2. Excursions in Modern Mathematics, 7e: 3.1 - 2Copyright © 2010 Pearson Education, Inc. 3 The Mathematics of Sharing 3.1Fair-Division Games 3.2Two Players: The Divider-Chooser Method 3.3 The Lone-Divider Method 3.4The Lone-Chooser Method 3.5The Last-Diminsher Method 3.6The Method of Sealed Bids 3.7The Method of Markers

3. Excursions in Modern Mathematics, 7e: 3.1 - 3Copyright © 2010 Pearson Education, Inc. The underlying elements of every fair- division game are as follows: Basic Elements of a Fair-Division Game The goods (or “booty”). This is the informal name we will give to the item or items being divided. Typically, these items are tangible physical objects with a positive value, such as candy, cake, pizza, jewelry, art, land, and so on.

4. Excursions in Modern Mathematics, 7e: 3.1 - 4Copyright © 2010 Pearson Education, Inc. In some situations the items being divided may be intangible things such as rights– water rights, drilling rights, broadcast licenses, and so on– or things with a negative value such as chores, liabilities, obligations, and so on. Regardless of the nature of the items being divided, we will use the symbol S throughout this chapter to denote the booty. Basic Elements of a Fair-Division Game

5. Excursions in Modern Mathematics, 7e: 3.1 - 5Copyright © 2010 Pearson Education, Inc. The players. In every fair-division game there is a set of parties with the right (or in some cases the duty) to share S. They are the players in the game. Most of the time the players in a fair- division game are individuals, but it is worth noting that some of the most significant applications of fair division occur when the players are institutions (ethnic groups, political parties, states, and even nations). Basic Elements of a Fair-Division Game

6. Excursions in Modern Mathematics, 7e: 3.1 - 6Copyright © 2010 Pearson Education, Inc. The value systems. The fundamental assumption we will make is that each player has an internalized value system that gives the player the ability to quantify the value of the booty or any of its parts. Specifically, this means that each player can look at the set S or any subset of S and assign to it a value – either in absolute terms (“to me, that’s worth $147.50”) or in relative terms (“to me, that piece is worth 30% of the total value of S”). Basic Elements of a Fair-Division Game

7. Excursions in Modern Mathematics, 7e: 3.1 - 7Copyright © 2010 Pearson Education, Inc. Like most games, fair-division games are predicated on certain assumptions about the players. For the purposes of our discussion, we will make the following four assumptions: Basic Assumptions

8. Excursions in Modern Mathematics, 7e: 3.1 - 8Copyright © 2010 Pearson Education, Inc. Rationality Each of the players is a thinking, rational entity seeking to maximize his or her share of the booty S. We will further assume that in the pursuit of this goal, a player’s moves are based on reason alone (we are taking emotion, psychology, mind games, and all other non-rational elements out of the picture.) Basic Assumptions

9. Excursions in Modern Mathematics, 7e: 3.1 - 9Copyright © 2010 Pearson Education, Inc. Cooperation The players are willing participants and accept the rules of the game as binding. The rules are such that after a finite number of moves by the players, the game terminates with a division of S. (There are no outsiders such as judges or referees involved in these games – just the players and the rules.) Basic Assumptions

10. Excursions in Modern Mathematics, 7e: 3.1 - 10Copyright © 2010 Pearson Education, Inc. Privacy Players have no useful information on the other players’ value systems and thus of what kinds of moves they are going to make in the game. (This assumption does not always hold in real life, especially if the players are siblings or friends.) Basic Assumptions

11. Excursions in Modern Mathematics, 7e: 3.1 - 11Copyright © 2010 Pearson Education, Inc. Symmetry Players have equal rights in sharing the set S. A consequence of this assumption is that, at a minimum, each player is entitled to a proportional share of S – then there are two players, each is entitled to at least one- half of S, with three players each is entitled to at least one-third of S, and so on. Basic Assumptions

12. Excursions in Modern Mathematics, 7e: 3.1 - 12Copyright © 2010 Pearson Education, Inc. Given the booty S and players P 1, P 2, P 3,…, P N, each with his or her own value system, the ultimate goal of the game is to end up with a fair division of S, that is, to divide S into N shares and assign shares to players in such a way that each player gets a fair share. This leads us to what is perhaps the most important definition of this section. Fair Share

13. Excursions in Modern Mathematics, 7e: 3.1 - 13Copyright © 2010 Pearson Education, Inc. Fair Share Suppose that s denotes a share of the booty S and that P is one of the players in a fair-division game with N players. We will say that s is a fair share to player P if s is worth at least 1/Nth of the total value of S in the opinion of P. (Such a share is often called a proportional fair share, but for simplicity we will refer to it just as a fair share.) FAIR SHARE

14. Excursions in Modern Mathematics, 7e: 3.1 - 14Copyright © 2010 Pearson Education, Inc. We can think of a fair-division method as the set of rules that define how the game is to be played. Thus, in a fair-division game we must consider not only the booty S and the players P 1, P 2, P 3,…, P N (each with his or her own opinions about how S should be divided), but also a specific method by which we plan to accomplish the fair division. Fair Division Methods

15. Excursions in Modern Mathematics, 7e: 3.1 - 15Copyright © 2010 Pearson Education, Inc. There are many different fair-division methods known, but in this chapter we will only discuss a few of the classic ones. Depending on the nature of the set S, a fair- division game can be classified as one of three types: continuous, discrete, or mixed, and the fair-division methods used depend on which of these types we are facing. Fair Division Methods

16. Excursions in Modern Mathematics, 7e: 3.1 - 16Copyright © 2010 Pearson Education, Inc. Continuous In a continuous fair-division game the set S is divisible in infinitely many ways, and shares can be increased or decreased by arbitrarily small amounts. Typical examples of continuous fair-division games involve the division of land, a cake, a pizza, and so forth. Types of Fair Division Games

17. Excursions in Modern Mathematics, 7e: 3.1 - 17Copyright © 2010 Pearson Education, Inc. Discrete A fair-division game is discrete when the set S is made up of objects that are indivisible like paintings, houses, cars, boats, jewelry, and so on. (One might argue that with a sharp enough knife a piece of candy could be chopped up into smaller and smaller pieces. As a semantic convenience let’s agree that candy is indivisible, and therefore dividing candy is a discrete fair- division game.) Types of Fair Division Games

18. Excursions in Modern Mathematics, 7e: 3.1 - 18Copyright © 2010 Pearson Education, Inc. Mixed A mixed fair-division game is one in which some of the components are continuous and some are discrete. Dividing an estate consisting of jewelry, a house, and a parcel of land is a mixed fair-division game. Types of Fair Division Games

19. Excursions in Modern Mathematics, 7e: 3.1 - 19Copyright © 2010 Pearson Education, Inc. Fair-division methods are classified according to the nature of the problem involved. Thus, there are discrete fair- division methods (used when the set S is made up of indivisible, discrete objects), and there are continuous fair-division methods (used when the set S is an infinitely divisible, continuous set). Mixed fair-division games can usually be solved by dividing the continuous and discrete parts separately, so we will not discuss them in this chapter. Types of Fair Division Games