I was born on my mother’s traditional lands and raised as a child between two households. My first home was my parents’, situated in the suburban sprawl of fibro ovens in treeless estates in western Sydney where we lived with a mix of Koori families, imports from war-torn Europe and the debris of Sydney’s post-colonial white society under the banner of ‘housos’, residents of the public housing estates of Bass Hill and Chester Hill. My second home was the coolness, calmness and stability of the loving arms and hugs of my mother’s mother and her Koori family in Harbord, Curl Curl and the Manly–Narrabeen Lakes areas, all traditional country of the Gai-mariagal. We swam, fi shed, prawned, regularly harvested swan and duck eggs, and ate possum and the odd goanna to supplement a diet rich in chayote, rhubarb, warrigal greens or anything else that was easy to grow or scavenge or that fell off the back of a truck.

I had an idyllic childhood, exploring Cowan Creek and the Hawkesbury River, camping with my uncles and cousins and having no knowledge of poverty or racism; then my cousins started to vanish. One by one they were plucked away, and the adults never spoke of them. We stopped visiting each other’s houses: card games that would last all night, the laughter and warmth of wider family, with Dad sharing long necks and Gran her blackberry or rhubarb pies – this all ceased abruptly. Then in ‘59 the camp at Narrabeen was bulldozed and the Elders trucked off, some to a mission place at Rooty Hill that I visited when my big sister took supplies to one of our Elders.

We represent the oldest culture in the world; we are also diverse and have managed to persevere despite the odds because of our adaptability and our survival skills and because we represent an evolving cultural spectrum inclusive of traditional and contemporary practices. At our best, we bring our traditional principles and practices - respect, generosity, collective benefi t and collective ownership - to our daily expression of our identity and culture in a contemporary context. When we are empowered to do this, and where systems facilitate this reclamation, protection and promotion, we are healthy, well and successful and our communities thrive (Dr Ngiare Brown, 2012 ). My name is Raelene Ward; I’m an Aboriginal woman originating from Cunnamulla in south-west Queensland. I am a descendant of the Kunja people on my grandfather’s side and the Kooma people on my grandmother’s side; both groups are in south-west Queensland. I was born and raised in Cunnamulla as well as a number of other smaller rural and remote communities in far south-west Queensland. I have worked in a number of mainstream hospitals and mental health institutions and community settings across the region. I have extensive knowledge of the Aboriginal community-controlled health sector, having worked in the sector for approximately eight years. From 2007 to 2010 I coordinated a number of suicide prevention projects across south-west Queensland, one of those being a three-year project to the value of $1.5 million. I have been a practising clinical nurse for the past 24 years, mainly in these contexts: suicide prevention, Aboriginal health, social and emotional wellbeing and mental health, all in the broader area of social determinants of health.

The basic principle of fairness for a game is easy to articulate: Each player should have an equal opportunity to win. For a social choice, the meaning of fairness is a far richer question. The criteria introduced in Chapter 12 focus on the suitability or unsuitability of certain candidates to be the social choice. In this chapter, we introduce various new fairness criteria that focus on the effect of changes in the preference table. We begin with a notion called Monotonicity concerning the invariance of election outcomes when voters increase their preferences for the winning candidate. We then introduce two criteria, Anonymity and Neutrality, that capture two fundamental democratic principles. These criteria focus on the existence of election biases, either for particular voters or for candidates. We prove May's Theorem (Theorem 18.8) to the effect that the only voting method for two-candidate elections satisfying these three fairness criteria is the Simple Majority Method.

We then turn to the notion of an impossibility theorem, an idea made famous by Arrow's thesis. We introduce Arrow's criterion of Independence and prove a simplified version of his impossibility theorem for social choice methods (Theorem 18.12) proved by Alfred Mackay.

Arrow's original theorem focuses on social welfare methods as opposed to social choice methods. We introduce Arrow's Independence Criterion for social welfare methods as well as the Pareto Criterion in this setting. We then state Arrow's Impossibility Theorem (Theorem 18.19). We conclude by proving an impossibility theorem formulated by Amartya Sen that reveals a basic incompatibility between individual liberties and collective rights.

Before beginning, we recall that we do not allow ties in either social choice or social welfare outcomes. We follow Convention 5.9 and use alphabetical order to break all ties arising in the final outcome of an election. The problem of ties will be a more substantive issue in this chapter. Indeed, we will see that our convention leads directly to certain violations of fairness criteria.

From a philosophical perspective, the idea of randomness is a somewhat mysterious concept. Mathematically, the study of random events leads to a calculable and widely applicable mathematical theory. Although we can never fully predict the outcome of a random process, we can nonetheless compute probabilities with great precision.

The foundations of probability theory closely parallel those of basic logic. An event plays the role of the mathematical statement. The first laws of probability theory are formulas for the probabilities that arise from applying connectives (and, or) and negation (not) to events. A conditional statement in mathematics translates, in turn, to the central notion of a conditional probability. The Law of Conditional Probability (Theorem 11.17) underlies a method for calculating the probabilities arising from sequential actions in games. We used this result extensively in Part I of this book in the form of the Law of the Probability Tree (Theorem 3.6).

We explore these foundational aspects of probability theory in this chapter. We begin with the idea of a sample space. This notion allows for the translation of the probability to arithmetic. We use the example of rolling a pair of dice to motivate the basic laws of connectives and negations. Drawing cards motivates a useful counting technique we call the Combinations Formula (Theorem 11.12). We next formulate the definition of a conditional probability and give the proof of Theorem 11.17. We conclude with a discussion of the proofs of the Law of the Probability Tree (Theorem 3.6) and the Linearity of Expectation (Theorem 3.13).

Sample Spaces

What does it mean to assign a probability to a random event? If the event involves human agency, perhaps the only reasonable approach is to gather empirical evidence. How else can we assess the likelihood that, say, a basketball player makes a free throw other than by consulting the player's past record? For such events, probabilities are not so much calculated as observed. A probability based on experimental evidence is called an empirical probability.

When I was in Grade 4, I was told I did not exist. This was repeated many times over many years. ‘There are no Tasmanian Aborigines left – we shot them all’, the father of a friend told me with glee; ‘they’ve all been wiped out’. As the Tasmanian Aboriginal community mobilised and became more visible and vocal, the mantra changed to ‘you’re not a real Aborigine’ or ‘you’re only part-Aboriginal’. However, if we behaved in a way that offended people, or was against the law, it was because we were those ‘bastard Aboriginals [sic]’. After being the subject of a sickening racist incident led by a teacher at my high school, my mother sought the support of workers at the Tasmanian Aboriginal Centre. Although the teacher was only reprimanded, it gave me considerable satisfaction and although he never apologised, he knew he was being watched. This planted the seed for my activism and instead of pursuing my goal of a career in midwifery, I went to work for the Tasmanian Aboriginal Centre and subsequently for the Australian Public Service in Canberra, in Aboriginal and Torres Strait Islander policy and program areas. I continued to battle the stereotypes others have of Aboriginal people, and to advocate and educate. In the face of denial, I was expected to justify and explain my own existence and identity until eventually, after one racist episode too many, I could not cope any more. I was diagnosed with post-traumatic stress disorder and a co-morbid major depression. I received medication and psychiatric treatment for more than fi ve years before I was deemed well enough to re-enter the workforce, although I was, and still sometimes am, very raw. I became increasingly disillusioned with political rhetoric and began to ponder my future. This led to me revisiting my goal of becoming a midwife and I enrolled in the Bachelor of Midwifery at the University of Canberra.

In their great variety, from contests of global significance such as a championship match or the election of a president down to a coin flip or a show of hands, games and elections share one common feature: each game or election offers the possibility of a final, decisive result obtained according to well-established rules, a public outcome. Mathematics offers a similar possibility. In mathematics, the rules are founded in the laws of logic and represent a formalization of our basic common sense. An outcome in mathematics, to pursue the analogy, is a theorem, a statement that can be proven true. The outcome of a game or election may be surprising or expected. Similarly, a theorem can either defy intuition or confirm a well-evidenced conjecture. Just as a game or an election separates winners from losers, a proven theorem distinguishes true statements from false ones, creating a new fact-of-the-matter about mathematics.

This book is an introduction to the mathematical theory of games and elections. We pursue the analogy between analyzing a game or election and developing a mathematical theory somewhat further before turning to our main topics. First, just as a game or an election creates a new language of specialized terminology, a mathematical theory begins with the formulation of definitions. A mathematical definition is a precise and verifiable description of an object of study. We adopt the following convention for definitions: when we define a new term, we use boldface. Thus the terms “outcome,” “theorem,” and “definition” were defined earlier. (Of course, subsequent definitions will be more technical than these.) We use italics to indicate we are mentioning a term that has not been defined yet but will be defined later. For example, we mentioned the term “definition” before subsequently defining it earlier.

The best way to learn to play a new game is often to just give it a try. In mathematics, the corresponding hands-on method of learning is the study of examples. By an example, we mean a specific instance of a definition or, alternately, a particular consequence of a theorem. A great advantage of our chosen topics is the wealth of examples.

How students are grouped in learning activities can affect the variation in the teaching environment and the dynamics of the learning process. Careful planning is required to ensure students benefit from the groups in which they work. Group work also introduces more unpredictability in teaching, since groups may approach tasks and solve problems in novel, interesting ways. This can be refreshing for instructors (Carnegie Mellon n.d.).

This chapter looks at types of groups, the advantages of groups, the benefits of different ways of grouping and how to divide the class into groups and then bring the class back together again.

The success of any group work in a class is based on the formation of groups based upon a comprehensive knowledge of each student in the class - their academic ability, social skills, and likes and dislikes.

Types of groups

There are many types of groups that can be used in teaching and learning. Each form has benefits for the students and the teacher, and the reasons for using one type of group over another must be considered carefully. Having a clear understanding of the different advantages and disadvantages of groups should ensure that students work in the most productive way with their classmates.

We have seen the role that strategy can play in making a social choice. What role is there for chance? In a democracy, we do not expect chance to be involved in the mechanism of an election. Yet, despite the seeming disconnect between chance and social choice, modern electoral politics has become a showcase for probability theory. Each major election is extensively analyzed as experts compete to correctly predict voter preferences, turnout, and ultimately, the final outcome. The U.S. Electoral College is perhaps the most famous example of a weighted voting system. We introduce weighted voting systems in this chapter and explore the connection between these voting systems and probability theory. We define the key notion of the power index, which provides a measure of the power of voting blocs in a weighted voting system. We also consider a direct connection between choice and chance. We prove a special case of Condorcet's Jury Theorem.

We begin with a motivating example inspired by the 2012 U.S. presidential election between Barack Obama and Mitt Romney.

The class of strategic games presents examples ranging from simple decision games such as Rock-Paper-Scissors, to models for economic and political interactions, to parlor games such as Checkers and Chess. Mathematically, the focus on strategy as opposed to chance represents a paradigm shift. Whereas games of pure chance are described by numbers (probabilities and expected values), the analysis of strategic games involves various notions of “solution.” We introduce three related solution concepts in this chapter.

Given the large variety of examples of games of strategy, it is natural to introduce some groupings. Focusing on the payoffs, we obtain one useful division. A total-conflict game is a two-player game in which the sum of the payoffs of the players is always the same amount at every outcome. In a total-conflict game, one player's gain is another's loss. Examples include win-lose games and zero-sum games, as introduced in Chapter 2. A partial-conflict game is any game that is not a total-conflict game. We have seen an example of a two-player partial conflict game with Battle of the Sexes (Example 2.6).

We begin with simultaneous-move partial-conflict games. We introduce the famous Prisoner's Dilemma as a representative example, giving both the standard presentation and a version arising in economics. We also analyze an example of an auction that serves to motivate our first solution concept, a dominated strategy solution. We next consider an example of a sequential-move partial-conflict game and introduce the notion of a backward induction solution.

Turning to games of total conflict, we apply our solution concepts to the case of zero-sum games and identify the maximin strategy. Finally, we consider the class of win-lose sequential-move games called combinatorial games. We introduce the Binary Labeling Rule to solve certain examples of games in this class. Our partial results here motivate Zermelo's Theorem (Theorem 4.17), which asserts the existence of a solution to all finite combinatorial games.

Partial-Conflict Games

Games in which the players are not in total conflict with one another have proven extremely useful for modeling social and political interactions.

Have you been online lately and bought something – through eBay or another source? One of the things you may look for is the feedback given about the person or establishment selling the item or providing the service. The rating may influence your choices. When you try on clothes in a shop, you immediately ask your friend or the shop assistant ‘How do I look? Does it suit me?’ Whatever the question, you are seeking information about how you look – and maybe an excuse to buy the item. In the workforce, employees are required to have an annual performance review, which involves giving and receiving feedback. Feedback is present in all facets of our life: work, social and home life.

We all need and want feedback to know how we are going and how we should improve. In teaching, for each learning activity you need to decide what you want the students to achieve so you can give feedback to support that learning. Each lesson you plan and write is based upon an objective that directs the teaching and learning and the feedback you give should relate to this.

What is feedback?

Feedback is information about reactions to a product, a person’s performance of a task and so on, which is used as a basis for improvement. It is different from evaluation. Many researchers have examined feedback, and this chapter brings together a range of ideas that will inform you about ways to work with individual students as well as the whole class.

The mathematical analysis of voting methods on the basis of fairness traces back to a debate between the Chevalier de Borda and the Marquis de Condorcet, two prominent eighteenth-century French intellectuals. Borda introduced the voting method we now call the Borda Count for elections with multiple candidates. He argued for the superiority of his method over the Plurality Method using a fairness criterion. Soon after Borda's work was published, Condorcet published a paper revealing a basic violation of fairness by the Borda Count. Condorcet observed that it is possible for a candidate to have the support of the majority of the voters and still lose the Borda Count, a violation of the Majority Winner Criterion. In this chapter, we introduce several fairness criteria: the Majority Winner, Condorcet Winner, Majority Loser, and Condorcet Loser Criteria. We examine the voting methods introduced in Chapter 5 using these criteria.

The study of voting methods by fairness criteria provides an excellent introduction to the dichotomy between proofs and counterexamples in mathematics, as discussed in Chapter 10. The fairness criteria that we present here are each universal statements about voting methods. The question of whether the voting method satisfies or violates the given fairness criterion becomes the prototypical mathematics question: is there a proof or is there a counterexample?

Fairness Criteria

To deserve the name, a fairness criterion should identify a property that corresponds to a “fair” outcome of an election. Devising fairness criteria is part art and part science. Fairness is something of a subjective question, and yet we need a precise, mathematical description of the criteria if we expect to use them to analyze voting methods. The fairness criteria we present will be structured as follows:

Definition 12.1

A fairness criterion is a universal statement of the following form: for all preference tables of a particular type, a voting method produces a social choice with a given property.

We say that a given voting method satisfies the fairness criterion if the universal statement is true for this voting method.

Kenneth Arrow's 1951 doctoral thesis launched a new wave of research on the mathematics of social choice. As with Nash's Equilibrium Theorem in game theory, Arrow's Impossibility Theorem has become the centerpiece of modern social choice theory, inspiring countless extensions, variants, alternate proofs, and responses.

We prove Arrow's Theorem (Theorem 18.19) in this chapter following the elegant argument given by Amayarta Sen. We begin by reviewing the basic ingredients of the statement (social welfare methods and their fairness criterion). Next, we introduce the idea of a dictating set and characterize the Pareto Criterion in terms of this notion. We then state Arrow's Principle, which asserts that dictating power passes from a set of voters to some proper subset. The Impossibility Theorem is then deduced as a direct consequence of Arrow's Principle. Our main work consists of proving Arrow's Principle. We prove this result using two further principles called the Power Contraction and Power Expansion Principles.

We conclude with some developments in social choice theory appearing after Arrow's thesis. We prove a theorem of Sen, called the Sen Coherence Theorem, that gives conditions on a preference table that ensure that the social preference relations are transitive. When the voters have Sen-coherent preferences, Condorcet's ideal of majority rule can be realized. We also introduce two voting methods that have attracted recent interest, Approval Voting and Majority Judgment. These methods do not fit neatly into our framework for voting methods. We briefly discuss the fairness properties of these methods.

Social Welfare and Fairness

Recall that a social welfare method is a mechanism that takes a preference table and produces a ranking of the candidates called the social welfare. We assume that ties are broken (see Convention 5.9) so that the ranking is strict, first to last. Recall that each of our social choice methods extend to a social welfare methods.