Probabilistic assignment: an extension approach

We study the problem of allocating objects using lotteries when agents only submit preferences over objects. A standard approach is to “extend” agents’ preferences over objects to preferences over lotteries, using (first-order) stochastic dominance, or the sd-extension. Following (Cho, Games Econ Behav 95:168–177, 2016a), we complement this approach with two alternative extensions, the dl- and ul- extensions, that give rise to lexicographic preferences (dl stands for “downward lexicographic” and ul for “upward lexicographic”) and apply all three of them in tandem to probabilistic assignment. Each property of rules now has three versions that vary with the extension chosen. We introduce a family of rules that generalizes the probabilistic serial rule. Then we study their behavior, as well as that of the random priority rule, in terms of efficiency, no-envy, and strategy-proofness.

[1]  Gabriel D. Carroll,et al.  An efficiency theorem for incompletely known preferences , 2010, J. Econ. Theory.

[2]  M. Utku Ünver,et al.  Two axiomatic approaches to the probabilistic serial mechanism , 2014 .

[3]  M. Pycia,et al.  Ordinal Efficiency, Fairness, and Incentives in Large Markets , 2016 .

[4]  A. Peterson,et al.  Advances in Dynamic Equations on Time Scales , 2012 .

[5]  Gabriel D. Carroll.  When Are Local Incentive Constraints Sufficient , 2012 .

[6]  Kazuo Murota,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS Recent Developments in Discrete Convex Analysis , 2008 .

[7]  S. Shapiro,et al.  Mathematics without Numbers , 1993 .

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

[9]  M. Manea,et al.  Asymptotic ordinal inefficiency of random serial dictatorship , 2009 .

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

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

[12]  Anna Bogomolnaia,et al.  Random assignment: Redefining the serial rule , 2015, J. Econ. Theory.

[13]  J. Chipman.  The Foundations of Utility , 1960 .

[14]  W. Cho,et al.  Equivalence of Efficiency Notions for Ordinal Assignment Problems , 2016 .

[15]  José A. Silva-Reus,et al.  Allocating via Priorities , 2013 .

[16]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[17]  D. Foley.  Resource allocation and the public sector , 1967 .

[18]  Fuhito Kojima,et al.  Incentives in the probabilistic serial mechanism , 2010, J. Econ. Theory.

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

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

[21]  Özgür Yilmaz,et al.  The probabilistic serial mechanism with private endowments , 2010, Games Econ. Behav..

[22]  José Alcalde.  Random Housing with Existing Tenants , 2013 .

[23]  Felix Brandt,et al.  On the Incompatibility of Efficiency and Strategyproofness in Randomized Social Choice , 2014, AAAI.

[24]  Yeon-Koo Che,et al.  Asymptotic Equivalence of Probabilistic Serial and Random Priority Mechanisms , 2008 .

[25]  Fuhito Kojima,et al.  Random assignment of multiple indivisible objects , 2009, Math. Soc. Sci..

[26]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

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

[28]  Mihai Manea,et al.  A constructive proof of the ordinal efficiency welfare theorem , 2008, J. Econ. Theory.

[29]  Paul R. Milgrom,et al.  Designing Random Allocation Mechanisms: Theory and Applications , 2013 .

[30]  W. Cho.  When is the probabilistic serial assignment uniquely efficient and envy-free? , 2016 .

[31]  Martin Bohner,et al.  Risk aversion and risk vulnerability in the continuous and discrete case , 2012 .

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

[33]  Eun Jeong Heo,et al.  Probabilistic Assignment of Objects: Characterizing the Serial Rule , 2011, J. Econ. Theory.

[34]  A. Gibbard.  Manipulation of Schemes That Mix Voting with Chance , 1977 .

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

[36]  K. Arrow.  Essays in the theory of risk-bearing , 1958 .

[37]  Yoichi Kasajima,et al.  Probabilistic assignment of indivisible goods with single-peaked preferences , 2013, Soc. Choice Welf..

[38]  Shin Sato,et al.  A sufficient condition for the equivalence of strategy-proofness and nonmanipulability by preferences adjacent to the sincere one , 2013, J. Econ. Theory.

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

[40]  J. Sethuraman,et al.  A note on object allocation under lexicographic preferences , 2014 .

[41]  Shin Sato.  Strategy-proofness and the reluctance to make large lies: the case of weak orders , 2013, Soc. Choice Welf..

[42]  Vijay V. Vazirani,et al.  Allocation of Divisible Goods Under Lexicographic Preferences , 2012, FSTTCS.

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

[44]  Wonki Jo Cho,et al.  Incentive properties for ordinal mechanisms , 2016, Games Econ. Behav..

[45]  Jay Sethuraman,et al.  A solution to the random assignment problem on the full preference domain , 2006, J. Econ. Theory.