Conclusion

time, thenNis can be computed in polynomial time. Since Pn

j=1Ni = ω,ωcan be

computed in polynomial time. ut

As a corollary, we strengthen the complexity results for two other network games which are representations of simple games and answer open questions about the complexity of a host of skill based games:

Corollary 3.30.Computing Shapley value is#P-complete for

1. Threshold Network Flow Games [26] 2. Vertex Connectivity Games [27]

3. STSG (Single Task Skill Game), TCSG (Task Count Skill Game), WTSG (Weighted Task Skill Game), TCSG-T (Task Count Skill Game with thresh- olds) and WTSG-T (Weighted Task Skill Game with thresholds) [25]

Proof. For the given games, computing Banzhaf values is #P-complete. It is easy to see that the games Threshold Network Flow Games, Vertex Connectivity Games, STSG (Single Task Skill Game), TCSG-T (Task Count Skill Game with thresholds) and WTSG-T (Weighted Task Skill Game with thresholds) are sim- ple games with reasonable representations. Also, TCSG (Task Count Skill Game) and WTSG (Weighted Task Skill Game) are generalizations of the STSG (Single Task Skill Game). ut

As we will see later, the proof technique of Theorem 3.29 will be used in the proof of Proposition 12.5.

3.7 Conclusion

A summary of results has been listed in Table 3.1. A question mark indicates that the specified problem is still open. It is conjectured that it is NP-hard to compute Banzhaf indices for a simple game represented by (N,Wm). It is found that al- though WVG, MWVG and even (N,Wm_{) are relatively compact representations}

of simple games, some of the important information encoded in these representa- tions can apparently only be accessed by unraveling these representations. There

52 3 Complexity of comparison of influence of players in simple games

is a need for a greater examination of transformations of simple games into com- pact representations.

Table 3.1.Complexity of comparing players

(N,W) (N,Wm_{) WVG MWVG}

IDENTIFY-DUMMIES P linear NP-hard NP-hard

IDENTIFY-VETOERS linear linear linear linear

IDENTIFY-PASSERS linear linear linear linear

IDENTIFY-DICTATOR linear linear linear linear

CHOW PARAMETERS linear #P-complete #P-complete #P-complete

IS-LINEAR P P (Always linear) NP-hard

DESIRABILITY-ORDERING P P P NP-hard

STRICT-DESIRABILITY P P NP-hard NP-hard

BANZHAF-VALUES P #P-complete #P-complete #P-complete

BANZHAF-INDICES P ? NP-hard NP-hard

SHAPLEY-SHUBIK-VALUES P #P-complete #P-complete #P-complete SHAPLEY-SHUBIK-INDICES P #P-complete #P-complete #P-complete

HOLLER-INDICES P linear NP-hard NP-hard

4

Classification of computationally tractable weighted

voting games

In order to distinguish what is most simple from what is complex, and to deal with things in an orderly way, what we must do, whenever we have a series in which we have directly deduced a number of truths one from another, is to observe which one is most simple, and how far all the others are removed from this—whether more, or less, or equally.

- Descartes (Rule VI, Rules for the Direction of the Mind)

Abstract It is well known that computing Banzhaf indices in a weighted voting

game is #P-complete. We give a comprehensive classification of weighted voting games which can be solved in polynomial time. Among other results, we provide a polynomial (O(k(n_k)k)) algorithm to compute the Banzhaf indices in weighted voting games in which the number of weight values is bounded by k. Compu- tational results concerning weighted voting games with special distributions of weights are also presented.

4.1 Introduction

4.1.1 Motivation and outline

The Banzhaf index is considered the most suitable power index by voting power theorists ([132] and [82]). As mentioned before in Chapter 3, the computational complexity of computing Banzhaf indices in WVGs is well studied. Prasad and

54 4 Classification of computationally tractable weighted voting games

Kelly [179] show that the problem of computing the Banzhaf values of players is #P-complete. It is even NP-hard to identify a player with zero voting power or two players with same Banzhaf indices [151].

Klinz and Woeginger [121] devised the fastest exact algorithm to compute Banzhaf indices in a WVG. In the algorithm, they applied a partitioning approach that dates back to Horowitz and Sahni [108]. However the complexity of the algo- rithm is stillO(n2₂n_{2). In this chapter, we restrict our analysis to exact computation}

of Banzhaf indices instead of examining approximate solutions. We show that al- though computing Banzhaf indices of WVGs is a hard problem in general, it is easy for various classes of WVGs, e.g., for WVGs with a bounded number of weight values, an important sub-class of WVGs.

The outline of the chapter is as following. Section 4.2 identifies WVGs in which Banzhaf indices can be computed in constant time. In Section 4.3, we ex- amine WVGs with a bounded number of weight values, and provide algorithms to compute the Banzhaf indices. Section 4.4 examines WVGs with special weight distributions. Section 4.5 considers WVGs with integer weights. Section 4.6 pro- vides a survey of approximate approaches to computing power indices in WVGs. We conclude with some open problems in the final section.

Generally, 1₂P

1≤i≤nwi < q ≤ P1≤i≤nwi so that there can be no two disjoint

winning coalitions. Such weighted voting games areproper.

The problem of computing the Banzhaf indices of a WVG can be defined formally as following:

Name: BI-WVG

Instance: WVG,v=[q;w1, ...,wn]

Question: What are the Banzhaf indices of the players?

Here, we will suppose that arithmetic operations on O(n)-digit numbers can be done in constant time.