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Dynamic Position Auctions with Consumer Search Scott Duke Kominers Harvard University Algorithmic Aspects in Information and Management June 16, 2009 Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 1 / 17 Background

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SLIDE 1

Dynamic Position Auctions with Consumer Search

Scott Duke Kominers

Harvard University

Algorithmic Aspects in Information and Management June 16, 2009

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 1 / 17

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SLIDE 2

Background

What are position auctions?

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 3

Background

What are position auctions?

Definition

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 4

Background

What are position auctions?

Definition

In a position auction,

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 5

Background

What are position auctions?

Definition

In a position auction,

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 6

Background

What are position auctions?

Definition

In a position auction, individuals submit bids for M positions

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 7

Background

What are position auctions?

Definition

In a position auction, individuals submit bids for M positions

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 8

Background

What are position auctions?

Definition

In a position auction, individuals submit bids for M positions, which are allocated via an auction rule.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 9

Background

What are position auctions?

Definition

In a position auction, individuals submit bids for M positions, which are allocated via an auction rule.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 10

Background

What are position auctions?

Definition

In a position auction, individuals submit bids for M positions, which are allocated via an auction rule. Position auctions are used to allocate sponsored search links to advertisers!

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 2 / 17

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SLIDE 11

Background

Why study position auctions?

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 3 / 17

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SLIDE 12

Background

Why study position auctions?

Sponsored search is a multibillion-dollar industry

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 3 / 17

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SLIDE 13

Background

Why study position auctions?

Sponsored search is a multibillion-dollar industry The mechanisms used are relatively new

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 3 / 17

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SLIDE 14

Background

Why study position auctions?

Sponsored search is a multibillion-dollar industry The mechanisms used are relatively new Welfare implications not well-understood

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 3 / 17

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SLIDE 15

Background

Previous Position Auction Models

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 16

Background

Previous Position Auction Models

Exogenous Click-through Rates

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 17

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 18

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 19

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 20

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 21

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 22

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 23

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 24

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 25

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Cary et al. (2008) dynamic extension

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 26

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Cary et al. (2008) dynamic extension ∼ convergence

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 27

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Cary et al. (2008) dynamic extension ∼ convergence

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 28

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Cary et al. (2008) dynamic extension ∼ convergence

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 29

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Cary et al. dynamic extension ∼ convergence

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 30

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Our dynamic extension ∼ convergence

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 31

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Our dynamic extension ∼ convergence

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 32

Background

Previous Position Auction Models

Exogenous Click-through Rates

Aggarwal, Goel, and Motwani (2006) Varian (2007) Edelman, Ostrovsky, and Schwarz (2007)

Endogenous Click-through Rates

Chen and He (2006) Athey and Ellison (2008)

Our dynamic extension ∼ convergence

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 4 / 17

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SLIDE 33

Our Model

Framework & Conventions

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 34

Our Model

Framework & Conventions

Athey and Ellison (2008) Model

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 35

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 36

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

value per-click: qπ (π = 1, 2, . . . , N)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 37

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

value per-click: qπ (π = 1, 2, . . . , N)

Interpretation: “probability of meeting a consumer’s need”

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 38

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

quality: qπ (π = 1, 2, . . . , N)

Interpretation: “probability of meeting a consumer’s need”

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 39

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

quality: qπ (π = 1, 2, . . . , N)

Interpretation: “probability of meeting a consumer’s need” Distribution: F(·)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 40

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

quality: qπ (π = 1, 2, . . . , N)

Interpretation: “probability of meeting a consumer’s need” Distribution: F(·) (public)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 41

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

quality: qπ (π = 1, 2, . . . , N)

Interpretation: “probability of meeting a consumer’s need” Distribution: F(·) (public) Sorted: q1 ≥ q2 ≥ · · · ≥ qN.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 42

Our Model

Framework & Conventions

Athey and Ellison (2008) Model N advertisers

quality: qπ (π = 1, 2, . . . , N)

Interpretation: “probability of meeting a consumer’s need” Distribution: F(·) (public) Sorted: q1 ≥ q2 ≥ · · · ≥ qN.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 43

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 44

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 45

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 46

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously

continuum of consumers

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 47

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously

continuum of consumers

search cost si per-click

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 48

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously

continuum of consumers

search cost si per-click

search until need is met

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 49

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously

continuum of consumers

search cost si per-click

search until need is met or until expected benefit < si

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 50

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously

continuum of consumers

search cost si per-click

search until need is met or until expected benefit < si Distribution: G(·)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 51

Our Model

Framework & Conventions

Athey and Ellison (2008) Model M < N positions

Awarded in a Generalized Second-Price Auction click-through rate: determined endogenously

continuum of consumers

search cost si per-click

search until need is met or until expected benefit < si Distribution: G(·) (public)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 5 / 17

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SLIDE 52

Our Model

Framework & Conventions

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 53

Our Model

Framework & Conventions

Our Dynamic Model

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 54

Our Model

Framework & Conventions

Our Dynamic Model Extends Athey and Ellison (2008)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 55

Our Model

Framework & Conventions

Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 56

Our Model

Framework & Conventions

Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting

Sequential rounds

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 57

Our Model

Framework & Conventions

Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting

Sequential rounds Synchronous updating

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 58

Our Model

Framework & Conventions

Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting

Sequential rounds Synchronous updating Advertisers play a “best-response” strategy

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 59

Our Model

Framework & Conventions

Our Dynamic Model Extends Athey and Ellison (2008) Dynamic setting

Sequential rounds Synchronous updating Advertisers play a “best-response” strategy Consumers ignorant of dynamics

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 6 / 17

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SLIDE 60

Our Model

Framework & Conventions

Balanced Bidding

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 61

Our Model

Framework & Conventions

Balanced Bidding Given the bids of the other advertisers, advertiser π

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 62

Our Model

Framework & Conventions

Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 63

Our Model

Framework & Conventions

Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 64

Our Model

Framework & Conventions

Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 65

Our Model

Framework & Conventions

Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 66

Our Model

Framework & Conventions

Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Athey and Ellison (2008) Envy-Free Equilibrium

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 67

Our Model

Framework & Conventions

Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 68

Our Model

Framework & Conventions

Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 69

Our Model

Framework & Conventions

Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 70

Our Model

Framework & Conventions

Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j among the positions below the current position which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 71

Our Model

Framework & Conventions

Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j among the positions below the current position which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 72

Our Model

Framework & Conventions

Restricted Balanced Bidding Given the bids of the other advertisers, advertiser π

targets the position j among the positions below the current position which maximizes utility chooses a bid bπ to satisfy the envy-free condition: G(¯ qj) · (1 − qπ) · (qπ − bπj+1) = G(¯ qj−1) · (qπ − bπ)

Unique fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 7 / 17

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SLIDE 73

Our Model

Results

Main Result

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

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SLIDE 74

Our Model

Results

Main Result

Theorem (Convergence Theorem)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

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SLIDE 75

Our Model

Results

Main Result

Theorem (Convergence Theorem)

If all advertisers play the Restricted Balanced Bidding strategy

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

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SLIDE 76

Our Model

Results

Main Result

Theorem (Convergence Theorem)

If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

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SLIDE 77

Our Model

Results

Main Result

Theorem (Convergence Theorem)

If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

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SLIDE 78

Our Model

Results

Main Result

Theorem (Convergence Theorem)

If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient. The dynamic model is “well-approximated” by the static model.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 8 / 17

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SLIDE 79

Our Model

Results

Parameters

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 80

Our Model

Results

Parameters γj(q) = (1 − q) G(¯

qj) G(¯ qj−1)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 81

Our Model

Results

Parameters γj(q) = (1 − q) G(¯

qj) G(¯ qj−1)

γ∗(q) = (1 − q) maxj>0

  • G(¯

qj) G(¯ qj−1)

  • Scott Duke Kominers (Harvard)

Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 82

Our Model

Results

Parameters γj(q) = (1 − q) G(¯

qj) G(¯ qj−1)

γ∗(q) = (1 − q) maxj>0

  • G(¯

qj) G(¯ qj−1)

  • γ∗∗ = max1≤π≤N γ∗(qπ)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 83

Our Model

Results

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 84

Our Model

Results

Lemma

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 85

Our Model

Results

Lemma

At every round t > t1 = 2 + logγ∗∗((1 − γ∗∗)(qM − qM+1)/qM+1):

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 86

Our Model

Results

Lemma

At every round t > t1 = 2 + logγ∗∗((1 − γ∗∗)(qM − qM+1)/qM+1):

  • bπ > qM+1

π < M + 1, bπ = qπ π ≥ M + 1.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 87

Our Model

Results

Lemma

At every round t > t1 = 2 + logγ∗∗((1 − γ∗∗)(qM − qM+1)/qM+1):

  • bπ > qM+1

π < M + 1, bπ = qπ π ≥ M + 1. Within t1 rounds, the N − M lowest-quality advertisers “drop out” of contention.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 9 / 17

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SLIDE 88

Our Model

Results

Convergence of the M Positions

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

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SLIDE 89

Our Model

Results

Convergence of the M Positions By the Lemma

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

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SLIDE 90

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions converge to the fixed point after round t1.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

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SLIDE 91

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions converge to the fixed point after round t1.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

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SLIDE 92

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

slide-93
SLIDE 93

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

slide-94
SLIDE 94

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Set of stable positions: P = {p + 1, . . . , M}

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

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SLIDE 95

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Set of stable positions: P = {p + 1, . . . , M} Set of advertisers in positions of P: π(P)

Scott Duke Kominers (Harvard) Dynamic Position Auctions June 16, 2009 10 / 17

slide-96
SLIDE 96

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Set of stable positions: P = {p + 1, . . . , M} Set of advertisers in positions of P: π(P) Next round, all advertisers in π(P) repeat their bids.

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slide-97
SLIDE 97

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Set of stable positions: P = {p + 1, . . . , M} Set of advertisers in positions of P: π(P) Next round, all advertisers in π(P) repeat their bids.

If π(P) = {1, . . . , M}, then we are done.

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slide-98
SLIDE 98

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Set of stable positions: P = {p + 1, . . . , M} Set of advertisers in positions of P: π(P) Next round, all advertisers in π(P) repeat their bids.

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slide-99
SLIDE 99

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Set of stable positions: P = {p + 1, . . . , M} Set of advertisers in positions of P: π(P) Next round, all advertisers in π(P) repeat their bids. Look at the advertiser π ∈ π(P) with the lowest bid.

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slide-100
SLIDE 100

Our Model

Results

Convergence of the M Positions By the Lemma, we need only show that the M positions stabilize after round t1.

Set of stable positions: P = {p + 1, . . . , M} Set of advertisers in positions of P: π(P) Next round, all advertisers in π(P) repeat their bids. Look at the advertiser π ∈ π(P) with the lowest bid.

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slide-101
SLIDE 101

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

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slide-102
SLIDE 102

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 1: π targets position p

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SLIDE 103

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 1: π targets position p ⇒ P′ = P ∪ {p} is stable

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slide-104
SLIDE 104

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 2: π targets position ˆ p > p

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SLIDE 105

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 2: π targets position ˆ p > p ⇒ P′ = {ˆ p, . . . , M} is stable

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SLIDE 106

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 2: π targets position ˆ p > p ⇒ P′ = {ˆ p, . . . , M} is stable

Depends upon the specific functional form of γˆ

p(q)

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SLIDE 107

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 2: π targets position ˆ p > p ⇒ P′ = {ˆ p, . . . , M} is stable

Depends upon the specific functional form of γˆ

p(q)

(Significant divergence from Cary et al. (2008))

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SLIDE 108

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 3: π targets position ˆ p < p

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slide-109
SLIDE 109

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 3: π targets position ˆ p < p ⇒ P remains stable

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slide-110
SLIDE 110

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 3: π targets position ˆ p < p ⇒ P remains stable minimum bid of advertisers not in π(P) increases

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slide-111
SLIDE 111

Our Model

Results

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SLIDE 112

Our Model

Results

Lemma

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SLIDE 113

Our Model

Results

Lemma

Let ǫ = G(¯

qM) 2G(¯ q1)(1 − γ∗∗) minφ=φ′ |qφ − qφ′|

M

j=1(1 − qj)

  • .

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SLIDE 114

Our Model

Results

Lemma

Let ǫ = G(¯

qM) 2G(¯ q1)(1 − γ∗∗) minφ=φ′ |qφ − qφ′|

M

j=1(1 − qj)

  • .

At most log1/γ∗∗((q1 − qM+1)/ǫ) consecutive instances of Case 3 may occur.

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SLIDE 115

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 3: π targets position ˆ p < p ⇒ P remains stable minimum bid of advertisers not in π(P) increases

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SLIDE 116

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 3: π targets position ˆ p < p ⇒ P remains stable minimum bid of advertisers not in π(P) increases after finitely many rounds, Case 1 or 2 must occur

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SLIDE 117

Our Model

Results

Convergence of the M Positions Set of stable positions: P = {p + 1, . . . , M} Advertiser π ∈ π(P) with the lowest bid

Case 3: π targets position ˆ p < p ⇒ P remains stable minimum bid of advertisers not in π(P) increases after finitely many rounds, Case 1 or 2 must occur ⇒ QED

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SLIDE 118

Our Model

Results

We have proven:

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SLIDE 119

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy

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slide-120
SLIDE 120

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy, then the M positions stabilize in finitely many rounds.

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slide-121
SLIDE 121

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy, then the M positions stabilize in finitely many rounds.

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slide-122
SLIDE 122

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point in finitely many rounds.

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slide-123
SLIDE 123

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point in finitely many rounds.

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slide-124
SLIDE 124

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point in finitely many rounds .

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slide-125
SLIDE 125

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient .

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slide-126
SLIDE 126

Our Model

Results

We have proven: If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient.

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slide-127
SLIDE 127

Our Model

Results

We have proven:

Theorem (Convergence Theorem)

If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient.

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SLIDE 128

Our Model

Results

We have proven:

Theorem (Convergence Theorem)

If all advertisers play the Restricted Balanced Bidding strategy, then their bids converge to the fixed point; this convergence is efficient. This also yields probability-1 efficient convergence in an asynchronous bidding model.

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SLIDE 129

Discussion

Possible Generalizations

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SLIDE 130

Discussion

Possible Generalizations

Our method ≈ Cary et al. (2008)’s method

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SLIDE 131

Discussion

Possible Generalizations

Our method ≈ Cary et al. (2008)’s method; its applicability is na¨ ıvely surprising.

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SLIDE 132

Discussion

Possible Generalizations

Our method ≈ Cary et al. (2008)’s method

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slide-133
SLIDE 133

Discussion

Possible Generalizations

Our method ≈ Cary et al. (2008)’s method Three key steps: Three key conditions:

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SLIDE 134

Discussion

Possible Generalizations

Our method ≈ Cary et al. (2008)’s method Three key steps:

1

restriction of the strategy space Three key conditions:

1

unique envy-free equilibrium

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slide-135
SLIDE 135

Discussion

Possible Generalizations

Our method ≈ Cary et al. (2008)’s method Three key steps:

1

restriction of the strategy space

2

analysis of low-quality advertisers’ behaviors Three key conditions:

1

unique envy-free equilibrium

2

low-quality advertisers drop out efficiently

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slide-136
SLIDE 136

Discussion

Possible Generalizations

Our method ≈ Cary et al. (2008)’s method Three key steps:

1

restriction of the strategy space

2

analysis of low-quality advertisers’ behaviors

3

proof that the M positions stabilize Three key conditions:

1

unique envy-free equilibrium

2

low-quality advertisers drop out efficiently

3

monotone equilibrium strategy

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SLIDE 137

Discussion

Conclusion

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SLIDE 138

Discussion

Conclusion

Convergence should be demonstrable in dynamic position auction models with sufficiently well-behaved static equilibrium strategies.

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SLIDE 139

QED

Questions?

kominers@fas.harvard.edu

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