In this chapter, we digress from our study of games and elections to focus more directly on the logic of mathematical proof. Our goal is to identify some proof methods that we have seen in the first part of the text and to introduce some that we will use in Parts II and III. We begin with the idea of a mathematical statement and the basic problem of constructing proofs and finding counterexamples. We then introduce proof methods for two types of mathematical statements: conditional statements and universal statements. We conclude with an introduction to the method of proof by induction. We apply this method to prove a version of Zermelo's Theorem (Theorem 4.17) for takeaway games.

Mathematical Statements

Like a coin flip, which offers two possible outcomes – a basic branching in a game of chance – the building blocks for a mathematical theory are statements with two possible values: true or false.

Definition 10.1

A mathematical statement is a statement that, by its wording, only admits two possible interpretations. The statement is true or the statement is false.

Mathematical statements arise frequently in ordinary language. For example,

Both are statements that can only be interpreted as true or false. Of course, there may be gray areas in such phrasings. Here are some examples of mathematical statements related to our topics of study:

There is no gray area for these statements. Statements 1 is a false statement. We can quickly check, for example, that the Democratic candidate D is the Borda Count social choice for the Third-Party Candidate Election (Example 6.9), whereas the Republican candidate R has the most first-place votes.

The shift of focus from games to elections represents a rather dramatic change in the objects of our study. We turn from probabilities, payoffs, and winning strategies to questions of the fairness, equity, and correctness of electoral outcomes. We have seen, however, that an election is not so different from a game at the most basic level. Recall in Definition 2.7, we defined an election as a game producing an outcome called the social choice. In this chapter, we study the mechanism for producing this outcome, the voting method.

We begin with a discussion to motivate the idea of voter preferences. The summary table of voter preferences is called a preference table. We introduce this important device along with a number of voting methods that will be studied throughout the text. We conclude with a discussion of social preferences and social welfare.

Preference Rankings

In practice, most elections do not require voters to rank all the candidates. Usually a “vote” consists of simply an indication of first choice. As we have mentioned, the use of voter preferences is assumed to facilitate our mathematical analysis of elections. That said, we observe that there are elections that do explicitly involve preference rankings of the candidates. We give two “controversial” examples, both occurring in the year 1990.

The world of sports provides a number of examples of elections in which voters submit preference rankings for the candidates. The Heisman Trophy in college football and the Most Valuable Player (MVP) awards and Hall of Fame elections for many professional sports are notable examples. Our example is from the National Basketball Association (NBA).

Combinatorial games offer an excellent example of the divide between theory and practice. On the one hand, we have Zermelo's Theorem (Theorem 4.17) asserting that one player can guarantee victory in any such game by following a particular series of moves. On the other hand, this theorem offers no practical help in executing the move-by-move play of a long and complicated game. While the theory of combinatorial games might be said to begin with Zermelo's Theorem, it certainly does not end with this result.

Recall that we have defined a combinatorial game to be a sequential-move finite win-lose game of perfect information. We begin this chapter by extending our proof of Zermelo's Theorem for takeaway games (Theorem 10.9) to the general case of combinatorial games. We then introduce an example of a Nim-like game called Northcott's Game and apply Bouton's Theorem (Theorem 17.11) to determine the winner and winning strategy. Each takeaway game gives rise to a new combinatorial game called the Misère version. We analyze the Misère version of Two-Pile Nim using both the Binary and Ordinal Labeling Rules. Finally, we introduce the idea of strategy stealing for takeaway games and apply this argument to give weak solutions for the games Chomp and Hex.

Zermelo's Theorem

Mathematically, a combinatorial game is modeled by a game tree. We recall a game tree is a rooted tree with end nodes labeled with the winning player (Definition 2.3). For takeaway games, the end-node labels are unnecessary: the player who moves to an end node on his or her turn is the winner. In Chapter 10, we used the Binary Labeling Rule (Definition 4.16) and the Principle of Induction for Trees (10.2) to prove Theorem 10.9. The idea was simply to label the game tree and then read off the winner and the winning strategy from the labels.

Here we extend this argument to prove Zermelo's Theorem for combinatorial games (Theorem 4.17).

In this chapter we formulate mathematical definitions of a game and an election and give examples. The examples of games will indicate some of the categories we will study and will also serve to introduce the two main devices we will use to represent games: the game tree and the payoff matrix. For elections, we introduce the notions of true preferences and insincere voting. We consider the famous example of the 2000 presidential election in the state of Florida as an illustration of the connection between games and elections.

We begin by thinking a bit about how to define the notion of a game. The term has broad meanings. What properties should we use to limit the scope? We can probably agree that a game should involve an act of competition with roles for skill, ingenuity, and maybe luck. Should a game be fun? Should it be fair? Fun is, of course, a matter of taste. Fairness also seems difficult to ensure. A grandmaster can play a game of Chess against a novice, but surely this this is not a “fair” game. However, allowing a weaker player to have a handicap destroys the intrinsic fairness of a game but may be the only way to create a good match.

We define a game in a manner that is quite inclusive, making no commitment either way as to the questions of fun, fairness, or other such qualities. We focus instead on what a game produces. This is the outcome, by which we mean some particular, well-defined final status such as “win,” “loss,” or “tie.” Whatever the method or means of play, when the dust settles and the game is over, we expect to have a well-defined final result.

Further, we often want to know the score of the game or the winnings for each player. We will refer to these as the payoffs. Precisely, the payoffs are numeric quantities assigned to each player for each particular outcome. We can think of the payoff to a player as either points or a dollar amount.

The Web has an enormous number of free sites with games, blogs, and articles relevant to our study. We mention some sites featuring future topics.

Apps

Games of chance including Poker, Blackjack, and Craps can be played at the site wizardofodds.com.

We will introduce the Iterated Prisoner's Dilemma in Chapter 16. You can play the game at the website s3.boskent.com/prisoners-dilemma/fixed.html. This app implements state-of-the-art strategies for the game discovered by William Press and Freeman Dyson. There is a link to the article as well.

Nim can be played at www.kongregate.com/games/ricardorix/nim.

We will introduce a number of Nim-like games in Part II. You can try some of these now. The site www.cut-the-knot.org has, in particular, versions of Northcott's Game (Example 20.3) and Turning Turtles (Exercise 20.5).

We discuss the combinatorial games Chomp and Hex in Chapter 20. These games are available online as well. Chomp can be played www.math.ucla.edu/∼tom/Games/chomp.html. Hex is available at www.mazeworks.com/hex7/.

The site egwald.ca/operationsresearch/cooperative.php has a number of partial- conflict and zero-sum games.

The site www.270towin.com has interactive electoral maps.

Content

The Stanford Encyclopedia of Philosophy at plato.stanford.edu has extensive, accessible articles on game theory and social choice theory with references.

The organization FairVote.org is devoted to promoting fair elections. It advocates for the use of Instant Runoff Voting (the Hare Method).

The site rangevoting.org advocates for Range Voting. The site contains helpful information about other voting methods as well.

This book was written to teach an accessible and authentic mathematics course to a general audience. Games and elections are ideal topics for such a course. The motivating examples are familiar and engaging: from the 2000 U.S. presidential election, to economic pricing wars, to gambling games such as Poker and Blackjack. The elementary theory of these topics involves manageable computations and emphasizes analytic reasoning skills. Most important, the study of games and elections has produced a wealth of beautiful theorems, many within the last century. Games and elections offer a wide selection of elegant results that can be approached without extensive technical background. A course on these topics provides a view of mathematics as a powerful and thriving modern discipline.

There are excellent texts covering aspects of the theory of games and elections, such as those of Straffin [48] and Taylor and Pacelli [49]. The topics are also frequently covered as chapters in liberal arts mathematics texts. This text is pitched at a level between these two types of treatments. The book is designed to be more mathematically ambitious than the general textbook but to allow for a less demanding course than one offered using a more specialized text.

A novelty of this book is the integrative approach taken to the three subjects: probability theory, game theory, and social choice theory. This approach highlights the mix of ideas occurring in seminal results on games and elections such as the Jury Theorem, the Minimax Theorem, and the Gibbard-Satterthwaite Theorem. On a practical level, the integrative approach allows for a more gradual development of material. Rather than taking a steep descent in one area, the chapters follow a spiraling path from chance to strategy to social choice and back again, exploring examples and developing techniques and intuitions in all areas while delving deeper into the theory of games and elections.

Structure of the Book

This book is divided into three parts. Part I introduces the main devices used in the text.

We have seen that gambling games and zero-sum games are described by a real number, the expected value E. For a gambling game, E is the average payoff for a round of the game. For a zero-sum game, E is the expected value at the mixed-strategy equilibrium (Definition 15.4).

In this chapter, we consider a class of strategic games described by an ordinal number, by which we mean a nonnegative integer n = 0,1,2, …. Recall that a takeaway game is a finite two-player sequential-move strategic game with perfect information for which the last player to move wins. In Chapter 10, we proved Zermelo's Theorem for this class of games (Theorem 10.9), establishing that one player has a strategy that guarantees victory. Our proof of Theorem 10.9 depended on the Binary Labeling Rule (Defintion 4.16) and the Principle of Induction for Trees. In this chapter, we introduce a refinement of the Binary Labeling Rule we call the Ordinal Labeling Rule. This new rule assigns an ordinal number called the Sprague–Grundy number to a takeaway game. Like the Binary Labeling Rule, the Ordinal Labeling Rule determines the winner and winning strategy for a takeaway game.

We then revisit the prototypical takeaway game: Nim. With many piles and many chips, the game Nim is too large to solve directly by labeling rules. Charles Bouton produced an elegant solution to the general game in 1901 involving binary numbers. We introduce binary numbers here and use them to prove Bouton's Theorem. We apply Bouton's Theorem, in turn, to address Question 1.4 from Chapter 1.

Sprague–Grundy Numbers

We introduce a new labeling system for game trees of takeaway games. Given a finite set of ordinal numbers S, the number Mex (S) is defined to be the smallest ordinal not in S. The term Mex(S)stands for the minimum excluded number of S.

A gambler may not know, off the top of his or her head, the exact probability of the winning cards in a hand of Poker or a doctor the exact likelihood of a particular diagnosis. They both may have a gut feeling for how likely or unlikely such an event is to happen. Calculating a probability provides, in this sense, a precise quantification for our intuition about the likelihood of an uncertain event.

In this chapter, we pursue a class of examples designed to stretch intuition and test beliefs about the idea of probability. Many of the examples presented here and in the exercises are commonly referred to as paradoxes. A paradox is a statement or problem leading to contradictory conclusions or solutions. True paradoxes in mathematics are rare and significant phenomena. For example, a simple paradox about “truth” known as the Liar Paradox (see Exercise 10.6) ultimately led to Gödel's remarkable Incompleteness Theorem in mathematical logic. In our own study, the Condorcet Paradox is the basis for Arrow's Impossibility Theorem (Theorem 18.19) and the central example in social choice theory. While we retain the term “paradox” in the names when this is common use, the question of which examples to follow are truly paradoxical is an interesting one and left to your interpretation. We begin with what is perhaps the most popular example of a probability puzzle.

In this chapter, we delve deeper into the idea of an election as a game of strategy (Definition 2.9). The implications of this definition are somewhat unsettling. Is a popular election a game to be won? In a democracy, we expect social choice to be a faithful accounting of the true preferences of the people rather than an outcome arrived at by the strategic cleverness of certain of the voters. Unfortunately, the ideal of a strategy-proof, democratic election is an impossible dream. We state the Gibbard-Satterthwaite Theorem (Theorem 6.4), which asserts that with three or more candidates, every nondictatorial voting method is vulnerable to strategic manipulation. We verify the theorem for each of the democratic voting methods introduced in Chapter 5. We also show how strategic considerations in social choice can elect a third-party candidate in a popular election. The latter result is an example of the Paradox of the Chair. Finally, we give an example of the No-Show Paradox, a scenario in which voters in an election better serve their own interests by staying home as opposed to participating in the election.

The Condorcet Paradox

The developments in this chapter will all flow out of arguably the central example in social choice theory. We introduce this important example by means of the following simple puzzle:

In this chapter, we explore the debate over voting methods that marked the birth of social choice theory as an academic field. Let us briefly recall the history and point of contention. On one side, we have the Chevalier Jean Charles de Borda, a mathematician and military scientist. Borda proposed his voting method in a lecture to the French Academy of Sciences urging the body to adopt his voting method for member elections. Borda's method, as we know, converts voter preferences into a single ranking, the social welfare. While ties do occur, they are not so common. In 1780, Borda published the first mathematical proof of fairness. He proved that the Borda Count can never elect a candidate ranked last by a majority of the voters (a Majority Loser). We proved this result ourselves in Theorem 12.15. In fact, Borda proved a stronger theorem. Answering the open question from Chapter 12, we state the following theorem:

Theorem 21.1 Borda's Theorem The Borda Count satisfies the Condorcet Loser Criterion.

On the other side of the debate is the Marquis Nicolas de Condorcet, a prominent intellectual of the Enlightenment period in France. Condorcet also had mathematical evidence supporting his political ideals for elections. Condorcet had a proof that the best chance for achieving the “correct” result in an election is to obey the majority opinion. Condorcet's Jury Theorem (Theorem 7.10) was stated in Chapter 7. For Condorcet, the majority opinion is expressed through head-to-head Simple Majority elections. As we have seen, the Borda Count does not always select the majority winner as the social choice and so violates a basic principle of fairness.

It was Condorcet himself who pointed out the principal obstacle to his own majoritarian ideals. When there are three or more candidates, the majority opinion, as expressed through head-to-head elections, can be fundamentally conflicted. We can have a Condorcet Paradox. Moving ahead almost 200 years to 1951, Kenneth Arrow used the Condorcet Paradox to prove his Impossibility Theorem, beginning the modern era of social choice theory.