On the fair and efficient allocation of indivisible commodities
Auctions and fair division problems are situations in which commodities are to be allocated fairly and efficiently. While a variety of schemes exist for fairly allocating finely divisible homogeneous commodities, most schemes are not applicable to the problem of allocating indivisible items. This paper considers the problem of fairly allocation sets of indivisible objects. 'Dollars, ' a finely divisible, homogeneous, transferrable commodity, are used to evaluate individuals preferences and to transfer value among individuals. This introduction of dollars has several implications; the main result is that fair allocation problems may be viewed as two smaller problems. First auction the goods among the individuals and then divide the resulting revenue according to the chosen definition of fairness. Several existing fair allocation schemes are reviewed; examples illustrate some difficulties associated with their use. Kuhn's definitions of 'fairness' are presented and two extensions are considered for the case where individuals have different shares in the collection of goods.
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additive in dollars allocation a_ assumed assumption auction scheme bidder bidding honestly chapter coalition defined definition of fairness discount functions divide an estate divide and choose divisible commodities dynamic programming entire estate equal shares equilibrium estate consisting example III.7 fair allocation scheme fair share final allocation finely divisible functions are additive greedy algorithm greedy heuristic honest bids implies individually-reasonable allocations individuals indivisible items iteration lemma maximum minimax strategy mittens model IV.l non-negative overall-reasonable allocation schemes painting Pareto optimal allocations partitioning problem permutation player receives player's bid player's value possible Proof proportional allocations pure strategies real numbers reasonable allocations remaining players requires restriction resulting revenue revenue maximizing assignment sealed bid auction second player security level sequential auction set of players sold subadditive subsets theorem IV.2 third player tion true values utility functions value functions verified vN M+$D zero