Maximizing Revenue in Expectation

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Maximizing Revenue in Expectation

Aaron Roth

University of Pennsylvania

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Overview

I We’ve spent a lot of time discussing welfare maximization.

I But many auctions have a more pecuniary goal. What if we want to maximize revenue?

I What does that mean? What is our benchmark?

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Overview

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Overview

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Overview

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Reasonable Benchmarks?

I The VCG mechanism was remarkable: we could always maximize welfare ex-post.

I What about for revenue? Not so simple.

I Consider a single bidder, single item auction. Offering a fixed price p is always dominant strategy truthful.

I Revenue is p if vi ≥ p, 0 otherwise.

I So ex-post, the revenue-optimal auction sets p = vi... But

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Reasonable Benchmarks?

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Reasonable Benchmarks?

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Reasonable Benchmarks?

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Reasonable Benchmarks?

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The Average Case

I Suppose we know that bidders have valuations vi ∼ D for

some distribution D.

I We know D, but we don’t know vi...

I In a single item, single bidder auction, a fixed price p yields expected revenue:

Rev (p) = p · (1 − F (p)) Where F (p) = Prv ∼D[v ≤ p].

I E.g. if D is uniform on [0, 1], then F (p) = p and: max p Rev (p) = 1 2 · (1 − 1 2) = 1 4

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The Average Case

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The Average Case

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The Average Case

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Average Case: Many Bidders

I One item, many bidders.

I We want to design a truthful mechanism (X , P) that maximizes: Ev ∼Dn " _n X i =1 Pi(v ) #

I For truthfulness, we need X to be monotone non-decreasing...

I And we know:

Pi(v ) = vi· Xi(v ) −

Z vi

0

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Average Case: Many Bidders

I And we know:

Pi(v ) = vi· Xi(v ) −

Z vi

0

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Average Case: Many Bidders

I And we know:

Pi(v ) = vi· Xi(v ) −

Z vi

0

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Average Case: Many Bidders

I And we know:

Pi(v ) = vi· Xi(v ) −

Z vi

0

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Myserson Optimal Auctions

I Lets assume monotonicity for now, and use our expression for P to derive the optimal X .

I If we are lucky and derive a monotone X , we will be done!

I Plan: Find X to maximize: Ev ∼Dn " _n X i =1 Pi(v ) # = n X i =1 Ev−i∼Dn−1[Evi∼D[Pi(vi, v−i)]] I Notation: f (p) is the pdf of D. F (p) = Pr v ∼D[v ≤ p] = Z p 0 f (v )dv

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

Consider the inner term: Evi[Pi(v )] = Evi vi· Xi(vi, v−i) − Z vi 0 Xi(z, v−i)dz = E [vi· Xi(vi, v−i)] − E Z vi 0 Xi(z, v−i)dz = Z 1 0 vi· Xi(vi, v−i) · f (vi)dvi− Z 1 0 f (vi) Z vi 0 Xi(z, v−i)dzdvi = Z 1 0 vi· Xi(vi, v−i) · f (vi)dvi− Z 1 0 Xi(z, v−i) Z 1 z f (vi)dvidz = Z 1 0 vi· Xi(vi, v−i) · f (vi)dvi− Z 1 0 Xi(vi, v−i) Z 1 vi f (z)dzdvi = Z 1 0 vi· Xi(vi, v−i) · f (vi)dvi− Z 1 0 Xi(vi, v−i)(1 − F (vi))dvi = Z 1 0 vi− (1 − F (vi)) f (vi) X (vi, v−i)f (vi)dvi = Evi vi− (1 − F (vi)) f (vi) X (vi, v−i)

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

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Myserson Optimal Auctions

So: We want to maximize

Ev ∼Dn " _n X i =1 φ(vi) · X (v ) # φ(vi) = vi − (1 − F (vi)) f (vi) | {z } “Virtual Value”

I Our objective looks just like welfare with values replaced by virtual values.

I (Pointwise) optimal allocation rule: Give the item to the bidder i with highest φ(vi) if it’s positive. Otherwise give the

item to nobody.

I This is a monotone allocation rule if D is regular: φ(vi) is

monotone.

I e.g. if D is uniform, φ(vi) = vi− (1 − vi) = 2vi− 1

I Note that φ−1(0) recovers the optimal p = 1/2 for a single bidder.

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Myserson Optimal Auctions

item to nobody.

monotone.

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Myserson Optimal Auctions

item to nobody.

monotone.

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Myserson Optimal Auctions

item to nobody.

monotone.

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Myserson Optimal Auctions

What do revenue maximizing auctions look like? (when vi drawn

iid from regular D)

I We give the item to bidder i∗= arg maxiφ(vi) when

φ(vi∗) ≥ 0.

I Because φ is monotone, i∗ = arg maxivi: the item goes to the

highest bidder when φ(vi∗) ≥ 0. I Winner pays vi∗−Rvi ∗

p∗ 1 = p∗, where:

p∗= max(max

i 6=i∗vi, φ

−1₍₀₎₎

I i.e. its just a Vickrey auction with a reserve price of φ−1(0)!

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Myserson Optimal Auctions

iid from regular D)

φ(vi∗) ≥ 0.

highest bidder when φ(vi∗) ≥ 0.

I Winner pays vi∗−Rvi ∗

p∗= max(max

i 6=i∗vi, φ

−1₍₀₎₎

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Myserson Optimal Auctions

iid from regular D)

φ(vi∗) ≥ 0.

p∗= max(max

i 6=i∗vi, φ

−1₍₀₎₎

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Myserson Optimal Auctions

iid from regular D)

φ(vi∗) ≥ 0.

p∗= max(max

i 6=i∗vi, φ

−1₍₀₎₎

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Myserson Optimal Auctions

iid from regular D)

φ(vi∗) ≥ 0.

p∗= max(max

i 6=i∗vi, φ

−1₍₀₎₎

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Extensions/Limitations

I Can be made to work even when D is not regular.

I Have to “iron” φ(v ) to make it monotone.

I Analysis required vi’s be drawn independently, but not

identically. Each bidder can have their own distribution Di.

I Each bidder has their own virtual valuation function φi(vi).

I Auction no longer so natural. e.g. high bidder no longer necessarily wins.

I Doesn’t extend beyond single parameter domains...

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Extensions/Limitations

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Extensions/Limitations

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Extensions/Limitations

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Extensions/Limitations

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One more thing

I If we care about revenue, should we give up on welfare?

I The Vickrey auction yields no revenue selling to a single bidder, whereas when D is uniform over [0, 1] we can get expected revenue 1/4.

I What about a Vickrey auction with 2 bidders?

I

Rev (VA) = Ev1,v2∼D[min(v1, v2)] = 1/3 I So we might be better off maximizing welfare with more

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One more thing

I

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One more thing

I

Rev (VA) = Ev1,v2∼D[min(v1, v2)] = 1/3

I So we might be better off maximizing welfare with more bidders...

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One more thing

I

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The Bulow/Klemperer Theorem

Theorem

Consider bidders drawn i.i.d. from a regular distribution D. For any n ≥ 1, the Vickrey auction with n + 1 bidders has higher expected revenue than the revenue optimal auction with n bidders.

So recruiting just one extra bidder is worth more than optimizing revenue for the current population.

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The Bulow/Klemperer Theorem

Theorem

Consider bidders drawn i.i.d. from a regular distribution D. For any n ≥ 1, the Vickrey auction with n + 1 bidders has higher expected revenue than the revenue optimal auction with n bidders. So recruiting just one extra bidder is worth more than optimizing revenue for the current population.

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The Bulow/Klemperer Theorem

Consider the hypothetical auction A for n + 1 bidders:

1. Run the revenue optimal auction for the first n bidders.

2. If the auction fails to allocate the item, give it to bidder n + 1 for free.

Observations:

1. The revenue of A is exactly equal to the optimal revenue obtainable from n bidders.

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The Bulow/Klemperer Theorem

Observations:

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The Bulow/Klemperer Theorem

Observations:

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But...

I Claim: The Vickrey mechanism is obtains the maximum revenue amongst all mechanisms that always allocate the item.

I Recall that Ev[PiPi(v )] = E[Piφi(vi) · Xi(v )].

I We can maximize the RHS (subject to always allocating the item) by always allocating to arg maxiφ(vi).

I Since D is regular, φ is monotone: this is arg maxivi — the

Vickrey allocation!

I So: The Vickrey-auction with n + 1 bidders has only higher revenue than the optimal n bidder auction.

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But...

Vickrey allocation!

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But...

Vickrey allocation!

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But...

Vickrey allocation!

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But...

Vickrey allocation!

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Maximizing Revenue in Expectation