Part I Introduction
2.2 Computational complexity
Definition 2.10.([63, 48]) For a simple game v, theChow parameters, CHOW(v)
are given by (|W1|, . . .|Wn|;|W|) where Wi = {S ∈ W : i ∈ S}. |W|and |Wi| are
also denoted byωandωi.
Apart from the Banzhaf and Shapley-Shubik indices, there are other indices which are also used. Both the Deegan-Packel index [56] and the Holler in- dex [106] are based on the notion of minimal winning coalitions. Minimal win- ning coalitions are significant with respect to coalition formation [55]. The Holler index,Hiof a playeriin a simple game is similar to the Banzhaf index except that
only swings in minimal winning coalitions contribute toward the Holler index.
Definitions 2.11.Let Mi be{S ∈Wm :i ∈S}. We define theHoller valueas|Mi|.
TheHoller index(also called thepublic good index) is defined by
Hi(v)=
|Mi| P
j∈N|Mj|
.
TheDeegan Packel indexfor player i in voting game v is defined by
Di(v)= 1 |Wm| X S∈Mi 1 |S|.
Compared to the Banzhaf index and the Shapley-Shubik index, both the Holler index and the Deegan-Packel index do not always satisfy the monotonicity con- dition.
2.2 Computational complexity
O time! thou must untangle this, not I; It is too hard a knot for me to untie! - William Shakespeare
There is no greater harm than that of time wasted. - Michelangelo
Thecomputational complexityof problems related to simple games is central to this thesis. Computational complexity may refer to time complexity or space
28 2 Preliminaries
complexity. We will normally refer to the time complexity of a problem as the complexity of the problem. The time complexity of a problem is the number of steps required to solve an instance of the problem as a function of the size of the input (measured in bits), using the most efficient algorithm. The big Onotation is a standard way to describe computational complexity. Let f(x) and g(x) be functions defined on some subset of the real numbers. Then
f(x)=O(g(x)) for allx→ ∞
if and only if there exists a positive real numberMand a real numberx0such that
|f(x)| ≤ M|g(x)|for all x> x0.
We define some basic computational complexity classes in lay terms for read- ers not familiar with computational complexity.
Definition 2.12.A problem is in complexity class P if it can be solved in time
which is polynomial in the size of the input. A problem is in the complexity class
EXPif it can be solved in time exponential in the size of the input. A problem is in the complexity classNPif its solution can be verified in time which is polynomial in the size of the input of the problem. A problem is in complexity classco-NPif and only if its complement is inNP. A problem is in the complexity classNP-hard
if any problem inNPis polynomial time reducible to that problem.NP-complete
problems are inNP and are as hard as the hardest problems inNP. A counting problem is in complexity class#Pif the objects being counted can be verified in polynomial time. A #P-hardproblem is a counting problem which is as hard as the counting version of any NP-hardproblem. A counting problem which in #P
and is#P-hardis#P-complete
Polynomial time algorithms are desirable because they ‘scale’ well and finish in a reasonable time compared to exponential time algorithms.
ThePartition Problemis an example of a classic NP-complete problem which we will use at times in the thesis:
2.2 Computational complexity 29
Name: PARTITION
Instance: A set ofkinteger weightsA={a1, . . . ,ak}.
Question: Is it possible to partition A, into two subsets P1 ⊆ A, P2 ⊆ A so that
P1∩P2 =∅andP1∪P2 = AandPai∈A1ai =
P
ai∈A2ai?
Readers unfamiliar with computational complexity may ask what is the use of this concept. Computational complexity is an inherent mathematical property of a problem irrespective of the model of computer. Some may still ask that why would bad news of a problem being NP-hard be of any use in real life. Of course one would prefer that a problem has an algorithm which can be run in time poly- nomial of its input. However, NP-hardness of a problem implies that no polyno- mial time algorithm is possible unless P=NP, i.e. the computational classes P and NP coincide, which is generally considered unlikely.
The theory of parameterized complexity is motivated by the fact that several NP-hard problems (for which no polynomial time algorithm is known) are solv- able in a time that is polynomial in the input size and exponential in a (small) parameter k. Any problemτ can be defined in its corresponding parameterized form where the parameterized problem is the original problemτalong with some parameterk.
Definition 2.13.A parameterized problemτwith an input instance n and param-
eter k is calledfixed-parameter tractableif there is an algorithm which can solve
τin O(f(k)nc)where c > 0and f is a computable function depending solely on k. The class of all fixed-parameter tractable problems is calledFPT.
Part II
3
Complexity of comparison of influence of players in
simple games
The mathematical study (under different names) of pivotal agents and in- fluences is quite basic in percolation theory and statistical physics, as well as in probability theory and statistics, reliability theory, distributed com- puting, complexity theory, game theory, mechanism design and auction theory, other areas of theoretical economics, and political science. - Kalai and Safra, [116]
Not everything that counts can be counted, and not everything that can be counted counts.
- Einstein
Abstract In this chapter, the complexity of comparison of influence between
players in coalitional voting games is characterized. The possible representations of simple games considered are by winning coalitions, minimal winning coali- tions, weighted voting game or multiple weighted voting games.
It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. As a corollary, we settle the complexity of computing the Shapley value of a number of network games.
34 3 Complexity of comparison of influence of players in simple games
3.1 Introduction
3.1.1 Overview and outline
John von Neumann and Morgenstern [213] observe that minimal winning coali- tions are a useful way to represent simple games. A similar approach has been taken in [88]. We examine the complexity of computing the influence of players in simple games represented by winning coalitions, minimal winning coalitions, weighted voting games and multiple weighted voting games.
In Section 3.2, we outline different representations and properties of simple games. In Section 3.3, compact representations of simple games are considered. After that, the complexity of computing the influence of players in simple games is considered from the point of view of player types (Section 3.4), desirability ordering (Section 3.5), power indices and Chow parameters (Section 3.6). The final Section 3.7 includes a summary of results and some open problems.
3.2 Background
We first provide some important definitions and facts needed for the chapter.
3.2.1 Definitions
Definition 3.1.A coalition S isblockingif its complement(N\S)is losing. For a simple game G = (N,W), there is adual gameGd = (N,Wd)where Wd contains
all the blocking coalitions in G.
Definitions 3.2.A WVG [q;w1, . . . ,wn] is homogeneous if w(S) = q for all
S ∈ Wm. A simple game (N,v) is homogeneous if it can be represented by a homogeneous WVG. A simple game (N,v)is symmetric if v(S) = 1, T ⊂ N and
|S|=|T|implies v(T)= 1.
It is easy to see that symmetric games are homogeneous with a WVG represen- tation of [k; 1, . . . ,1
| {z } n
] for somek. That is the reason they are also calledk-out-of-n
3.2 Background 35
We will often use the following lemma.
Lemma 3.3.For a simple game(N,W), Wmcan be computed in polynomial time. Proof. For everyS ∈ W, check ifS \ {i}is winning for all i ∈ S. If yes for any suchi, thenS <Wm. OtherwiseS ∈Wm. This takes time|input|2
. ut
3.2.2 Desirability relation and linear games
The individual desirability relations between players in a simple game date back at least to Maschler and Peleg [150].
Definitions 3.4.In a simple game(N,v),
• A player i is more desirable/influentialthan player j (iD j) if v(S ∪ {j}) =
1⇒ v(S ∪ {i})=1for all S ⊆ N\ {i, j}.
• Players i and j areequally desirable/influentialorsymmetric(i∼D j) if v(S ∪
{j})=1⇔v(S ∪ {i})=1for all S ⊆ N\ {i, j}.
• A player i is strictly more desirable/influential than player j (i D j) if i is
more desirable than j, but i and j are not equally desirable.
• A player i and j are incomparable if there exist S , T ⊆ N \ {i, j} such that v(S ∪ {i})=1, v(S ∪ {j})=0, v(T ∪ {i})=0and v(T ∪ {j})=1.
Linear simple games are a natural class of simple games:
Definitions 3.5.A simple game islinearwhenever the desirability relationDis
complete, that is, any two players i and j are comparable (i D j, j D i or
i∼D j).
For linear games, the relationR∼ divides the set of votersN into equivalence classesN/R∼D = {N1, . . . ,Nt}such that for anyi∈ Npand j ∈Nq,i D jif and
only if p< q.
Definitions 3.6.A simple game v is swap robust if an exchange of two players
from two winning coalitions cannot render both coalitions losing. A simple game istrade robustif any arbitrary redistributions of players in a set of winning coali- tions does not result in all coalitions becoming losing.
36 3 Complexity of comparison of influence of players in simple games
It is easy to see that trade robustness implies swap robustness. Taylor and Zwicker [202] proved that a simple game can be represented by a WVG if and only if it is trade robust. Moreover they proved that a simple game being linear is equivalent to it being swap robust.
Taylor and Zwicker [202] show in Proposition 3.2.6 thatvis linear if and only if Dis acyclic which is equivalent toDbeing transitive. This is not guaranteed
in other desirability relations defined over coalitions [64].
Proposition 3.7.A simple game with three or fewer players is linear.
Proof. For a game to be non-linear, we want to players 1 and 2 to be incompa- rable, i.e., there exist coalitions S1,S2 ⊆ N \ {1,2} such that v({1} ∪S1) = 1,
v({2} ∪S1) = 0, v({1} ∪S2) = 0 andv({2} ∪S2) = 1. This is clearly not possi-
ble for n = 1 or 2. For n = 3, without loss of generality, vis non-linear only if
v({1} ∪ ∅)=1,v({2} ∪ ∅)= 0,v({1} ∪ {3})= 0 andv({2} ∪ {3})= 1. However the fact thatv({1} ∪ ∅)=1 andv({1} ∪ {3})= 0 leads to a contradiction. ut
In Example 3.17, we present a 4-player simple game which is not linear.