Crowdsourced Bayesian Auctions MIT Pablo Azar Jing Chen Silvio Micali ♦ TexPoint fonts used in EMF. ♦ Read the TexPoint manual before you delete this box.: A

Agenda 1. Motivation for Crowdsourced Bayesian 2. Our Model 3. What We Can Do In-Principle in Our Model 4. What We Constructively Do in Our Model Tools ♦ Richer Strategy Spaces (again!) ♦ New Solution Concept (mutual knowledge of rationality) 5. Comparison

1. Motivation for Crowdsourced Bayesian

Mechanism Design: Leveraging the Players’ Knowledge and Rationality to obtain an outcome satisfying a desired property Wanted Property: “Good” revenue in auctions

Auctions in General n players a set of goods Valuation (for subsets) ({ })= 310 Allocation: Outcome: allocation (A 0, A 1, …, A n ) + prices (P 1, …, P n ) Utility: : { } Revenue: :

Bayesian : designer [Myerson’81]: optimal revenue for single-good auctions  4, D players  n, D  3, D  2, D  1, D D Centralized Bayesian : Very Strong! Designer knows D further assumes: Independent distribution

 4, D  n, D  3, D  2, D  1, D D Bayesian Nash further assumes: Still Strong! ignorant players know each other better than designer knows them, D Bayesian : ♦ D common knowledge to players

ignorant  4, D  n, D  3, D  2, D  1, D, D I know that he knows that I know that he knows that I know that Bayesian : Bayesian Nash further assumes: ♦ D common knowledge to players

ignorant ♦ (Hidden:) Each i knows ≥ and ≤  4, D  n, D  3, D  2, D  1, D, D Bayesian : !!! E.g., [Cremer and McLean’88] Bayesian Nash further assumes: ♦ D common knowledge to players

2. Our Crowdsourced Bayesian Model

Crowdsourced if: ignorant ♦ Each i individually knows ≥ ♦ No common knowledge required  2, D |S 2  1, D |S 1  3, D |S 3  4, D |S 4  n, D |S n Bayesian : ♦ Designer ignorant  2, D |S 2  1, D |S 1  3, D |S 3  4, D |S 4  n, D |S n ♦ D : iid, independent, correlated…

Our Crowdsourced Bayesian Assumption Each player i knows an arbitrary refinement of D |θ i Players’ knowledge to be leveraged! θ : Si1Si1 Si2Si2 Si3Si3  i, D |S i 2 No requirement on higher-order knowledge Ignorant Designer  Mechanism gets players’ strategies only i knows D |θ i and refines as much as he can

Can We Leverage? Yes, with proper tools!

Tool 1: Richer Strategy Spaces Each i’s strategy space ♦ Classical Revealing Mechanism: ♦ Our Revealing Mechanism: “richer language” for player i

Tool 2: Two-Step DST Recall (informally): DST mechanism Define (informally): Two-Step DST mechanism θ i is the best strategy regardless what the others do θ i is the best regardless what the others do D |S i is the best given first part actions = θ regardless i’s second part action regardless the others’ second part actions DST = Dominant Strategy Truthful,,,,,, θiθi, θnθn, θ1θ1, θiθi D |S i i

Tool 2: Two-Step DST ♦ Mutual Knowledge of Rationality ♦ A special case of CM’s solution concept DST = Dominant Strategy Truthful Define (informally): Two-Step DST mechanism θ i is the best regardless what the others do D |S i is the best given first part actions = θ regardless i’s second part action regardless the others’ second part actions

3. What We Can Do In-Principle in Our Model

Revenue In General Auctions optimal DST revenue under centralized Bayesian Hypothetical benchmark ♦ Not asymptotic ♦ n=1000? 100? Wonderful! ♦ n=2? “Tight” (even for single-good auctions)!

Mechanism [B’50]: ♦ Choose a player i uniformly at random 1. Player i announces 2. Each other player j announces ♦ Run the optimal DST mechanism M with ♦ Reward i using Brier’s Scoring Rule for -i Allegedly: Player i gets nothing and pays nothing bounded in [-2, 0] to a real number expectation maximized if

♦ Black-box usage of the optimal DST mechanism [Myerson’81]  “almost optimal” for single-good auction with independent distribution under crowdsourced Bayesian ♦ An existential result Mechanism Remarks ♦ Leverage one player’s knowledge about the others

4. What We Constructively Do in Our Model

Revenue In Single-Good Auctions ♦ Our Star Benchmark : [Ronen’01]  the monopoly price for given the others’ knowledge p, Y/N?

Mechanism ♦ Aggregate all but ’s knowledge ♦ Loses δ fraction in revenue for 2-step strict DST ♦ Is NOT of perfect information Remarks Only Crucial: The other players must not see otherwise nobody will be truthful

5. Comparison

Mechanism ♦ [Caillaud and Robert’05]: single good auction, ignorant designer, for independent D common knowledge to players, Bayesian equilibrium ♦ Ours: for n=2 under crowdsourced Bayesian “Tight” for 2-player, single-good, independent D Separation between the two models ( For General Auctions, )

♦ [Ronen’01]: under centralized Bayesian Mechanism ( For Single-Good Auctions, ) ♦ Ours: under crowdsourced Bayesian

♦ [Segal’03], [Baliga and Vohra’03]: as When ♦ Ours: for any n≥2 under crowdsourced Bayesian Mechanism Prior-free: Doesn’t need anybody to know D ( For Single-Good Auctions, )

In Sum ignorant designer  4, D |S 4 informed players  n, D |S n  3, D |S 3  2, D |S 2  1, D |S 1 2-Step DST Crowdsourced Bayesian

Thank you!

Complete Information 1 2 …n1 2 …n informed players ignorant designer MR’88 JPS’94 AM’92 GP’96 CHM’10 ACM’10

2-Step Dominant-Strategy Truthful Recall: DST mechanism Define: 2-Step DST mechanism Each i’s strategy space

Mechanism Analysis BSR [B’50]: ♦ Choose a player i uniformly at random 1. Player i announces 2. Each other player j announces ♦ Run the optimal DST mechanism with ♦ Reward i using Brier’s Scoring Rule

Mechanism Analysis: 2-Step DST (b) Brier’s SR [B’50]: ♦ Choose a player i uniformly at random 1. Player i announces 2. Each other player j announces ♦ Run the optimal DST mechanism M with ♦ Reward i using Brier’s Scoring Rule for -i (a) M DST  announcing is dominant for j≠i Allegedly: Player i gets nothing and pays nothing  announcing is 2-step DST for i

Mechanism Analysis: Revenue Convex mechanism M : for any partition P of the valuations space, M is convex  ♦ Choose a player i uniformly at random 1. Player i announces 2. Each other player j announces ♦ Run the optimal DST mechanism M with ♦ Reward i using Brier’s Scoring Rule for -i M is optimal 

Generalization ♦ Recall ♦ Generalization

Incomplete Information Centralized Bayesian Assumption: Designer knows D But: Why should the designer know?  Mechanism gets players’ strategies and D Bayesian:

Crowdsourced Bayesian ignorant  4, … informed players  n, …  3, …  2, …  1, … designer

Strong Crowdsourced Bayesian Assumption: D is common knowledge to the players Crowdsourced Bayesian  Mechanism gets players’ strategies only Knowledge is distributed among individual players Each player i has no information about θ -i beyond D |θ i More information  incentive to deviate Indeed very strong I knows that he knows that I knows that he knows that … Bayesian Nash equilibrium requires even more: We require even less …

Single-parameter games satisfying some property Dhangwatnotai, Roughgarden, and Yan’10: approximate optimal revenue when n goes infinity

Mechanism [B’50]: ♦ Choose a player i uniformly at random 1. Player i announces 2. Each other player j announces ♦ Run the optimal DST mechanism M with ♦ Reward i using Brier’s Scoring Rule for -i Allegedly: Player i gets nothing and pays nothing bounded in [-2, 0] to a real number expectation maximized if

♦ Choose a player i uniformly at random 1. Player i announces 2. Each other player j announces ♦ Run the optimal DST mechanism M with ♦ Reward i using Brier’s Scoring Rule for -i Remarks ♦ Black-box usage of any DST mechanism M [Myerson’81]  “almost optimal” for single-good auction with independent distribution ♦ Works for any n≥2 ♦ An existential result Mechanism