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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
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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
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1. Motivation for Crowdsourced Bayesian
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Mechanism Design: Leveraging the Players’ Knowledge and Rationality to obtain an outcome satisfying a desired property Wanted Property: “Good” revenue in auctions
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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: :
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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
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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
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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
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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
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2. Our Crowdsourced Bayesian Model
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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…
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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
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Can We Leverage? Yes, with proper tools!
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Tool 1: Richer Strategy Spaces Each i’s strategy space ♦ Classical Revealing Mechanism: ♦ Our Revealing Mechanism: “richer language” for player i
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Tool 2: Two-Step DST Recall (informally): DST mechanism Define (informally): Two-Step DST mechanism 1. 2. 3. θ i is the best strategy regardless what the others do 1. 2. 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
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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 1. 2. 3. 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
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3. What We Can Do In-Principle in Our Model
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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)!
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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
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♦ 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
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4. What We Constructively Do in Our Model
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Revenue In Single-Good Auctions ♦ Our Star Benchmark : [Ronen’01] the monopoly price for given the others’ knowledge p, Y/N?
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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
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5. Comparison
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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, )
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♦ [Ronen’01]: under centralized Bayesian Mechanism ( For Single-Good Auctions, ) ♦ Ours: under crowdsourced Bayesian
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♦ [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, )
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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
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Thank you!
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Complete Information 1 2 …n1 2 …n informed players ignorant designer MR’88 JPS’94 AM’92 GP’96 CHM’10 ACM’10
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2-Step Dominant-Strategy Truthful Recall: DST mechanism Define: 2-Step DST mechanism Each i’s strategy space 1. 2. 3.
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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
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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
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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
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Generalization ♦ Recall ♦ Generalization
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Incomplete Information Centralized Bayesian Assumption: Designer knows D But: Why should the designer know? Mechanism gets players’ strategies and D Bayesian:
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Crowdsourced Bayesian ignorant 4, … informed players n, … 3, … 2, … 1, … designer
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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 …
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Single-parameter games satisfying some property Dhangwatnotai, Roughgarden, and Yan’10: approximate optimal revenue when n goes infinity
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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
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♦ 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
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