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.

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.

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.

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.

Part I provides prerequisite information for using optimization in practice. Specifically, we need to use software and we need to have some basic knowledge of mathematics. This book uses Matlab as a computational tool. If you are new to Matlab or do not know it yet, Chapter 1 will provide you with the preparatory information to get started. If your mathematics knowledge is a bit rusty, Chapter 2 will provide you with the preparatory knowledge for optimization application.

Specifically, the topics presented, with the chapter numbers, are given below: