MA 111 Agenda - Friday, 11.18.201111.18.1 Introduction to Fair-Division Games — Adaption of Notes by Elizabeth Weaver• In this chapter, we will discuss several ways that something can be divided among competing partiesin a way that ensures that each party receives what they consider to be a fair share.• We will think of fair division in terms of games — with players, goals, and rules.• Basic Elements of Fair-Division Games• The set of goods to be divided, denoted by S.• The players who have a right to the goods, denoted by P1, P2, ..., PN.• The value systems that give each player the ability to quantify the value of the goods.• Basic Assumptions• Rationality — Each of the players acts rationally.• Cooperation — The players are willing participants and accept the rules of the game as binding.• Privacy — The players know nothing of the other players’ value systems.• Symmetry — Every player is entitled to an equal share of the goods.• Fair Division: A fair division of S is a division in which every player gets a fair share.• Fair Share: Suppose that s is a share of the goods to be divided, S, and that P is one of theplayers in a fair-division game with N players. We say that s is a fair share to player P if s is worthat least 1/Nth of the total value of S in the opinion of P.• Example: Three players (DJ, Stephanie, and Michelle) are sharing a cake. Suppose that the cakeis divided into three slices (s1, s2, and s3). The following table gives the value of each slice in theeyes of the players.s1s2s3DJ $3.00 $5.00 $4.00Stephanie $4.00 $4.50 $6.50Michelle $4.50 $4.50 $4.50(a) Which of the three slices are fair shares to DJ?(b) Which of the three slices are fair shares to Stephanie?(c) Which of the three slices are fair shares to Michelle?(d) Find a fair division of the cake using s1, s2, and s3.(e) Explain why there is only one such fair division possible.• There are three different types of fair-division games.• A fair division game is continuous if the set S is divisible in infinitely many ways. Typicalexamples involve the division of land, pizza, cake, etc.• A fair division game is discrete if the set S is made of objects that are not divisible like houses,jewelry, art, etc.• A fair division game is mixed if the set S is one in which S is made up of components that areboth continuous and discrete. In these types of games, the continuous and discrete componentscan be divided up separately.• Example: Jerry and George jointly buy the half chocolate-half vanilla cake for $24. Suppose thatJerry feels that 25% of the value of the cake lies in the vanilla half and 75% of the value lies in thechocolate half. Find the dollar value to Jerry for each of the following pieces.(a) The vanilla half of the cake.(b) The chocolate half of the cake.(c) A slice which is 1/3 of the vanilla and 2/3of the chocolate.(d) A slice which is 1/4 of the chocolate and 3/4of the vanilla.(e) How would Jerry slice the cake into two fairshares?11.18.2 The Divider-Chooser Method• This method can be used any time the fair-division game involves two players and a continuous setS.• Divider-Chooser Method1. One player, called the divider, divides S into two shares that he considers to be fair.2. The second player, called the chooser, picks the share that he wants, leaving the other shareto the divider.• Why is the divider guaranteed a fair share in this method? Why is the chooser guaranteed a fairshare?• Why will the divider always split S in a way he determines to be 50/50?• Example: Diane and Carla want to split a cheesecake that is halfchocolate and half vanilla. Diane likes chocolate and vanilla equallywell. To her, both halves of the cake, shown below, are equal invalue. Carla, however is allergic to chocolate, so for her the chocolatehalf has 0% of the value and the vanilla half has 100% of the value.Suppose Diane cuts the cake into the two pieces shown below. Whichpiece will Carla choose? Why is this arrangement fair? Who getsthe better deal?• Is it better to be the divider or the chooser?11.18.3 After this, you should be able to...• understand the concept of a fair-division game.• use the notations and ideas introduced in the basic elements of fair-division games.• state the basic assumptions of our fair-division games.• determine whether a fair-division game is continuous, discrete, or mixed.• determine a share’s worth to a player given a certain set up.• determine whether shares are fair shares to certain players given a certain set-up.• state the steps in the divider-chooser method.• determine the outcome of the divider-chooser method given a certain set-up.• solve problems in parts A and B of pages 103–107. Begin with problems #1, 3, 5, 7, 11, 15, 17,