The probabilistic serial mechanism with private endowments

A random assignment is ordinally efficient if it is not stochastically dominated with respect to individual preferences over sure objects. When there are no private endowments, the set of ordinally efficient random assignments is characterized by the eating algorithm (Bogomolnaia and Moulin, 2001). When there are private endowments, the main requirement is individual rationality; however, the eating algorithm fails to deliver this property. Our contribution is the natural generalization of the eating algorithm for this general class of problems. The family of this generalized eating algorithm characterizes the set of individually rational and ordinally efficient random assignments. A special solution in this family, the individually rational probabilistic serial (PSIR), also achieves a new fairness axiom, no justified-envy. However, it is not immune to strategic manipulation. We show that individual rationality, no justified-envy and strategy-proofness are incompatible.

[1]  Tayfun Sönmez,et al.  Ordinal efficiency and dominated sets of assignments , 2003, J. Econ. Theory.

[2]  S. Pápai,et al.  Strategyproof Assignment by Hierarchical Exchange , 2000 .

[3]  Alvin E. Roth,et al.  Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis , 1990 .

[4]  Atila Abdulkadiroglu,et al.  RANDOM SERIAL DICTATORSHIP AND THE CORE FROM RANDOM ENDOWMENTS IN HOUSE ALLOCATION PROBLEMS , 1998 .

[5]  S. Vajda,et al.  Contribution to the Theory of Games , 1951 .

[6]  Hervé Moulin,et al.  Scheduling with Opting Out: Improving upon Random Priority , 2001, Oper. Res..

[7]  Jinpeng Ma.  Strategy-proofness and the strict core in a market with indivisibilities , 1994 .

[8]  A. Roth.  Incentive compatibility in a market with indivisible goods , 1982 .

[9]  F. Kojima,et al.  STRATEGY-PROOFNESS OF THE PROBABILISTIC SERIAL MECHANISM IN LARGE RANDOM ASSIGNMENT PROBLEMS , 2007 .

[10]  Andrew McLennan,et al.  Ordinal Efficiency and the Polyhedral Separating Hyperplane Theorem , 2002, J. Econ. Theory.

[11]  John von Neumann,et al.  1. A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem , 1953 .

[12]  L. Shapley,et al.  On cores and indivisibility , 1974 .

[13]  Özgür Yilmaz,et al.  Random assignment under weak preferences , 2009, Games Econ. Behav..

[14]  Lars-Gunnar Svensson.  Strategy-proof allocation of indivisible goods , 1999 .

[15]  Atila Abdulkadiroglu,et al.  HOUSE ALLOCATION WITH EXISTING TENANTS , 1999 .

[16]  D. Gale.  A theorem on flows in networks , 1957 .

[17]  Alvin E. Roth,et al.  Pairwise Kidney Exchange , 2004, J. Econ. Theory.

[18]  Onur Kesten,et al.  Why do popular mechanisms lack efficiency in random environments? , 2009, J. Econ. Theory.

[19]  P. Hall.  On Representatives of Subsets , 1935 .

[20]  Lin Zhou.  On a conjecture by gale about one-sided matching problems , 1990 .

[21]  Hervé Moulin,et al.  A New Solution to the Random Assignment Problem , 2001, J. Econ. Theory.

[22]  Atila Abdulkadiroglu,et al.  School Choice: A Mechanism Design Approach , 2003 .

[23]  U. Rothblum,et al.  Truncation Strategies in Matching Markets-in Search of Advice for Participants , 1999 .

[24]  Andrew Postlewaite,et al.  Weak Versus Strong Domination in a Market with Indivisible Goods , 1977 .

[25]  H. Moulin,et al.  A simple random assignment problem with a unique solution , 2002 .

[26]  Tayfun Sönmez,et al.  House Allocation with Existing Tenants: An Equivalence , 2001, Games Econ. Behav..

[27]  R. Zeckhauser,et al.  The Efficient Allocation of Individuals to Positions , 1979, Journal of Political Economy.

[28]  Eric V. Denardo,et al.  Flows in Networks , 2011 .