Informationally Robust Auction Design

Introduction Model Results Conclusion

Benjamin Brooks Songzi Du UChicago UCSD

SIGecom Winter Meeting

February 25, 2021

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Informationally robust auction design (`ala Bergemann and Morris): Ignore the correlation in signals Focus on the correlation in values Maximizes the worst-case revenue across all info structures

Introduction Model Results Conclusion

Introduction

Bob Wilson and Paul Milgrom: Equilibrium bidding in standard auctions w/. common or interdependent values

Next step: optimal auction design Correlated values → Correlated signals → Cr´emerand McLean full surplus extraction

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion

Introduction

Bob Wilson and Paul Milgrom: Equilibrium bidding in standard auctions w/. common or interdependent values

Next step: optimal auction design Correlated values → Correlated signals → Cr´emerand McLean full surplus extraction

Informationally robust auction design (`ala Bergemann and Morris): Ignore the correlation in signals Focus on the correlation in values Maximizes the worst-case revenue across all info structures

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Info structure I = ({Si }, σ): Bidder i observes si ∈ Si σ ∈ ∆(V × S), marginal of σ over V is µ.

Mechanism M = ({Ai }, {qi }, {ti }): P i qi (a) ≤ 1 for all a

∃0 ∈ Ai , ti (0, a−i ) = 0 for all a−i

Bidder i gets vi qi (a) − ti (a)

Bayes-Nash equilibrium given (M, I)

Introduction Model Results Conclusion

Model

Single unit for sale, N bidders

N Seller knows µ ∈ ∆(V ), V = R+.

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Mechanism M = ({Ai }, {qi }, {ti }): P i qi (a) ≤ 1 for all a

∃0 ∈ Ai , ti (0, a−i ) = 0 for all a−i

Bidder i gets vi qi (a) − ti (a)

Bayes-Nash equilibrium given (M, I)

Introduction Model Results Conclusion

Model

Single unit for sale, N bidders

N Seller knows µ ∈ ∆(V ), V = R+.

Info structure I = ({Si }, σ): Bidder i observes si ∈ Si σ ∈ ∆(V × S), marginal of σ over V is µ.

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Bayes-Nash equilibrium given (M, I)

Introduction Model Results Conclusion

Model

Single unit for sale, N bidders

N Seller knows µ ∈ ∆(V ), V = R+.

Info structure I = ({Si }, σ): Bidder i observes si ∈ Si σ ∈ ∆(V × S), marginal of σ over V is µ.

Mechanism M = ({Ai }, {qi }, {ti }): P i qi (a) ≤ 1 for all a

∃0 ∈ Ai , ti (0, a−i ) = 0 for all a−i

Bidder i gets vi qi (a) − ti (a)

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion

Model

Single unit for sale, N bidders

N Seller knows µ ∈ ∆(V ), V = R+.

Info structure I = ({Si }, σ): Bidder i observes si ∈ Si σ ∈ ∆(V × S), marginal of σ over V is µ.

Mechanism M = ({Ai }, {qi }, {ti }): P i qi (a) ≤ 1 for all a

∃0 ∈ Ai , ti (0, a−i ) = 0 for all a−i

Bidder i gets vi qi (a) − ti (a)

Bayes-Nash equilibrium given (M, I)

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Nature’s minmax info structure solves: inf sup sup Π(M, I, β), (2) I M β∈B(M,I)

Strong Minimax Theorem Suppose we restrict to ﬁnite mechanisms and info structures, then

sup inf inf Π(M, I, β) = inf sup sup Π(M, I, β). M I β∈B(M,I) I M β∈B(M,I)

Introduction Model Results Conclusion Solution concept Common value Heuristic Solution concept

Seller’s maxmin mechanism solves: sup inf inf Π(M, I, β), (1) M I β∈B(M,I) B(M, I) is the set of equilibria, P Π(M, I, β) is the expected proﬁt (i.e., i ti )

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Strong Minimax Theorem Suppose we restrict to ﬁnite mechanisms and info structures, then

sup inf inf Π(M, I, β) = inf sup sup Π(M, I, β). M I β∈B(M,I) I M β∈B(M,I)

Introduction Model Results Conclusion Solution concept Common value Heuristic Solution concept

Seller’s maxmin mechanism solves: sup inf inf Π(M, I, β), (1) M I β∈B(M,I) B(M, I) is the set of equilibria, P Π(M, I, β) is the expected proﬁt (i.e., i ti ) Nature’s minmax info structure solves: inf sup sup Π(M, I, β), (2) I M β∈B(M,I)

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion Solution concept Common value Heuristic Solution concept

Strong Minimax Theorem Suppose we restrict to ﬁnite mechanisms and info structures, then

sup inf inf Π(M, I, β) = inf sup sup Π(M, I, β). M I β∈B(M,I) I M β∈B(M,I)

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Maxmin mechanism is proportional auction: Ai = R+ for each bidder i, a a q (a) = i · Q(Σa), t (m) = i · T (Σa), i Σa i Σa PN where Σa = i=1 ai , ( Σa/x Σa < x, Q(Σa) = 1 Σa ≥ x.

Price per unit is T (Σa)/Q(Σa) Reminiscent of a Tullock contest, and of Voucher auction in Russia in 1990s.

Introduction Model Results Conclusion Solution concept Common value Heuristic Common value setting

Suppose bidders have common value: v1 = v2 = ··· = vN . Suppose the distribution of the common value is “single-crossing.”

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Price per unit is T (Σa)/Q(Σa) Reminiscent of a Tullock contest, and of Voucher auction in Russia in 1990s.

Introduction Model Results Conclusion Solution concept Common value Heuristic Common value setting

Suppose bidders have common value: v1 = v2 = ··· = vN . Suppose the distribution of the common value is “single-crossing.”

Maxmin mechanism is proportional auction: Ai = R+ for each bidder i, a a q (a) = i · Q(Σa), t (m) = i · T (Σa), i Σa i Σa PN where Σa = i=1 ai , ( Σa/x Σa < x, Q(Σa) = 1 Σa ≥ x.

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion Solution concept Common value Heuristic Common value setting

Suppose bidders have common value: v1 = v2 = ··· = vN . Suppose the distribution of the common value is “single-crossing.”

Maxmin mechanism is proportional auction: Ai = R+ for each bidder i, a a q (a) = i · Q(Σa), t (m) = i · T (Σa), i Σa i Σa PN where Σa = i=1 ai , ( Σa/x Σa < x, Q(Σa) = 1 Σa ≥ x.

Price per unit is T (Σa)/Q(Σa) Reminiscent of a Tullock contest, and of Voucher auction in Russia in 1990s. Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Truth telling is an equilibrium of the proportional auction under I: Proportional auction is the proﬁt maximizing direct mechanism on I I is the proﬁt minimizing correlated equilibrium on proportional auction.

Introduction Model Results Conclusion Solution concept Common value Heuristic Common value setting

Minmax info structure I has Si = R+, si i.i.d. exponentially distributed The interim expected value ( C exp(Σs)Σs < x, [v | s] = E −1 H (GN (Σs)) Σs ≥ x,

where H is the CDF of v, and GN is the CDF of Σs (Gamma distribution).

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion Solution concept Common value Heuristic Common value setting

Minmax info structure I has Si = R+, si i.i.d. exponentially distributed The interim expected value ( C exp(Σs)Σs < x, [v | s] = E −1 H (GN (Σs)) Σs ≥ x,

where H is the CDF of v, and GN is the CDF of Σs (Gamma distribution). Truth telling is an equilibrium of the proportional auction under I: Proportional auction is the proﬁt maximizing direct mechanism on I I is the proﬁt minimizing correlated equilibrium on proportional auction.

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion Solution concept Common value Heuristic Common value setting

Proposition The optimal proﬁt guarantee of the proportional auction converges to the full surplus [v] as N → ∞ at the rate of √1 . E N

0.7

0.6

0.5

0.4 Optimal profit guarantee

0.3 for uniform distribution

0.2

0.1

0.0 N 0 20 40 60 80 100 Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion Solution concept Common value Heuristic Why Ai = S i = R+?

Suppose Ai = Si = R+. The Lagrangian for seller’s proﬁt maximization problem, ﬁxing σ:

X Z L = ti (a)σ(da, dv) i A×V X Z Z + αi (ai ) [vi ∇i qi (a) − ∇i ti (a)] σ(da, dv) i Ai A−i ×V " # Z X + γ(a) 1 − qi (a) da A i Z Z + λ(v) σ(da, dv) − µ(dv) V A Local IC: Z qi (ai , a−i ) − qi (ai − , a−i ) ti (ai , a−i ) − ti (ai − , a−i ) vi − σ(da, dv) ≥ 0 A−i ×V

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion Solution concept Common value Heuristic Why Ai = S i = R+?

Suppose Ai = Si = R+. The Lagrangian for Nature’s proﬁt minimization problem, ﬁxing (q, t):

X Z L = ti (a)σ(da, dv) i A×V X Z Z + αi (ai ) [vi ∇i qi (a) − ∇i ti (a)] σ(da, dv) i Ai A−i ×V " # Z X + γ(a) 1 − qi (a) da A i Z Z + λ(v) σ(da, dv) − µ(dv) V A Local obediance: Z qi (ai + , a−i ) − qi (ai , a−i ) ti (ai + , a−i ) − ti (ai , a−i ) vi − σ(da, dv) ≤ 0 N A−i ×{0,1}

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion Solution concept Common value Heuristic Why Ai = S i = R+?

X Z L = ti (a)σ(da, dv) i A×V X Z Z + αi (ai ) [vi ∇i qi (a) − ∇i ti (a)] σ(da, dv) i Ai A−i ×V " # Z X + γ(a) 1 − qi (a) da A i Z Z + λ(v) σ(da, dv) − µ(dv) V A

The same Lagrangian for both seller and Nature:

N Seller choose direct mechanism on A = R+ to maximize L N Nature choose info structure on S = R+ to minimize L A standard zero-sum game

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021 Introduction Model Results Conclusion

Conclusion

A novel auction format (proportional auction) to sell common value good

Informationally robust auction design is especially appealing when values are highly correlated

Brooks and Du (UChicago & UCSD) Informationally Robust Auction Design February 25, 2021