Core-Selecting Auctions with Incomplete Informationbaranov/job_market_website/...Example from...

Motivation Pricing Rules Model Main Results Extensions Conclusion

Core-Selecting Auctions with IncompleteInformation

Lawrence M. Ausubel and Oleg V. Baranov

University of Maryland

NBER Market Design Workshop

October 2010

Ausubel and Baranov (UMD) Core-Selecting Auctions with Incomplete Information


Vickrey-Clarke-Groves (VCG) mechanism

The VCG mechanism has the attractive property that truthfulbidding is a dominant strategy, implying efficient outcomes.

VCG mechanism:

1 Bidders submit values for every subset of items.2 Allocation Rule: Assign the items so as to maximize social

welfare (relative to reported values)3 Pricing Rule: Each bidder receives a payoff equaling the

incremental value she brings.



VCG practical drawbacks

VCG weaknesses in environments with complementarities:

Low (or zero) revenue

Non-monotonicity of the seller’s revenue

Vulnerability to collusion and shill bidding

Example from Ausubel and Milgrom (2002)

Bidder A B A and B1 0 0 2

2 2 0 2

3 0 2 2

VCG assigns A and B to thebidders 2 and 3 at zero prices!Bidder 1 and seller can block this

outcome.

Reason:

Sometimes the VCG mechanism produce a payment vector whichlies outside of the core.










2 2 0 2

3 0 2 2


outcome.

Reason:











2 2 0 2

3 0 2 2


outcome.

Reason:











2 2 0 2

3 0 2 2


outcome.

Reason:




Core-selecting auctions:

Central Idea:

Given the submitted bids, a core-selecting auction identifiesefficient allocation (i.e an allocation which maximizes total valuewith respect to submitted bids) and chooses payments which areassociated with a core payoff vector.

Minimum-revenue core (MRC) pricing (Day and Milgrom (2009)):

...“minimize” bidders’ incentives to deviate from “truthful bidding”

Flow of the MRC pricing procedure:

1 Efficient allocation

2 MRC pricing

3 In case MRC prices are not unique, choose one which usingsome selection criteria




Central Idea:






2 MRC pricing





Central Idea:






2 MRC pricing





Central Idea:






2 MRC pricing




Some notation:


Two units are offered for sale

Two bidders have unit demands- i.e. local bidders

One bidder demands both unitsas perfect complements- i.e. the global bidder

b1, b2 - bids for different unitsby the local bidders

B - a package bid by the globalbidder


Some notation:








Some notation:








Some notation:








Core-Selecting Auctions: Global side wins the auction



Core-Selecting Auctions: Global side wins the auction



Core-Selecting Auctions: Local side wins the auction



Core-Selecting Auctions: Local side wins the auction



Literature review:

Krishna and Rosenthal (1995)

Ausubel and Milgrom (2002)

Parkes and Ungar (2000), Parkes (2001)

Ausubel, Cramton and Milgrom (2006)

Day and Raghavan (2007)

Day and Milgrom (2008)

Day and Cramton (2008)

Cramton (2009)

Erdil and Klemperer (2009)

Goeree and Lien (2009)

Sano (2010)



Our contribution:

Incomplete information analysis

Partial-correlation model for bidders’ values

Closed-form solutions for all pricing rules

Comparison of different pricing rules in terms of revenue andefficiency

Very different conclusions for independence vs. highcorrelation



Proxy Pricing Rule

“Ascending Proxy Auction” - Ausubel and Milgrom (2002)

p(b1, b2,B) =

(12B, 1

2B, 0) if min{b1, b2} ≥ 12B

(b1,B − b1, 0) if 2b1 < B < b1 + b2

(B − b2, b2, 0) if 2b2 < B < b1 + b2

(0, 0, b1 + b2) if B ≥ b1 + b2



Proxy Pricing Rule



Proxy Pricing Rule



Nearest-Vickrey Pricing Rule

Day and Raghavan (2007), Day and Cramton (2008)

Two UK Spectrum Auctions (second stage) - Cramton (2009)

Announced for use in the FAA airport slot auction

p(b1, b2,B) =

{(V1 + ∆,V2 + ∆, 0) if B < b1 + b2

(0, 0, b1 + b2) if B ≥ b1 + b2

where∆ = B−V1−V2

2V1 = max{0,B − b2}V2 = max{0,B − b1}









Proportional Pricing Rule

Very simple to explain and compute

p(b1, b2,B) =

{(b1

b1+b2B, b2

b1+b2B, 0

)if B < b1 + b2

(0, 0, b1 + b2) if B ≥ b1 + b2









Nearest-Bid Pricing Rule

Strongly-related to bidders’ own bids

Sounds reasonable

Easy to explain to bidders

p(b1, b2,B) =

(b1 −∆, b2 −∆, 0) if b̄ − b < B < b1 + b2

(B, 0, 0) if b1 ≥ B + b2

(0,B, 0) if b2 ≥ B + b1

(0, 0, b1 + b2) if B ≥ b1 + b2

where∆ = b1+b2−B

2b̄ = max(b1, b2)b = min(b1, b2)









Valuations:

The global bidder’s value u is drawn according to the uniformdistribution on [0,2], independently of the local bidders’ values

Local bidders’ value model: (vi is the value of the local bidder i , i = 1, 2)

o

γzztttttttttt

(1−γ)##GG

GGGG

GGG

Perfect Correlationv1 = v2 = vv ∼ F (.)

Independencev1, v2

i .i .d F (.)

Valuation vector of the local bidders (v1, v2) is drawn from thedistribution on [0, 1]× [0, 1] described by the cdf

H(x1, x2) = γF (min[x1, x2]) + (1− γ)F (x1)F (x2) γ ∈ [0, 1]

where F (x) = xα, α > 0.



Valuations:



o

γzztttttttttt

(1−γ)##GG

GGGG

GGG


Independencev1, v2

i .i .d F (.)






Valuations:



o

γzztttttttttt

(1−γ)##GG

GGGG

GGG


Independencev1, v2

i .i .d F (.)






Valuations:



o

γzztttttttttt

(1−γ)##GG

GGGG

GGG


Independencev1, v2

i .i .d F (.)






Local Value Model: Conditional CDF

Conditional CDF of vj given vi is:

H(xj |vi = s) =

{(1− γ)F (xj ) xj < s(1− γ)F (xj ) + γ xj ≥ s

i 6= j

First-Order Stochastic Dominance, not Affiliation





H(xj |vi = s) =


i 6= j






H(xj |vi = s) =


i 6= j




Pivotal Pricing Property


An auction satisfies the pivotal pricing property with respect to a given bidderif, whenever the bidder’s bid is pivotal, the price that she pays (if she wins)equals her bid.

Examples

Standard single-item auctions: (bi is pivotal if bi = b̄−i )

1 First-Price Auction Satisfied

2 Second-Price Auction Satisfied

3 Third-Price Auction Not Satisfied

Lemma 1

Every core-selecting auction satisfies the pivotal pricing property with respectto all bidders.






Examples


1 First-Price Auction

Satisfied



Lemma 1







Examples



2 Second-Price Auction

Satisfied


Lemma 1







Examples




3 Third-Price Auction

Not Satisfied

Lemma 1







Examples





Lemma 1







Examples





Lemma 1




Optimality conditions 1:

Lemma 2

For any bidder satisfying the pivotal pricing property, the optimality conditionsare given by:

(v − b)∂Pr(Win)

∂b≤ E

(∂Pay

∂b

)b ≥ 0

b

[(v − b)

∂Pr(Win)

∂b− E

(∂Pay

∂b

)]= 0

.



Minimum-Revenue Core with Proxy Pricing Rule:

Proposition 1:

The equilibrium bid function of local bidders (in symmetricBayesian-Nash equilibria) under the Proxy Pricing Rule is given by

β(v) =

{0 v ≤ d(γ)

1 + ln(γ+(1−γ)v)1−γ v > d(γ)

if γ < 1

andβ(v) = v if γ = 1,

where d(γ) = exp(−(1−γ))−γ1−γ > 0 ∀γ < 1.



Equilibrium Bid Function: Proxy Pricing Rule















Minimum-Revenue Core with Nearest-VCG Pricing Rule:

Proposition 2:

The equilibrium bid function of local bidders under theNearest-VCG Pricing Rule is given by

β(v) =

{0 v ≤ d(γ)

k(γ)v − d(γ) v > d(γ)if γ < 1

andβ(v) = 2

3v if γ = 1,

where k(γ) = 22+γ and d(γ) =

3k−2√

(3k−1)

3k−2 ∀γ < 1.



Equilibrium Bid Function: Nearest-VCG Pricing Rule















Minimum-Revenue Core with Proportional Pricing Rule:

Proposition 3:

The equilibrium bid function of local bidders under theProportional Pricing Rule is given by

β(v) =

{0 v ≤ d(γ)

k(γ)v − d(γ) v > d(γ)if γ < 1

andβ(v) = 2

3v if γ = 1,


3k−2√

(3k−1)

3k−2 ∀γ < 1.

Driving Forces:

1 Uniform Distribution of the Global Bidder’s value

2 Number of Local Markets



Minimum-Revenue Core with Proportional Pricing Rule:

Proposition 3:

The equilibrium bid function of local bidders under theProportional Pricing Rule is given by

β(v) =

{0 v ≤ d(γ)

k(γ)v − d(γ) v > d(γ)if γ < 1

andβ(v) = 2

3v if γ = 1,


3k−2√

(3k−1)

3k−2 ∀γ < 1.

Driving Forces:

1 Uniform Distribution of the Global Bidder’s value

2 Number of Local Markets



Minimum-Revenue Core with Nearest-Bid Pricing Rule:

Proposition 4:

The equilibrium bid function of local bidders under the Nearest-BidPricing Rule is given by

β(v) = 11−γ [ln(2)− ln(2− (1− γ)v)] if γ < 1

andβ(v) = 1

2v if γ = 1.



Equilibrium Bid Function: Nearest-Bid Pricing Rule















Revenue and Efficiency:



Summary Statistics:

γ Statistics VCG Proxy N-VCG N-Bid Pay-as-Bid*

Revenue 0.5833 0.5360 0.5327 0.5 0.5471γ = 0 Efficiency 1 0.8679 0.8431 0.8069 0.8754

Profit Global 0.2916 0.4642 0.4673 0.5 0.4267Profit Local 0.2087 0.1342 0.1335 0.1253 0.1498

Revenue 0.5417 0.5852 0.52 0.4521 0.5414γ = 0.5 Efficiency 1 0.9261 0.8356 0.7739 0.9036


Revenue 0.5 0.6667 0.5185 0.4167 0.5411γ = 1 Efficiency 1 1 0.8334 0.75 0.9049


* - Simulated using numerical solutions from Baranov (2010)



Extensions:

Some Extensions:

Global vs. Local asymmetry (α 6= 1)

More Local markets (N > 2)

Robustness: Continuous Correlation Model


Skip to Conclusion




Skip to Conclusion

α changes the expected value of the Local Side(u, v1, v2 - value draws of the Global Bidder and Local Bidders)

Side α = 0.5 α = 1 α = 2

Global E(u) 1 1 1

Local E(v1 + v2) 2/3 1 4/3




Skip to Conclusion


Side α = 0.5 α = 1 α = 2

Global E(u) 1 1 1

Local E(v1 + v2) 2/3 1 4/3

Two effects:

1 Change in the overall size of the “pie”2 Change in the relative probability of winning under full efficiency




Skip to Conclusion


Side α = 0.5 α = 1 α = 2

Global E(u) 1 1 1

Local E(v1 + v2) 2/3 1 4/3

Two effects:


Some of the pricing rules continue to have (almost) closed-form solutions,while others need to be purely computational




Skip to Conclusion


Side α = 0.5 α = 1 α = 2

Global E(u) 1 1 1

Local E(v1 + v2) 2/3 1 4/3

Two effects:


Some of the pricing rules continue to have (almost) closed-form solutions,while others need to be purely computational

Surprising closed-forms:For α = 2 and γ < 1 and Proxy Pricing Rule

β(v) =

{0 v ≤ d(γ)

1√γ(1−γ)

arctan(√

1−γγ

v)

+ C v > d(γ)


Example: Proxy Pricing Rule


Skip to Conclusion


Revenue and Efficiency: α = 2


Skip to Conclusion


Revenue and Efficiency: α = 0.5


Skip to Conclusion


Extensions:

Some Extensions:





Skip to Conclusion


Proportional Pricing Rule: N = 2, 3 and 5


Skip to Conclusion


Extensions:

Some Extensions:





Skip to Conclusion




M - a common unknown distributional factor distributed with density fM (m) on [0,1] (uniform)

v1, v2 are drawn independently from the truncated logit distribution on [0,1] with parameters (m, σ)

fL(vj |vi = s) =

∫ 10

fL(vj |m)fL(s|m)fM (m)dm∫ 10

fL(s|m)fM (m)dm

Skip to Conclusion




Skip to Conclusion

Numerical approximations: All bids are qualitatively the same


Conclusion:

Comparison of different package bidding protocols requires anincomplete-information analysis

A simple environment where each package pricing rule issolvable

Parametric family that allows independence to perfectcorrelation

Correlation has enormous impact on the revenue/efficiencycomparisons

Correlation has different effects on different pricing rules

No “Silver Bullet”, but some rankings seem robust


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