that agent Y is currently allocated

resource R;

(iii) ¬has(Y ,R), for R ∈ R, indicating x’s belief that agent Y is not currently allocated resource R;

(iv) goal(Y ,G), for G ∈ G, indicating x’s belief that G is the goal of agent Y ; (v) fulfils(R,G), for R ∈ R and G ∈ G, representing that resource R fulfils goal

Note that each agent has a single goal but several agents may have the same goal. Note also that each agent may have several resources. Given AS = (G, R, A), let names(AS) = {x | hx, , , i ∈ A}.1 _{We will assume that names(AS) ∩ G = {}, names(AS) ∩ R = {} and}

G ∩ R = {} (namely, the constants used to refer to agents, goals and resources are distinct). We will also assume that resources held by agents do not overlap. Namely: Res(x) ∩ Res(y) = ∅ for x, y ∈ names(AS) where x 6= y, and that S

x∈names(AS)Res(x) = R. An allocation of resources

in an agent system then is a partitioning of R amongst the agents in A, as follows:

Lemma 3.1 (allocation in an agent system) Given an agent system AS = (G, R, A), let P : A → 2R

such that P (hx, , , i) = Res(x). Then, P is an allocation of resources (in the sense of Definition 2.1).

We will refer to the allocation P as Res, and say that AS contains the allocation Res.

The lemma above implies that, in our agent systems, resources are indivisible (i.e. agents cannot receive fractions of resources) and non-sharable (i.e. a resource cannot be allocated to two or more agents at the same time). Note also that, by using constants to distinguish resources, these are single-unit (i.e. there is only one copy of each resource in our agent systems).

In our agent systems it may be the case that two (or more) agents have the same goal. Each goal may be achieved by acquiring any of a number of resources, as indicated by the fulfils beliefs in the agents’ belief sets. Note that we assume, for simplicity, that single resources suffice for the fulfilment of goals. Each resource fulfilling a goal may be seen as a plan for that goal.

Definition 3.2 (agent’s beliefs about plans in an agent system) An agent x’s beliefs about plans in an agent system AS is defined as follows: P lAS(x) = { fulfils(R,G) |

fulfils(R,G) ∈ B(x) }. We denote all beliefs about plans in an agent system as follows: P lAS =

S

x∈names(AS)P lAS(x).

1

3.1. Agent System 53 With an abuse of terminology, from now on we will refer to beliefs about plans simply as plans. Note that it may be the case that there are no (beliefs about) plans for a goal in an agent system.

In general, in our agent systems, agents may have incorrect and only partial beliefs of the resources and goals of the other agents in the system. In this thesis we assume initial agent systems such that agents are aware of the existence of all their fellow agents in the agent system and have beliefs as to which resources fulfil which goals but have no beliefs (initially) about the resources held (or not held) by other agents, nor any beliefs about the goals of other agents, as follows:

Definition 3.3 (initial agent system) For an initial agent system AS = (G, R, A), all agents hx, Res(x), G(x), B(x)i ∈ A are such that:

• isAgent(y) ∈ B(x) for all y ∈ names(AS) where y 6= x; • P lAS(x) = P lAS(y) for all agents y ∈ names(AS) where y 6= x;

• has( , ), ¬has( , ), goal( , ) /∈ B(x).

We will refer to the allocation that the initial agent system AS contains as the initial allocation.

Agents would come to hold beliefs about the resources and goals held by other agents by means of dialogue (see Chapter 5).

To summarise:

In this thesis we assume that single resources suffice for the fulfilment of goals, that agents know which resources fulfil which goals, and that agents are aware of the existence of all their fellow agents in the agent system.

As a consequence of assuming that agents know which resources fulfil which goals, we have that P lAS = P lAS(x) for any x ∈ names(AS).

g1 @ @@ r1 r2 g2 r1 g3 @ @@ r2 r3

Figure 3.1: Plans P l in Example 3.1.