Matching Markets with Ordinal Preferences 25 minutes TIFR, May 2013

Matching Markets 𝜋 1 𝜋 2 𝜋 3 Problem Statement. Fish out references. N agents, N items, N complete preferences. Outcome: Agent-Item Matching

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

Random Serial Dictatorship Agents arrive in a random permutation and pick their best unallocated item. 𝜋= (2,1,3) Choice 1 Choice 2 Choice 3 Definition. Strongly truthful. Uniqueness. … 𝜋= (3,2,1)

Rank Maximal Matching Maximize #(top choice), then Maximize #(top 2),... Polytime computable. RSD and PS have bad rank approximation Good Rank approximation for RMM Irving, 2003 Irving, Kavitha, Melhorn, Michail, Paluch, 2004.

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. How would we define social welfare? (Break into two slides?) Pareto Optimality Cardinality. Why is it not desirable? 1. Bad numbers. 2. Arrow Two notions: Condorcet? Ordinal Welfare Factor? Rank Approximation – connection to cardinal welfare

Social Welfare Cardinal Welfare Each pair associated with cardinal number. Social welfare = Sum of utilities. What to do when no numbers are known? How would we define social welfare? (Break into two slides?) Pareto Optimality Cardinality. Why is it not desirable? 1. Bad numbers. 2. Arrow Two notions: Condorcet? Ordinal Welfare Factor? Rank Approximation – connection to cardinal welfare

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

Ordinal Welfare Factor (OWF) Randomization. A distribution 𝑴 is 𝛼-efficient, if for any other distribution 𝑴′, Mechanism has OWF 𝛼 if it returns an 𝛼-efficient distribution. 𝐄𝐱 𝐩 𝑀←𝑴, 𝑀 ′ ← 𝑴 ′ [#agents with (𝑀≥ 𝑀 ′ )]≥𝛼𝑁

Symmetric “Bad” Example Every agent has same preference order. 𝑴 is uniform over all matchings. Fix matching 𝑀 ′ ={ 1,1 , 2,2 …, 𝑁,𝑁 }, 𝑴 is 1 2 + 1 2𝑛 -efficient. ∀𝑖, 𝐏 𝐫 𝑀←𝑴 𝑀 ≥ 𝑖 𝑀 ′ ≥ 𝑖 𝑁

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.

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

Rank Approximation Let 𝑀 𝑖 ∗ maximize #(agents getting top i) 𝑛 𝑖 ≔ 𝑀 𝑖 ∗ 𝑀 is 𝛼-rank approximate if #(agents getting top 𝑖 in 𝑀) ≥𝛼 𝑛 𝑖 . Mechanism has 𝛼-rank approximation if it returns an 𝛼-rank approximate matching. At the end of this slide state that there’s no relation between the two.

Connection to Cardinal Welfare Homogenous agents: Each agent has same cardinal profile 𝑢 1 > 𝑢 2 >⋯> 𝑢 𝑁 𝑀 is 𝛼-rank approximate implies 𝛼-approximation for homogenous agents.

Performance of Mechanisms Theorem. RMM has ½-rank approximation. - Maximal/Maximum ≥ 1 2 - Optimal. RSD is not 𝛼-approximate for any constant 𝛼. ≈ 𝑁 Choice 1

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.

Truthfulness If an agent lies, he gets a worse item. If an agent lies, he doesn’t get a better item. Issues with randomized mechanisms. What are worse and better distributions? Hierarchy of truthfulness. Definition of truthfulness (small slide)

Randomization vs Truthfulness Universally Truthful. Distribution over deterministic mechanisms Strongly Truthful. (Gibbard, 77) Lying gives a stochastically dominated allocation. Lex Truthful. (?) Lying gives a lexicographically dominated allocation. Various notions under randomization. Stochastic dominance Weak SD Lex. Weakly Truthful. (Bogomolnaia-Moulin, 01) Lying can’t give stochastically dominating allocation.

Lex Truthful Implementation A deterministic algorithm A can be 𝜖-lex-truthful implemented if there is a randomized mechanism M such that M is Lex Truthful. With probability > (1-𝜖), outcome of M is same as that of A Theorem. Any pseudomonotone algorithm A is 𝜖-lex-implementable, for any 𝜖>0. C, Swamy 2013

Pseudomonotonicity A 𝜋 𝑖 , 𝜋 −𝑖 =𝑀 A 𝜋′ 𝑖 , 𝜋 −𝑖 =𝑀′ 𝜋 𝑖 𝜋 𝑖 𝜋′ 𝑖 M(i) M’(i) b M’(i) M(i) M’(i) is below M(i) in 𝜋 𝑖 𝜋′ 𝑖 b ADD a proof slide? RMM satisfies pseudomonotonicity; proof of the theorem. or there’s b above M’(i) in 𝜋 𝑖 which has been demoted.

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 𝜖-LT implemeneted.

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?

Recall RSD and OWF Agents arrive in a random order 𝜎. At time 𝑡, agent 𝜎[𝑡] chooses his best item among the unallocated items. Induces a distribution on matchings. Fix any matching 𝑀’. 𝑀′(𝑎): item allocated to agent a in M’. Theorem: Expected number of agents a getting item M’(a) or better is ≥𝑁/2. Bhalgat, C, Khanna 2011.

OWF of RSD 𝑀′(𝑎): item allocated to agent a in M’. Call 𝑎 dead at time 𝑡 if 𝑀′(𝑎) allocated by 𝑡 and 𝑎 hasn’t arrived by time t. 𝐷 𝑡 : expected dead agents at time t. 𝐴𝐿 𝐺 𝑡 : expected number of “happy” agents at 𝑡. 𝐴𝐿 𝐺 𝑡+1 −𝐴𝐿 𝐺 𝑡 ≥1 − 𝐷 𝑡 𝑁−𝑡 Claim: 𝐷 𝑡 ≤ 𝑡+2 𝑁−𝑡 𝑁+1 ≈ 𝑡 𝑁−𝑡 𝑁

Proof of Claim Claim: 𝐷 𝑡 ≤ 𝑡+2 𝑁−𝑡 𝑁+1 ≈ 𝑡 𝑁−𝑡 𝑁 𝐸 𝑎 :𝑎 hasn’t arrived by time t. 𝐷 𝑡 = 𝒂 𝐏𝐫 𝐸 𝑎 ∧ 𝐹 𝑎 𝐹 𝑎 :𝑀′(𝑎) allocated by time t. 𝐏𝐫 𝐸 𝑎 = 𝑁−𝑡 𝑁 a 𝐏𝐫 𝐹 𝑎 = 𝑖 𝐏𝐫[ Item 𝑖 allocated by time 𝑡]=𝑡 Independent, then done. ≈ Negatively correlated.