Welfare Maximization and Truthfulness in Mechanism Design with Ordinal Preferences Chaitanya Swamy University of Waterloo Joint work with Deeparnab Chakrabarty Microsoft Research, India

Ordinal mechanism design n players, m outcomes (O = outcome set) Each player i has a private (strict) preference relation >i over O: o >i o’ º i prefers o to o’ Contrast with cardinal mechanism design: player has a utility function specifying his value for each outcome (and expected value for a distribution over outcomes) Why ordinal settings? Less informational burden on players: only need to compare outcomes More apt and natural in certain settings, e.g., elections, allocating dorm rooms to students

Goals and challenges {(>1,..., >n)} A mechanism is a social-choice function (SCF) M : set of preference profiles → outcome-set O Typical mechanism-design goal: maximize social welfare Natural to define social welfare in cardinal settings as sum of players’ utilities Economics literature focuses on Pareto optimality We seek a more refined and quantitative measure How to measure social welfare in ordinal settings? What is the target SCF that we should aim for?

Goals and challenges (contd.) other players’ preferences We want to design a truthful mechanism M, i.e., for all i, >i, and all >-i, M(>i, >-i) >i M(>’i, >-i) (so no player has an incentive to lie) BUT, Gibbard-Satterthwaite (GS) theorem: deterministic truthful mechanisms are extremely restrictive (dictatorial) So we move to randomized mechanisms How to define truthfulness for randomized mechanisms? How to extend player preferences over O to preferences over lotteries (i.e., distributions) over O ?

Goals and challenges (contd.) How to define truthfulness for randomized mechanisms? How to extend player preferences over O to preferences over lotteries (i.e., distributions) over O ? Typical approach: extension using stochastic dominance Strong truthfulness: truth stochastically dominates any lie Gibbard’s theorem: strongly truthful mechanisms are also quite limited Weak truthfulness: no lie stochastically dominates truth. No impossibility result, but not so easy to leverage Seek a reasonable truthfulness notion that allows ample flexibility in mechanism design

Our contributions Introduce rank-approximation as a metric for measuring social welfare appealing, robust notion that can be motivated in various ways allows us to evaluate the worst-case performance of ordinal mechanisms Propose lex-truthfulness (LT) as a truthfulness notion for randomized ordinal mechanisms LT is sandwiched between strong and weak truthfulness avoids Gibbard’s impossibility results for strong truthfulness is more amenable to work with than weak truthfulness: we characterize LT algorithmically – exploit this to devise lex-truthful mechanisms for various ordinal settings

Our contributions (contd.) We design lex-truthful mechanisms achieving good rank-approximation for various ordinal settings: (one-sided) matching markets, matroid markets scheduling markets general ordinal settings In many cases, our rank-approximation factors are tight.

Rank approximation Let maxrankr(>1,...,>n) = max o∈O |{i: o is a top-r outcome for i}| = maximum no. of players who can be assigned one of their top-r outcomes An outcome o has rank-approximation a for profile (>1, ..., >n) if: for all positions r = 1,...,m, |{i: o is a top-r outcome for i}| ≥ maxrankr(>1,...,>n)/a Example: A mechanism M has rank-approximation a if: for all (>1,...,>n), outcome M(>1,...>n) has rank-approximation a for (>1,...,>n). a choice 1 maxrank1 = 2 (a) maxrank2 = 3 (c) maxrank3 = 3 Rank-approx. – of a is 2/3 – of c is 1/2 b choice 2 choice 3 c

Rank approximation properties Let M be an a-rank-approximation mechanism. Let (>1,...,>n) be any preference profile, o = M(>1,..., >n) Then, for every cardinal-utility profile U = (U1,...Un) that is: consistent with (>1,...>n): Ui(o1)>Ui(o2) iff o1 >i o2 homogeneous: for all r, all Ui’s assign same value to their rank-r outcome total utility of o under U ≥ (maximum social welfare for U)/a Theorem: M simultaneously yields an a-approximation to the optimal social welfare for all consistent, homogenous utility profiles. (In some sense, strongest connection to consistent cardinal utilities one can hope for: without homogeneity, no mechanism can obtain any non-trivial simultaneous approximation.)

Rank approximation properties Let M be an a-rank-approximation mechanism. Theorem: M simultaneously yields an a-approximation to the optimal social welfare for all consistent, homogenous utility profiles. Scoring rule is a SCF that assigns non-decreasing scores to positions, and returns outcome with highest total score Equivalent Theorem: M simultaneously yields an a-approximation to all scoring rules: for every scoring rule g and every (>1,...,>n), total score of M(>1,...,>n) ≥ (total score of outcome returned by g)/a. (Total score of o = ∑r (score of r).(no. of players for which o is ranked r) )

Rank approximation properties Let M be an a-rank-approximation mechanism. Theorem: M simultaneously yields an a-approximation to the optimal social welfare for all consistent, homogenous utility profiles. Equivalent Theorem: M simultaneously yields an a-approximation to all scoring rules. Are any meaningful rank-approximation bounds possible? (I.e., is simultaneous approximation possible?) Matching markets: devise 2-rank-approximation (with a lex-truthful mechanism), and this is best possible General ordinal settings: get O(log n)-rank-approximation in expectation, and this is best possible YES!

Lex-truthfulness Extend preferences over O to preferences over lotteries (i.e., distributions) over O via lexicographic ordering. Let p, q be two lotteries. p lex-dominates q with respect to preference relation >, if there is some position r such that: po > qo where o is outcome at position r in >, and po’ = qo’ for all outcomes o’ > o (Lex-dominance imposes a total order on lotteries.) A randomized mechanism M is lex-truthful if: for all i, all >i, >’i, and all >-i, M(>i, >-i) lex-dominates M(>’i, >-i) with respect to >i. (M strongly truthful Þ M lex-truthful Þ M weakly truthful.) Independently, Cho and Schulman-Vazirani also considered lex-truthfulness (for different purposes).

e-lex truthful implementation A randomized mechanism M e-lex-truthfully (LT) implements an SCF f if: Pr[M(>1,...,>n) = f(>1,...,>n)] ≥ 1-e for all (>1,...,>n), M is lex-truthful. A family {Me} of mechanisms fully-LT-implements an SCF f if Me e-LT-implements f for all e>0. We isolate an algorithmic condition called pseudomonotonicity that completely characterizes the SCFs that are fully-LT-implementable.

Pseudomonotonicity An SCF f satisfies pseudomonotonicity if for all i, all >i, >’i, and all >-i, the following holds. Let o = f(>i, >-i), o’ = f(>’i, >-i). Either o ≥i o’ OR

Pseudomonotonicity An SCF f satisfies pseudomonotonicity if for all i, all >i, >’i, and all >-i, the following holds. Let o = f(>i, >-i), o’ = f(>’i, >-i). Either o ≥i o’ OR for every position r, if >i and >’i agree on their top r outcomes then o’ is not a top-(r+1) outcome under >i. choice 1 choice m >i a b c d e f g h = o a choice 1 c b h >’i e g d = o’ f choice m

Pseudomonotonicity An SCF f satisfies pseudomonotonicity if for all i, all >i, >’i, and all >-i, the following holds. Let o = f(>i, >-i), o’ = f(>’i, >-i). Either o ≥i o’ OR for every position r, if >i and >’i agree on their top r outcomes then o’ is not a top-(r+1) outcome under >i. a a choice 1 choice 1 b c c b d = o’ h >i >’i e e g f = o g d = o’ h choice m f choice m

Pseudomonotonicity An SCF f satisfies pseudomonotonicity if for all i, all >i, >’i, and all >-i, the following holds. Let o = f(>i, >-i), o’ = f(>’i, >-i). Either o ≥i o’ OR for every position r, if >i and >’i agree on their top r outcomes then o’ is not a top-(r+1) outcome under >i. a a choice 1 choice 1 b c demoted c b d = o’ h >i >’i e e g f = o g d = o’ h choice m f choice m So for i to (strictly) benefit by lying, when moving from >i to >’i, he must have demoted an outcome ranked higher in >i than o’.

Pseudomonotonicity and full-LT-implementation An SCF f satisfies pseudomonotonicity if for all i, all >i, >’i, and all >-i, the following holds. Let o = f(>i, >-i), o’ = f(>’i, >-i). Either o ≥i o’ OR for every position r, if >i and >’i agree on their top r outcomes then o’ is not a top-(r+1) outcome under >i. So for i to (strictly) benefit by lying, when moving from >i to >’i, he must have demoted an outcome ranked higher in >i than o’. Characterization Theorem (a) If f is pseudomonotone, then f is fully-LT-implementable. (b) If f is e-LT-implementable for any e<0.5, then f is pseudomonotone.

Results for ordinal settings Matching markets Players have strict ordering over items; outcomes are matchings Plenty of work. Well-known mechanisms are: random serial dictatorship top-trading-cycle algorithm (Gale) probabilistic serial (Bogomolnaia-Moulin) Bhalgat et al. showed that ordinal-welfare factor is 2. Theorem: There is a pseudomonotone 2-rank-approx. SCF Þ get a (2-e)-rank-approximation LT mechanism for any e>0. Theorem: Rank-approximation lower bounds of: (a) 2 for any SCF (b) W(log log n/log log log n) for any deterministic truthful SCF choice 1 choice 2 rank-approx. W(√n) at least lex-truthful log log n log log log n W

Results for ordinal settings Matroid markets generalization of matching markets; results easily extend Scheduling markets players are jobs, items are machines; makespan constraint for each machine obtain O(log n)-rank-approx., and this is tight General ordinal settings obtain tight O(log n)-rank-approx., non-LT mechanism; plurality and other “top-choice” SCFs are pseudomonotone simultaneous O(log n)-approx. for every scoring SCF contrasts with Q(√m)-approximation of Procacia even for plurality rule via strongly truthful mechanisms

Matching markets 2-rank approximation, pseudomonotone algorithm MaxMatch (Top-r graph: bipartite graph where each player has edges to his top r items) Choose a maximal matching in the top-1 graph For r=2,…,m, augment matching in the top-(r-1) graph to a maximal matching in the top-r graph Proof of rank approximation: maxrankr = size of the maximum matching in the top-r graph MaxMatch maintains a maximal matching in the top-r graph, and every maximal matching has size ≥ 0.5(size of maximum matching) Pseudomonotonicity follows from iterative description.

Conclusions and open questions Rank-approximation and lex-truthfulness provide a versatile framework for studying ordinal mechanism design We obtain various results for resource-allocation settings (assigning players to items subject to packing constraints) Open direction: What are other settings where one can good rank-approximation bounds? Open direction: What about other extensions to preferences over lotteries? Can one get similar positive results? What makes one type of extension more amenable than other?

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