### Dagstuhl Seminar Proceedings 07261 Fair Division

Walter Stromquist
2007 unpublished
David Gale asked in [8, 1993] whether, when a pie is to be divided among n claimants, it is always possible to find a division that is both envy free and undom-inated. The pie is cut along n radii and the claimants' preferences are described by separate measures. We answer Gale's question in the negative for n = 3 by exhibiting three measures on a pie such that, when players have these measures, no division of the pie can be both envy free and undominated. The measures assign positive values to
more » ... pieces with positive area. 1 Cutting pies Mathematicians study cake cutting as a metaphor for more general fair division problems. Early papers on cake cutting include [7] and [10], and many more references can be found in the recent books by Brams and Taylor [5], Robertson and Webb [9], and Barbanel [1]. A recent survey is by Brams [2]. A cake is cut by planes parallel to a given plane. It can be represented as an interval [0, m] with possible cuts corresponding to points in the interval and possible pieces to subin-tervals [a, b] ⊆ [0, m]. We assume n players (claimants) whose preferences are represented by nonatomic measures on the interval. Letting v i stand for the i-th player's measure, we write v i (S) or just v i (a, b) for the i-th player's valuation of the piece S = [a, b]. We always assume that v i (S) = 0 if S has length 0. By an allocation we mean a partition of the pie into n pieces together with an assignment of pieces to players. An allocation is called envy free if v i (S i) ≥ v i (S j) for every i and j, where S i represents the i-th player's piece. This means that no player prefers another player's piece to its own. A player i is satisfied (non-envious) if v i (S i) ≥ v i (S j) for all j, so that an allocation is envy free precisely when all players are satisfied. An allocation {S i } is dominated by another allocation {T i } if v i (T i) ≥ v i (S i) for each i with strict inequality for at least one i. This means that the allocation {T i } makes at least one player better off without making any player worse off. We say that an allocation is undominated (or, synonymously, efficient or Pareto optimal) if it is not dominated by any other allocation. In this paper we are looking for allocations that are both envy free and undominated. Note that each of these properties is defined without interpersonal comparisons of values.