Crowdsourced Bayesian Auctions

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

2. 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)

3. 1. Motivation for Crowdsourced Bayesian

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

5. Auctions in General n players a set of goods Valuation (for subsets) ({ }) = 310 Allocation: : : { } Outcome: allocation (A0, A1, …, An) + prices (P1, …, Pn) Utility: Revenue:

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

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

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

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

10. 2. Our Crowdsourced Bayesian Model

11. Bayesian : Crowdsourced if: Designer ignorant No common knowledge required D: iid, independent, correlated… Each i individually knows ≥ 2,D|S2 2,D|S2 4,D|S4 4,D|S4 n,D|Sn n,D|Sn 1,D|S1 1,D|S1 ignorant 3,D|S3 3,D|S3

12. Our Crowdsourced Bayesian Assumption Each player i knows an arbitrary refinement of D|θi : iknows D|θi and refines as much as he can i, D|Si2 θ Si2 Ignorant Designer  Mechanism gets players’ strategies only Si1 Si3 No requirement on higher-order knowledge Players’ knowledge to be leveraged!

13. Can We Leverage? Yes, with proper tools!

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

15. Tool 2: Two-Step DST DST = Dominant Strategy Truthful Recall (informally): DST mechanism Define (informally): Two-Step DST mechanism 1. 1. 2. θi is the best regardlesswhat the others do θi is the best strategy regardlesswhat the others do 2. regardlessi’s second part action 3. D|Siis the best given first part actions = θ regardlessthe others’ second part actions , , , , , , θ1 i , , θi θi D|Si , θn

16. Tool 2: Two-Step DST DST = Dominant Strategy Truthful Define (informally): Two-Step DST mechanism 1. 2. θi is the best regardlesswhat the others do regardlessi’s second part action 3. D|Siis the best given first part actions = θ regardlessthe others’ second part actions Mutual Knowledge of Rationality A special case of CM’s solution concept

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

18. Revenue In General Auctions Hypothetical benchmark optimal DST revenue under centralized Bayesian Notasymptotic n=1000? 100? Wonderful! n=2? “Tight” (even for single-good auctions)!

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

20. 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

21. 4. What We Constructively Do in Our Model

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

23. 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

24. 5. Comparison

25. 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

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

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

28. In Sum ignorant designer 2-Step DST Crowdsourced Bayesian informed players 4,D|S4 2,D|S2 n,D|Sn 3,D|S3 1,D|S1

29. Thank you!

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

31. 2-Step Dominant-Strategy Truthful Recall: DST mechanism Each i’s strategy space Define: 2-Step DST mechanism 1. 2. 3.

32. 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]:

33. Mechanism Choose a player i uniformly at random 1. Player i announces Allegedly: 2. Each other player j announces Allegedly: Run the optimal DST mechanism M with for -i Player i gets nothing and pays nothing Reward i using Brier’s Scoring Rule 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

34. 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 for -i Reward i using Brier’s Scoring Rule Analysis: Revenue Convex mechanism M: for any partition P of the valuations space, M is convex  M is optimal 

35. Generalization Recall Generalization

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

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

38. 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 require even less … Bayesian Nash equilibrium requires even more: I knows that he knows that I knows that he knows that … Each player i has no information about θ-i beyond D|θi More information  incentive to deviate

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

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

41. 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 for -i Reward i using Brier’s Scoring Rule 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