Review Vocabulary

Adjusted winner procedure A fair-division procedure introduced by Steven Brams and Alan Taylor in 1993. It works only for two players and begins by having each player independently spread 100 points over the items to be divided so as to reflect the relative worth of each object to that player. The allocation resulting from this procedure is equitable, envy-free, and Pareto-optimal. It requires no cash from either player, but one of the objects may have to be divided or shared by the two players. (p. 540)

Bottom-up strategy A strategy under an alternating procedure in which sophisticated choices are determined by working backward. (p. 549)

Cake-division procedure A fair-division procedure that uses a cake as a metaphor. Such procedures involve finding allocations of a single object that is finely divisible, as opposed to the situation encountered with either the adjusted winner procedure or the Knaster inheritance procedure. In a cake-division procedure, each player has a strategy that will guarantee that player a piece with which he or she is “satisfied,” even in the face of collusion by the others. (p. 552)

Convention of the Law of the Sea An agreement based on divide-and-choose that protects the interests of developing countries in mining operations under the sea. This is also referred to as the Law of the Sea Treaty. (p. 551)

Divide-and-choose A fair-division procedure for dividing an object or several objects between two players. This method produces an allocation that is both proportional and envy-free (the two being equivalent when there are only two players). (p. 551)

Envy-free A fair-division procedure is said to be envy-free if each player has a strategy that can guarantee him or her a share of whatever is being divided that is, in the eyes of that player, at least as large (or at least as desirable) as that received by any other player, no matter what the other players do. (p. 543)

Equitable A fair-division procedure like adjusted winner is said to be equitable if each player believes he or she received the same fractional part of the total value. (p. 543)

Knaster inheritance procedure A fair-division procedure for any number of parties that begins by having each player (independently) assign a dollar value (a “bid”) to the item or items to be divided so as to reflect the absolute worth of each object to that player. The allocation resulting from this procedure leaves each party feeling that he or she received a dollar value at least equal to his or her fair share (and often more so). It never requires the dividing or sharing of an object, but it may require that the players have a large amount of cash on hand. (pp. p. 544 p. 546)

Last-diminisher method A cake-division procedure introduced by Stefan Banach and Brønislaw Knaster. It works for any number of players and produces an allocation that is proportional but not, in general, envy-free. (p. 554)

Lone-divider method A cake-division procedure introduced by Hugo Steinhaus. It works only for three players and produces an allocation that is proportional but not, in general, envy-free. (p. 554)

Pareto-optimal A fair-division procedure is said to be Pareto-optimal if it produces an allocation with the property that no other allocation achieved by any means whatsoever can make any one player better off without making some other player worse off. (p. 543)

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Point ratio The fraction in which the numerator is the number of points one party placed on an object and th denominator is the number of points the other party placed on the object. (p. 541)

Preference lists Rankings of the items to be allocated, from best to worst, by each of the participants. (p. 550)

Proportional A fair-division procedure is said to be proportional if each of players has a strategy that can guarantee that player a share of whatever is being divided that he or she considers to be at least of the whole in size or value. (p. 553)

Selfridge-Conway procedure A cake-division procedure introduced independently by John Selfridge and John Conway. It works only for three players but produces an allocation that is envy-free (as well as proportional). (p. 556)

Taking turns A fair-division procedure in which two or more parties alternate selecting objects. (p. 548)

Vickrey auction A sealed-bid auction in which the high bidder wins but pays only the amount of the second-highest bid. (p. 558)