In this section, the classical definitions of stability solution concepts for characteristic function games are detailed, where the grand coalition is assumed to form. In Section2.2.3, these clas- sical definitions are modified for the possible formation of different coalition structures within characteristic function games. In Section2.3, some solution concepts for other coalitional game types are discussed.
This section begins with the following general definition of stability, taken from [136]: Definition 12: A coalitional game is stable if all the agents have no valid objection to the coalitional game outcome. Stability is a necessary but not sufficient condition for a payoff vector (or coalition structure) to be agreed, because there could be multiple possible stable outcomes.
The idea of a valid objection changes depending on which stability concept is used. Perhaps the most intuitive stability based solution concept is known as thecore. The core was introduced in [58] (defined below) and is the set of stable outcomes where no subset of agents have an incentive to deviate, i.e., there is no payoff distribution that would make at least one member of a deviating coalition better off, without negatively affecting the other members of the deviating coalition.
Definition 13:Thecore:- For a set of agentsN, a payoff vectorx=(x1, ..., xn)for the grand coalition is in the core iff:
X i∈N xi =v(N) X i∈C xi≥v(C),∀C⊂N
The payoff vector must be feasible and efficient. The payoff vector must be feasible in the sense that the total value of the payoff vector should not be above the value of the grand coalition and the payoff vector must be efficient in the sense that the grand coalition payoff should be totally shared between the agents of the coalitional game (i.e. x(N) =v(N)). Ad- ditionally there must be noblocking coalition, meaning there should be no coalition who could deviate from the grand coalition and get more payoff by dividing v(C) between themselves (i.e. P
i∈Cxi ≥v(C),∀C ⊂N)1. For the core solution concept with a coalition structure see Section2.2.3.
Example 4: Consider a characteristic function gameG =hN, viwhereN = {1,2,3,4}and the characteristic functionv gives the following valuations: v({1,2}) = 12, v({3,4}) = 10, v({2,3}) = 8, v({1,2,3,4}) = 24 andv(C) = 0 for every other possible coalition C. In the traditional definition of the core, the grand coalition{1,2,3,4}forms. A payoff vector in the core has to give every potential deviating coalitionC0 a payoffx(C0)greater than or equal to its valuev(C0). Example payoff vectors that satisfy these restrictions are: (i)x(6,6,6,6); (ii)x(7,7,5,5); and (iii)x(7,6,2,9). Example payoff vectors that violate these restrictions are: (iv)x(5,5,7,7)becausex({1,2})< v({1,2}); (v)x(8,8,4,4)becausex({3,4})< v({3,4}); and (vi)x(10,4,2,8)becausex({2,3})< v({2,3}).
A problem with the core is that it can sometimes be empty. To tackle this issue, the concept of the-coreis introduced, where the-core is a more general case of the core. Two different definitions of the-corewere introduced in [114], named thestrong-coreand theweak-core. Both concepts rely on the idea of coalitional excess:
Definition 14:Thestrong excessof a coalitionCfor a payoff vectorxis denoteds(C, x)and calculated by:
s(C, x) =v(C)−X i∈C
xi
Definition 15: Theweak excessof a coalitionCfor a payoff vectorxis denotedw(C, x)and calculated by:
w(C, x) = v(C)−
P i∈Cxi |C|
Positive excess for a potential coalitionCmeans thatChas a higher utility value (i.e.v(C)) than the combined payoff the members ofC currently receive (i.e. P
i∈Cxi). In this case, the members ofCwill benefit ifCforms (providing there is no penalty on forming a new coalition). Negative excess for a potential coalitionC, means thatC has a lower utility value (i.e. v(C)) than the combined payoff the members ofC currently receive (i.e. P
i∈Cxi). In this case, the 1Remember that sometimes in the thesis, notation will be abused by usingx(C)to denoteP
members ofCwill not benefit ifCis formed. The maximum strong or weak excess, denoteds andw respectively, of all the possible coalitionsC⊂N, is formalised as [114]:
Definition 16: Themaximum strong excesssof a characteristic function game is the follow- ing: s= max C⊂N v(C)−X i∈C xi
Definition 17:Themaximum weak excesswof a characteristic function game is the follow- ing: w = max C⊂N v(C)−P i∈Cxi |C|
The -cores are useful as they can relax the strict conditions of the core solution concept. The-cores are the set of stable outcomes where no subset of agents has an incentive to deviate whenthe subset has to pay a penalty for deviation [114]:
Definition 18: Thestrong-core:- For a characteristic function gameG=hN, viand a value , a payoff vectorx=(x1, ..., xn)for the grand coalition is in the strong-core iff:
X i∈N xi=v(N) (2.3) X i∈C xi ≥v(C)−, ∀C ⊂N (2.4)
Definition 19:Theweak-core:- For a characteristic function gameG =hN, viand a value, a payoff vectorx=(x1, ..., xn)for the grand coalition is in the weak-core iff:
X i∈N xi=v(N) (2.5) X i∈C xi ≥v(C)− |C|, ∀C ⊂N (2.6)
The difference between the strong and weak -cores is that under the weak definition, the penalty of forming a new coalition is dependent on the size of that coalition, while the penalty for forming a new coalition under the strong definition is a fixed amount for any coalition.
When= 0, the-core definitions are the same as the core. Ifis positive then the-cores are a wider version of the core whereas ifis negative then they are tighter versions of the core. Again like the core, an -core payoff vector must be feasibleand efficient(i.e. 2.3 and 2.5). Additionally there must be no blocking coalition, which means there should be no coalition who could deviate from the grand coalition,pay the deviation penaltyand get more payoff by dividing the remaining payoff between themselves (i.e. 2.4 and 2.6). Like core solutions,-core solutions can only be defined over the grand coalition, for the -core solution concept with a coalition structure see Section2.2.3.
Example 5: Consider a characteristic function gameG =hN, viwhereN = {1,2,3,4}and the characteristic functionv gives the following valuations: v({1,2}) = 16, v({3,4}) = 14, v({2,3}) = 8,v({1,2,3,4}) = 24andv(C) = 0for every other possible coalitionC. In the traditional definition of the-cores, the grand coalition{1,2,3,4}forms. A payoff vector in the -core has to give every potential deviating coalitionC0a payoffx(C0)greater than or equal to its valuev(C0)minus the deviation penalty. Example payoff vectors (that are indexed via their superscript) in different-cores are: (i)x1(10,0,1,13)where s = 7 becausex1({2,3}) + 7 = v({2,3}), whilew = 3.5because v({2,3}|{)−2,x3}|1({2,3}) = 3.5; (ii)x2(8,8,4,4)wheres = 6 because x2({3,4}) + 6 = v({3,4}), while w = 3 because v({3,4}|{)3−,4x}|2({3,4}) = 3; and (iii)x3(8,4,4,8) where = 4because x3({1,2}) + 4 = v({1,2}), whilew = 2because
v({1,2})−x3({1,2})
|{1,2}| = 2.
There are an infinite number of possible-cores. Theleast coreis the smallest, non-empty -core, and so again there are two different definitions [114]:
Definition 20:Thestrong least core:- For a characteristic function gameG=hN, vi, a payoff vectorx=(x1, ..., xn)is in the strong least core iff:
x is in the strong-core
∀0 < , the strong0-core is empty
Definition 21:weak least core:- For a characteristic function gameG=hN, vi, a payoff vector x=(x1, ..., xn)is in the weak least core iff:
x is in the weak-core
∀0 < , the weak0-core is empty
A payoff vector in the least cores minimises the maximum dissatisfaction of any coalition. The least cores therefore contain the payoff vectors that are ‘least objectionable’, under the strong or weak-core definition of objectionable. If the core of a characteristic function game is empty, the value of the least cores can be viewed as the minimum deviation penalty that is needed to stop a potential coalition from deviating from the grand coalition under the strong or weakcore definition [109]. For the least core with coalition structure, see Section2.2.3. Example 6: Consider a characteristic function gameG =hN, viwhereN = {1,2,3,4}and the characteristic functionv gives the following valuations: v({1,2}) = 12, v({3,4}) = 10, v({2,3}) = 8, v({1,2,3,4}) = 20 andv(C) = 0 for every other possible coalition C. In the traditional definition of the least-cores, the grand coalition {1,2,3,4} forms. A payoff vector in the least cores minimises the maximum strong or weak excess every potential deviating coalitionC0 has. Example payoff vectors that satisfy these restrictions are: (i)x(6,5,5,4); (ii) x(4,7,2,7); and (iii)x(5.5,5.5,4.5,4.5), all of which give the minimal s = 1orw = 0.5
because: (a)s({1,2}) =s({3,4}) = 1andw({1,2}) =w({3,4}) = 0.5; (b) any transfer of payoff from an agent of coalition {1,2} to an agent of coalition{3,4}(or vice versa) will result in the excess value of one potential coalition increasing; and (c) any transfer of payoff between the agents of the coalition{1,2}(or between the agents of the coalition {3,4}) will not result in the excess value of either coalition decreasing. Example payoff vectors that are not in the least core are: (iv)x(2,5,7,6)becauses({1,2}) = 12−(2 + 5) = 5 > 1and w({1,2}) = 12|{−1(2+5),2}| = 2.5 > 0.5; (v)x(6,6,4,4)becauses({3,4}) = 10−(4 + 4) = 2 > 1 andw({3,4}) = 10|{−3(4+4),4}| = 1 > 0.5; and (vi)x(10,0,0,10)becauses({2,3}) = 8−(0 + 0) = 8>1andw({2,3}) = 8−(0+0)
|{2,3}| = 4>0.5.
The nucleolus, introduced by Schmeidler [108], is a further refinement of the strong least core (even though it could easily be modified to be a refinement of the weak least core). The nu- cleolus is a single payoff vector and is referred to as the “most stable point” inside the core/strong least core. To find the nucleolus, as detailed in [36], all of the strong excesses of each coalition can be organised into andeficit vectorδ(x) = (s(x, C1), ..., s(x, C2n)), where the coalitions
C1, ..., C2n is the ordering of all the possible coalitions from largest excess to smallest, i.e.
s(x, Ci)≥s(x, Cj)wherei < j. The nucleolus is the payoff vector that has the lexicograph- ically smallest deficit vector. A deficit vectorδ(x)for the payoff vectorxis lexicographically smaller then the deficit vector δ(y) for the payoff vector y, i.e. δ(x) <lex δ(y) if the first i ∈ {0, ...,2n−1} entries of the deficit vectors are equal and then the i+ 1 entry of δ(x) is smaller than thei+ 1entry ofδ(y).
Definition 22: Thenucleolus:- For a characteristic function gameG =hN, vi, a payoff vector x=(x1, ..., xn)for the grand coalition is the nucleolus of the game iff:
x∈Imp(N, v)
δ(x)<lexδ(y)for ally∈Imp(N, v)wherex6=y
The first condition states that the nucleolus has to be an imputation payoff vector (one that satisfies individual rationality for all agents). The second condition states that the nucleolus must be lexicographically smaller than any other possible imputation payoff vector. If the same definition is used, except with all possible payoff vectors (i.e. not just imputations) then the pre-nucleolusis found.
Example 7: Consider a characteristic function game G = hN, vi where N = {1,2,3} and the characteristic functionv gives the following valuations: v({1,2}) = 12, v({1,3}) = 10, v({2,3}) = 8,v({1,2,3}) = 15andv(C) = 0for every other possible coalitionC. In the tra- ditional definition of the nucleolus, the grand coalition{1,2,3}forms. There is only one payoff vector (that is also an imputation) in the nucleolus. The nucleolus for this game is: x(7,5,3)
that gives the strong least core values = 0and the deficit vector ofδ(x) = (s(x,{1,2}) = 0, s(x,{1,3}) = 0, s(x,{2,3}) = 0, s(x,{3}) = −3, s(x,{2}) = −5, s(x,{1}) = −7). It can be seen that x is in fact the nucleolus because any payoff transfer from any xi to an- otherxjwill raise the strong excess value of the coalition includingiand notjover the current
maximum excess value of zero.
This thesis uses solution concepts based on the core, -cores and least cores, yet does not use the solution concept of the nucleolus. The reason for this decision is that the nucleolus is significantly more computationally intensive to find due to the deficit vector containing all C ⊆N coalitions, compared to the core/-cores/least cores solution concepts that need to only consider the coalition at the start of the deficit vector. Therefore, in decentralised environments, with the possibility of agents not having full coalitional value knowledge, finding the nucleolus will require much more communication in the majority of cases compared to searching for a variation of the core or least-core.
The reasoning for not using the nucleolus, also applies for not using the mostfaircooperative game solution concept, named the Shapley value [113]. Like the nucleolus the Shapley value is computationally intensive to find, as the Shapley value requires multiple calculations per coalition (after the coalition value has been found), to calculate the marginal contribution of each agent to each potential coalition it is a member of. Additionally the Shapley value may not even assign a payoff vector that is individually rational [49], which self-interested agents should not agree with.