Iterative voting and multi-mode control in preference ... - Math Unipd

Universit` a degli Studi di Padova Facolt`a di Scienze MM.FF.NN. Corso di Laurea Magistrale in Informatica

Tesi di laurea

Iterative voting and multi-mode control in preference aggregation

Relatore

Laureando

Prof.ssa Francesca Rossi

Andrea Loreggia Matricola 1014235

Controrelatore

Prof. Piotr Faliszewski

Anno Accademico 2011-2012

Contents 1 Introduction

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

The research . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

The results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Background

7

2.1

Brief history of Election Systems . . . . . . . . . . . . . . . .

7

2.2

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

Voting Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4

2.5

2.6

2.3.1

Voting rules . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.2

Tie-breaking rules . . . . . . . . . . . . . . . . . . . . . 19

2.3.3

Voting rules properties . . . . . . . . . . . . . . . . . . 20

2.3.4

Arrow’s theorem . . . . . . . . . . . . . . . . . . . . . 22

Strategic behaviour . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1

Strategical actions . . . . . . . . . . . . . . . . . . . . 22

2.4.2

Gibbard-Satterthwaite theorem . . . . . . . . . . . . . 25

2.4.3

Computational Complexity . . . . . . . . . . . . . . . . 25

2.4.4

Immunity, resistance and vulnerability . . . . . . . . . 26

Single and multi control with separate budget . . . . . . . . . 27 2.5.1

Results on single control actions . . . . . . . . . . . . . 27

2.5.2

Results on multimode control actions . . . . . . . . . . 28

Convergence in Iterative Voting . . . . . . . . . . . . . . . . . 30 2.6.1

Game, Manipulation and Nash Equilibrium . . . . . . 30 i

ii

CONTENTS 2.6.2

Research study . . . . . . . . . . . . . . . . . . . . . . 31

3 Equal budget in multimode control results 33 3.1 Multimode control attacks with equal-budget . . . . . . . . . . 33 4 Convergence of iterative voting 4.1 Iterative Veto with non-linear tie-breaking 4.2 Loop searching in iterative voting . . . . . 4.3 Iterative Copeland divergence . . . . . . . 4.4 Iterative Copeland convergence . . . . . . 4.5 Iterative STV convergence . . . . . . . . . 4.6 Iterative Approval convergence . . . . . . . 4.7 Iterative Maximin convergence . . . . . . . 4.8 Iterative Cup Rule convergence . . . . . .

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5 Multimode control and iterative voting 5.1 Iterative Voting and control actions . . . . . . . . . . . . . 5.2 Iterative Plurality Voting and control actions . . . . . . . . 5.3 Iterative Voting with manipulation and single control . . . 5.4 Iterative Voting with manipulation and multimode control 5.5 Iterative Plurality with antagonistic multicontrol . . . . . .

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45 45 47 50 56 57 57 59 60

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65 65 66 67 68 69

6 Conclusions 73 6.1 Further works . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

List of Figures 2.1 2.2

Nicolas de Condorcet . . . . . . . . . . . . . . . . . . . . . . . Oxford trafficlight management using SCATS - screenshot . .

4.1 4.2 4.3

The cycle of Veto non convergence (top-left is the truthful state) Objects’ diagram . . . . . . . . . . . . . . . . . . . . . . . . . The cycle of Copeland non convergence (top-left is the truthful state) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The cycle of Copeland Majority Graph . . . . . . . . . . . . . 4.5 The cycle of Copeland non convergence (top-left is the truthful state) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The cycle of Copeland Majority Graph . . . . . . . . . . . . . 4.7 The cycle of Copeland non convergence (top-left is the truthful state) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 The cycle of STV non convergence (top-left is the truthful state) 4.9 The cycle of Cup non convergence (top-left is the truthful state) 4.10 The sequence of Cup trees non convergence (top-left is the truthful state) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2

8 9 46 48 51 51 52 53 55 58 61 63

The cycle of Plurality non convergence (top-left is the truthful state) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 The cycle of Plurality non convergence (top-left is the truthful state) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

iii

iv

LIST OF FIGURES

List of Tables 2.1 2.2 2.3 2.4 2.5

6.1 6.2 6.3

Plurality and Copeland report to constructive control . . . . Plurality and Condorcet report to destructive control . . . . Combination of different kind of control actions . . . . . . . Plurality convergence - C stays for converge . . . . . . . . . Convergence of scoring rules - C stays for Converge and NC for Not Converge . . . . . . . . . . . . . . . . . . . . . . . .

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28 29 29 31

. 32

Multimode control, equal-budget results . . . . . . . . . . . . 73 Iterative voting results . . . . . . . . . . . . . . . . . . . . . . 74 Multimode control and iterative voting results . . . . . . . . . 75

v

vi

LIST OF TABLES

Chapter 1 Introduction It does not matter how slow you go, as long as you don’t stop. Confucius

1.1

Introduction

A democratic society bases every decision on community expressions. Each person should express his preference over a set of choices, whatever they are, e.g. political candidates, the restaurant where a group of friends wants to have dinner and so on. In such a situation everyone is asked to express a vote, a kind of representation of his preferences. The winner is the best trade-off between all the voters’ preferences. And it is the same in a society of artificial agents, where every agent needs to coordinate its efforts with other agents, and where the final decision is the best trade-off between all the single agents preferences. But what happens when the voters are not sincere? When they will lie in order to change the election restults? Or when someone is willing to pay to change the voters’ preferences? We can label each strategic action based on who is acting to change the election result. We speak about manipulation when a voter or a coalition of voters is acting untruthfully, because they know that this behaviour will get 1

CHAPTER 1. INTRODUCTION them a better result. On the other hand, when someone is willing to pay to change some voters’ preferences we speak about bribery. It can also happen, that the same organizers of the election are interested in the final outome. So, as they are in charge of the organization of the election, they can make some specific candidate or prevent one specific candidate to win, acting on the structure of the election. For example, they can delete or add candidates, because this induces some voters to change their preferences and so the results. This type of strategical action is called control. Consider the following examples. Your friends are deciding where to have dinner together this evening. Everyone can cast his vote for a specific restaurant, you will go to the one which gets the most votes. You already know that your preferred restaurant will never win, and you also know that two others restaurants will compete for the first place and that normally these two restaurants are joint winners. One is an awful place, the place where you promised to never set foot again. The other one is a normal restaurant. So you decide to cast your vote not for your preferred restaurant, but for the normal one that compete for the first place, so it will be chosen by your friends. You have just exploited manipulation to obtain a better result for you. Consider now the case in which you give your friends different choices where to go. Your chosen alternatives are your preferred pub, the well-known awful place and the normal restaurant. As you are deciding the choices, you are not permitted to cast your vote. You already know the two restaurants that compete for the first place. But you also know that some of your friends, who will vote for the terrible place, can shift their preferences if you give them the chance to choose another specific pub. In this way, the awful place will receive less votes and once again you will have dinner at the same normal restaurant, where you and your friends go every weekends. This time you exploited a kind of control action. There are different types of control actions. The chair could use them individually or combine them in various ways. For instance, you could replace the awful place with some other pub or you could simply not choose it as 2

1.2. THE RESEARCH a choice. We speak about single control, when you are permitted to use only the same one type of control action. On the other hand, when you are permitted to use several types of control, we speak about multimode control. Normally, these strategic actions are considered to be negative, because they change the results to whom is playing truthfully. But there exist some cases where manipulation is permitted, and it plays a central role. An instance of such a systems could be websites used to coordinate dates for an event, one of the most famous is www.doodle.com. Initially, everyone specifics its preference. If some voters are not satisfied from the final result, some participants, one at a time, can change their preference. This type of elections is called iterative voting, and it is interesting to understand if this type of election converges with everyone satisfied or goes on forever in a permanent changing.

1.2

The research

Previous works have studied how different single kinds of control can affect the outcome of an election on different voting rules, using a bounded amount of resources [2, 4, 6]. Also iterative voting election has already been studied in the past [9, 10]. The goal of this thesis is to answer important questions that have been left open. For example, multimode control actions have been studied in settings where the chair can exploit them using separate budgets [5], and each control action has a distinct bounded amount of resources. Moreover, iterative voting systems have been studied only for some voting rules [9, 10]. The aim of this work is threefold: 1. To understand on how multimode control actions can affect some election systems in different ways, when there is one agent or a coalition of agents that tries to overtake the common will, using different configurations of budget; 2. To expand the knowledge about iterative voting systems to other important and widely used voting rules; 3

CHAPTER 1. INTRODUCTION 3. To describe how the combination of multimode control actions could force some iterative voting procedures not to converge to a stable state.

1.3

The results

We analyze two different scenarios, which can be seen as two different facets of the same problem. Multimode control actions can be considered as a malicious strategic action. To calculate the robustness of the election w.r.t. such actions, we study how difficult it is to exploit this actions on a system with equal budgets. In an equal budget configuration, actions have different separated resources, but these have to be used equally. In particular, this can model the replace control action, where the chair has to delete as many candidates/voters as he adds. We prove some new results on two well-known voting rules: Plurality and Copeland. The other studied scenario is iterative voting elections in which manipulation naturally occurs. We study the stability of voting rules not yet considered. We also try to identify some specific restrictions which allow some specific voting rules to always converge. After that, we consider single and multiple control actions within iterative voting, in order to understand if the malicious side of control can break stability of iterative voting systems.

1.4

Thesis structure

The thesis is organized as follow: • Chapter 2 introduces the readers to the main concepts of voting theory. We start from a brief history of election systems and then we present basic notions of voting theory and how these are used in computational social choice. We consider different voting rules and their properties. We also report the results based on research on Voting Theory, the impossibility theorems that characterize election systems, the different types of controls an agent can use to affect the outcome of an election, 4

1.4. THESIS STRUCTURE and the concept of computational complexity used to protect an election. Lastly, we briefly summarize the results present in the literature about the computational complexity of single types of control in different voting rules. We introduce the problem of multimode control attacks and describe previous results on iterative voting. • Chapter 3 reports new results on multimode control with equal budget. • Chapter 4 reports new results on the convergence of iterative voting for some voting rules. • Chapter 5 applies multimode control in iterative voting systems and studies how the convergence of stable voting systems is affected by it. • Chapter 6 briefly summarizes the results of our research.

5

CHAPTER 1. INTRODUCTION

6

Chapter 2 Background Et vero in dissensione civili, cum boni plus quam multi valent, expendendos cives, non numerandos. Cicerone De Repubblica

This chapter describes the background of the studies. It introduces the terminology of Voting Theory and the behaviour of the analyzed voting rules, using also some examples. Later, we describe the different strategic actions that an agent can exploit to affect an election. The chapter ends with a series of computational complexity results about different election systems.

2.1

Brief history of Election Systems

For thousands of years, elections have played a central role in human societies. For example, in ancient Greece people were asked to play an active role in the poleis, where a group of elected people had to make decisions for the community, so those decisions had to be chosen to satisfy the majority. Nowadays we could report thousands of examples of election systems from sports to politics, from religion to friendship relationships. A deep study of the elections was considered in the thirteenth century 7

CHAPTER 2. BACKGROUND by Ramon Lull. He was a luminary in many disciplines. Nowadays he is considered a pioneer in computational and election theories. He studied elections to produce an efficient system to elect abbesses, abbots, bishops and so on, modelling an election system that chooses the best winner in an efficient and robust way. As it often happens, his genius was not understood and his election system never gained public acceptance and it was forgotten for a long time. A mathematical approach of these systems began only a few hundreds years ago, with the works of Borda and Condorcet.

Figure 2.1: Nicolas de Condorcet

Nicolas de Condorcet, also known as the Marquis de Condorcet, was a French mathematician and philosopher of the 18th century. His early studies in political science state that in elections with more than two candidates, even if all voters have rational preferences, their aggregations could be irrational. This means that if a generic candidate wins every head-to-head contest against the other candidates, then he should be the winner of the election. This type of winner is called Condorcet Winner and it is surprisingly similar to the concept of Lull winner.

2.2

Applications

There are many different Artifical Intelligence applications using voting systems to solve decision problems. 8

2.2. APPLICATIONS An example is Web Meta Search [3], which tries to tackle the Web spam problem also called spamdexing. Web spam is a malicious manipulation technique of the search engine indexes; the relevance of some resources is modified to improve their quality and relevance. This study tries to develop a set of technique for rank aggregation. The idea is to use different “general purpose” search engines results to rank a specific query, Web Meta Search aggregates the different general purpose results to produce a more reliable indexing of the resources involved in the query.

Figure 2.2: Oxford trafficlight management using SCATS - screenshot

Other examples can be found in multiagent systems, where we need to aggregate the preferences from several agents. In [13] an instance of the problem is found in SCATS (the Sydney Coordinated Adaptive Traffic System), a complex system used to control the traffic lights in Sydney and in many other cities. Each intersection has a computer that manages the traffic using a preferred plan. Each intersection has also sensors to monitor how the traffic flow change during the day. All the computers have to be synchronized to allow for better management of the city traffic. So each one casts a vote on its preferred plan, the choice is take by the local computer based on the traffic situation captured by the sensors. Votes are collected by the central computer, that select the new plan to upload to the traffic lights, aggregating the different preferences. One more example can be found in collaborative filtering used by some recommender systems or even to plan the development of a computer system. 9

CHAPTER 2. BACKGROUND They are both examples of the use of the voting systems. DEVOTEE is a vote tracking system of Debian Project [11], it uses a variant of the Condorcet Method to make formal decisions, such as the designation of project leaders.

2.3

Voting Theory

In this chapter we will introduce the main concepts of the Voting Theory in terms of Computer Science. Nowadays the study of election systems and their computational properties is of primary importance in multiagent systems.

2.3.1

Voting rules

Formally an election system E = (C, V ) is formed by a set C of m alternatives or candidates and a set V of n voters that express their preferences over the candidates. The voting rule R is the method used by the election system to aggregate all the preferences and choose the winner. The chair is the agent in charge of the organization of the election; it could choose which candidate and/or voter can participate in the election, and which voting rule to use. There are many different voting rules. We report only few different types: • Positional scoring rules • Approval • Single Transferable Voting • Condorcet-consistent methods For each voting type we will report an example to describe how it works. Positional scoring rules In Positional Scoring rules each candidate receives points based on the ranked position by a voter. The sum of the points gives the total score of the candidate. The one with the highest score is the winner. 10

2.3. VOTING THEORY There are many types of Positional Scoring rules, some of them very famous, and used everyday in lots of situations. They differ from one another in the way the voters give points to a candidate. This difference is expressed by the scoring vector. Formally it is v =< s1 , . . . , sm > where si expresses how many points the voter gives to a candidate in position i of his ranking. Typical Positional Scoring rules and their scoring vector are: • Plurality v =< 1, 0, . . . , 0 > • Veto v =< 1, 1, . . . , 1, 0 > • K-Approval v =< 1, 1, . . . , 1, 0, 0, . . . , 0 > | {z } k

• Borda v =< m − 1, m − 2, . . . , 0 > Scoring rule examples Let us consider an election with 3 candidates {a, b, c} and 5 voters with preferences: v1 : a  b  c v2 : b  a  c v3 : c  b  a v4 : a  b  c v5 : a  b  c Where vi : a  b  c represent the preference of the ith voter. This means that the specified voter most prefers a, then b and the last choice is c. In an election with those candidates and those preferences the final result could be different depending on the scoring rule the system is using. The score is calculated using the scoring vector previously described. With Plurality candidate a is the winner because he gets 3 points, b and c get 1 point. We could also express the score of each candidate using the following syntax score(x) = n 11

CHAPTER 2. BACKGROUND where x is the candidate and n is an integer that expresses the total point got by the candidate. If the used voting rule is Veto the candidate scores are score(a) = 4, score(b) = 5 and score(c) = 1 and so b is the most preferred alternative. Using Borda the candidate scores are score(a) = 7, score(b) = 6 and score(c) = 2 and so a is the most preferred alternative. Using k-Approval, with k = 2 the candidate scores are score(a) = 4, score(b) = 5 and score(c) = 1 and so b is the most preferred alternative.

Approval In the Approval rule, each voter approves a set of candidates, whose cardinality could differ from voter to voter. Each of these candidates gets one point. The total score of a candidate is the sum of the points. The winner is the candidate with the highest score. Let E = (C, V ) be an election with 3 candidates C = {a, b, c} and a collection V of 5 voters that approve the following alternatives: v1 : a, b v2 : b v3 : c, b v4 : a, c v5 : a, b, c The ballot will have the following results: score(a) = 3 score(b) = 4 score(c) = 3 hence the winner is b. 12

2.3. VOTING THEORY Single Transferable Voting Single Transferable Voting ranks all the candidates based on the preference of each voter, using the same model as Plurality. If one candidate gets a strict majority, i.e. over 50% of the voters, then he is the winner, otherwise the candidate with the fewest votes gets eliminated from the competition and his votes are transferred to other candidates, so the rankings are shifted. Let E = (C, V ) be an election system with 4 candidates C = {a, b, c, d} and collection V of 10 voters with the following preferences: 3 voters : a  bcd 3 voters : b  a  d  c 1 voter : a  b  c  d 2 voters : c  d  a  b 1 voter : d  b  c  a By considering the choices, no candidate gets more than 50% of the votes. The alternative with the smallest number of votes has to be eliminated from the competition. score(a) = 4 score(b) = 3 score(c) = 2 score(d) = 1 Hence d is the candidate that is excluded from the election. Precluding d from the competition the new preferences are: 3 voters : a  b  c 3 voters : b  a  c 1 voter : a  b  c 2 voters : c  a  b 13

CHAPTER 2. BACKGROUND 1 voter : b  c  a As we can see, the only particular change is that the vote for d is transfered to the second choice of the last voter. So the new scores are: score(a) = 4 score(b) = 4 score(c) = 2 There still is no candidate that gets more than 50% of the votes. Hence c is the candidate with smallest number of votes and it is excluded from the election. Now, once again we can transfer votes in voters’ preferences: 3 voters : a  b 3 voters : b  a 1 voter : a  b 2 voters : a  b 1 voter : b  a Since there are only two alternatives the election becomes a Majority one, where the votes are score(a) = 6 score(b) = 4 Candidate a has more than 50% of votes and it is thus the winner of the election.

Condorcet-consistent methods Condorcet-consistent methods depend on the pair-wise comparison of candidates. They always elect the Condorcet winner, if one exist. These rules differ one from another by the way they arrange this head-to-head competi14

2.3. VOTING THEORY tions. They also assign different points to the winner of each head-to-head competition or the points they get in the case of tie. They are based on the Majority graph. It is a complete graph where each node corresponds to a candidate and each edge between candidates can be directed or not. For example, this is a majority graph: a

b

d

c

The edge from d to c is d → c. This means that a majority of voters prefer d over c. All other edges are not directed, which means the pairwise competitions between candidates is a tie. Edges could be labeled with numbers that express the entity of the majority, that is how many voters prefer one candidate over the other one. Here are some examples of Condorcet-consistent methods. Copelandα We define this Condorcet-consistent method using the definition of [6] to be consistent with results reported in other chapters. Let α be a rational number 0 ≤ α ≤ 1. In Copelandα election, each candidate gets: • one point for each head-to-head contest won. An head-to-head contest is won by a candidate when it is preferred over the other one by a majority of voters; • zero points for each head-to-head contest lost; • α points for each head-to-head contest that ends in a tie, i.e. 50% of voters prefer one and 50% prefer the other one. Let E = (C, V ) be an election with 4 candidates C = {a, b, c, d} and collection V of 4 voters with the following preferences: v1 : a  b  c  d v2 : a  b  c  d 15

CHAPTER 2. BACKGROUND v3 : a  c  d  b v4 : c  d  b  a The majority graph based on the voters’ preferences is

a

b

d

c

We can calculate the score of each candidate using the majority graph. Each outer arrow is one point, each inner arrow is zero points and each undirected edge corresponds to α points. Hence: score(a) = 3 score(b) = 2α score(c) = 1 + α score(d) = α Since 0 ≤ α ≤ 1, then a is the alternative with the highest score so it is the winner. In this case a is also the Condorcet winner, i.e. the candidate that defeats all others in head-to-head competitions. Cup Rule In Cup rule the winner is the candidate who wins a series of head-to-head contests. The lists of the competitions are represented as a binary tree in which the leaf are candidates. Each parent node is labeled with the winner of the majority election between the children. This binary tree is called the agenda. At the end of the election the candidate associated with the root is the final winner. Consider the following election with 4 candidates C = {a, b, c, d} and the following preferences: 16

2.3. VOTING THEORY v1 v2 v3 v4

:abcd :acbd :acdb :cdba

For the sake of simplicity we report the majority graph based on the voters’ preferences. It will help us to determine the winner in the pair wise contests.

a

b

d

c

The starting agenda is as follow: −

−

a

−

d

b

c

17

CHAPTER 2. BACKGROUND The first competitions are between a, d and b, c. From the majority graph we can see that a defeats d, and so does c with b. We can complete the second level of the binary tree as follow: −

a

a

c

d

c

b

Once again we can see from the majority graph that a defeats c in a head-to-head contest. Now the election ends, beacause all the pair-wise competition took place. a

a

a

c

d

b

c

The winner is the one in the root label. Hence a is the winner. Maximin We define this Condorcet-consistent method using the definition of [9] to be consistent with results reported in other chapters. For every pair of candidates x, y ∈ C, N (x, y) is the number of voters who prefer x over y. Each candidate gets a score that is Sx = min N (x, y). The y∈C−{x}

winners are the candidates with the maximal score. Consider the following election with 4 candidates C = {a, b, c, d} and the following preferences: v1 : a  b  c  d v2 : a  b  c  d 18

2.3. VOTING THEORY v3 : a  c  d  b v4 : c  d  b  a The weighted majority graph based on the voters’ preferences is 3

a

b

2 3

3 d

2 4

c

We can derive the score of each candidate from the majority graph: score(a) = min N (a, y) = 3 y∈C−{a}

score(b) = 1 score(c) = 1 score(d) = 0 Hence a is the winner.

2.3.2

Tie-breaking rules

It could happen that the election ends in a tie, that is more than one candidates have the same score. In such a situation, it is still important to be able to determine a winner. The way used to break the tie and decide which alternatives is the best one is called tie-breaking rule. There are many different tie-breaking rules. For instance, we can choose randomly the winner by tossing a coin or by drawing a card. A tie-breaking rule is defined in [9] and [10] as a function that, given a set of elements in C, chooses one of them as the (unique) winner. We consider two different rules: 1. Deterministic: every time we use the same input to determine the winner we get the same output. A special case is the linear tie-breaking 19

CHAPTER 2. BACKGROUND rule. It decides the winner based on some linear preference order over C. This means that if a, b ∈ D ⊆ C and t(D) = a, then if a, b ∈ D0 ⊆ C, then t(D0 ) 6= b. 2. Randomized: it randomly decides a winner, choosing it in the subset of the alternatives with the highest score. This type of tie-breaking rule is considered neutral, because no candidate or voter is preferred over another

2.3.3

Voting rules properties

It is possible to define a set of significant properties that describe the voting rules. They are not possessed by each voting rule, but they are very important because they characterize some important features of the voting rules. In the following sections, we report a brief description of these properties. Unanimity and weak Pareto condition A voting rule R is unanimous or Pareto efficient if it elects the candidate c ∈ C whenever all voters express that c is the best alternative. On the other hand, the weak Pareto condition holds if, whenever an alternative y ∈ C is dominated by some other one x ∈ C for every voters, then y cannot win the election. For example, Plurality satisfies the unanimity property. Surjective A voting rule is surjective if for every candidate there exist a profile that produces that candidate as a winner. All of the voting rules described above satisfy the surjective property. Neutrality A voting rule R satisfies the neutrality property if, whenever the candidates are permuted in the voters preferences, then the winner alternative selected by the voting rule changes accordingly. All of the voting rules described above satisfy the neutrality property. 20

2.3. VOTING THEORY Anonymity A voting rule R is anonymous if it treats the voters symmetrically. This means that if we switch the preferences of two voters then the winner does not change, for example, when the voters have the same weight in the election. All of the voting rules described above satisfy the anonymity property. Condorcet Consistency A voting rule R is said to be Condorcet Consistent when it elects the Condorcet winner, whenever there is one. The Condorcet winner is the candidate that defeats all other candidates in the head-to-head contest. Dictatorship A voting rule R is dictatorial if there exists a dictator, that is a voter such that the winner is always his top-ranked alternative. Participative A voting rule R is participative when there is no incentive for voters to abstain. This means that the addition of a voter in a profile leads to a result that is equally preferred or more preferred for this voter. Independence of Irrelevant Alternatives A voting rule R is Independence of Irrelevant Alternatives (or IIA) if, whenever x ∈ C is a winner and y ∈ C is not, and the relative position between these two in the voters’ preferences does not change, then y cannot be a winner. This means that the final relationship between x and y depends only on the individual preference between x and y. Monotonicity A voting rule R is monotone if for each election E = (C, V ), it holds that if some candidate ci ∈ C is a winner of E, then, without changing the relative order of all the remaining alternatives, ci is also a winner of an election 21

CHAPTER 2. BACKGROUND 0

E which is identical to E, except that some voters rank ci higher in his preferences.

2.3.4

Arrow’s theorem

In 1951, K.J. Arrow demonstrated in his PhD thesis that if a social welfare function with 3 or more alternatives is at the same time weak Pareto and independent of irrelevant alternatives then it is also dictatorial [1]. This means that there exists a voters called the dictator that can always decide the final result of the election. We can state that there does not exist a voting rule that is really fair. Indeed from the Arrow’s results follow that voting rules are susceptible to strategic voting. It means that voters may have incentive to lie about their real preference against a set of alternatives to reach the best results in the ballot.

2.4

Strategic behaviour

In the last decades a lot of efforts were posed to characterize the election systems. The following sections report the main results about the voting theory. They are the reason for the improvement of the studies in this area.

2.4.1

Strategical actions

In this section we would like to introduce the different types of controls that someone could use to change the outcome of an election. These kind of controls have different names depending on who is trying to change the results. They are also named constructive and destructive. We speak about constructive control, when the set of control actions try to make a candidate win who is different from the current one. On the other hand, we will speak about destructive control when the set of control actions try to make the current winner lose. In the next sections we assume that there is a complete knowledge about the election system, i.e. every agent has complete knowledge of: 22

2.4. STRATEGIC BEHAVIOUR • the chosen voting rule R • the candidates set C The knowledge of the preferences of each voter may not be complete, it depends on the system we are talking about. For example, when we analyze the Iterative Voting systems the preferences of each voters are not available, but they are collected during the iterative process.

Manipulation In this type of control, a subset K ⊆ V of voters could express untruthful preferences, these voters are named the manipulators. They normally work together. We called constructive manipulation when their aim is to make their preferred candidate the winner, otherwise we named destructive manipulation when they ensure that a particular candidate never be a winner.

Bribery In this type of control, an external agent named the briber attempts to affect the election by paying some of the voters to change their preferences. Normally, he has a limited budget to bribe the voters. The briber is normally considered an external agent, that is not in the set V of voters, nor in the set C of candidates and he is not the chair either. Formally, let E be an election system, given C the set of candidates and V the collection of the voters specified via their preference lists over C, a candidate p ∈ C and a nonnegative integer k, the constructive bribery problem E -bribery tries to make p a winner of the E election resulting from (C, V ) by modifying the preference list of at most k voters, actually this is not the only way to use the budget but it is just an example. The destructive form of the problem E -destructive-bribery is very similar, except that the briber tries to ensure that p is not a winner of the E election. 23

CHAPTER 2. BACKGROUND Control In this type of control, the chair is interested in changing the election results in favour of some candidate. This type of strategic action was first introduced by Bartholdi, Tovey and Trick in [2]. Normally he could affect the election trying to add/delete candidates or trying to add/delete voters. We consider 4 basic type of control, these are: 1. AC: the chair can add a bounded number of candidates to the election 2. DC: the chair can delete a bounded number of candidates from the election 3. AV : the chair can add a bounded number of voters to the election 4. DV : the chair can delete a bounded number of voters from the election Formally, let E be an election system, given two disjoint set of candidates C and A, two disjoint collections of the voters V and W , specified via their preference lists over C ∪ A, a candidate p ∈ C and a nonnegative integer k, we can define the following control problems: E − CCAC: tries to make p a winner of the E election, adding at most k candidates from the set A E − DCAC: tries to ensure that p is not a winner of the E election, adding at most k candidates from the set A E − CCDC: tries to make p a winner of the E election, deleting at most k candidates from the set C E − DCDC: tries to ensure that p is not a winner of the E election, deleting at most k candidates other than p from the set C E − CCAV : tries to make p a winner of the E election, adding at most k voters from the collection W of voters E − DCAV : tries to ensure that p is not a winner of the E election, adding at most k voters from the collection W of voters 24

2.4. STRATEGIC BEHAVIOUR E − CCDV : tries to make p a winner of the E election, deleting at most k voters from the collection V of voters E − DCDV : tries to ensure that p is not a winner of the E election, deleting at most k voters from the collection V of voters

2.4.2

Gibbard-Satterthwaite theorem

It is really important to work with voting rules where voters are encouraged to play truthfully is. Such a rules are named strategy proof. The open question is why do we not work with such voting rules? The answer comes from the theorem of Gibbard-Satterthwhaite, it says that in a system with 3 or more alternatives where the voting rule is strategy proof and surjective then it is also dictatorial [7, 14]. The conclusion is that does not exist a voting rule that is fair and not manipulable. So voters, the chair or some other agent could have some incentives to act strategically. One of the ways to protect the system and the fairness of the result is make any speculative action really difficult to elaborate and execute.

2.4.3

Computational Complexity

We speak about computational complexity in terms of how much work is required to solve a well-defined problem. Formally a problem is a class of instances, which share the same characteristics in terms of input and output data. We are interested in the amount of work required to solve an instance of the problem, that will be expressed in terms of time useful to determine a solution of the problem. If the time useful to the determination of the solution grows as a polynomial in the size of the instance, then the method used to identify the solution is considered fast, otherwise if it requires time that grows as an exponential function, then it is considered slow. This is due to the fact that an exponential function grows faster that a polynomial one, so an algorithm that use such an exponential function would be awkward. For 25

CHAPTER 2. BACKGROUND these reasons, we could label two classes of problems, that are of particular interest: NP problems: is a set of problems whose main characteristic is that every “yes/no” question about a potential solution of the instance could be verified in polynomial time, even though the solution can be computed in time which is not polynomial P problems: is a set of problems whose solution can always be computed in polynomial time We are interested in the computational complexity of a problem, because in the case of election systems, it could be a shield that protects the election system against any malicious actions a potential controller could use to change the outcome. For these, a lot of papers were written to control if a single control action belongs to the class of P problems or the NP one, or to be more precise to the class of the NP-complete problems. In computational complexity theory a problem p is NP-complete if it is NP and it is also NP-hard, that is each problem in NP can be converted in p just by a transformation of its inputs in polynomial time.

2.4.4

Immunity, resistance and vulnerability

As we just said, the computational complexity is of main importance for the election system, because it could protect the systems against speculative actions. So, we could categorize an election E = (C, V ) in two main different ways. If an election system cannot be affected by a particular control type, we say that the system is immune to that type of control, otherwise we say that it is susceptible to that type of control. If the system is susceptible to a control type, we could be, in some cases, more accurated. Indeed, studying the complexity of the problem, we could say if it is resistant or vulnerable. We say that a problem is resistant to a control action if it is NP-complete, otherwise, if it is P, then we say that the problem is vulnerable. 26

2.5. SINGLE AND MULTI CONTROL WITH SEPARATE BUDGET

2.5

Single and multi control with separate budget

Arrow and Gibbard Satterthwaite theorems demonstrate that there does not exist a voting system that is purely fair. So it is important to understand the limitations of the rules that we are using. Scientists started to describe formally the failures of the voting rules from the most used ones. Moreover, the first studies involve only one type of control action. As we will see later, the chair may have different types of speculative actions available everytime, and the problem becomes more difficult to study. In the following sections we report some results about the complexity of different types of control applied to different voting rules. We use the same abbreviations used in the subsection 2.4.1 to indicate the control type, while we use the following abbreviations to refer concepts in section 2.4.4: • I to indicate that the voting system is immune • S to indicate that the voting system is susceptible, that means it is not immune • R to indicate that the voting system is resistant • V to indicate that the voting system is vulnerable

2.5.1

Results on single control actions

The following results report that it is easy to the chair make win or lose a specific candidates p just adding/deleting, if the system is using the Plurality rule. While it is difficult adding/deleting or partitioning candidates with the same rule. If the voting rule is Copeland, then it becomes easy to the chair make win or lose a specific candidates p just adding/deleting, while it is difficult adding/deleting with the same rule. In the next sections we assume that there is a complete knowledge about the election system, i.e. every agent has complete knowledge of: 27

CHAPTER 2. BACKGROUND • the chosen voting rule R • the candidates set C • the preferences of each voters Plurality and Copeland resistance to constructive control In 1992 Bartholdi, Tovey and Trick published a work [2] that describes how hard it is to control an election which uses Plurality or Condorcet voting rule. In the same way Faliszewski, Hemaspaandra, Hemaspaandra and Rothe in 2009 describe the resistance to constructive control in elections which use Copeland voting rules [6]. Table 2.1 briefly describes their works and reports their conclusions on the voting rules, we study in this thesis, for different constructive control types. Control Type

Plurality

Copeland

AC DC AV DV

R R V V

R R R R

Table 2.1: Plurality and Copeland report to constructive control

Plurality and Condorcet resistance to destructive control In the same paper of Faliszewski, Hemaspaandra, Hemaspaandra and Rothe in 2009 [6], they describe the resistance to destructive control in elections which use Plurality or Copeland voting rules. Table 2.2 briefly describes their work and reports their conclusions on these voting rules for different destructive control types.

2.5.2

Results on multimode control actions

In many cases the chair could exploit different kind of control action at the same time. In such a situation he could choose the action that brings the better results. 28

2.5. SINGLE AND MULTI CONTROL WITH SEPARATE BUDGET Control Type

Plurality

Copeland

AC DC AV DV

R R V V

V V R R

Table 2.2: Plurality and Condorcet report to destructive control

In this section we report some results about complexity of multiple different types of control applied to different voting rules. As in the previous section, we use the same abbreviation used in subsection 2.4.1 to indicate the control type and to indicate if it is susceptible or not to the combination. The main result is from [5], which studied the combination of control actions with separated budget. That’s means that each action is bounded differently. Table 2.3 summarizes the results of the research for the combination of 2 control actions. The results could be easily expanded to more than 2 types of control. In the table C1 and C2 are generic control types.

C1

C2

C1 + C2

R R R I I V

R V I I V V

R R R I V S

Table 2.3: Combination of different kind of control actions

Results reported in table 2.3 state that if the voting system is resistant to a specific control C1 and it is also resistant to another control C2 , then it is resistant to their combination C1 + C2 , in the separated budget version. Same explanation for other combination of control action could be derived using data in table 2.3. 29

CHAPTER 2. BACKGROUND

2.6

Convergence in Iterative Voting

In multiagent systems each agent could have different preferences over the alternatives, so it is important to aggregate their preferences in such a way that each agent is satified and no one wants to change to a better situation. That is called a stable state or rather a state where no agent wants to change its vote. The Iterative Voting is a process in which each agent has no knowledge about the preferences of the others. Such a situation can happen when there are agents that do not trust one another and they are restive to share their preferences. One of the solutions is to vote strategically so as to collect information about other agent preferences. We restrict the system so only one voter at a time could change its vote to reach a situation that is better for it and satisfies its preferences. The question is if there exists a state in which each agent is satisfied and so no one wants to change its vote.

2.6.1

Game, Manipulation and Nash Equilibrium

As we already mentioned in section 2.3.4, the Gibbard-Satterthwaite result states that if the voting rule is not dictatorial, then there exists the possibility that voters acting strategically could change the outcome of the election. Normally in the previous sections we consider single types of manipulation or control actions, or, in the case of multiple agents, coalition in which every agent has the same intention. We would like to consider environments where each agent may have different aims and so they could be antagonistic to one another. In this situation, as we have already said, they may not share their preferences. In iterative voting they may start voting truthfully, and then at each round the agents could analyze the outcome. Now, they could control if by manipulating their vote, the results could change in a more satisfiable situation. Each agent participating in a game has different strategies available to reach their goal. But we are interested to know if the process sometimes 30

2.6. CONVERGENCE IN ITERATIVE VOTING stops, because no agent has incentive to change its vote again. This stable solution is called Nash Equilibrium. There may exit many different and sometimes trivial Nash Equilibriums.

2.6.2

Research study

We will not use Game Theory terminology in our results, because we preferred to analyze the system in a simpler way. But we defined all these terms to better understand the results reported in [10] and in [9]. Convergence of Plurality Voting The first work, made in [10], designs a framework to study the convergence of Plurality Voting rules. Basically it states that Plurality voting always converge under the assumption that: • agents are not weighted, • begin from truthful state • use best-response moves. Table 2.4 summarizes the results. Tie Breaking

Agent weight

Result

Deterministic Random

1 1

C C

Table 2.4: Plurality convergence - C stays for converge

Convergence of other voting rules The studies conducted in [9] analyze the convergence of other iterative voting rules. All studies are under the assumption that agents use best-response moves.

31

CHAPTER 2. BACKGROUND

Voting Rule

Tie-breaking

Borda any Veto linear K-Approval, k ≥ 3 linear Maximin deterministic

Agent weight

Result

any unweighted any any

NC C NC NC

Table 2.5: Convergence of scoring rules - C stays for Converge and NC for Not Converge

32

Chapter 3 Equal budget in multimode control results It is better to be vaguely right than exactly wrong. Carveth Read

This chapter presents the first results of our studies. We prove the resistance or the vulnerability of Plurality and Copeland to some types of multimode control actions that use different budget settings.

3.1

Multimode control attacks with equal-budget

We are interested to expand the results of 2.3 to control actions with equalbudget. In equal-budget settings, the chair can use two new forms of control, that we introduce in the following theorems: 1. replacing candidates (RC) 2. replacing voters (RV) This is derived from the combination of two basic control actions with the constraint that we have to delete as many candidates/voters as we add, or that the two budget have to be used in the same way. In practice the chair can replace some alternatives or some voters to change the outcome. 33

CHAPTER 3. EQUAL BUDGET IN MULTIMODE CONTROL RESULTS The results reported in this chapter follow via reductions from exactcover-by-3-sets (also called X3C) and hitting set problems. We define these problems before using them in the following proofs. Exact cover by 3-sets It is a decision problem concerned with an exact cover, it is also a well-know NP-complete problem. Given a set B = {b1 , . . . , b3k }, a positive number k ≥ 1 and a collection of sets S = {S1 , . . . , Sn } such that Si ⊆ B and |Si | = 3, for each i, with 1 ≤ i ≤ n. The question is if there exists a set A ⊆ {1, . . . , n}, |A| = k such S that Si = B. i∈A

If such a set A exist then it is called an exact cover of B. It is called a cover because every item in B occurs in some Si , it is called exact because Si ∩ Sj = ∅, for each i, j ∈ A, i.e. each element belongs to exactly one set in the cover. Hitting Set Given a set B = {b1 , . . . , bm }, a positive number k ≤ m and a collection S = {S1 , . . . , Sn } of subsets of B. The question is if there is a subset B 0 ⊆ B with |B 0 | ≤ k such that B 0 contains at least one element from each subset of S. Plurality voting - Constructive Control We already know that Plurality is vulnerable to single control attacks such as adding (deleting) voters, and it is resistant to single control by adding (deleting) candidates. We now prove that their combination, using equalbudget setting, maintain the same characteristics. Theorem 1. Plurality is vulnerable to constructive control via replacing voters (CCRV). Proof. We prove the vulnerability of Plurality to this type of control reporting a polynomial algorithm. The preferred chair candidate p can win if it has 34

3.1. MULTIMODE CONTROL ATTACKS WITH EQUAL-BUDGET enough resources to defeats all other candidates, or if it is possible to reduce the actual winner’s score below the one of candidate p. Let E = (C, V ) be an election system, where C = {p, b1 , . . . , bm }, V is the set of the qualified voters and A is the set of voters that the chair can potentially use to replace some voter in V . The budget available for the replace control is k = kD = kA , where kD is the budget available for the deletion and kA is the budget available for the addition. First of all calculate dif f (bi ) = score(bi ) − score(p), this is the number of voters that prefer bi to p, clearly dif f (bi ) > 0 for each candidate that receives more votes than p. While dif f (bi ) < 0 for each candidate that receives less votes than p and dif f (bi ) = 0 for each candidate that receives the same amount of votes of p. This operation can be done in O(m). Now sort candidates based on dif f (bi ), from the highest to the lowest one. Calculate kp that is the number of voters in A which has p as its top ranked choice, this can be done in O(kAk). Now, if dif f (bwinner ) < k and at the same time dif f (bwinner ) < kp , then p is very close to be the winner and there are also enough resources to make it win the election. So, we can just add dif f (bwinner ) + 1 voters who prefer p and then delete any dif f (bwinner ) + 1 voters who do not prefer p. At the end of this process if it is the new winner, then the replacing control is done and the algorithm can stop. Otherwise, we have not enough voters, who prefer p, to add. Calculate scorenew (p) = score(p) + kmax , where kmax is the maximum number of voters that prefer p, with respect to kmax ≤ k and kmax ≤ kp . This operation can be done in O(m). Now, update dif f (bi ) = score(bi ) − scorenew (p) and sort again candidates based on dif f (bi ), from the highest to the lowest one. Let W ⊂ C be the set of candidates with score(x) > scorenew (p), and denote them with wi ∈ W . Our preferred candidate can win the election if and only if X score(wi ) < k wi ∈W

that is the number of voters that prefer others candidates is lower than the budget, so delete all those candidates until the score of p is the highest one. After that, we can add or delete as many candidates as we need to make equal the two spent budget. At the end of this process if it is the new winner, then 35

CHAPTER 3. EQUAL BUDGET IN MULTIMODE CONTROL RESULTS the replacing control is done and the algorithm can stop. If p is not the new winner, then it cannot be the winner using replacing control with the available budget and profile. All the above operations can be exploited in O(kAk + m). Bartholdi, Tovey and Trick proved that Plurality is resistant to constructive control by adding candidates (CCAC) [2]. Since it is also anonymous and unanimous, we could derive from the theorem ?? that it is also resistant to the combination of AC + DC. Theorem 2. Plurality is resistant to constructive control via replacing candidates (CCRC). Proof. We prove the NP-hardness of the problem via reduction from X3C problem. Let (B, S ) be an X3C instance, where B = {b1 , . . . , b3k } and S = {S1 , . . . , Sn } is a finite set of collection of three-elements of B, that is Sj = {b1j , b2j , b3j }, bij ∈ B. From any instance of the X3C problem we can contrive an election E = (C, V ), where C is composed by the following candidates:

p,

: the intended winner

w,

: the antagonist

di ∈ D,

i = 1, . . . , k

: disturbers

bj ∈ B, j = 1, . . . , 3k : corresponding to elements in B

Furthermore, for each Sj ∈ S , there exists one unqualified candidate sj , that could be used by the chair to replace some qualified alternatives. sl ∈ S, l = 1, . . . , n : corresponding to Sl ∈ S

The collection of voters is specified with the list of preferences. So V is 36

3.1. MULTIMODE CONTROL ATTACKS WITH EQUAL-BUDGET composed by the following ones: 1 voter : D  w  B  S  p n − k + 2 voters : D  p  w  . . . 1 voter for each sl ∈ S : D  sl  w  . . . 1 voter for each sl ∈ S : D  sl  b1l  . . . 1 voter for each sl ∈ S : D  sl  b2l  . . . 1 voter for each sl ∈ S : D  sl  b3l  . . .

Let us call li the number of voters, of the last three types, that rank bi after D and sl . Then, there are others voters in V with the following preferences for each bi :

n − li + 1, ∀bi ∈ B : D  bi  . . .

We claim that the X3C instance has a solution if and only if we can replace no more than k candidates in this election so that p is the winner. First of all we need to delete all the disturbers, the profile is unanimous and so there is no way to make p the winner without eliminating all di candidates. We cannot replace less than k candidates otherwise the winner remains one of the alternatives in D. After deleted all these alternatives we need to add k candidates to the profile.

We can replace all the di ∈ D alternatives with sl ∈ S corresponding to the cover. So after the disturbers’ deletion, the score of the remaining 37

CHAPTER 3. EQUAL BUDGET IN MULTIMODE CONTROL RESULTS alternatives are as follow: score(w) = n + 1 score(p) = n − k + 2 score(bi ) = n + 1

So the winner could be anyone in B or w. Now suppose there exist an exact3-cover, then adding those candidates sl corresponding to the Sl in the cover, w will lose k points corresponding to voters which move their preferences to these alternatives, bj cannot get points from voters other than the final group and p does not decrease its score. This way the final scores are as follow: score(w) = n − k + 1 score(p) = n − k + 2 score(bi ) = n − k + 1 score(sl ) = 4

hence p is the unique winner of the election. If p can become a winner adding k candidates then it means that this addition decrease w’s score of k points, but at the same time we assure that bj alternatives cannot get a score equal or higher than p. Since bj could happen in at most n Sj subsets than it could be gain n points, but at the same time it means that a cover will move k points from bj to sl alternatives. We need to cover all bj alternatives to secure that no one of those have a score equal or higher than p. But this means that Si must comprise a cover.

Plurality voting - Destructive Control Once again, we already know that Plurality is vulnerable to single control attacks by adding (deleting) voters in destructive form. It is resistant to 38

3.1. MULTIMODE CONTROL ATTACKS WITH EQUAL-BUDGET single control by adding (deleting) candidates. We now prove that their combination, using equal-budget setting, maintain the same characteristics. Theorem 3. Plurality is vulnerable to destructive control via replacing voters (DCRV). Proof. We prove the vulnerability of Plurality to this type of control reporting a polynomial algorithm. The idea is trying to make win one candidate that can become winner because there are enough resources available. Let E = (C, V ) be an election system, where C = {p, b1 , . . . , bm }, V is the set of the qualified voters and A is the set of voters that the chair can potentially use to replace some voter in V . The budget available for the replace control is k = kD = kA , where kD is the budget available for the deletion and kA is the budget available for the addition. First of all calculate dif f (p, bi ) = score(p) − score(bi ), this is the number of voters that cannot permit bi to be the winner, clearly every dif f (p, bi ) > 0. Now calculate ∆bi = dif f (p, bi ) − 2k, the alternatives that could be winners are that ones whose ∆bi < 0, this can be done in O(m). We need to identify which one of the possible winners have enough resources. Sort all the bi , arranging them from the lowest ∆bi to the higher one, so the first one is the alternative closer to be a winner other than p. For each bi ∈ C with ∆bi < 0, calculate kbi that is the number of voters in A which rank bi higher, this can be done in O(kAk). Each bi with ∆bi < 0 could be a winner if the chair can add enough voters, that is if and only if dif f (p, bi ) − k ≤ kbi or rather after the deletion of at most k voters, that prefer p to it, the chair has enough electors to add that prefer bi . Last step permit to understand how to spend the budget, the chair can exploit the replacing control if one of the following cases is true for at least one bi with ∆bi < 0 and dif f (p, bi ) − k ≤ kbi : 1. k ≤ kbi : the chair has enough budget and enough voters available, so he can replace dif f (p, bi )/2 voters that prefer p with dif f (p, bi )/2 voters which prefer bi to make it the new winner 2. k > kbi : the chair can add enough voters to make bi the new winner, but not enough to match the one spent in deleting voters, so he can add voters preferring other bi candidates 39

CHAPTER 3. EQUAL BUDGET IN MULTIMODE CONTROL RESULTS All the above operations can be exploited in O(kAk + m). If no bi with ∆bi < 0 alternatives respect one of the above cases the control is not applicable to the election system with the available budget.

Theorem 4. Plurality is resistant to destructive control via replacing candidates (DCRC). Proof. From any instance of Hitting set problem we can derive an election system E = (C, V ). There is also another set of unqualified candidates B = {b1 , . . . , bm }, the chair could use to replace some qualified alternative. The election system have the following characteristics: C = {w, p} ∪ A, where A is a set of qualified candidates with |A| ≥ |B|, w is the actual winner while p and ai are other qualified candidates. V is a collection of 4(m − k) + 6n(k + 1) + 9 voters, whose preferences are the following: • 2(m − k) + 2n(k + 1) + 4 voters prefer w  p  A  B • 2(m − k) + 2n(k + 1) + 5 voters prefer p  w  A  B • for each Si there are 2(k+1) voters that prefer Si  w  p  A  S−Si So the idea is to replace the right number of candidates, if the chair does not exercise any control then w wins the election with 2(m−k)+4n(k +1)+4 votes, while p has 2(m − k) + 2n(k + 1) + 5 votes. In such a system we could replace any ai ∈ A with bj ∈ B. Candidates w lose one vote for each bj that participates to the election. If there exist an hitting set B 0 with |B 0 | ≤ k then replacing any ai ∈ A with candidate bj ∈ B 0 , w will lose 2n(k + 1) votes and the final scores are score(w) = 2(m − k) + 2n(k + 1) + 4 score(p) = 2(m − k) + 2n(k + 1) + 5 for each bi ∈ B 0 score(bi ) ≤ 2n(k + 1) 40

3.1. MULTIMODE CONTROL ATTACKS WITH EQUAL-BUDGET so p is the new winner. Now suppose that w lose the election in favour of p after replacing qualified alternatives with the ones in B 0 . This can happen if the alternatives used by the chair, decrease the score(w) and the only voters that can switch their preferences from w to bi are the ones corresponding to each Si . So each of these sets must contained at least one of candidates put into the system, hence B 0 must be an hitting set. If B 0 is not an hitting set then there still exist some 2(k + 1) voters that prefer w  p  A, in this situation score(w) = 2(m − k) + (2n + x)(k + 1) + 4 score(p) = 2(m − k) + 2n(k + 1) + 5 where x is the number of groups of 2(k + 1) who still prefer w over all alternatives. Hence w is still the actual winner cause score(w) > score(p) So the only way to make w lose the election is that B 0 is an hitting set.

Copeland voting We know that Copeland is resistant to single control attack by adding (deleting) candidates. We now prove that their combination, using equal-budget setting, maintain the same characteristics in constructive form. Theorem 5. Let α be a rational number such that 0 ≤ α ≤ 1. Copelandα is resistant to constructive control via replacing of voters (CCRV). Proof. We prove the theorem via reductions from X3C problem. This is a modification of the proof used by [2]. Fo any instance of the X3C problem create an election E = (C, V ), where C = {c} ∪ {b1 . . . , bm }. Let V consist of 2 ∗ (m/3) − 3 voters vi ∈ V , all with the following preference order: v i : b1  . . .  bm  c 41

CHAPTER 3. EQUAL BUDGET IN MULTIMODE CONTROL RESULTS So b1 is the Condorcet winner with score(b1 ) = (m − 1) + 1 = m All bi candidates beat c with V SE (bi , c) = 2 ∗ (m/3) − 3 Where V SE (x, y) is the number of voters that prefer candidate x to y and score(x) is the number of candidates defeated by x in the pairwise contests. Let W contain unregistered voters wj ∈ W , one voter for each Sj ∈ S, its preferences are wj : b1j  b2j  b3j  c  . . . where bkj , with k = (1, 2, 3) corresponds to the bi ∈ Sj . Alternatives after c are in arbitrary order. We claim that there is a solution to X3C if and only if (m/3) or fewer voters wj ∈ W can be added to V and at the same time (m/3) voters vi ∈ V are be deleted, so that c become a Condorcet winner. First of all, assume that there exists an exact-3-cover. Including the corresponding voters wj ∈ W in the election the scores change as follows: • each bi gains 1 vote against c V SE (bi , c) = 2 ∗ (m/3) − 2 • c increases its score against each bi V SE (c, bi ) = (m/3) − 1

The candidate c is still loosing all the head-to-head contest against each bi . So the only way to make c the Condorcet winner is to delete (m/3) voters vi ∈ V . This deletion change the score as follow: V SE (bi , c) = (m/3) − 2 42

3.1. MULTIMODE CONTROL ATTACKS WITH EQUAL-BUDGET

V SE (c, bi ) = (m/3) − 1 Thus c becomes the Condorcet winner as it wins all the contests against each bi . Assume that c can become a Condorcet winner by adding (m/3) or fewer voters after (m/3) voters vi ∈ V are be deleted. There cannot exist more than 1 added voter who prefers bx to c, otherwise bx would gain 2 vote, so the scores against c would be V SE (bx , c) = (m/3) − 1

V SE (c, bx ) = (m/3) − 2 this means that c looses against bx and it is not the Condorcet winner. So each bi is preferred to c by 0 or 1 added voters. In the last case c must be preferred by (m/3) − 1 added voters. Since some voter must be added it must be exactly m/3. On the other hand, if no voters prefer bi to c and since each of the added voters have 3 alternatives above c then, by the pigeonhole principle, some bi is ranked above c by more than 1 voter. This is a contradictions, because we assume that cannot exist more than 1 added voter who prefers bx to c. Therefore each bi is preferred to c exactly 1 of the m/3 added voters and so they correspond to an exact-3-cover of the bi .

43

CHAPTER 3. EQUAL BUDGET IN MULTIMODE CONTROL RESULTS

44

Chapter 4 Convergence of iterative voting In the following sections we will consider iterative elections. This is a semplification of the iterative voting schemas studied in [9] and [10] and reported in section 2.6. Here we will not use any concepts linked to game theory. We proved stability of some voting rules and report counterexamples for the ones that do not converge.

4.1

Iterative Veto with non-linear tie-breaking

As we already mention in section 2.6.2, iterative Veto converge to a stable state, under the specific assumption, that the used tie-breaking rule is linear. We report a counterexample which proves the non-convergence of the iterative voting rule, when such a kind of tie-breaking is not used. Theorem 6. An iterative Veto election with non-linear tie-breaking, even for unweighted voters that use best-response and start from truthful state, does not always converge. Proof. We shall use 2 voters (v1 , v2 ) and 4 candidates (a, b, c, d). The real preference of the voters are: v1 : a  b  c  d v2 : d  c  b  a

45

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING c wins

b wins

v1 : a  b  c  d v2 : d  c  b  a

v1 : a  b  d  c v2 : d  c  b  a

v1 : a  b  c  d v2 : d  c  a  b

v1 : a  b  d  c v2 : d  c  a  b

a wins

d wins

Figure 4.1: The cycle of Veto non convergence (top-left is the truthful state)

We define the Tie-breaking rule as follows: a=b→a a=c→a a=d→d b=c→c b=d→b c=d→c Let we start from a truthful state b, c are tied with score(b) = score(c) = 2, hence c is the winner for tie-break. Voter v1 realizes that vetoing candidate c could make b the winner, and so it does. Now v2 manipulates its preferences and candidate d wins. Once again v1 lies on its real preference and vetos d, in this way a could win the election. The system enter in a loop when v2 vetos a to make c the winner again. Voters inside a rectangle highlight which one is manipulating its preferences because it is unhappy with the results.

46

4.2. LOOP SEARCHING IN ITERATIVE VOTING

4.2

Loop searching in iterative voting

We found the counterexamples that prove the non convergence of Veto. But, due to the combinatorial explosion of some voting rules, we designed and developed a software that searches for a loop in iterative voting. It could use different types of tie-breaking rules and different types of best-response moves. The software is written in Java Language and it is composed by 6 classes and 1 interface. These are the 4 classes: • Voter: it represents the single voter and its preferences • Copeland: it implements the Copeland rule • Cup: it implements the Cup rule • Stv: it implements the STV rule • Manipulation: it is the class that implements the manipulation on the voting system, using the desired best-response moves and search for a cycle • TestVoting: it is the class that brute force the voting system The interface is: • VotingRule: it rapresents the voting rule to use The relashionship between the different object is depicted in figure 4.2. The most important class is Manipulation. It implements the method isCiclo that, given an initial profile, that is considered the truthful one, it recursively searches for a cycle that rapresents the possibility that the considered voting system does not always converge to a stable state. We briefly report and describe the algorithm used by the method. 47

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING

Figure 4.2: Objects’ diagram

48

4.2. LOOP SEARCHING IN ITERATIVE VOTING

Algorithm 4.2.1: isCiclo(prof ile, listP rof ile) w ← current winner scorew ← current winner score unhappy ← store unhappy voters cycle ← F ALSE pref ← store all possible preferences for each x ∈ unhappy do   dummy ← pref.pop()     prof ileold ← prof ile      while(cycle = F ALSE and pref 6= ∅)       replace x with dummy in prof ileold           if (RBR(prof ileold , x, w, scorew ))           then       if (prof ile      ( old ∈ listP rof ile)        do   then print(listP rof ile)            ( cycle ← T RU E          listP rof ile ← listP rof ile ∪ prof ileold         else           n cycle ← isCiclo(prof ileold , listP rof ile)       else dummy ← pref.pop() return (cycle)

The algorithm recursevely looks for a repeated profile. It performs a brute force search, enumerating all the possibile combinations of valid preferences. At each step only the unhappy voters can perform a manipulation. Unhappy voters are the voters whose first choice is different from the current winner and so they are not satisfied from the actual results, so they are the only ones boosted to change their preferences to make win a different candidates that is better than the current one. For each unsatisfied voters the algorithms goes through all the possible valid preferences and control which ones is a 49

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING best-response. When such a preference is found, it controls if the profile is already been used in the past or not. If the profile is present in the list, it means we found a loop, otherwise it means we could recursevely search for some other solutions, until all the combinations are visited.

4.3

Iterative Copeland divergence

Theorem 7. An iterative Copelandα election with a linear tie-breaking, even for unweighted voters that use best-response and start from truthful state, does not always converge. Proof. We shall use 2 voters (v1 , v2 ) and 3 candidates (a, b, c). The real preferences of the voters are: v1 : a  b  c v2 : c  a  b We use a linear-tie-break like the following one: a=b→b a=c→c b=c→c Let we start from a truthful state where a is the winner. Voter 2 manipulates the outcome changing its preferences and making c the winner. Voter 1 does the same thing, making b the winner. Once again voter 2 changes its preferences and makes c the winner. Voter 1 now changes again its preferences making a the winner. And we enter a loop. The process is depicted in figure 4.3. In the figure, voters inside a rectangle highlight which one is manipulating its preferences.

Theorem 8. An iterative Copelandα election with every type of tie-breaking, even for unweighted voters that use best-response and start from truthful state, does not always converge. 50

4.3. ITERATIVE COPELAND DIVERGENCE

c wins

a wins

v1 : a  b  c v2 : c  a  b

v1 : a  b  c v2 : c  b  a

v1 : b  a  c v2 : c  a  b

v1 : b  a  c v2 : c  b  a b wins

c wins

Figure 4.3: The cycle of Copeland non convergence (top-left is the truthful state)

a

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Figure 4.4: The cycle of Copeland Majority Graph

51

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING c wins

b wins

v1 : a  b  c  d v2 : c  d  b  a

v1 : b  a  d  c v2 : c  d  b  a

v1 : a  b  c  d v2 : d  c  a  b

v1 : b  a  d  c v2 : d  c  a  b

a wins

d wins

Figure 4.5: The cycle of Copeland non convergence (top-left is the truthful state)

Proof. We shall use 2 voters (v1 , v2 ) and 4 candidates (a, b, c, d). The real preferences of the voters are: v1 : a  b  c  d v2 : c  d  b  a We do not need a tie-breaking rule. As reported in the following example, the election never gets stuck in a tie. Let we start from a truthful state where c is the winner. Voter 1 manipulates the outcome changing its preferences and making b the winner. Voter 2 does the same thing, making d the winner. Once again voter 1 changes its preferences and makes a the winner. Voter 2 now changes again its preferences making c the winner. And we enter a loop. The process is depicted in figure 4.5. In the figure, voters inside a rectangle highlight which one is manipulating its preferences.

Definition 1. Let be scorei (x) the score of the candidate x in the election 52

4.3. ITERATIVE COPELAND DIVERGENCE a

b

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Figure 4.6: The cycle of Copeland Majority Graph

state Gi . The score of the winner in the election state Gi is max(Gi ), while the upper bound of the score is |C| − 1, that is the score of the Condorcet winner, if any exists. Definition 2. A restricted best-response or RBR move implies a change in voter vi preferences such that the new one rewards a better candidate for voter vi but it does not decrease the score of the current winner. Let w be the winner in the state i of the iterative election. After a restricted best-response the new winner is p. As we use a restricted bestresponse scorei+1 (p) ≥ scorei (w) but this also implies that scorei+1 (w) ≥ scorei (w). Theorem 9. An iterative Copelandα election with a non-linear-tie-breaking, even for unweighted voters that use restricted best-response moves and start from truthful state, may not converges to a stable state. Proof. We shall use 6 voters (v1 , v2 , v3 , v4 , v5 , v6 ) and 4 candidates (a, b, c, d). The real preferences of the voters are: v1 : a  b  c  d v2 : d  a  b  c v3 : d  a  b  c v4 : d  b  a  c v5 : a  b  c  d 53

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING v6 : a  b  c  d We use a non-linear-tie-break like the following one: a=b=c=d→a a=b=d→a a=b=c→c a=c=d→a b=c=d→d d=c→d d=b→d d=a→a c=b→c c=a→c b=a→b The process is depicted in figure 4.7. In the figure, voters inside a rectangle highlight which one is manipulating its preferences.

54

4.3. ITERATIVE COPELAND DIVERGENCE

a wins v1 : a  b  c  d v2 : d  a  b  c v3 : d  a  b  c v4 : d  b  a  c v5 : a  b  c  d v6 : a  b  c  d

b wins v1 : b  d  a  c v2 : d  a  b  c v3 : d  a  b  c v4 : a  b  d  c v5 : a  b  c  d v6 : a  b  c  d

b wins v1 : a  b  c  d v2 v3 v4 v5 v6

:dabc :dabc :bdac :abcd :abcd

v1 : b  d  a  c v2 : d  a  b  c v3 : d  a  b  c v4 : b  d  a  c v5 : a  b  c  d v6 : a  b  c  d a wins d wins

v1 : b  d  a  c v2 : d  a  b  c v3 : d  a  b  c a wins v4 : a  b  d  c v5 : b  c  d  a v6 : a  b  c  d

v1 : a  b  d  c v2 v3 v4 v5 v6

:dabc :dabc :abdc :bcda :abcd b wins

v1 : b  d  a  c v2 v3 v4 v5 v6

:dabc :dabc :bdac :bcda :abcd

v1 : a  b  d  c v2 : d  a  b  c v3 : d  a  b  c v4 : b  d  a  c v5 : b  c  d  a v6 : a  b  c  d a wins

Figure 4.7: The cycle of Copeland non convergence (top-left is the truthful state)

55

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING

4.4

Iterative Copeland convergence

We could change the behaviour of the iterative Copeland election, just changing together the tie-breaking rule and the meaning of the best-response moves. Theorem 10. An iterative Copelandα election with a linear tie-breaking, even for unweighted voters that use restricted best-response moves and start from truthful state, always converges to a stable state. Proof. Suppose that the tie-breaking rule is made up of k different states. As it is a linear order we could infer a total order between states. When a voter changes its preferences, it cannot decrease the score of the actual winner, because voters are using restricted best-response. So after an RBR, the score of the new winner p must be equal or higher of the previous winner w. scorei+1 (p) ≥ scorei (w) and at the same time scorei+1 (w) ≥ scorei (w) and so scorei+1 (p) ≥ scorei+1 (w) ≥ scorei (w) If the score is equal this means that p defeats the actual winner w for tiebreaking rule, otherwise p is the new winner in the iterative state Gi+1 . If the manipulation creates always winner alternatives with equal score, sometimes we end the tie-breaking rule states and the only way to make some candidates the winner is to increase its score. The process ends when someone gets the Condorcet winner score, as no one can decrease its score the process ends in a stable state because no voters can manipulate the preferences without decreasing the actual winner score. If no alternatives reach the Condorcet winner score, at each step of the iterative election we could go through all the k tie-break steps before increase the score. The iterative election will converge in at most O(kn) steps, where n is the number of voters. 56

4.5. ITERATIVE STV CONVERGENCE

4.5

Iterative STV convergence

Theorem 11. An iterative STV election with linear tie-breaking rule, unweighted voters that use best-response moves and start from truthful state, does not always converge to a stable state. Proof. We shall use 6 voters (v1 , v2 , v3 , v4 , v5 , v6 ) and 3 candidates (a, b, c). The real preferences of the voters are: v1 : c  a  b v2 : c  a  b v3 : b  c  a v4 : b  c  a v5 : a  b  c v6 : a  b  c We use a linear tie-breaking like the following one: a > b > c The process is depicted in figure 4.8. In the figure, voters inside a rectangle highlight which one is manipulating its preferences.

4.6

Iterative Approval convergence

Theorem 12. An iterative Approval election with linear-ordered tie-breaking rule, even for un- weighted voters that use best-response moves and start from a truthful state, does not always converge. Proof. We prove the theorem just reporting that k-Approval rule is a special case of Approval. In such a situation we can imagine a profile where all voters decide to approve all the same number k of candidates. As we reported in section 2.6.2, Lev and Rosenschein [9] already proved that k-Approval, with k ≥ 3, does not always converge to a stable state. So, to prove that Approval does not always converge, we could use the same example used in [9] and fix that each voter has to approve the same number k of alternatives.

57

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING

b wins v1 : c  a  b v2 v3 v4 v5 v6

:cab :bca :bca :abc :abc

v1 : c  a  b v2 : c  a  b v3 : c  a  b v4 : b  c  a v5 : a  b  c v6 : a  b  c c wins

a wins v1 : b  c  a v2 : c  a  b v3 : b  c  a v4 : b  c  a v5 : a  b  c v6 : a  b  c

v1 : b  c  a v2 v3 v4 v5 v6

:cab :cab :bca :abc :abc b wins

Figure 4.8: The cycle of STV non convergence (top-left is the truthful state)

58

4.7. ITERATIVE MAXIMIN CONVERGENCE

4.7

Iterative Maximin convergence

As we already mention in section 2.6.2, iterative Maximin does not always converge to a stable state, when the tie-breaking rule is deterministic. Now, we prove that it is possible to make iterative Maximin rule always converges, when some restrictions are applied to the system. Theorem 13. An iterative Maximin election with a linear tie-breaking, unweighted voters that use restricted-best-response moves and starts from truthful state, always converges to a stable state. Proof. The proof is very similar to the one used for theorem 10. For Maximin the score of each candidate is calculated as follow score(x) = min N (x, y) y∈C−{x}

and the winners are candidates with max(score(x)). x∈C

Since each voter can never decrease the score of the actual winner, they can only try to change the result by making win a candidate that is better for the tie-breaking rule or making a candidate win with a higher score. As in the theorem 10 scorei+1 (p) ≥ scorei (w) and at the same time scorei+1 (w) ≥ scorei (w) and so scorei+1 (p) ≥ scorei+1 (w) ≥ scorei (w) Now suppose that the tie-breaking rule is made up of k different states. If the score is equal this means that p defeats the actual winner w for tiebreaking rule, otherwise p is the new winner in the iterative state Gi+1 . The process ends in two different ways. When someone gets the Condorcet winner score, as no one can decrease its score the process ends in a stable state because no voters can manipulate the preferences without decreasing 59

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING the actual winner score. The worst case of the manipulation is when voters always create winner alternatives with equal score. Sometimes we end the k tie-breaking rule state and the only way to make some candidate the winner is to increase its score. If no alternatives reach the Condorcet winner score, at each step of the iterative election we could go through all the k tie-break steps before increase the score. The iterative election will converge in at most O(kn) steps.

4.8

Iterative Cup Rule convergence

In the following section, we report the results of our studies about Cup rule. We prove with a counterexample the non-convergence of Cup rule. We also define a restriction that always assures the convergence of the rule. Theorem 14. An iterative Cup election with a linear tie-breaking, unweighted voters that use best-response moves and start from truthful state, does not always converge to a stable state. Proof. We shall use 4 voters (v1 , v2 , v3 , v4 ) and 6 candidates (a, b, c, d, e, f ). The real preferences of the voters are: v1 : a  b  c  d  e  f v2 : a  b  c  d  e  f v3 : f  e  d  c  a  b v4 : d  f  e  c  b  a We use a linear-tie-break like the following one: d > e > f > c > b > a The process is depicted in figure 4.9. In the figure, voters inside a rectangle highlight which one is manipulating its preferences. As we already describe in subsection 2.3.1, the lists of the competitions are represented as a binary tree in which the leaf are candidates. Each parent node is labeled with the winner of the majority election between the children. The winner is the candidate that labels the root node of the tree. Figure 4.10 describes the sequence of binary trees generated by the iterative Cup. Each binary tree corresponds to the profile in the same position. 60

4.8. ITERATIVE CUP RULE CONVERGENCE d wins v1 : a  b  c  d  e  f v2 : a  b  c  d  e  f v3 : f  e  d  c  a  b v4 : d  f  e  c  b  a

f wins v1 : a  b  c  d  e  f v2 : a  b  c  d  e  f v3 : f  e  c  d  b  a v4 : d  f  e  c  b  a

e wins v1 : e  d  f  c  b  a v2 : a  b  c  d  e  f v3 : e  f  d  c  b  a v4 : d  f  e  c  b  a

d wins

v1 : c  d  f  e  b  a v2 : a  b  c  d  e  f v3 : e  f  d  c  b  a v4 : d  f  e  c  b  a

v1 : e  d  f  c  b  a v2 : a  b  c  d  e  f v3 : f  e  c  d  b  a

d wins

v4 : d  f  e  c  b  a

v1 : c  d  f  e  b  a v2 : a  b  c  d  e  f v3 : f  e  c  d  b  a v4 : d  f  e  c  b  a

f wins

Figure 4.9: The cycle of Cup non convergence (top-left is the truthful state)

As you can see from the previous example, the manipulation gives more support to candidates that are not more preferred by the voters acting strategically. This allows to these candidates to win head-to-head competitions that are in the lower part of the tree. Once they climb the tree, they can compete with other candidates that are more preferred by the voters, so these better alternatives can win the majority contest. Definition 3. A voter does a worsening flips (WP) when it brings a more preferred candidate below a less preferred candidate. For example, in figure 4.9 voter 3 starts acting the manipulation through 61

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING two worsening flips. It brings a below b and d below c. It also does not change other candidates’ relative positions. This way make candidates b and c win, who are weaker than f . This is enough to make win candidate f . Theorem 15. An iterative Cup rule with a linear tie-breaking, unweighted voters that can use a bounded number k of worsening flips and starts from truthful state, always converges to a stable state. Proof. We start proving the theorem from the observation that in Cup rule a voter cannot give more support to a candidate that it already prefers to another one. If voter vi already prefers alternatives a to b, then it already has a over b in its preferences. So it cannot change the majority in the contest between these two candidates. The only thing, it could do, is trying to change the majority to some less preferred candidates. These candidates can win one or more competitions that bring them to compete and lose with an alternative more preferred to vi . So this more preferred alternative can win the election. For each worsening flip, voters can make a flip that bring back two candidates to their original relative positions. If each strategical voter can use only a bounded number of worsening flips, then sometime they cannot make any other strategical actions to indirectly help some more preferred candidates. The iterative cup rule converge in at most O(kn) steps, where k is the number of worsening flips allowed to the n manipulators.

62

4.8. ITERATIVE CUP RULE CONVERGENCE

f

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Figure 4.10: The sequence of Cup trees non convergence (top-left is the truthful state)

63

CHAPTER 4. CONVERGENCE OF ITERATIVE VOTING

64

Chapter 5 Multimode control and iterative voting We are interested in applying control actions on converging iterative-voting, such as Plurality, to observe if its behaviour changes. We prove that single control actions with bounded budget do not affect the behaviour of the voting rule, while multimode control actions with specific features can change the system, so that it does not converge to a stable state.

5.1

Iterative Voting and control actions

In the following sections we study how iterative voting systems react to single control or to multimode control actions. Before that, we need to define some settings about this new scenarios. For example, suppose we are in the middle of iterative voting and the chair adds a new candidate. We have some voters who are voting strategically at this point. In this new augmented election, how do these voters rank the new candidates? We consider itertive voting systems in which voters always express their preferences over all candidates, even that ones that are unqualified. This means that during all the iterative process, voters give their preferences over the alternatives that the chair has already deleted or not yet added. If these unqualified alternatives are never been yet added, then during 65

CHAPTER 5. MULTIMODE CONTROL AND ITERATIVE VOTING each iterative step voters rank them in the real position. For instance, if an unqualified alternatives a is ranked in the third position in the truthful preferences of some vi , then it must keep ranking a in the third posistion until the chair add it, so vi cannot move a in its preferences while it is unqualified. Clearly, if the iterative voting system is using a positional scoring rule, such candidates are not concerned in the score. For example, if an unqualified candidate is top-ranked in the majority it cannot win the election, because it is not really participating to it. On the other hand, during the iterative process each voter can exploit manipulation and change its real preferences, as a result alternatives are ranked in different way than the truthful one. When the chair delete an alternative, this must be kept in the same last position where each voter rank it. Unqualified alternatives are freezed in their position until the chair add them again.

5.2

Iterative Plurality Voting and control actions

Theorem 16. Using Adding (Deleting) Candidates control action to change an iterative Plurality election with deterministic tie-breaking, where voters have weight 1 and use best-response moves, always converges to a stable state. Proof. We shall use theorem 3 of [10] to prove the theorem. Let GD the Plurality election, C the set of initial alternatives and A the limited set of candidates that the chair can add to the election. The voters always express their preferences over the set C ∪ A. Each time the chair adds a candidate to the election, he creates a new Plurality election with deterministic tiebreaking, where the voters have weight 1 and use best-response moves. So for theorem 3 of [10] this new election will converge to a stable state. Since the chair can add a limited number of candidates to the election, he can create a bounded number of different Plurality elections, each of them always reach a stable state.

66

5.3. ITERATIVE VOTING WITH MANIPULATION AND SINGLE CONTROL Theorem 17. Using Adding (Deleting) Voters control action to change a Plurality election with deterministic tie-breaking always converges to a stable state from any state, under the assumption that voters have weight 1 and use best-response moves. Proof. The proof is very similar to the previous one. We shall use theorem 3 of [10] to prove the theorem. Let GD the Plurality election, C the set of alternatives, V the collection of the voters and A the limited set of voters that the chair can add to the election. The voters always express their preferences over C. Each time the chair adds a voter to the election, he creates a new Plurality election with deterministic tie-breaking, where the voters have weight 1 and use best-response moves. So for theorem 3 of [10] this new election will converge to a stable state. Since the chair can add a limited number of voters to the election, he can create a bounded number of different Plurality elections, each of them always reach a stable state.

5.3

Iterative Voting with manipulation and single control

Using only one kind of basic control action, see section 2.4.1, does not affect the convergence of the system. Theorem 18. If an iterative voting rule GD with deterministic tie-breaking, in which all voters have weight 1 and use best-response moves, always converges to a stable state only from truthful state, then also the iterative rule GD +X always converges, where X is one of the 2 basic control types deleting candidates, deleting voters. Proof. Control X has a limited budget, by definition in 2.4.1. This means that sometimes, applying all the possible actions, we will end the budget and 0 we will stop in a new election GD that could be different in the number of voters or candidates, but it uses always the same voting rule and it has the characteristics of the theorem. Since this is only eliminating some possible 67

CHAPTER 5. MULTIMODE CONTROL AND ITERATIVE VOTING iteration paths from the initial truthful state, so if all of them were converging before, the remaining ones will be converging now. Theorem 19. If an iterative voting rule GD with deterministic tie-breaking, in which all voters have weight 1 and use best-response moves, always converges to a stable state from any state, then also the iterative rule GD + X always converges, where X is one of the 4 basic control types (adding/deleting candidates, adding/deleting voters). Proof. Control X has a limited budget, by definition in 2.4.1. This means that sometimes, applying all the possible actions, we will end the budget and 0 we will stop in a new election GD that could be different in the number of voters or candidates, but it uses always the same voting rule and it has the characteristics of the theorem. Since this is only eliminating some possible iteration paths from the initial truthful state, so if all of them were converging before, the remaining ones will be converging now. 0

so as GD always converges to a stable state so GD do.

5.4

Iterative Voting with manipulation and multimode control

Our first result shows that elections normally converge, under specific assumption, do not converge anymore if the chair can control their outcome with different action. The chair can prevent to reach a stable state if he uses control types with set of candidate/voters that are not disjoint. Theorem 20. If an iterative voting rule GD with deterministic tie-breaking, in which all voters have weight 1 and use best-response moves, always converges to a stable state from any state, then also the iterative election GD + X1 +X2 always converges, where X1 , X2 are two different kind of basic control types whose sets of control actions are disjoint. Proof. It follows from the theorem 19. Since X1 , X2 have limited budget, by definition in 2.4.1, and the set of alternatives are disjoint, applying all the 68

5.5. ITERATIVE PLURALITY WITH ANTAGONISTIC MULTICONTROL possible actions, we will end the budget and we will stop in a new election 0 GD that could be different in the number of voters or candidates, but it uses always the same voting rule and it has the characteristics of the theorem, so 0 as GD always converges to a stable state so GD do.

5.5

Iterative Plurality with antagonistic multicontrol

We already know that Plurality always converges to a stable state, under some specific assumption that we reported in section 2.6.2. Now, we prove that this behaviour could be changed if we exploit antagonistic control actions on the system. This means that we could use different control actions whose effects cancel the ones produced by precedent actions. Theorem 21. An iterative Plurality election with deterministic tie-breaking, even for unweighted voters that use best-response and start from truthful state, does not converge if the chair can control the ballot with multimode control AC + DC, where the subset A ⊆ C of candidates that can be added to the system and the subset D ⊆ C of candidates that can be deleted are not disjoint, i.e. A ∩ D 6= ∅. Proof. We shall use 2 voters (v1 , v2 ) and 4 candidates (a, b, c, d). The real preferences of the voters are: v1 : a  b  c  d v2 : d  c  b  a Also the chair, that are going to control the election, has his own preferences: acbd We define the Tie-breaking rule as follows: a=b→a a=c→a 69

CHAPTER 5. MULTIMODE CONTROL AND ITERATIVE VOTING d wins

b wins

v1 : a  b  c  d v2 : d  c  b  a

DC(a) v1 : a  b  c  d v2 : d  c  b  a

v1 : a  b  c  d v2 : c  d  b  a

v1 : a  b  c  d AC(a) v2 : c  d  b  a

a wins

c wins

Figure 5.1: The cycle of Plurality non convergence (top-left is the truthful state)

a=d→d b=c→c b=d→b c=d→c Let we start from a truthful state where a, d are tied at the top, hence d is the winner for tie-break. The chair is not satisfied from the result, so decides to delete candidate a, now b, d are tied at the top and b is the winner for tie-break. Voter v2 tries to make c the winner and succeeds. Now the chair add candidate a again so it could be the winner, but once again v2 tries to make d the winner and we enter in a loop. The process is depicted in figure 5.1. In the figure, candidates with gray background do not participate to the ballot, because deleted or not added by the chair, while voters inside a rectangle highlight which one is manipulating its preferences.

Theorem 22. An iterative Plurality election with any type of tie-breaking, even for unweighted voters that use best-response and start from truthful 70

5.5. ITERATIVE PLURALITY WITH ANTAGONISTIC MULTICONTROL state, does not converge if the chair can control the ballot with multimode control AV + DV , where the subset A ⊆ V of voters that can be added to the system and the subset D ⊆ V of voters that can be deleted are not disjoint, i.e. A ∩ D 6= ∅. Proof. We shall use 3 voters (v1 , v2 , v3 ) and 4 candidates (a, b, c, d). The real preferences of the voters are: v1 : a  b  c  d v2 : d  b  c  a v3 : b  c  a  d The third voter v3 is not present when the election starts. Also the chair, that are going to manipulate the election, has his own preferences: dbac We define the Tie-breaking rule as follows: a=b=d→d a=b=c→a a=c=d→a b=c=d→c a=c→a a=d→a b=c→c b=d→d c=d→c Let we start from a truthful state where a, d are tied at the top, hence a is the winner for tie-break. The chair is not satisfied from the result, so decides to add a voter v3 , now a, b, d are tied at the top and d is the winner for tie-break. Voter v1 change its preferences and move b at the top of it, so now b is the winner. Now the chair delete v3 so d could be the winner again, but once again v1 changes its ballots moving a at the top of its preferences, in that way it is winner and we enter in a loop. The process is depicted in 71

CHAPTER 5. MULTIMODE CONTROL AND ITERATIVE VOTING a wins

d wins

v1 : a  b  c  d v2 : d  b  c  a v3 : b  c  a  d

AV (v3 ) v1 : a  b  c  d v2 : d  b  c  a v3 : b  c  a  d

v1 : b  a  c  d v2 : d  b  c  a v3 : b  c  a  d

v1 : b  a  c  d v2 : d  b  c  a DV (v3 ) v : b  c  a  d 3

d wins

b wins

Figure 5.2: The cycle of Plurality non convergence (top-left is the truthful state)

figure 5.2. In the figure, voters with gray background do not participate to the ballot, because deleted or not added by the chair, while voters inside a rectangle highlight which one is manipulating its preferences.

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Chapter 6 Conclusions In the last chapters, we formalized the results of our studies. We tackle the multicontrol actions problem on Plurality and Copeland, using equal budget. This allowed to expand the knowledge of these types of malicious controls. We study the problem of control actions in an equal budget configuration, and we prove that this problem has the same computational complexity of the one with single control action with separate budget. Plurality is vulnerable to the single control actions linked to voters and it is still vulnerable to their combination. While it is resistant to the single control actions linked to candidates and it is still resistant to their combination. These and other results are reported in Table 6.1, which also shows the correspoding theorem numbers. Control Type

Plurality

Copeland

CCRV CCRC DCRV DCRC

V R V R

R (Th. 5) ? ? ?

(Th. (Th. (Th. (Th.

1) 2) 3) 4)

Table 6.1: Multimode control, equal-budget results

Although we have answered several open questions, some still remain on our agenda. We still miss some results about Copeland. These missing results are marked with a ? in Table 6.1. 73

CHAPTER 6. CONCLUSIONS In iterative voting scenarios, manipulation is not considered to be a negative action. We study iterative voting scenarios not yet considered. Some iterative voting rules do not converge to a stable state. For some of them, we indentify which constraints allow for an iterative election to always converge, even combining some restrictions. Results are reported in Table 6.2. The last column on the right reports number of the corresponding theorem. Rule Veto Copeland Copeland Copeland Copeland STV Approval Maximin Cup Rule Cup Rule

Restriction

Tie-break

NON-Linear Linear Any RBR NON-Linear RBR Linear Linear Linear Linear Linear bounded WPs Linear

Result

Theorem

NC NC NC NC C NC NC C NC C

Th. 6 Th. 7 Th. 8 Th. 9 Th. 10 Th. 12 Th. 12 Th. 13 Th. 14 Th. 15

Table 6.2: Iterative voting results

After that, we use single and multiple control actions within iterative voting, in order to understand if the malicious side of control can overtake the stability of iterative voting systems. We prove that voting rules, which normally converge to a stable state, may not maintain the same behaviour, if we allow the chair to use the control on the ballot. Results are reported in table 6.3. The last column on the right reports the number of the theorem that demonstrates the results.

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75

Plurality Plurality Any Any Any Plurality Plurality

Rule dis. control joint control joint control

Restric. AC/DC AV/DV DC/DV AC/DC/AV/DV X1 + X2 AC+DC AV+DV

Control type Det. Det. Det. Det. Det. Any Any

Tie-break

Table 6.3: Multimode control and iterative voting results

Any Any Truthful Any Any Any Any

Conv. only from state C C no change no change no change NC NC

Result

Th. Th. Th. Th. Th. Th. Th.

16 17 18 19 20 21 22

Theorem

CHAPTER 6. CONCLUSIONS

6.1

Further works

A lot of work remains to be done. A natural continuation is completing the missing results about Copeland in equal budget. We also plan to study new iterative scenarios, which use voting rules not yet considered. In these scenarios and in the previous ones, it is very interesting to identify different ways to bind iterative voting to convergence, also when we are exploiting multimode control on the system.

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Bibliography [1] Kenneth J. Arrow. Social Choice and Individual Values. Ed. by Cowles Foundation. John Wiley & Sons Inc., New York London Sydney, 1963. [2] John J. Bartholdi, Craig A. Tovey, and Michael A. Trick. “How hard is it to control an election”. In: Mathematical and Computer Modeling (1992), pp. 27–40. [3] Cynthia Dwork et al. Rank Aggregation Methods for the Web. Paper presented at the 10th International World Wide Web Conference. 2001. [4] Faliszewski and Procaccia. “AI’s war on manipulation: are we winning?” In: AI Magazine (2010), pp. 53–64. [5] Piotr Faliszewski, Edith Hemaspaandra, and Lane A. Hemaspaandra. “Multimode Control Attacks on Elections”. In: Journal of Artificial Intelligence Research (2011), pp. 305–351. [6] Piotr Faliszewski et al. “Llull and Copeland voting computationally resist bribery and control”. In: Journal of Artificial Intelligence Research (2009), pp. 275–341. [7] A. Gibbard. “Manipulation of voting schemes: A general result”. In: Econometrica 41 (1973), pp. 587–601. [8] R. Karp. “Reducibility among combinatorial problems”. In: Complexity of Computer Computations. Ed. by R. Miller and J. Thatcher. Plenum Press, 1972, pp. 85–103. [9] Omer Lev and Jeffrey J. Rosenschein. “Convergence of iterative voting”. In: Proceedings of the 11th International Conference and Multiagent Systems (2012). 77

BIBLIOGRAPHY [10] Reshef Meir et al. “Convergence to equilibria in Plurality Voting”. In: Proc 24th National Conference on AI (AAAI) (2010), pp. 823–828. [11] Debian Project. Debian Voting Information. Oct. 2011. url: http : //www.debian.org/vote/. [12] Javad Khazaei Reyhaneh Reyhani Mark Wilson. “Coordination via Polling in Plurality Voting Games under Inertia”. In: COMSOC (2012). [13] Francesca Rossi, Kristen Brent Venable, and Toby Walsh. A short introduction to preferences. Between Artificial Intelligence and Social Choice. Morgan and Claypool publishers, 2011. [14] Mark. A. Satterthwaite. “Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions”. In: Journal of Economic Theory 10 (1975), pp. 187–217.

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Ringraziamenti Voglio ringraziare sinceramente: la prof.ssa Rossi per l’inesauribile pazienza, la continua presenza e i consigli che mi hanno guidato nel realizzare questo lavoro. La prof.ssa Venable e il prof. Walsh per il supporto e i commenti costruttivi. Il prof. Faliszewski per l’interessamento dimostrato. I compagni di corso, come sempre senza i loro appunti non sarei riuscito ad arrivare fin qui. La mia famiglia, gli amici e i colleghi, per la pazienza e il supporto. Gabriella per avermi appoggiato, sopportato, incitato, sostenuto, tutto questo `e anche merito tuo. Andrea Loreggia

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