Part I Introduction
2.1 Simple coalitional games
Definition 2.1.Acooperative game with transferable utilityis a pair(N,v)where N = {1, . . . ,n} is a set of players and v : 2N 7→ R is acharacteristic/valuation
functionthat associates, for each coalition S ⊆N, a payoffv(S)which the coali- tion members may distribute among themselves.
Throughout the thesis, when we refer to a cooperative game, we assume such aTU-cooperative game with transferable utility which can be freely transferred among players.
24 2 Preliminaries
Definitions 2.2.A simple coalitional game/simple voting game is a pair (N,v)
with v : 2N → {0,1} where v(∅) = 0, v(N) = 1 and v(S) ≤ v(T) whenever S ⊆ T . A coalition S ⊆ N is winning if v(S) = 1 and losing if v(S) = 0. A simple voting game can alternatively be defined as(N,W)where W is the set of winning coalitions. This is called theextensive winning form. Aminimal winning coalition(MWC) of a simple game v is a winning coalition in which defection of any player makes the coalition losing. The set of minimal winning coalitions of a simple game v can be denoted by Wm(v). A simple voting game can be defined as
(N,Wm). This is called theextensive minimal winning form.
For the sake of brevity, we will abuse the notation to sometimes refer to game (N,v) asv.
Definitions 2.3.For each player x∈ N, let x have weight wx. The simple voting
game(N,v)where W = {X ⊆ N,P
x∈Xwx ≥ q}is called aweighted voting game(WVG). A weighted
voting game is denoted by[q;w1,w2, ...,wn]where wi is the non-negative voting
weight of player i. Usually, wi ≥wj if i< j.
For many of the algorithms, our assumption that the weights of the WVG are non-negative is essential. Of course, any computational hardness results that hold for WVG with non-negative weights also hold for WVGS which have both negative and positive weights.
We now define multiple weighted voting games [2] which are an extension of weighted voting games.
Definitions 2.4.An m-multiple weighted voting game (MWVG) is the simple
game (N,v1 ∧ · · · ∧vm) where the games (N,vt) are the WVGs [qt;wt1, . . . ,wtn]
for1≤ t≤m. Then v =v1∧ · · · ∧vmis defined as:
v(S)= 1,if vt(S)= 1,∀t,1≤t≤ m, 0,otherwise.
Thedimensionof(N,v)is the least k such that there exist WVGs(N,v1), . . . ,(N,vk)
2.1 Simple coalitional games 25
We now define some of the important properties of simple games:
Definition 2.5.A simple game is
• properif the complement of every winning coalition is losing.
• strongif the complement of every losing coalition is winning.
• dual-comparableif it is proper or strong.
• decisiveif it is both proper and strong.
The Banzhaf index [31] and Shapley-Shubik index [192] are two classic and popular indices to gauge the voting power of players in a simple game. They are used in the context of weighted voting games, but their general definition makes them applicable to any simple game.
Definition 2.6.A player i iscriticalin a coalition S when S ∈W and S \ {i}<W. For each i ∈ N, we denote the number of swingsor the number of coalitions in which i is critical in game v by the Banzhaf value ηi(v). The Banzhaf index of
player i in a simple game v is
βi(v)=
ηi(v) P
i∈Nηi(v)
.
Theprobabilistic Banzhaf index (or Penrose index)of player i in game v is equal to
β0
i(v)=ηi(v)/2n−1.
Intuitively, the Banzhaf value is the number of coalitions in which a player plays a critical role and the Shapley-Shubik index is the proportion of permuta- tions for which a player is pivotal. For a permutation π of N, the π(i)th player is pivotal if coalition{π(1), . . . , π(i−1)}is losing but coalition{π(1), . . . , π(i)}is winning.
Definitions 2.7.TheShapley-Shubik value is the function κ that assigns to any
simple game(N,v)and any voter i a valueκi(v)where
κi(v)= X
X⊆N
26 2 Preliminaries
TheShapley-Shubik indexof i is the functionφdefined by
φi(v)=
κi(v)
n! .
TheShapley valueis a generalization of the Shapley-Shubik index. It has the same definition as the Shapley-Shubik index but is also applied to non-simple cooperative games. The Banzhaf index and the Shapley-Shubik index are the normalized versions of the Banzhaf value and the Shapley-Shubik value respec- tively. Since the denominator of the Shapley-Shubik index is fixed, computing the Shapley-Shubik index and Shapley-Shubik value have the same complexity. This is not necessarily true for the Banzhaf index and Banzhaf value. Only fact known is that if Banzhaf values can be computed, then they can be used to compute the Banzhaf indices.
Example 2.8.Consider WVG [v = 51; 50,49,1] where the players are {A,B,C}. Then the winning coalitions are {A,B,C}, {A,B} and {A,C}. Players A and B
are critical in{A,B}, AandC are critical in {A,C}and Ais critical in {A,B,C}. Therefore ηA(v) = 3, ηB(v) = 1 and ηC(v) = 1 which means that βA(v) = 3/5,
βB(v)= 1/5,βC(v)=1/5.
For the Shapley-Shubik index, we consider permutations. We identify the piv- otal player in each of the following permutations. PlayerBis pivotal in ABCbe- cause{A}is not winning but{A,B}is winning. SimilarlyCis pivotal forACBand
Ais pivotal for BAC, BCA,CABandCBA. ThereforeφA(v) = 2/3, φB(v) = 1/6
andφC(v)= 1/6.
In voting games, another relevant consideration is the ease with which a deci- sion can be made. This concept was introduced by Coleman in [49]:
Definition 2.9.Coleman’s power of the collectivity to act, A, is defined as the ratio of the number of winning coalitions|W|to2n: A=|W|/2n.
Both Coleman’s power of the collectivity to act and the probabilistic Banzhaf index (or Penrose index) will be used in Chapter 7. Chow parameters are another important parameters of a simple game.