Crowdsourced Bayesian Auctions

MIT

Pablo Azar Jing Chen Silvio Micali

Agenda1. Motivation for Crowdsourced Bayesian2. Our Model3. What We Can Do In-Principle in Our Model4. 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 Rationalityto 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 (A0, A1, …, An) + prices (P1, …, Pn)

Utility:

: { }

Revenue:

:

Bayesian :

designer

[Myerson’81]: optimal revenue for single-good auctions

4, D

players

n, D3, D2, D

1, D

D

Centralized Bayesian :

Very Strong!Designer knows D further assumes: Independent distribution

4, Dn, D3, D

2, D1, D

D

Bayesian Nash further assumes:

Still Strong!

ignorant

players know each other better than designer knows them

, D , D, D , D , D

Bayesian :

♦ D common knowledge to players

ignorant4, Dn, D3, D

2, D1, D

, D , D, D , 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, Dn, D3, D

2, D1, D

, D , D, D , 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|S2

1, D|S1 3, D|S3

4, D|S4

n, D|Sn

Bayesian :

♦ Designer ignorant

2, D|S2

1, D|S1 3, D|S3

4, D|S4

n, D|Sn

♦ D: iid, independent, correlated…

Our Crowdsourced Bayesian Assumption

Each player i knows an arbitrary refinement of D|θi

θ:

Si1

Si2

Si3

i, D|Si2

Ignorant Designer Mechanism gets players’ strategies only

i knows D|θi 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 mechanismDefine (informally): Two-Step DST mechanism1.

2.

3.

θi is the best strategy regardless what the others do

1.

2. θi is the best regardless what the others do

D|Si is the best given first part actions = θ

regardless i’s second part action

regardless the others’ second part actions

DST = Dominant Strategy Truthful

,

,

,

,

,

,θi

,θn

,θ1

,θi D|Sii

Tool 2: Two-Step DST

♦ Mutual Knowledge of Rationality

DST = Dominant Strategy TruthfulDefine (informally): Two-Step DST mechanism1.

2.

3.

θi is the best regardless what the others do

D|Si 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:Allegedly:

Player i gets nothing and pays nothing

bounded in [-2, 0]

to a real numberexpectation 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

MechanismRemarks

♦ 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

ignorantdesigner

4, D|S4

informedplayers

n, D|Sn3, D|S3

2, D|S2

1, D|S1

2-Step DSTCrowdsourced

Bayesian

Thank you!

Complete Information

1 2 …n

informedplayers

ignorantdesigner

MR’88JPS’94AM’92GP’96CHM’10ACM’10

2-Step Dominant-Strategy TruthfulRecall: DST mechanism

Define: 2-Step DST mechanism

Each i’s strategy space

1.

2.

3.

Mechanism

AnalysisBSR [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:Allegedly:

Player i gets nothing and pays nothing

announcing is 2-step DST for i

Mechanism

Analysis: RevenueConvex 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

ignorant4, …

informedplayers

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|θi

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 propertyDhangwatnotai, 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:Allegedly:

Player i gets nothing and pays nothing

bounded in [-2, 0]

to a real numberexpectation 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