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Matching Markets with Ordinal Preferences

Matching Markets with Ordinal Preferences. TIFR , May 2013. Matching Markets. N agents, N items, N complete preferences. Outcome: Agent-Item Matching. Outline of Talk. Mechanisms Random Serial Dictatorship (RSD) Rank Maximal Matching (RMM) Welfare Ordinal Welfare Factor

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Matching Markets with Ordinal Preferences

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  1. Matching Markets with Ordinal Preferences TIFR, May 2013

  2. Matching Markets • N agents, N items, N complete preferences. • Outcome: Agent-Item Matching

  3. Outline of Talk • Mechanisms • Random Serial Dictatorship (RSD) • Rank Maximal Matching (RMM) • Welfare • Ordinal Welfare Factor • Rank Approximation • Truthfulness • Dealing with randomness.

  4. Random Serial Dictatorship • Agents arrive in a random permutation and pick their best unallocateditem. (2,1,3) … (3,2,1) Choice 1 Choice 2 Choice 3

  5. Rank Maximal Matching • Maximize #(top choice), then Maximize #(top 2),... • Polytime computable. Irving, 2003Irving, Kavitha, Melhorn, Michail, Paluch, 2004.

  6. Social Welfare • Pareto Optimality.No other outcome makes everyone happier. • RMM leads to a Pareto Optimal outcome. • RSD leads to ex-post Pareto Optimal outcome.

  7. Social Welfare • Cardinal WelfareEach pair associated with cardinal number. Social welfare = Sum of utilities. • What to do when no numbers are known?

  8. Ordinal Welfare Factor (OWF) • Outcome is -efficient, if for any , • Problem: Everyone has same ordering. #agents with (1, 1) (2, 2) (3, 3) … (N,N) (1, N) (2, 1) (3, 2) … (N,N-1) M = M’ =

  9. Ordinal Welfare Factor (OWF) • Randomization.A distribution is -efficient, if for any otherdistribution , • Mechanism has OWF if it returns an -efficient distribution. agents with

  10. Symmetric “Bad” Example • Every agent has same preference order. • is uniform over allmatchings. • Fix matching , • is -efficient.

  11. Performance of Mechanisms • Theorem. RSD has OWF 1/2 • RMM is deterministic.Many agents can be made better off at the expense of one agent. Bhalgat, C, Khanna 2011.

  12. Strengths and Weaknesses • Comparative Measure. • Notion of “approximation”. Quantify mechanisms. • Not good for deterministic mechanisms. • No notion of “how much better off”.

  13. Rank Approximation • Let maximize #(agents getting top i) • is -rank approximate if #(agents getting top in ) . • Mechanism has -rank approximation if it returns an -rank approximate matching.

  14. Connection to Cardinal Welfare • Homogenous agents: Each agent has same cardinal profile • is -rank approximate implies -approximation for homogenous agents.

  15. Performance of Mechanisms • Theorem. RMM has ½-rank approximation.- Maximal/Maximum - Optimal. • RSD is not -approximate for any constant . Choice 1

  16. Strengths and Weaknesses • Deterministic mechanisms can have good rank approximation. • Cardinal welfare for homogenous agents. • Could improve many while hurting only a few. • No good rank appx known in non-matching setting.

  17. Truthfulness • If an agent lies, he gets a worseitem.If an agent lies, he doesn’t get a better item. • Issues with randomized mechanisms.What are worse and better distributions? • Hierarchy of truthfulness.

  18. Randomization vs Truthfulness Universally Truthful. Distribution over deterministicmechanisms Strongly Truthful. (Gibbard, 77)Lying gives a stochastically dominated allocation. Lex Truthful. (?) Lying gives a lexicographically dominated allocation. Weakly Truthful. (Bogomolnaia-Moulin, 01) Lying can’t give stochastically dominating allocation.

  19. Lex Truthful Implementation • A deterministic algorithm A can be -lex-truthful implemented if there is a randomized mechanism M such that • M is Lex Truthful. • With probability > (1-), outcome of M is same as that of A Theorem. Any pseudomonotone algorithm Ais -lex-implementable, for any C, Swamy 2013

  20. Pseudomonotonicity A A M(i) M’(i) bM’(i) M(i) M’(i) is below M(i) in b or there’s b above M’(i) in which has been demoted.

  21. Performance of Mechanisms • RSD is Universally Truthful.Under certain conditions, it is the only strongly truthful mechanism. (Larsson, 94) • RMM satisfies pseudomonotonicity.Therefore, it can be -LT implemeneted.

  22. Summary • Welfare definitions unclear in ordinal settings.Saw two notions.Generalizes to Social choice settings. • Truthfulness of randomized mechanisms also tricky. Hierarchy of truthfulness. • Can results be extended to general settings?

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