Thesis outline

The approach of the thesis is algorithmic. For many problems in cooperative game theory and social choice theory, there are mathematical results such as the existence or non-existence of properties. However, there is a need for an algorith- mic study of these topics so that efficient constructive methods can be devised to test different properties of games. For various computational problems asso- ciated with simple coalitional games, polynomial time exact algorithms, pseudo- polynomial algorithms, approximation algorithms and parameterized algorithms are presented. In other cases, a proof is provided that the problem is, for instance, NP-hard or #P-complete.

1.3 Prerequisites

The thesis presupposes familiarity with combinatorial optimization and computa- tional complexity. For readers unfamiliar with these areas, the following excellent book is recommended: [169]. In Section 2.2, non-technical definitions of funda- mental complexity classes are given. One may also refer to Section 1.2 of [173] which outlines some basics of discrete optimization.

1.4 Thesis outline

The thesis is one of the first unified treatments of simple coalitional games from a computational perspective.

Chapter 3: In this chapter, the complexity of comparison of influence be- tween players in simple games is characterized. The chapter is based on [10]. The influence of players is gauged from the viewpoint of basic player types, de- sirability relations and classical power indices such as the Shapley-Shubik index, Banzhaf index, Holler index, Deegan-Packel index and Chow parameters. Among other results, it is shown that for a simple game represented by its set of minimal winning coalitionsWm, although it is easy to verify whether a player has voting power zero or one, computing the Banzhaf value of the player is #P-complete. Moreover, it is proved that for multiple weighted voting games, it is NP-hard to

18 1 Introduction

verify whether the game is linear or not. For a simple game on n players and represented byWm, aO(n.|Wm|+n2log n) algorithm is presented which returns ‘no’ if the game is non-linear and returns the strict desirability ordering other- wise. 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. The complexity of transforming simple games into compact representa- tions is also examined.

Chapter 4: It is well known that computing Banzhaf indices in a weighted voting game is #P-complete. We give a comprehensive classification of those 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 in- dices in weighted voting games in which the number of weight values is bounded byk. The chapter is based on [16].

Chapter 5: We study the mathematical and computational aspects of multiple weighted voting games which are an extension of weighted voting games. We analyse the structure of multiple weighted voting games and some of their com- binatorial properties especially with respect to dictatorship, veto power, dummy players and Banzhaf indices. An illustrative Mathematica program to compute voting power properties of multiple weighted voting games is also provided. The chapter is based on the following publication: [15].

Chapter 6: The calculation of voting powers of players in a weighted voting game has been extensively researched in the last few years. However, the inverse problem of designing a weighted voting game with a desirable distribution of power has received less attention. We present an efficient algorithm which uses generating functions and interpolation to compute an integer weight vector for target Banzhaf power indices. This algorithm has better performance than any other known to us. It can also be used to design egalitarian two-tier weighted voting games and a representative weighted voting game for a multiple weighted

1.4 Thesis outline 19

voting game. The results in this chapter are based on paper [19] written with my supervisors.

Chapter 7: This chapter is based on a paper [137] jointly written with Dennis Leech. We tested a heuristic on the real life case-study of the EU constitution. The Double Majority rule in the Reform Treaty agreed in Rome in September 2004 is claimed to be simpler, more transparent and more democratic than the existing rule. We use voting power analysis to examine these questions against the democratic ideal that the votes of all citizens in whatever member country should be of equal value. We also consider possible future enlargements involving candidate countries and then a number of hypothetical future enlargements. We find the Double Majority rule fails to measure up to the democratic ideal in all cases. We find the Jagiellonian compromise to be very close to this ideal.

Chapter 8: An important aspect of mechanism design in social choice pro- tocols and multiagent systems is to discourage insincere behaviour. Manipu- lative behaviour has received increased attention since the famous Gibbard- Satterthwaite theorem. We examine the computational complexity of manipula- tion in weighted voting games, which are ubiquitous mathematical models used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. It is a natural question to check how changes in a weighted voting game may affect the overall game. The tolerance and amplitude of a weighted voting game signify the possible variations in a weighted voting game which still keep the game unchanged. We characterize the complexity of comput- ing the tolerance and amplitude of weighted voting games. Tighter bounds and results for the tolerance and amplitude of key weighted voting games are also provided. Results from this chapter were published in [17].

Chapter 9: We examine the computational complexity of false-name manip- ulation in weighted voting games. This includes checking how much the Banzhaf index of a player increases or decreases if it splits up into sub-players. A pseudo- polynomial algorithm to find the optimal split is also provided. In the chapter, we also examine the cases where a player annexes other players or merges with them to increase their Banzhaf index or Shapley-Shubik index payoff. We characterize

20 1 Introduction

the computational complexity of such manipulations as well as providing limits to the manipulation. TheAnnexation Non-monotonicity paradox is also discovered in the case of the Banzhaf index. The results give insight into coalition formation and manipulation. The chapter is based on a paper [18] co-authored with Mike Paterson.

Chapter 10: This chapter is based on the following paper: [11]. Length and width are important characteristics of coalitional voting games which indicate the efficiency of making a decision. Duality theory also plays an important role in artificial intelligence. In this chapter, the complexity of problems concerning the length, width and minimal winning coalitions of simple games is analysed. The complexity of questions related to duality of simple games such as DUAL, DUALIZE and SELF-DUAL is also examined. Since susceptibility to manipula- tion is a major issue in multiagent systems, it is observed that the results obtained have direct bearing on susceptibility to optimal bribery in simple games.

Chapter 11: In this chapter, cooperative games and cooperative game solu- tions are introduced. The trend of using computational tractability as a criterion for cooperative game solutions is both recent and prevalent in the mathematics of operations research and theoretical computer science. In this chapter, the com- putational aspects of various cooperative game solutions in simple games are examined. Questions considered include the following: 1) for solution setX and simple game v, isX ofv empty or not, 2) compute an element inX ofv and 3) verify if a payoffis inXofv. Some representations taken into account are simple games represented byW,Wm, weighted voting games and multiple weighted vot- ing games. The cooperative solutions considered are the core,-core, least-core, nucleolus, prekernel, kernel, bargaining set and stable sets. The complexity of checking the stability of the core of simple games is also examined. A theorem from the paper “The nucleolus and kernel for simple games or special valid in- equalities for 0−1 linear integer programs” by Wolsey is corrected. Finally, the relation between cost of stability and the least core is examined. A natural and desirable solution called the super-nucleolus is also proposed.

1.4 Thesis outline 21

Chapters 12 and 13 concern spanning connectivity games (SCGs). They are based on joint work with Oded Lachish, Mike Paterson and Rahul Savani.

Chapter 12:

We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values is #P- complete and computing Shapley-Shubik indices or values is NP-hard for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in poly- nomial time. Among other results, it is proved that Banzhaf indices can be com- puted in polynomial time for graphs with bounded tree-width. Results from this chapter were published in [13].

Chapter 13: We consider the least core imputations and the nucleolus of SCGs. For any least core imputation, we refer to thevalueof SCGs as the payoff of any coalition with the worst excess. We show that the value is equal to the reciprocal of thestrengthof the underlying graph.

We efficiently compute a unique partition of the edges of the graph, called the prime-partition, and find the set of coalitions which always get the worst excess for every least core imputation. We define a partial order on the elements of the prime-partition which allows us to compute the nucleolus.

We also consider the problem of maximizing the probability of hitting a strate- gically chosenhidden networkby placing a wiretap on a single link of a commu- nication network. This can be seen as a two-player win-lose (zero-sum) game that we call the wiretap game. The nucleolus turns out be the unique maxmin strat- egy which satisfies certain desirable properties. Results from the chapter will be published in the following paper: [14].

2

Preliminaries

The advanced reader who skips parts that appear too elementary may miss more than the reader who skips parts that appear too complex.

- G. Polya

The beginning of wisdom is the definition of terms. - Socrates

A definition is the enclosing of a wilderness of idea within a wall of words. - Samuel Butler, Notebooks (1912)

Abstract In this chapter, the preliminary definitions concerning simple coali-

tional games and computational complexity are presented.