Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Welfare Maximization via Posted Prices Michal Feldman Tel Aviv University and MSR Based on Joint work with: Nick Gravin, Brendan Lucier, SODA 2015 Vincent Cohen-Addad, Alon Eden, Amos Fiat, ACM EC 2016 Paul Duetting, Thomas Kesselheim, Brendan Lucier 2016

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Mechanism Design Design of protocols for aligning incentives of strategic agents (e.g., pricing, auctions,…) Internet Algorithmic Mechanism Design New domains of problems Adword auctions Social networks New orientations toward solutions Peer-to- peer Spectrum auctions 70’s late 90’s New opportunities Cloud computing Algorithmic mechanism design NOBELs: 1996 Vickrey 2007 Hurwicz, Maskin, Myerson 2012 Roth, Shapley

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Complex settings Spectrum auctionsAdword auctionsCloud computing SIMPLE COMPLEX First decade of algorithmic mechanism design: truthful mechanisms (sometimes complex)

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Simple mechanisms SIMPLE COMPLEX truthful Recent years focus on simple, non-truthful mechanisms generalized second price auctions [Edelman Ostrovsky Schwarz 05, Varian 07, Lucier Paes Leme 10, 11, Lucier Paes Leme Tardod 12...] simultaneous item auctions [Christodoulou Kovacs Schapira 08, Bhawalkar Roughgarden 12, Feldman Fu Gravin 13, Hassidim Kaplan Mansour Nisan 11, …] Evaluated at equilibrium (price of anarchy) non-truthful

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Simple and truthful mechanisms SIMPLE COMPLEX truthful non-truthful Posted price mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Posted prices

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Posted prices SIMPLE COMPLEX truthful non-truthful How well do posted price mechanisms perform? Posted price mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Recent work on posted prices Various objective functions (welfare/revenue/makespan/…) Various information structures of values (full-info/Bayesian/online/…) Various pricing structures (static/dynamic/…) E.g., Revenue (& welfare) maximization in combinatorial auctions [Chawla Hartline Kleinberg 07, Chawla Malec Sivan 10, Chawla Hartline Malec Sivan 10, Kleinberg Weinberg 12, …] Makespan minimization in machine scheduling [Feldman Fiat Roytman 16]

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Talk outline Scenario 2Scenario 1 Submodularunit-demandValuations BayesianFull informationInformation structure staticdynamicPricing structure OPTWelfare guarantee Problem: welfare maximization in combinatorial markets Posted pricing in two scenarios: General framework for pricing (and connection to price of anarchy)

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Model: combinatorial markets

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Posted price mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Posted price mechanisms utilityvalueprice

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Scenario 1 Matching markets

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Matching markets

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Matching markets B C A 1 2 3

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Walrasian pricing B C A 1 2 3

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Walrasian pricing B A 1 2

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Walrasian pricing B A 1 2

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Walrasian equilibrium existence Every market with gross substitutes valuations admits a Walrasian equilibrium [Kelso Crawford 82, Gul Stacchetti 99] Hierarchy due to [Lehman Lehman Nisan 06]

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 What’s wrong? Walrasian prices cannot coordinate the market alone 1 2 A B Alice comes first

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 What’s wrong? Walrasian prices cannot coordinate the market alone 1 2 A B Alice comes first Minimal Walrasian prices always lead to over demand Remark: over demand is low assuming genericity of valuations and many copies of each item [Hsu Morgenstern Rogers Roth Vohra ‘16]

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 What’s wrong? If Bob comes first, he may take item 1 leading to SW of 2 Remark: under some genericity condition, overdemand is low in minimal Walrasian equilibrium [Hsu Morgenstern Rogers Roth Vohra 16] 1 2 B A Maybe a special Walrasian pricing will work? E.g., minimal one?

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 What’s wrong? Even if there is a unique Walrasian pricing, SW can be 0 B A 1 Alice and Bob may both buy nothing

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Other (non-Walrasian) pricing? Theorem: no pricing gives more than 2/3 of optimal welfare B C A 1 2 3

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 A fix: dynamic pricing B C A

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Dynamic item pricing Repeat until all buyers arrived: 1.Seller sets item prices for remaining items (before knowing the next buyer’s identity) 2.An arbitrary buyer arrives and chooses an arbitrary item in demand Theorem [Cohen-Addad Eden Feldman Fiat ‘16]: for any matching market, we give a poly-time dynamic pricing scheme that achieves the optimal social welfare

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Algorithm Combinatorial algorithm Invariant: every buyer picks an item allocated to her in some optimal allocation in the residual market Remark: result can also be derived by analyzing the assignment LP and its dual (using strict complementary slackness)

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August Fix an arbitrary max weight matching B C A D 1

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August Build a weighted directed graph of items B C A D

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August Delete 0 cycles B C A D

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August B C A D Delete 0 cycles 1 234

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August Decrease all weights by epsilon B C A D

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August Price using shortest paths B C A D

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August An arbitrary agent arrives and picks an item B CD 1 2 A

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Repeat the process on the residual graph B CD 1 More technical details on Wednesday, 9am, Session 4b in EC’16, by Alon Eden

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Observation: No negative cycles (by optimality) B C A D

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August B C A D Buyer D must take item

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August Buyer D must take item 4 d

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Optimal welfare cannot always be obtained via dynamic pricing (even in a generic game that admits Walrasian equilibrium) Main open problem: can one always obtain optimal welfare via dynamic pricing for gross substitutes valuations? Additional results [Cohen-Addad Eden Feldman Fiat ‘16]

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Partial answers: Optimal welfare for gross-substitutes with a unique optimum – Construction relies on Murota graph [Murota ‘96] and local improvement property [Gul Stachetti ‘99] Half of optimal welfare via static bundle pricing for general valuations – inspired by [Feldman Gravin Lucier ‘13] Additional results [Cohen-Addad Eden Feldman Fiat ‘16]

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Results extend to models with uncertainty Bounded rationality: seller and buyers know a perturbed instance of real valuations Bayesian: known distribution over user types, large population We give dynamic pricing schemes that obtain approximately optimal SW in both cases Beyond full information

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Scenario 2 Submodular valuations

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Submodular (&XOS) valuations sub- modular XOS unit- demand gross- substitutes

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Submodular (&XOS) valuations sub- modular XOS unit- demand gross- substitutes

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 XOS valuations

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Submodular valuations S T

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Pricing for submodular valuations Theorem [Feldman Gravin Lucier ‘15] For any market with XOS valuations, half of OPT welfare can be obtained via static item pricing, even in a Bayesian setting

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 A note on approximation Provides insight regarding tradeoff between simplicity and optimality E.g., anonymous prices vs price discrimination selling items individually vs bundling Obtained ratio should not be expected in practice (worst case) Exact optimization may be infeasible

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Construction (unit-demand, full info) B A 1 2 un- sold sold

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Proof (unit-demand, full info)

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Extension to submodular valuations

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Extension to Bayesian model

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Bayesian model Buyers Items B C A 1 2 3

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Algorithmic mechanism design 1.Economic efficiency: max social welfare 2.Computational efficiency: poly runtime 3.Dominant strategy incentive compatibility (DSIC) approx algorithms

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Algorithmic mechanism design 1.Economic efficiency: max social welfare 2.Computational efficiency: poly runtime 3.Dominant strategy incentive compatibility (DSIC)

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Algorithmic mechanism design 1.Economic efficiency: max social welfare 2.Computational efficiency: poly runtime 3.Dominant strategy incentive compatibility (DSIC) Question: does incentive compatibility impose additional welfare loss beyond loss already incurred due to computational constraints?

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Known results (XOS valuations) Sub-polynomial approximation requires exponentially many value queries [Dobzinski’11, Dughmi-Vondrak’11] Algorithmic DSIC mechanism NP-hard to solve optimally

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Known results (submodular & XOS valuations) Algorithmic DSIC mechanism NP-hard to solve optimally Major open problem: Is there a poly-time incentive compatible mechanism that achieves a constant approximation for submodular/XOS valuations? Our pricing scheme answers this question in the affirmative for Bayesian settings

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 For every approximation algorithm, the mechanism: 1.(approximately) preserves social welfare of algorithm 2.satisfies incentive compatibility Approximation ALG Mechanism Allocation Payments Input Black box paradigm Decouple algorithmic and economic perspectives

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Results

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Existential result

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Computational result

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Extension to complements Spectrum auctions

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Final remarks

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 General framework for posted prices In a recent working paper, we introduce a general framework for (non-anonymous) posted prices that give nearly optimal welfare in Bayesian settings [Duetting Feldman Kesselheim Lucier 16] Extension theorem: full information to Bayesian Composability of pricing Connection between price of anarchy and pricing – POA is often computed using smoothness framework [Roughgarden 09, Syrgkanis Tardos 13] – Many smooth mechanisms can be recast as posted price mechanisms with similar welfare guarantees – Prior dependent, computational burden offset to seller

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Non-truthful mechanisms SIMPLE COMPLEX truthful Price of anarchy (POA) POA = POA is often computed using smoothness framework [Roughgarden 09, Syrgkanis Tardos 13] – Apply equilibrium hypothesis for each player, with a clever hypothetical deviation, and sum over players – E.g. Simultaneous 1 st price auctions with XOS valuations [Hassidim Kaplan Mansour Nisan 11, Syrgkanis Tardos 13] non-truthful Observation: close connection between the hypothetical deviation and the posted prices

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 (informal) Results [Duetting Feldman Kesselheim Lucier 16] Many smooth mechanisms can be recast as posted price mechanisms with similar welfare guarantees Inherit nice properties of smooth mechanisms; i.e., composability Applications: – Combinatorial auctions – Sponsored search auctions (generalized second price) – General binary single parameter settings Can be interpreted as multi-dimension prophet inequality

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Summary Posted price mechanisms are simple, straightforward and strategyproof* Two scenarios where posted price mechanisms perform well – Matching markets: optimal welfare via dynamic prices – Combinatorial markets with submodular valuations: ½ of optimal welfare via static prices Connection between price of anarchy and welfare guarantees of posted price mechanisms Posted price mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Thank you! Posted prices Prophet inequality Smoothness Price of anarchy Truthful mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Proof idea

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 A note on simplicity [Dobzinski’07] Simple vs. optimal mechanisms Obviously Strategy-proof [Li’15] Posted price mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Related work Black box reduction from ALG to BIC mechanisms [Hartline Lucier 10, Hartline Kleinberg Malekian 11, Bei Huang 11] – BIC vs. DSIC, poly in support size of distributions Bayesian price of anarchy (simultaneous item auctions) [Christodoulou Kovacs Schapira 08, Bhawalkar Roughgarden 12, F. Fu Gravin 13, Hassidim Kaplan Mansour Nisan 11, …] – Posted prices are DSIC but prior dependent Revenue maximization – Sequential posted prices [Chawla Hartline Kleinberg 07, Chawla Malek Sivan 10, Chawla, Hartline, Malek, Sivan 10, Kleinberg, Weinberg 12] – Menu complexity [Hart Nisan 12,13, Babaioff Immorlica Lucier Weinberg 14, Yao 15, Rubinstein Weinstein 15]

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Can do ½, can’t do better than 2/3 Static item prices, full information – Identical items, decreasing valuations – Unit demand bidders 0/1 Static bundle prices, full information, general valuations CWE: simultaneous, but controlled tie breaking

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Results

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 Tight exampe

Michal Feldman – Tel Aviv University and Microsoft Research Becker Friedman Institute, University of Chicago, August 2016 sub- modular XOS unit- demand gross- substitutes