The research presented in this thesis has been developed under the supervision of Dr. Katie Atkinson, Dr. Terry Payne and Professor Trevor Bench-Capon. A summary of the publications (to date) relating to the work presented in this thesis is as follows:
• The contributions of theC-inqprotocol,C-pActprotocol and critical questions of Chap- ter3, builds on work undertaken with Katie Atkinson, Terry Payne and Elizabeth Black and was published as “An Implemented Dialogue System for Inquiry and Persuasion” in the first International Workshop on the Theory and Applications of Formal Argumentation (TAFA), in Barcelona, Spain, 2011, [101].
• The contributions of the argumentation scheme of Chapter3 and the application of the critical questions of [101] to coalition formation, builds on work published as “A Persua- sive Dialogue Game for Coalition Formation” in the inaugural Imperial College Student Workshop (ICCSW), in London, UK, 2011, [98].
• The contribution of the DCG algorithm for the distribution of coalition value calculations of Chapter4, initially builds on work undertaken with Terry Payne, Trevor Bench-Capon and Katie Atkinson and was published as an extended abstract “Distributing Coalition Value Calculations to Self-Interested Agents” in the thirteenth International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), in Paris, France, 2014, [102].
• The contribution of the DCG algorithm for the distribution of coalition value calculations of Chapter4additionally builds on work undertaken with Katie Atkinson, Paul Dunne and Terry Payne and was published as “Distributing Coalition Value Calculations to Coalition Members” in the twenty-ninth Conference on Artifical Intelligence (AAAI), in Austin, Texas, USA, 2015, [99].
Literature Review
In this literature review, the topics of coalitional games, cooperative game theory, coalition formation in multi-agent systems,agent communicationandargumentationare discussed. Dis- cussing these topics allows the background of coalition formation to be detailed and the limita- tions of current multi-agent system coalition formation methods to be described, thus showing where the motivation for this thesis’s research question occurs from.
The literature review commences by detailing the formal foundations of coalitional games in Sections2.1,2.2and2.3. Section2.4goes on to describe, with examples, how multi-agent systems operate within these formal foundations to form coalitions. Later in this thesis, Chap- ters4and5offer new methods for agents to form coalitions within these predefined quantitative coalitional game foundations, while Chapter6gives a new formal quantitative coalitional game definition.
The literature review concludes with Sections2.5and2.6that discuss models of agent com- munication and argumentation, including how agents can communicate effectively and reason logically over possibly conflicting qualitative information. Later in this thesis, these models are built on in Chapter3to give a new method for agents to form coalitions using these predefined qualitative foundations.
2.1
Characteristic Function Games
In the seminal work of [79], von Neumann and Morgenstern constructed the theory ofn-person cooperative games, where groups of agents can join together to form coalitions. They detailedn- person cooperative games in characteristic function form, where each coalition has an associated real numericutilityvalue that it can achieve:
Definition 2: A characteristic function game is denoted G = hN, vi, where N is the set of agents in the game andv is the characteristic function that maps every potential coalition C ⊆ N to a real numeric value (i.e. v(2N) → R). By default an empty coalition receives no
utility payoff (i.e.v(∅) = 0).
In the literature, some assumptions on characteristic function games are made (see Section 2.3for discussion on alternatives to these assumptions):
• The characteristic function formula is not detailed in the formalism.
• The characteristic function formula is agreed on by all the agents beforehand.
• All the agents have perfect information to use the characteristic function formula to get an accurate utility value for each coalition.
• The valuev(C)returned by the characteristic function is the largest value that coalition Ccould achieve.
• Each coalition’s utility value is independent of what non-members do.
• The value of each coalition doesnotchange over time.
• The value of each coalition can be distributed to its members in any manner. Games with this property are known astransferable utility games.
Additionally, it is traditionally assumed that in characteristic function games the coalition of all agents (known as thegrand coalition) forms.
Definition 3:A coalitionCis thegrand coalitionfor a characteristic function gameG=hN, vi iffC =N.
It is rational to assume that the grand coalition will form in superadditive games [106]: Definition 4:A characteristic function gameG=hN, viissuperadditiveif for any two disjoint coalitionsDandE, whereD, E ⊂N andD∩E =∅,v(D∪E)> v(D) +v(E).
Traditionally the grand coalition is assumed to form in any characteristic function game because it is argued that at worst the agents in the larger coalition can behave as if they were in smaller coalitions [104]. Yet [104] details the reasons that this classical assumption should not always be assumed in multi-agent systems: (i) there maybe a coordination overhead for forming the grand coalition, e.g. communication costs or anti-trust penalties; (ii) finding the optimal manner the grand coalition can work together maybe more costly than finding out how a group of smaller coalitions can optimally work together; and (iii) if time is limited, the agents may not be able to carry out the communication and computation required to coordinate effectively within the larger coalition. Therefore in multi-agent systems it may be beneficial to form a set of coalitions, known as a coalition structure [104], defined as:
Definition 5: A coalition structure(CS) is the set of coalitions in a system, denoted: CS =
{C1, ..., Ck}. In a characteristic function game G = hN, vi, the coalition structure has the following properties:
k [
j=1
Cj =N and (2.1)
The first condition states that the union of all the coalitions must equal the total set of agents of the game. The second condition states that each agent should only be a member of a single coalition. Overlapping coalitional games (e.g. [35,42]) relax the second condition, yet they are not addressed here as they are outside the scope of this thesis. Throughout the thesisv(CS)is written to denote the total value of a coalition structure CS, i.e. v(CS) = P
C∈CSv(C). A special type of coalition structure, measured by it’ssocial welfare(i.e. the sum of the values of all the coalitions in the coalition structure), is as follows:
Definition 6:Anoptimal coalition structure(CS∗) is a coalition structure that maximises social welfare, i.e.:¬∃CS0wherev(CS0)≥v(CS∗).
Finding anoptimal coalition structure(CS∗) that maximises social welfare is known as the coalition structure generationproblem [104], discussed in Section2.4.2.
Example 1: Consider the characteristic function gameG = hN, viwhere N = {1,2,3}and the characteristic functionv gives the following valuations: v({1}) = 2(i.e. the utility value, of agent 1 by itself, is 2),v({2}) = 1,v({3}) = 2,v({1,2}) = 10(i.e. the utility value of the coalition, containing agents 1 and 2, is 10),v({1,3}) = 8,v({2,3}) = 9andv({1,2,3}) = 10 (i.e. the utility value, of the grand coalition, is 10). Then CS∗ = {{1,2},{3}} because v(CS∗) =v({1,2}) +v({3}) = 10 + 2 = 12, and no other valid coalition structure can give a higher total value for the gameG.
Finding an acceptable coalition structure is only half the requirement in characteristic func- tion games. Agents also need to be motivated to join a coalition. In characteristic function games, this motivation comes from the transferable utility that each agent in the coalition re- ceives. Apayoff vectorxis used to distribute the total utility value of the coalition structure to the individual agents [79].
Definition 7:Apayoff vectoris denoted:x= (x1, ...xn)∈Rnwherexi ≥0for alli∈Nand xi corresponds to the utility payoff assigned to agenti. Throughout the thesis, notation will be abused by usingx(C)to denote the component of the payoff vectorxthat has individual payoffs only for agents of the coalitionC, i.e.x(C) =P
i∈Cxi.
Given a payoff vector, the worst payoff any agent can receive is zero. An example payoff vector isx(1,4,5,2), where the coalition {1,4}receivesx({1,4}) = x1 +x4 = 1 + 2 = 3 total payoff. Traditionally all the payoff of a coalition can only be divided between agents of that coalition, i.e. traditionally side payments between coalitions are not allowed [83]. Yet there exists some research in multi-agent systems that is starting to relax the assumption of this property (e.g. [2,66]).
A problem with some payoff vectors is that they may not satisfy the requirements of all the agents of the system. The most obvious requirement for a payoff vector is the following [83]: Definition 8: A payoff vector is individually rational if all agents receive at least as much utility payoff as they would receive by themselves, i.e.∀i∈N,xi ≥v({i}).
A payoff vector that satisfies individual rationality is known as animputationand is defined as [83]:
Definition 9: Animputationis a payoff vector that satisfies individual rationality. The full set of imputations of a characteristic function game is denotedImp(N, v).
Two different imputations may not be of the same quality, which motivates the next defini- tion [83]:
Definition 10: Given two payoff vector imputationsxp andxq, xp is said to dominate xq if there is a non-empty coalitionSsuch that∀i∈S,xpi > xiqandxp(S)≤v(S).
The notion of dominance is used by the coreandstable setsdefinitions (described in Sec- tion2.2), which detail the acceptable payoff vectors for self-interested agents.
Example 2:Consider the characteristic function gameG=hN, viwhereN ={1,2,3}and the characteristic functionvgives the following valuationsv({1}) = 1,v({2}) = 2,v({3}) = 2, v({1,2}) = 4, v({1,3}) = 6, v({2,3}) = 5 andv({1,2,3}) = 9. The optimal coalition structure of this game isCS∗={{1,2,3}}. The following payoff vectors, dividingv(CS∗) = 9
between the agents, are imputations: x1(3,3,3); and x2(2,5,2)(wherex1 dominates x2 via coalition{1,3}becausex11 > x21,x13 > x23 andx1({1,3}) ≤v({1,3})). The following payoff vectors are not imputations:x4(0,4,5)becausex41 < v({1});x5(4,1,4)becausex52< v({2}); andx6(0,9,0)becausex61 < v({1})andx63 < v({2}).
Given the above information, the outcome of a characteristic function game can now be detailed [104]:
Definition 11: An outcome of a characteristic function game G = hN, vi for n agents is a coalition structure and payoff vector pair, denoted: hCS, xiwhereCSis a set ofkcoalitions, denoted{C1, ..., Ck}andxis the payoff vector, denotedx(x1, ..., xn).
Example 3:Consider the characteristic function gameG=hN, viwhereN ={1,2,3}and the characteristic functionvgives the following valuations: v({1}) = 1,v({2}) = 2,v({3}) = 2, v({1,2}) = 4,v({1,3}) = 6,v({2,3}) = 5andv({1,2,3}) = 9. Example outcomes to this game are:h{{1,2,3}}, x(3,3,3)i;h{{1,2},{3}}, x(2,2,2)i; andh{{1,3},{2}}, x(2,2,4)i.
Some outcomes are favoured more than others for reasons of social welfare or individual self-interest. The next section looks at methods to reason over what are the ‘best’ characteristic function game outcomes.
In this thesis, characteristic function games are used in Chapters4and5. Chapter4details an algorithm to distribute the calculation of the coalition values around the agents of the system. Chapter5 describes a distributed algorithm, that finds characteristic function game outcomes that are ‘best’, given a specific definition on what a ‘best’ solution is.