[PPT] - Truth or Envy? (a theory for prior-free mechanism design) Jason D. PowerPoint Presentation

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Truth or Envy?

(a theory for prior-free mechanism design) Jason D. Hartline — Northwestern University (Joint work with Qiqi Yan) May 26, 2011

SLIDE 2

Truth

SLIDE 3

Mechanism Design

Mechanism Design: how can a social planner / optimizer achieve

bjective when participant preferences are private.

Challenge: designer does not know participant preferences, participants may strategize when reporting preference!

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 4

Incentive Compatibility

Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully.

TRUTH OR ENVY? – MAY 26, 2011

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Incentive Compatibility

Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully. Goal: Design IC mechanisms with good performance, e.g., profit.

TRUTH OR ENVY? – MAY 26, 2011

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Incentive Compatibility

Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully. Goal: Design IC mechanisms with good performance, e.g., profit. Main Complication: IC constraints bind across preference profiles.

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Incentive Compatibility

Definition: a mechanism is incentive compatible (IC) if truthful reporting is an equilibrium. I.e., given others report truthfully, an agent maximizes her utility by reporting truthfully. Goal: Design IC mechanisms with good performance, e.g., profit. Main Complication: IC constraints bind across preference profiles. Consequences:

no mechanism is optimal for all preference profiles.
with prior distribution, can trade-off revenue.

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Example

Question: For

k units of an item, and
n agents with values drawn i.i.d. from U[0, 1]

what auction maximizes the seller’s expected revenue?

TRUTH OR ENVY? – MAY 26, 2011

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Example

Question: For

k units of an item, and
n agents with values drawn i.i.d. from U[0, 1]

what auction maximizes the seller’s expected revenue? Answer: the k-item Vickrey auction with reserve 1/2

TRUTH OR ENVY? – MAY 26, 2011

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Example

Question: For

k units of an item, and
n agents with values drawn i.i.d. from

exponential with rate 1

U[0, 1]

what auction maximizes the seller’s expected revenue? Answer: the k-item Vickrey auction with reserve 1/2

TRUTH OR ENVY? – MAY 26, 2011

SLIDE 11

Example

Question: For

k units of an item, and
n agents with values drawn i.i.d. from

exponential with rate 1

U[0, 1]

what auction maximizes the seller’s expected revenue? Answer: the k-item Vickrey auction with reserve

1 1/2

TRUTH OR ENVY? – MAY 26, 2011

SLIDE 12

Example

Question: For

k units of an item, and
n agents with values drawn i.i.d. from

exponential with rate 1

U[0, 1]

what auction maximizes the seller’s expected revenue? Answer: the k-item Vickrey auction with reserve

1 1/2

Conclusion: optimal auction depends on prior distribution.

TRUTH OR ENVY? – MAY 26, 2011

SLIDE 13

The trouble with priors

The trouble with priors:

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The trouble with priors

The trouble with priors:

where does prior come from?

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The trouble with priors

The trouble with priors:

where does prior come from?
is prior accurate?

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The trouble with priors

The trouble with priors:

where does prior come from?
is prior accurate?
prior-dependent mechanisms are non-robust.

TRUTH OR ENVY? – MAY 26, 2011

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The trouble with priors

The trouble with priors:

where does prior come from?
is prior accurate?
prior-dependent mechanisms are non-robust.
what if one mechanism must be used in many scenarios?

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 18

The trouble with priors

The trouble with priors:

where does prior come from?
is prior accurate?
prior-dependent mechanisms are non-robust.
what if one mechanism must be used in many scenarios?

Goal: theory for prior-free mechanism design.

TRUTH OR ENVY? – MAY 26, 2011

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The trouble with priors

The trouble with priors:

where does prior come from?
is prior accurate?
prior-dependent mechanisms are non-robust.
what if one mechanism must be used in many scenarios?

Goal: theory for prior-free mechanism design. (one of the main contributions of AGT to GT/Econ)

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 20

Envy

SLIDE 21

Multi-unit Pricing

Problem: Multi-unit Pricing

n agents, values v1 ≥ . . . ≥ vn
k units of an item.

Goal: envy-free revenue-maximizing pricing.

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Multi-unit Pricing

Problem: Multi-unit Pricing

n agents, values v1 ≥ . . . ≥ vn
k units of an item.

Goal: envy-free revenue-maximizing pricing. First Attempt:

sell to top i at price vi gives revenue R(i) = ivi
pick j = argmaxi≤k R(i).

TRUTH OR ENVY? – MAY 26, 2011

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Multi-unit Pricing

Problem: Multi-unit Pricing

n agents, values v1 ≥ . . . ≥ vn
k units of an item.

Goal: envy-free revenue-maximizing pricing. First Attempt:

sell to top i at price vi gives revenue R(i) = ivi
pick j = argmaxi≤k R(i).

Note: can view as menu, a.k.a., pricing.

TRUTH OR ENVY? – MAY 26, 2011

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Example

Example: suppose we have

20 units of an item for sale,
90 interested agents:

– 10 with value 10, and – 80 with value 2.

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Picture

180 100 10 90 20 40

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Picture

180 100 10 90 20 40

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Picture

180 100 10 90 20 40 Question: can we do better?

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Example, revisited

Example: suppose we have

20 units of an item for sale,
90 interested agents:

– 10 with value 10, and – 80 with value 2.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 29

Example, revisited

Example: suppose we have

20 units of an item for sale,
90 interested agents:

– 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 30

Example, revisited

Example: suppose we have

20 units of an item for sale,
90 interested agents:

– 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries.

Sell to all 10’s at price 9.
Sell to 2’s with probability 1/8 at price 2.
Revenue = 9 × 10 + 2 × 10 = 110.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 31

Example, revisited

Example: suppose we have

20 units of an item for sale,
90 interested agents:

– 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries.

Sell to all 10’s at price 9.
Sell to 2’s with probability 1/8 at price 2.
Revenue = 9 × 10 + 2 × 10 = 110.

Is it envy-free?

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 32

Example, revisited

Example: suppose we have

20 units of an item for sale,
90 interested agents:

– 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries.

Sell to all 10’s at price 9.
Sell to 2’s with probability 1/8 at price 2.
Revenue = 9 × 10 + 2 × 10 = 110.

Is it envy-free?

10’s utility = 10 − 9 = 1.
10’s utility if swap = (10 − 2) × 1/8 = 1.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 33

Example, revisited

Example: suppose we have

20 units of an item for sale,
90 interested agents:

– 10 with value 10, and – 80 with value 2. Idea: price “randomized allocations”, a.k.a., lotteries.

Sell to all 10’s at price 9.
Sell to 2’s with probability 1/8 at price 2.
Revenue = 9 × 10 + 2 × 10 = 110.

Is it envy-free? Yes!

10’s utility = 10 − 9 = 1.
10’s utility if swap = (10 − 2) × 1/8 = 1.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 34

Picture

180 100 10 90 20 110

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 35

envy-free pricing

Definition: a pricing (and allocation) is envy free (EF) no agent wants to swap outcomes with another.

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envy-free pricing

Definition: a pricing (and allocation) is envy free (EF) no agent wants to swap outcomes with another. Main Simplification: EF constrains pricing outcome “pointwise”, nothing is required of pricing on different preference profiles.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 37

envy-free pricing

Definition: a pricing (and allocation) is envy free (EF) no agent wants to swap outcomes with another. Main Simplification: EF constrains pricing outcome “pointwise”, nothing is required of pricing on different preference profiles. Consequence: for any objective, there is an optimal envy-free pricing.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 38

Truth or Envy(-free)?

Thesis: envy freedom ≈ incentive compatibility

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SLIDE 39

Truth or Envy(-free)?

Thesis: envy freedom ≈ incentive compatibility Related Work:

1. IC = EF in limit [Jackson, Kremer ’07]

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SLIDE 40

Truth or Envy(-free)?

Thesis: envy freedom ≈ incentive compatibility Related Work:

1. IC = EF in limit [Jackson, Kremer ’07]
2. Core (related to EF) is more important than IC [Day, Milgrom ’07]

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 41

Truth or Envy(-free)?

Thesis: envy freedom ≈ incentive compatibility Related Work:

1. IC = EF in limit [Jackson, Kremer ’07]
2. Core (related to EF) is more important than IC [Day, Milgrom ’07]
3. “IC for price-takers” is good enough [Azevedo, Budish ’11]

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 42

Truth or Envy(-free)?

Thesis: envy freedom ≈ incentive compatibility Related Work:

1. IC = EF in limit [Jackson, Kremer ’07]
2. Core (related to EF) is more important than IC [Day, Milgrom ’07]
3. “IC for price-takers” is good enough [Azevedo, Budish ’11]

Our Perspective: small n is the interesting case.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 43

Truth or Envy(-free)?

Thesis: envy freedom ≈ incentive compatibility (and not just in the limit) Related Work:

1. IC = EF in limit [Jackson, Kremer ’07]
2. Core (related to EF) is more important than IC [Day, Milgrom ’07]
3. “IC for price-takers” is good enough [Azevedo, Budish ’11]

Our Perspective: small n is the interesting case.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 44

Truth or Envy(-free)?

Thesis: envy freedom ≈ incentive compatibility (and not just in the limit) Related Work:

1. IC = EF in limit [Jackson, Kremer ’07]
2. Core (related to EF) is more important than IC [Day, Milgrom ’07]
3. “IC for price-takers” is good enough [Azevedo, Budish ’11]

Our Perspective: small n is the interesting case. Goal: approximate optimal EF with prior-free IC mechanism. (generalize results for digital goods / multi-unit auctions)

TRUTH OR ENVY? – MAY 26, 2011

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Overview

1.

= ⇒

characterization and optimization: IC ≈ EF

2. revenue: IC ≈ EF
3. towards prior-free incentive-compatible mechanisms.

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Notation

Notation:

values: v1 ≥ . . . ≥ vn
outcome: x = (x1, . . . , xn); xi = probability i is served.
payments: p = (p1, . . . , pn); pi = i’s expected payment.
utility: ui = vixi − pi
profit:

i pi

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Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj

b b b b b b

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Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

b b b b b b

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 49

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

b b b b b b

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 50

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

Thm: EF ⇒ pi =

b b b b b b

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SLIDE 51

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

Thm: EF ⇒ pi =

b b b b b b

revenue curve: R(i) = ivi virtual value: ϕi = R(i)−R(i−1)

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 52

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

Thm: EF ⇒ pi =

b b b b b b

revenue curve: R(i) = ivi virtual value: ϕi = R(i)−R(i−1) Thm: revenue:

i ϕixi

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 53

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

Thm: EF ⇒ pi =

b b b b b b

revenue curve: R(i) = ivi virtual value: ϕi = R(i)−R(i−1) Thm: revenue:

i ϕixi

Thm: optimal pricing: maximize ϕs s.t. feasible and monotone.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 54

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

Thm: EF ⇒ pi =

b b b b b b

revenue curve: R(i) = ivi virtual value: ϕi = R(i)−R(i−1) Thm: revenue:

i ϕixi

Thm: optimal pricing: maximize ϕs s.t. feasible and monotone. ironed r.c.: ¯

R(·) = hull(R(·))

ironed v.v.: ¯

ϕi = ¯ R(i) − ¯ R(i − 1)

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 55

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

Thm: EF ⇒ pi =

b b b b b b

revenue curve: R(i) = ivi virtual value: ϕi = R(i)−R(i−1) Thm: revenue:

i ϕixi

Thm: optimal pricing: maximize ϕs s.t. feasible and monotone. ironed r.c.: ¯

R(·) = hull(R(·))

ironed v.v.: ¯

ϕi = ¯ R(i) − ¯ R(i − 1)

Thm: optimal pricing: maximize ¯

ϕs

s.t. feasible (random tie-breaking).

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 56

Characterization and Optimization: EF ≈ IC

Envy-freedom values: v1 ≥ . . . ≥ vn EF: vixi − pi ≥ vixj − pj Thm: EF ⇒ xi monotone in i

[Mu’alem ’09]

Thm: EF ⇒ pi =

b b b b b b

revenue curve: R(i) = ivi virtual value: ϕi = R(i)−R(i−1) Thm: revenue:

i ϕixi

Thm: optimal pricing: maximize ϕs s.t. feasible and monotone. ironed r.c.: ¯

R(·) = hull(R(·))

ironed v.v.: ¯

ϕi = ¯ R(i) − ¯ R(i − 1)

Thm: optimal pricing: maximize ¯

ϕs

s.t. feasible (random tie-breaking). Incentive Compatibility

[M’81;BR’89]

value distrib’n: F(z) = Pr[vi < z] IC: vixi(vi)−pi(vi)≥vixi(z)−pi(z) Thm: IC ⇒ xi(z) monotone in z Thm: IC ⇒ pi(z) =

rev. curve: R(q) = qF −1(1 − q)

virtual value: ϕi = R′(1 − F(vi)) Thm: revenue: Ev∼F [

i ϕixi(v)]

Thm: optimal auction: maximize ϕs s.t. feasible and monotone. ironed r.c.: ¯

R(·) = hull(R(·))

ironed v.v.: ¯

ϕi = ¯ R′(1 − F(vi))

Thm: optimal auction: maximize ¯

ϕs

s.t. feasible (random tie-breaking).

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Comparison

Contrast:

optimal IC mechanism based on prior distribution.
optimal EF pricing based on empirical distribution.

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Example: Position Pricing/Auction

Problem: Position Pricing/Auction

n agents, values v1 ≥ . . . ≥ vn
n positions, probabilities w1 ≥ . . . ≥ wn

Goal: revenue-maximizing pricing/auction. (models “sponsored search”)

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Example: Position Pricing/Auction

Problem: Position Pricing/Auction

n agents, values v1 ≥ . . . ≥ vn
n positions, probabilities w1 ≥ . . . ≥ wn

Goal: revenue-maximizing pricing/auction. (models “sponsored search”) Solution:

1. calculate R(·) and concave hull ¯

R(·).

2. Assign agents to positions greedily by ¯

ϕi (breaking ties randomly)

3. Revenue is

{i : ¯ ϕi≥0} ¯

ϕiwi.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 60

Example: Position Pricing/Auction

Problem: Position Pricing/Auction

n agents, values v1 ≥ . . . ≥ vn
n positions, probabilities w1 ≥ . . . ≥ wn

Goal: revenue-maximizing pricing/auction. (models “sponsored search”) Solution:

1. calculate R(·) and concave hull ¯

R(·).

2. Assign agents to positions greedily by ¯

ϕi (breaking ties randomly)

3. Revenue is

{i : ¯ ϕi≥0} ¯

ϕiwi.

Note: all that matters for allocation is partial order on ¯

ϕis.

TRUTH OR ENVY? – MAY 26, 2011

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Overview

1. characterization and optimization: IC ≈ EF

2.

= ⇒

revenue: IC ≈ EF

3. towards prior-free incentive-compatible mechanisms.

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Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders.

b b

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Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation:

b b

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Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation:

b b

2 10

10/80 1 TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 65

Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation:

b b

2 10

10/80 1

Payments:

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SLIDE 66

Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation:

b b

2 10

10/80 1

Payments:

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SLIDE 67

Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation:

b b

2 10

10/80 1

Payments:

plow = 2 × 10

80 = 1 4

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SLIDE 68

Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation:

b b

2 10

10/80 1

Payments:

plow = 2 × 10

80 = 1 4

phigh = 10 − 8 × 10

80 = 9

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 69

Envy-free Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation:

b b

2 10

10/80 1

Payments:

plow = 2 × 10

80 = 1 4

phigh = 10 − 8 × 10

80 = 9

Revenue: 10 × 9 + 80 × 1

4 = 110.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 70

IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

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IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation Rules:

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SLIDE 72

IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation Rules:

xlow(v) 2 10

10/80 1

xhigh(v) 2 10

11/81 1 TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 73

IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation Rules:

xlow(v) 2 10

10/80 1

xhigh(v) 2 10

11/81 1

Payments:

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 74

IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation Rules:

xlow(v) 2 10

10/80 1

xhigh(v) 2 10

11/81 1

Payments:

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 75

IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation Rules:

xlow(v) 2 10

10/80 1

xhigh(v) 2 10

11/81 1

Payments:

plow = 2 × 10

80 = 1 4

phigh = 10 − 8 × 11

81 ≈ 8.9

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SLIDE 76

IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation Rules:

xlow(v) 2 10

10/80 1

xhigh(v) 2 10

11/81 1

Payments:

plow = 2 × 10

80 = 1 4

phigh = 10 − 8 × 11

81 ≈ 8.9

Revenue: 10 × 8.9 + 80 × 1

4 = 109.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 77

IC Example

Example: 20 units, 10 high bidders, 80 low bidders. Proposed Mechanism: allocate items all high bidders, and remaining items to random low bidders.

Allocation Rules:

xlow(v) 2 10

10/80 1

xhigh(v) 2 10

11/81 1

Payments:

plow = 2 × 10

80 = 1 4

phigh = 10 − 8 × 11

81 ≈ 8.9

Revenue: 10 × 8.9 + 80 × 1

4 = 109.

Conclusion: IC revenue ≈ EF revenue.

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SLIDE 78

IC revenue ≈ EF revenue

Thm: for any virtual surplus maximizer: EF(v) ≥ IC(v) ≥ EF(v)/2.

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SLIDE 79

IC revenue ≈ EF revenue

Thm: for any virtual surplus maximizer: EF(v) ≥ IC(v) ≥ EF(v)/2.

first “≥” requires structural property (e.g., matroid)
second “≥” requires assumption on virtual value function

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SLIDE 80

Overview

1. characterization and optimization: IC ≈ EF
2. revenue: IC ≈ EF

3.

= ⇒

towards prior-free incentive-compatible mechanisms.

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SLIDE 81

Prior-Free IC mechanisms

1. prior-free benchmark: envy-free optimal revenue
2. digital goods (β approx).
3. multi-unit auctions (2β approx) by reduction
4. position auctions (2β approx) by reduction
5. matroid permutation auctions (2β approx) by reduction
6. downward closed permutation auctions (Θ(1) approx)

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 82

Prior-Free IC mechanisms

1. prior-free benchmark: envy-free optimal revenue
2. digital goods (β approx). E.g., β = 3.25 [H, McGrew ’05]
3. multi-unit auctions (2β approx) by reduction
4. position auctions (2β approx) by reduction
5. matroid permutation auctions (2β approx) by reduction
6. downward closed permutation auctions (Θ(1) approx)

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 83

Prior-Free IC mechanisms

1. prior-free benchmark: envy-free optimal revenue
2. digital goods (β approx). E.g., β = 3.25 [H, McGrew ’05]
3. multi-unit auctions (2β approx) by reduction
reduction: k-Vickrey + digital good auction.
4. position auctions (2β approx) by reduction
5. matroid permutation auctions (2β approx) by reduction
6. downward closed permutation auctions (Θ(1) approx)

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 84

Prior-Free IC mechanisms

1. prior-free benchmark: envy-free optimal revenue
2. digital goods (β approx). E.g., β = 3.25 [H, McGrew ’05]
3. multi-unit auctions (2β approx) by reduction
reduction: k-Vickrey + digital good auction.
4. position auctions (2β approx) by reduction
position auction = convex comb. of k-unit auctions
reduction: simulate convex comb. + Birkhoff–von Neumann
5. matroid permutation auctions (2β approx) by reduction
6. downward closed permutation auctions (Θ(1) approx)

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 85

Prior-Free IC mechanisms

1. prior-free benchmark: envy-free optimal revenue
2. digital goods (β approx). E.g., β = 3.25 [H, McGrew ’05]
3. multi-unit auctions (2β approx) by reduction
reduction: k-Vickrey + digital good auction.
4. position auctions (2β approx) by reduction
position auction = convex comb. of k-unit auctions
reduction: simulate convex comb. + Birkhoff–von Neumann
5. matroid permutation auctions (2β approx) by reduction
permutation is worst-case/bayesian middle ground
permutation: induces characteristic weights
6. downward closed permutation auctions (Θ(1) approx)

TRUTH OR ENVY? – MAY 26, 2011

20

SLIDE 86

Prior-Free IC mechanisms

1. prior-free benchmark: envy-free optimal revenue
2. digital goods (β approx). E.g., β = 3.25 [H, McGrew ’05]
3. multi-unit auctions (2β approx) by reduction
reduction: k-Vickrey + digital good auction.
4. position auctions (2β approx) by reduction
position auction = convex comb. of k-unit auctions
reduction: simulate convex comb. + Birkhoff–von Neumann
5. matroid permutation auctions (2β approx) by reduction
permutation is worst-case/bayesian middle ground
permutation: induces characteristic weights
6. downward closed permutation auctions (Θ(1) approx)
random sampling & EF analysis.

TRUTH OR ENVY? – MAY 26, 2011

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SLIDE 87

Conclusions

1. envy-free optimal pricings in single-dimensional settings.
2. connection between envy-free pricings and Bayesian mechanism

design.

3. envy-free pricings give “right” benchmark for IC approximation.

TRUTH OR ENVY? – MAY 26, 2011

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