Crowdsourced Bayesian Auctions Pablo Azar Jing Chen MIT

Crowdsourced Bayesian Auctions Pablo Azar Jing Chen MIT Silvio Micali

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 5. Comparison Tools ♦ Richer Strategy Spaces (again!) ♦ New Solution Concept (mutual knowledge of rationality)

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

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

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

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

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

2. Our Crowdsourced Bayesian Model

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

Our Crowdsourced Bayesian Assumption Each player i knows an arbitrary refinement of D|θi : i knows D|θi and refines as much as he can i, D|Si 2 θ i S Ignorant Designer 2 Mechanism gets players’ S i 1 strategies only S i 3

Can We Leverage? Yes, with proper tools!

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

Tool 2: Two-Step DST = Dominant Strategy Truthful Recall mechanism Define (informally): DST Two-Step DST mechanism 1. 2. θθi i isis the what othersthe doothers do the best regardless strategy regardless i’s second part action 3. D|Si is the best given first part actions = θ regardless the others’ second part actions i , , , θi , , , θ 1 , θ , D|Si i θn ,

Tool 2: Two-Step DST = Dominant Strategy Truthful Define (informally): Two-Step DST mechanism 1. 2. θi is the best regardless what the others do regardless i’s second part action 3. D|Si is the best given first part actions = θ regardless the others’ second part actions ♦ Mutual Knowledge of Rationality

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

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

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

Mechanism Remarks ♦ Leverage one player’s knowledge about the others ♦ 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

4. What We Constructively Do in Our Model

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

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

5. Comparison

Mechanism ( For General Auctions, ) ♦ [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

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

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

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

Thank you!

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

2 -Step Dominant-Strategy Truthful Recall: DST mechanism Each i’s strategy space Define: 2 -Step DST mechanism 1. 2. 3.

Mechanism ♦ 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 Analysis BSR [B’ 50]:

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

Mechanism ♦ 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 Analysis: Revenue Convex mechanism M: for any partition P of the valuations space, M is convex M is optimal

Generalization ♦ Recall ♦ Generalization

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

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

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

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

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

Mechanism ♦ 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 Remarks for -i ♦ Black-box usage of any DST mechanism M n≥ 2 “almost optimal” for single-good auction ♦ [Myerson’ 81] Works for any with independent distribution ♦ An existential result