So far we have discussed the traditional coalitional game type: characteristic function games. Yet characteristic function games make three big assumptions: (i) the value of the coalition can be split up in any manner between the agents of the coalition; (ii) the value of each coalition is the same for all the agents of the coalition; and (iii) a coalition’s value is not affected by the other coalitions that form. However these assumptions do not necessarily hold for certain types of coalitional games, described below. Assumption (i) is discarded innon-transferable utility games. Assumption (ii) is discarded incoalitional game models for uncertainty/valuation disagreements. Assumption (iii) is discarded inpartition function games. A variant of a non- transferable utility game, a coalitional game model for valuation disagreements and a partition function game is considered in Chapter3of this thesis. While another variant of a coalitional
game model for valuation disagreements, where assumptions (i) and (ii) hold, is considered in Chapter6of this thesis.
2.3.1 Non-Transferable Utility Games
When the utility of a coalition is not transferable between the agents, then this coalitional game is a non-transferable utility game. Examples include qualitative coalition games, coalition Boolean gamesandhedonic games[26,51,132,133]. Two non-transferable utility coalitional games that most relate to the work of Chapter3are now briefly discussed.
Firstly, qualitative coalitional games (QCGs) [132] model the payoff of a coalition in terms of goal sets that the coalition can achieve by performing a certain choice. The incentive for an agent to join a coalition in a QCG is to achieve a goal that it would not be able to achieve individually. In QCGs, agents have a set of goals they want to achieve (yet have no preference order over), and the formation of one coalition does not affect the goal sets that another coalition can achieve. A coalitionC is in the core of a QCG if: (i)Ccan successfully achieve the goal setG; (ii) this goal set satisfies all agents ofC; (iii)Cis minimal (otherwise some agents could defect to form other coalitions and achieve other goals at the same time asGis achieved).
Several approaches have extended the idea of qualitative coalition games by adding pref- erences to the goals to be achieved [28, 40, 52]. Rephrasing the stability definition of these approaches (to incorporate coalition structures): a coalition C, that can achieve the goal set G, is in the core of a QCG (with preferences), if: there does not exist a coalition S (where S∩C6=∅) that can achieve a goal setG0, which all the members ofSfavour compared to their current coalition goal sets.
In this thesis, Chapter3investigates a variant of QCGs where the goals that coalitions can achieve are linked to social-values (social-values are described in Section3.5). In Chapter3, un- like QCGs, the formation of a coalition can affect the goals that another coalition could achieve. Secondly, coalition resource games (CRGs) model where the choices of coalitions in QCGs come from. The choices of a coalition in a CRG depend on the resources available to the agents of that coalition, and the amount each agent has of these resources. CRGs are a modification of QCGs. CRGs can always be mapped to a QCG; whereas a QCG cannot always be mapped to a CRG [133]. In Chapter3, the choices of each coalition are the joint-actions that the coalition can undertake, given the current capabilities of its member agents.
2.3.2 Valuation Disagreements in Coalitional Games
In many systems, it is reasonable to believe that agents may only have partial, incomplete knowl- edge of their environment, including knowledge of the abilities of other agents, and on the potential effects of individual and coalitional actions. Therefore these environments require a coalitional game model that allows agents to hold different opinions on each coalition’s value. Finding solutions to a coalitional game in an environment with partial agent knowledge involves trying to satisfy all agents as much as possible, with respect to their own beliefs on the value of each coalition.
Various studies have introduced solution concepts that find stable coalition structures and/or payoff vectors for coalitional games with uncertain valuations or valuation disagreements, e.g: [33,34,65, 77, 78]. The use of mechanism design for coalition formation was introduced in [77], where the first Bayesian model of coalitional games was detailed. In the Bayesian model of [77], each agent has probabilistic beliefs regarding the capabilities of the other agents and so is uncertain regarding each coalition’s true value. This coalition formation process in this Bayesian model is completed via a centralised and trusted mediator that chooses the coalitions to form and the manner in which the payoff should be distributed. In this model it is essential that the agents only communicate with the mediator, otherwise information may be leaked to other agents, who may use this information to their own strategic advantage.
The most closely related coalitional game model, compared to the valuation disagreement coalitional game model introduced in this thesis, is found in [34]. Chalkiadakis et al. [34] proposed a transferable utilty coalitional game model that allows agents to have single point beliefs on the predicted coalition values. A single point belief represents an agent’s best guess on each coalition’s value but can lead to possible valuation disagreements between the agents. Another method of modeling beliefs involves assigning a probability to a distribution of differ- ent possible coalition values. Yet, as stated in [34], the single point belief assumption is helpful because: probabilistic beliefs may not be available; single point beliefs are easier to form; rea- soning over single point beliefs is computationally less complex; and single point beliefs are a natural assumption in many real-world situations.
In the model of [34],agent typesare used, where each agentihas a set of possible types, denotedΛi, where
− →
Λ denotes (Λ1, ...,Λn). An agent’s type encapsulates all the information possessed by the agent that is not common knowledge, e.g. the agent’s knowledge of its own type, its beliefs of its own payoff and its beliefs over other agents’ payoffs [119]. Each agenti has point beliefs on every other agent’s type, denotedbi = (bi(1), ..., bi(n))where
− →
b denotes
(b1, ..., bn). The utility function is denotedu. The utility value of a coalitionC, using agent i’s beliefs on the types of C, denoted bi(C), can be found byu(bi(C)) (which is written in this thesis simply as ui(C)). Given these preliminaries, the model presented in [34] can be formalised as:
Definition 26: A coalitional game with beliefs is Gb = (N;→−Λ ;u;−→b) where N is the set of agents, −→Λ is the collection of possible agent types, u is the utility function, and−→b is the collection of agent beliefs.
To find an acceptable outcome to a coalitional game with beliefs, fixed percentage-based demand vectors are used [33,37,122]. In this model, to make a demand from a coalitionC, an agent first finds the numeric valuexiit requires, then finds what percentagexiis of its valuation of the coalition (i.e. xi
ui(C)). This percentage value becomes the demand.
Definition 27: A demand vector, denoted d = hd1, ..., dni, for a coalition structure CS = {C1, ..., Ck}in a gameGb, satisfiesdi ≥0for alli∈N andPi∈Cdi = 1for eachC ∈CS. Usingd, the expected payoff for an agenti ∈ C ∈ CS is given bydi ×ui(C). The demand vector of a coalitionC is denotedd(C). The expected payoff fornagents usingdis denoted p= (p1, ..., pn).
Example 11: Consider a coalitional game with beliefs where N = {1,2,3} and the utility functionugives the following valuations: u3({3}) = 2,u1({1,2}) = 12andu2({1,2}) = 24.
In this example the beliefs of an agent iis encapsulated within i’s utility functionui. If the coalition structure{{1,2},{3}}formed, then a valid demand vector would bed1(0.25,0.75,1)
because for alli∈ N thendi >0, whiled11+d12 = 1andd13 = 1. This demand vector gives
the expected payoff for the agents of p(3,18,2)because u1({1,2}) ×d11 = 12×0.25 = 3, u2({1,2})×d12 = 24×0.75 = 18andu3({3})×1 = 2. An invalid demand vector would be d2(0.1,0.4,1)becaused21+d22 6= 1.
The solution concept proposed by Chalkiadakiset al.[34] to find stable coalition structures is theCS-core for Beliefs(CSB):
Definition 28: TheCS-core for Beliefsfor a gameGb consists of all pairshCS, di such that for any coalitionS /∈CS and an associated demand vectord0, there exists ani∈ Ssuch that d0i×ui(S)≤pi(whereuiis the utility function of agentiandpiis the expected payoff of agent igivend).
Therefore a pairhCS, diis in the CSB if no coalitionSnot inCScan provide a new demand vectord0 that gives a higher expected payoff compared todfor all agents ofS.
Example 12: Consider a coalitional game with beliefs where N = {1,2,3} and the utility functionugives the following valuations: u1({1}) = u2({2}) = u3({3}) = 2,u1({1,2}) =
12,u2({1,2}) = 24,u1({1,3}) = 8,u3({1,3}) = 20andui(C) = 0for any other coalitionC and agenti∈ {1,2,3}. In this example the beliefs of an agentiis encapsulated withini’s utility functionui. Thenh{{1,2},{3}}, d(0.75,0.25,1)i, giving the expected payoffp(9,6,2), would be in the CS-core for beliefs because: (i) agent 1 believes that it could earn no more than 9 in another coalition, as it believes 100% of the payoff of coalitions{1}and{1,3}would only give 2 and 8 respectively; (ii) agent 2 believes that it could not earn more than 6 in another coalition, as its only other option is{2}which gives a payoff of 2; and (iii) due to parts (i) and (ii), agent 3 cannot tempt any other agent into a coalition with it.
On the other hand, the solutionh{{1,2},{3}}, d(0.25,0.75,1)i, giving the expected payoff p(3,18,2), wouldnotbe in the CS-core for beliefs because: the coalition{1,3}can improve the expected payoff of both agents through, for example,d1 = 0.75andd3 = 0.25, giving the
expected payoffs ofp1 = 6andp3 = 5.
The CSB solution concept restricts the agents to one percentage demand per coalition (this is also the case in [33,37,122]). In Chapter6, this thesis investigates the advantages of allowing different percentage demands for different potential values of the same coalition.