Simple, Credible, and Approximately-OptimalAuctions

Constantinos Daskalakis Maxwell Fishelson Brendan LucierVasilis Syrgkanis Santhoshini Velusamy

January 27, 2020

Abstract

We establish the first credible mechanisms that achieves a constant factorof the optimal revenue in the multi-item auction setting with additive bidders.Adapting the duality framework of Cai et al [1], we extend the classical separatevs bundle revenue optimality results to a large class of auctions. We provethat either selling the items in separate reserve first price auctions (SFP) orcharging each bidder a grand entry fee to access separate first price auctions(EFP) achieves a constant factor of the optimal revenue. This is the firstknown winner-pays-bid revenue approximation result (Corollary 5.7). We provea parallel result for the separate first-price auction and the bundle entry fee all-pay auction that achieves credibility (Corollary 5.11). Our techniques are alsoused to show that either separate Myerson auctions (SMyer) or FIXED bundleentry fee VCG (EVCG) are approximately revenue-optimal (Corollary 5.2). Inall past literature, the entry fee was a function of the other bidder’s types,whereas our fixed entry fee is easy to implement via online learning.

1 Introduction

Recently, Google switched their ad auctions from second price auctions to firstprice auctions. cite this They decided to make this move for the convenience of theirusers, a seemingly nonsensical motivation as it is the second price auction that ac-tually maximizes consumer welfare. However, the first-price auction does have twoadvantages over the second-price auction: its simplicity and credibility. The paymentrule of the first-price auction is quite easy to understand, you simply pay what youbid in the event you win the auction. There is also the issue of credibility in thesecond-price auction. We have no guarantee that the auctioneer does not artificially

1

inject a bid ε below our bid. However, in the first price auction, there is no way forthe auctioneer to make a profitable deviation from the proposed mechanism withoutbeing detected.

2 Preliminaries

2.1 General Auction Theory

Throughout this paper, we consider the setting of the multi-item sealed bid auctionwith multiple additive bidders. We have a collection of n rational agents called biddersand a set of m items to be allocated to the bidders. Each item is indivisible and cannotbe allocated to multiple agents. Each bidder i has a private valuation tij for eachitem j. This value (or type) is sampled from a continuous distribution Dij, supportedon type space Tij ⊆ R ≥ 0. Each of these types is sampled independently, and thedistributions Dij are common knowledge. In the additive setting, the utility of abidder is the sum of their valuations over the items they are allocated minus theirpayment. We denote

Ti = ×jTij T−i = ×i∗ 6=iTi∗ T = ×iTiDi = ×jDij D−i = ×i∗ 6=iDi∗ D = ×iDi

In an auction, each bidder i observes their own type ti = (ti1, · · · , tim) and choosessome action ai (e.g. a bid to submit or a total contingency plan over a multi-roundauction). The auction mechanism is a mapping of these actions a = (a1, · · · , an) toallocations x(a) = (x1(a), · · · , xn(a)) and payments p∗(a) = (p∗1(a), · · · , p∗n(a)). Werefer to these as “expost allocations” and “expost payments” as they are in terms theactions of all the players, post-submission. In the literature, π is interim allocation,x is expost allocation, and p is sometimes interim payment and sometimes expostpayment. We want to disambiguate that here and the current solution is p is interimpayment and p∗ is expost. Maybe there’s better notation possible? Maybe greek let-ters for interim and english for expost? But the greek for p is π which is already used.What to do hmmmmmm The allocation of each bidder xi(a) = (xi1(a), · · · , xim(a))is a vector of their allocation for each item. In a deterministic auction, we havexij(a) = 0 or 1 for all i, j. It is an indicator of item j being allocated to bidder i. Inrandomized auctions, xij(a) could be fractional representing an allocation probability.For example, xij(a) = 1

2represents that, under actions a, bidder i is allocated item

j with probability 12. In either case, we always have

∑i xij(a) ≤ 1 for all items j.

There is inequality here as sometimes item j can be allocated to no bidder.

2

The utility obtained by a bidder is the value they get from the items they receiveminus the payment they must make. For additive bidders, their value for receivinga collection of items is the sum of their values for each of the items. Thus, we havethat the expost utility of bidder i is

u∗i (a) = xi(a) · ti − p∗i (a)

We define a “bid strategy” b = (b1, · · · , bn) to be a collection of mappings bi fromtypes ti to actions ai. We say that a bid strategy is an “equilibrium bid strategy” ifit forms a Pure Strategy Bayesian Nash Equilibrium. That is, if each bidder observestheir type ti and plays action bi(ti), no bidder has incentive to deviate

Et−i∼D−i [u∗i (bi(ti), b−i(t−i))] ≥ Et−i∼D−i [u∗i (a′i, b−i(t−i))] ∀a′iFor a specific bid equilibrium b, we can define the “interim” utility u, allocation π,and payment p of bidder i respectively as

ubi(ti) = Et−i∼D−i [u∗i (bi(ti), b−i(t−i))]πbi (ti) = Et−i∼D−i [xi(bi(ti), b−i(t−i))]pbi(ti) = Et−i∼D−i [p∗i (bi(ti), b−i(t−i))]

Similarly, we can abbreviate expost utility, allocation, and payment to be in terms oftype Bidders choose actions to maximize interim utility. Define Wel, Rev, Util

2.2 Sealed Bid Auctions with Monotone Allocation

Many of the most common single-item auctions (VCG, first price, all pay) arewhat are known as “sealed bid auctions”. In these auctions, the action ai is simplya non-negative real value, or bid, that is submitted at the start. Single-item sealedbid auctions take place over one round: bidders simultaneously submit a bid, andthe mechanism determines who gets allocated the item. Here, the bid is often a realnumber belonging to the type space. The fact that bids are real gives rise to thenotion of an auction with “monotone allocation”. That is, for each bidder i, fixingthe other bids a−i expost, we have that their allocation xi(ai, a−i) monotonicallyincreasing in their bid ai. In a deterministic mechanism, this can only occur if thereis some threshold value Bi(a−i) that is the inflection point about which bidder i isallocated:

xi(ai, a−i) =

{0 if ai ≤ Bi(a−i)

1 if ai > Bi(a−i)

For an equilibrium strategy b, we can define this threshold in terms of the types

Bbi (t−i) = Bi(b−i(t−i))

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The common allocation rule satisfying this property is to simply allocate to the high-est bidder, where the threshold is the max of the other bids. VCG, first-price, andall-pay all have this allocation rule. Where they differ is payment rule. Maybe brieflydescribe payments?

We also note that monotone expost allocation implies that interim allocation isalso monotone. For two possible bids ai, a

′i with a′i > ai we have: Write this analysis

2.3 Revenue Approximation

For single-item auctions, the mechanism optimizing revenue is Myerson discuss

However, in multi-item auctions, the revenue-optimal mechanism is extremelycomplex and unknown. Thus, much interest in simple, approx opt. Works of this guyand that guy all made progress in this direction. All unified by a paper “A Duality-Based Unified Approach to Bayesian Mechanism Design” [1] by Cai et. al. In thepaper, they prove the following theorem.

Theorem 2.1. [Theorem 31 [1]] Consider a multi-item auction setting with dis-crete type space T+ and discrete valuation distribution D+. For each v−i ∈ T+

−i,let R

v−i0 , R

v−i1 , · · · , Rv−i

m be an “upwards-closed” partition of the type space T+i . That

is, for all j 6= 0,

ti = (ti1, · · · , tij, · · · , tim) ∈ Rv−ij ⇒ (ti1, · · · , t∗ij, · · · , tim) ∈ Rv−i

j for all t∗ij > tij

Let M be any BIC mechanism with values drawn from D+ that has interim allo-cation and payment πM , pM in the truthful equilibrium. The expected revenue of M inthe truthful equilibrium is upper bounded by the expected virtual welfare of the sameallocation rule with respect to the canonical virtual value function Φi. In particular,

Rev(M) =∑i

∑ti∈T+

i

fi(ti)pMi (ti) ≤ VWel(M) =

∑i

∑ti∈T+

i

∑j

fi(ti)πMij (ti)Φij(ti)

≤∑i

∑ti∈T+

i

∑j

fi(ti)πMij (ti)

(tij · Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]+ ϕ̃ij(tij) · Pr

v−i∼D−i

[ti ∈ Rv−i

j

])where ϕ̃ij(tij) represents Myerson’s discrete ironed virtual value for the distributionD+ij .

By the revelation principle, allocations and payments achievable by any auction inany equilibrium are also achievable by a BIC auction in truthful equilibrium. Thus,

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we can use this theorem to upper bound the revenue of the revenue-optimal mecha-nism.

It is important to note that this theorem is proven specifically for discrete typespaces and discrete probability distributions. However, we want to work with con-tinuous probability distributions. We do extensive work with first-price auctions inthis paper, which often have exclusively mixed strategy equilibria when type spacesare discrete. This will make the pure equilibrium bid strategies discussed in the firstsubsection nonexistent. We make use of a discretization argument from [1] that willenable us to apply the theorem in the continuous setting.

In Appendix A, we prove

Theorem 2.2. Consider a multi-item auction setting with type space T ⊆ Rn×m,continuous atomless valuation distribution D, and additive bidders. Say that each Tijis bounded: Tij ⊆ [0, H].

For all i and each v−i ∈ T−i, let Rv−i0 , R

v−i1 , · · · , Rv−i

m be a “preference” partition ofthe type space Ti. That is, for each item j and each v−i, there exists a non-decreasingutility function Uv−ij : Tij → R ≥ 0 such that, for all j 6= 0

ti ∈ Rv−ij ⇔Uv−ij (tij) ≥ Uv−ik (tik) ∀k 6= j

Uv−ij (tij) > Uv−ik (tik) ∀k < j

Uv−ij (tij) > 0

andti ∈ Rv−i

0 ⇔ Uv−ij (tij) = 0 ∀jThat is, the region containing ti is determined by the item j which ti most prefers,breaking ties lexicographically. The 0 region contains all ti with 0 utility on all items.

Let πM represent a feasible interim allocation. That is, for all i, j, there ex-ists some mapping xMij : Rn×m → [0, 1] so that πMij (ti) = Et−i∼D−i

[xMij (ti, t−i)

]and∑

i xMij (t) ≤ 1 for all items j and t ∈ T . Let OPT be the revenue-optimal BIC mecha-

nism with values drawn from D. Then, the expected revenue of OPT is upper boundedby

Rev(OPT ) ≤∑i

Eti∼Di

[∑j

πMij (ti)

(tij · Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]+ ϕ̃∗ij(tij) · Pr

v−i∼D−i

[ti ∈ Rv−i

j

])]where ϕ̃∗ij(tij) = max(ϕ̃ij(tij), 0) and ϕ̃ij(tij) represents Myerson’s continuous ironedvirtual value for the distribution Dij.

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We note that πM doesn’t have to be the allocation of any specific mechanism.It just must satisfy the feasibility constraint. We also note that a partition being apreference partition implies that it is also upwards-closed. Additionally, in almostall of the previous literature using Theorem 2.1, the regions used also satisfy thisstronger preference condition.

They use this to prove bundle vs sep

In the Cai paper, they prove this virtual welfare bound for general upwards-closedregions, but then only end up applying it to a specific set of regions defined in termsof expost utility in VCG auctions. Crucial to our arguments will be a redefining ofthese regions in terms of INTERIM utility rather than ex-post.

2.4 Credibility

A credible auction is defined informally as an auction in which the auctioneercannot make a profitable deviation from the proposed mechanism without being de-tected. Clearly Sealed-bid VCG auction is not credible because it is possible for theauctioneer to report an arbitrarily large second highest price undetected. However,first-price auction (with reserve) is credible [6].

The following theorem by Hartline et al [2] shows that first-price auction withreserve is also approximately revenue-optimal.

Theorem 2.3. For an auction with regular type distributions

Rev(FP with reserve) ≥ e− 1

2eRev(Myerson)

Therefore, it follows that a constant-factor of the revenue of Separate Myersonauctions for each item, proposed in [1] can be achieved by a credible mechanism.However, it is not clear if the revenue of the proposed bundle-VCG mechanism in [1]can be approximately achieved in a credible mechanism. In this paper, we propose abundled entry-fee all-pay auction which is both credible and approximately-optimal.

3 Alternative virtual welfare function

In our arguments, we define our preference regions in terms of a general single-itemsealed-bid auction A. This could be first-price, VCG, all-pay, etc. We only requirethat A be deterministic with monotone allocation rule. For each item j, we consideran equilibrium bid strategy (b1j, · · · , bnj) that would arise if we were to sell item j

6

separately via an A-auction. This equilibrium would be in terms of the A mechanismand the distributions Dij for all i. We let b denote the collection of these separateequilibrium bid strategies over all the items. These equilibria will provide us with anotion of the interim utility that bidder i would receive in a separate item j auction:ubij(tij). This will be fundamental to the definition of our regions.

We define our preference regions in terms of interim utility, but over items thatare allocated expost. Each type vector ti of bidder i induces a ranking of the itemsbased on interim utility: j1, j2, · · · , jm such that

ubij1(tij1) ≥ ubij2(tij2) ≥ · · · ≥ ubijm(tijm)

breaking ties lexicographically. Then, we find the highest ranked item which wouldbe allocated to bidder i expost when facing types t−i. As A has a monotone allocationrule, expost allocation is determined by a bid threshold; whether or not bij(tij) exceedsBbij(t−ij). This determines the regions R

t−ij . Formally, we define

U t−ij (tij) = ubij(tij) · 1[bij(tij) > Bb

ij(t−ij)]

which gives rise to regions defined as such:

ti ∈ Rt−ij ⇔ bij(tij) > Bb

ij(t−ij)

and bik(tik) ≤ Bbik(t−ik) for all k satisfying ubik(tik) > ubij(tij)

and bik(tik) ≤ Bbik(t−ik) for all k satisfying ubik(tik) = ubij(tij) and k < j

We say ti ∈ Rt−i0 if it belongs to no other regions, i.e. bij(tij) ≤ Bb

ij(t−ij) for all j.This switch from upwards closed to preference regions introduces this thing where wecould be in R0 if interim util is 0, even if we are expost allocated. It is an irrelevantedge case but certainly something to back to and address

We want to apply Theorem 2.2 on these regions, and so we must verify thatU t−ij (tij) is non-decreasing. This is true because it is the product of two non-decreasingfunctions, as long as bij(tij) is non-decreasing in tij.

We see that ubij(tij) is monotonic in tij. Let tij, t′ij ∈ Tij with t′ij > tij. Since

bidder i bids to optimize utility, deviating to bid bij(tij) when his type is t′ij cannotincrease utility. Thus,

ubij(t′ij) = πbij(t

′ij) · t′ij − pbij(t′ij)

≥ πbij(tij) · t′ij − pbij(tij)≥ πbij(tij) · tij − pbij(tij)= ubij(tij)

7

Then, if bij(tij) is increasing in tij, the expost allocation 1[bij(tij) > Bb

ij(t−ij)]

isas well, due to the monotonic allocations of the A auction. This increasing bid is aproperty of all the auctions we consider: discuss how this is evident for VCG andall-pay and ref papers about this for FP

4 Decomposition of the virtual welfare

Before stating the main theorem, we introduce the following mechanisms. First,SMyer is simply the multi-item mechanism in which each item is sold in a separateMyersonian auction. Second, for a single-item auction mechanism A, we define themulti-item bundle-entry-fee auction EA as follows. First, each bidder i is given theoption to pay a fixed entry fee ri that is in terms of their type distribution over allitems Di. Then, the bidders who pay the entry fee have access to separate A-auctionson each of the items. The bidders who do not pay the entry fee are replaced by a“ghost” who bids in the auctions as the bidder would have had they paid the entryfee, according to some collection of equilibrium bid strategies b on each of the separateauctions. More specifically, observing their own type, bidder i will choose to pay theentry fee if it is less than their total interim utility over all the separate A-auctions.If bidder i does not pay the entry fee, a “ghost” is created with type sampled fromDi conditioned on the type achieving a total interim utility less than the entry fee.The ghost then participates in each of the auctions, playing according to b in eachauction. If it is to win an item, that item is allocated to nobody and no payment isreceived for that item. The key here is that, from the perspective of the bidders whodo pay the fee and participate in the auctions, their opponents in the auction havetype distributed according to D−i. They are unable to observe which opponents areghosts. All they know is that each opponent i′ has type sampled from Di′ and then,if ti′ happens to lie in the subset of Ti′ that gives interim utility less than the fee, ti′is rerolled from that subset again according to Di′ (conditioned).

Importantly, any collection of equilibria b over the separate A auctions forms anequilibria in the EA auction. Each bidder who pays the entry fee faces opponentdistributions D−i. The goal of each bidder is to simply maximize their utility oneach of the separate auctions independently. Their valuations are additive and thetype distributions of their opponents are independent between the items. So, bid-ding according to a separate A-auction equilibria in each of the auctions will also beNash in the EA auction. When we discuss the revenue of the EA auction through-out this paper, we assume that the auctioneer knows which equilibria b the biddersplay according to post-entering. This equilibria will determine which entry fees theauctioneer sets and how he samples the ghosts. This specific equilibria b will be tied

8

to our analysis too. Our interim regions will be defined in terms of it.

Discuss how, if bid equilibria unique, none of this is a concern. Talk about how itis often unique in the auctions we consider and cite some papers.

Theorem 4.1. Let A be a deterministic sealed-bid single-item auction with monotoneallocation satisfying the “type-loss trade off” property. That is, for any collection ofbidders participating in an A-auction, regardless of type distributions Di and equilib-rium played b, we have

Et∼D[maxiti(1− πbi (ti)

)]≤ c · Rev(Myer)

Essentially, this implies that bidders with large type are very likely to have high allo-cation probability in the A-auction.

Then, in a multi-item auction setting where bidder’s utilities are additive, therevenue achieved by the revenue-optimal mechanism OPT is upper bounded

Rev(OPT ) ≤ (c+ 5) · Rev(SMyer) + 2 · Rev(EA)

Proof. Since the A-auction is deterministic with a monotone allocation rule, the re-gions described in the previous section satisfy the preference condition, and we canapply Theorem 2.2.

Rev(OPT ) ≤∑i

Eti∼Di

[∑j

πMij (ti)

(tij · Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]+ ϕ̃∗ij(tij) · Pr

v−i∼D−i

[ti ∈ Rv−i

j

])]

We decompose this expected virtual welfare. Each term in the decomposition isthen upper bounded by either a constant factor of the revenue of a mechanism whichsells items separately (which is in turn bounded by SMyer), or the revenue of EA.

First, we bound

Prv−i∼D−i

[ti 6∈ Rv−i

j

]≤ Pr

v−i∼D−i

[bij(tij) ≤ Bb

ij(v−i)

∨ ∃k 6= j, s.t. ubik(tik) ≥ ubij(tij) and bik(tik) > Bbik(v−i)

]

≤ Prv−i∼D−i

[bij(tij) ≤ Bb

ij(v−i)

∨ ∃k 6= j, s.t. ubik(tik) ≥ ubij(tij)

]= Pr

v−i∼D−i

[bij(tij) ≤ Bb

ij(v−i)]· 1[∀k 6= j, ubij(tij) > ubik(tik)

]+ 1

[∃k 6= j, ubik(tik) ≥ ubij(tij)

]9

as the event that ∃k 6= j, s.t. ubik(tik) ≥ ubij(tij) is independent of v−i. Splitting theterm

1[∃k 6= j, ubik(tik) ≥ ubij(tij)

]=

(Pr

v−i∼D−i

[bij(tij) ≤ Bb

ij(v−i)]

+ Prv−i∼D−i

[bij(tij) > Bb

ij(v−i)])

1[∃k 6= j, ubik(tik) ≥ ubij(tij)

]we have

Prv−i∼D−i

[ti 6∈ Rv−i

j

]≤ Pr

v−i∼D−i

[bij(tij) ≤ Bb

ij(v−i)]

+ 1[∃k 6= j, ubik(tik) ≥ ubij(tij)

]Pr

v−i∼D−i

[bij(tij) > Bb

ij(v−i)]

So, we can decompose the virtual welfare

∑i

Eti∼Di

[∑j

πMij (ti)

(tij · Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]+ ϕ̃∗ij(tij) · Pr

v−i∼D−i

[ti ∈ Rv−i

j

])]

≤∑i

Eti∼Di

[∑j

πMij (ti)tij · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)

]Pr

v−i∼D−i

[bij(tij) > Bb

ij(v−i)]]

(NON-FAVORITE)

+∑i

Eti∼Di

[∑j

πMij (ti)tij Prv−i∼D−i

[bij(tij) ≤ Bb

ij(v−i)]]

(UNDER)

+∑i

Eti∼Di

[∑j

πMij (ti)ϕ̃∗ij(tij) Pr

v−i∼D−i

[ti ∈ Rv−i

j

]]

(SINGLE)

10

We further decompose NON-FAVORITE∑i

Eti∼Di

[∑j

πMij (ti)tij · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)] Prv−i∼D−i

[bij(tij) > Bb

ij(v−i)]]

=∑i

Eti∼Di

[∑j

πMij (ti) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)] · Ev−i∼D−i[tij · 1

[bij(tij) > Bb

ij(v−i)]]]

=∑i

Eti∼Di

∑j

πMij (ti) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)]

· Ev−i∼D−i[tij · 1

[bij(tij) > Bb

ij(v−i)]− p∗ij(tij, v−i) + p∗ij(tij, v−i)

]

=∑i

Eti∼Di

[∑j

πMij (ti) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)] · Ev−i∼D−i[u∗ij(tij, v−i) + p∗ij(tij, v−i)

]]where p∗ij and u∗ij are respectively the expost payment and expost utility of bidder ion a separate A-auction for item j. We have∑

i

Eti∼Di

[∑j

πMij (ti) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)] · Ev−i∼D−i[u∗ij(tij, v−i) + p∗ij(tij, v−i)

]]

=∑i

Eti∼Di

[∑j

πMij (ti) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)](ubij(tij) + pbij(tij)

)]

≤∑i

Eti∼Di

[∑j

πMij (ti)pbij(tij)

]

(OVER)

+∑i

Eti∼Di

[∑j

πMij (ti)ubij(tij) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)]

]

(SURPLUS)

We prove the following.

SINGLE ≤ SMyer

OVER ≤ SMyer

SURPLUS ≤ 3SMyer + 2EA

UNDER ≤ c · SMyer

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which gives that the revenue of the optimal multi-item auction is upper bounded by(c+ 5) · SMyer + 2 · EA, as desired.

SINGLE

∑i

Eti∼Di

[∑j

πMij (ti)ϕ̃∗ij(tij) Pr

v−i∼D−i

[ti ∈ Rv−i

j

]]

≤∑i

Eti∼Di

[∑j

πMij (ti)ϕ̃∗ij(tij)

]

=∑i

Eti∼Di

[∑j

Et−i∼D−i[xMij (ti, t−i)

]ϕ̃∗ij(tij)

]

=∑j

Et∼D

[∑i

xMij (t)ϕ̃∗ij(tij)

]≤∑j

Et∼D[maxiϕ̃∗ij(tij)

]=SMyer

OVER

∑i

Eti∼Di

[∑j

πMij (ti)pbij(tij)

]

≤∑i

Eti∼Di

[∑j

pbij(tij)

]=∑j

Revj(A)

≤SMyer

SURPLUS

12

First, rearranging the terms in SURPLUS, we have∑i

Eti∼Di

[∑j

πMij (ti)ubij(tij) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)]

]

≤∑i

Eti∼Di

[∑j

ubij(tij) · 1[∃k 6= j, ubik(tik) ≥ ubij(tij)]

]=∑i

∑j

Etij∼Dij[ubij(tij)Eti,−j∼Di,−j

[1[∃k 6= j, ubik(tik) ≥ ubij(tij)]

]]=∑i

∑j

Etij∼Dij[ubij(tij) Pr

ti,−j∼Di,−j

[∃k 6= j, ubik(tik) ≥ ubij(tij)

]]

Analyzing the relative size of each ubij(tij) (we abbreviate as ubij) will be fundamen-tal to bounding the surplus term. Intuitively, in the event that ubij is not too large,its contribution to the SURPLUS sum will be not too large and therefore boundable.When ubij is very large, bounding SURPLUS will still be possible due to the fact thatthe probability there exists an even larger ubik will be small. Fundamentally, we areanalyzing SURPLUS by splitting into an analysis of the CORE of the distributionand the TAIL. The pivotal point in our analysis is ubij ≥ rbi defined as follows.

rbij = maxx

(x Prtij∼Dij

[ubij(tij) ≥ x]

)rbi =

∑j

rbij

We decompose SURPLUS:∑i

∑j

Etij∼Dij[ubij(tij) Pr

ti,−j∼Di,−j

[∃k 6= j, ubik(tik) ≥ ubij(tij)

]]≤∑i

∑j

Etij∼Dij[ubij(tij) Pr

ti,−j∼Di,−j

[∃k 6= j, ubik(tik) ≥ ubij(tij)

]· 1[ubij ≥ rbi ]

](TAIL)

+∑i

∑j

Etij∼Dij[ubij(tij) · 1[ubij < rbi ]

](CORE)

TAILWe upper bound this term by rb =

∑i rbi . At the end of the analysis, we demonstrate

rb ≤ SMyer. First, by union bound

Prti,−j∼Di,−j

[∃k 6= j, ubik ≥ ubij

]≤∑k 6=j

Prtik∼Dik

[ubik ≥ ubij

]13

By the definition of rbik, we have rbik ≥ ubij Prtik∼Dik [ubik ≥ ubij], and so we can bound

TAIL ∑i

∑j

Etij∼Dij[ubij(tij) Pr

ti,−j∼Di,−j

[∃k 6= j, ubik(tik) ≥ ubij(tij)

]· 1[ubij ≥ rbi ]

]

≤∑i

∑j

Etij∼Dij

[1[ubij ≥ rbi ]

∑k 6=j

ubij(tij) Prtik∼Dik

[ubik ≥ ubij

]]

≤∑i

∑j

Etij∼Dij

[1[ubij ≥ rbi ]

∑k 6=j

rbik

]≤∑i

∑j

rbi · Etij∼Dij[1[ubij ≥ rbi ]

]≤∑i

∑j

rbi · Prtij∼Dij

[ubij ≥ rbi

]≤∑i

∑j

rbij

since rbij ≥ xPr[ubij ≥ x] for all x. Thus, TAIL ≤∑

i

∑j r

bij =

∑i rbi = rb, as desired.

COREWe show EA ≥ CORE

2− rb, which gives the bound 2rb + 2EA ≥ CORE. We let

cbij(tij) = ubij(tij) · 1[ubij(tij) < rbi ]

and rewriteCORE =

∑i

∑j

Etij∼Dij [cbij(tij)]

Now, we consider the EA mechanism with entry fee for bidder i

ebi =

(∑j

Etij [cbij(tij)]

)− 2rbi

This is a valid entry fee function as it is a constant: rbi is constant and cbij(tij) is onlyin terms of tij, but here we take expectation over tij.

We will show that each bidder i accepts the entry fee with probability at least 1/2.Bidder i accepts the entry fee iff his total interim utility over the auctions exceedsthe fee. Thus, if we can show

Prti∼Di

[∑j

ubij(tij) ≥ ebi

]≥ 1/2

14

then we know the expected revenue of EA (in equilibrium b) from entry fees alone isat least

1

2

∑i

ebi =1

2

∑i

(∑j

Etij [cbij(tij)]− 2rbi

)=

CORE

2− rb

as desired. We make use of the following lemma, originally proved in [3],

Lemma 4.2. Let x be a positive single dimensional random variable drawn from Fof finite support, such that for any number a, a · Prx∼F [x ≥ a] ≤ B where B is anabsolute constant. Then, for any positive number s, the second moment of the randomvariable xs = x · 1[x ≤ s] is upper bounded by 2B · s.

Applying this lemma with x = ubij(tij), B = rbij and s = rbi , we obtain E[(cbij(tij))2] ≤

2rbirbij. Since cbij’s are independent,

Var[∑j

cbij(tij)] =∑j

Var[cbij(tij)] ≤∑j

E[(cbij(tij))2] ≤ 2(rbi )

2

By Chebyshev, we know

Prti∼Di

[∑j

cbij(tij) ≤∑j

E[cbij(tij)]− 2rbi

]≤

Var[∑

j cbij(tij)

]4(rbi )

2≤ 1

2

and so ubij(tij) ≥ cbij(tij) gives the desired Pr[∑

j ubij ≥ ebi ] ≥ 1/2.

rWe can obtain revenue rb via selling the items separately in single-item EA auctions.For each item j, each bidder i can choose to pay an entry fee eij to access an A-auction on item j. Bidder i can choose whether or not to buy into the item j auctiontotally independently of his choice for the other auctions. We will again be usingghost bidders for all bidders who do not pay the entry fee. Thus, bidder i’s utility forentering the item j auction is ubij(tij), and he will pay the entry fee iff ubij(tij) ≥ eij.What is the maximum expected revenue we can obtain from this eij entry fee? It isexactly

maxeij

eij Prtij∼Dij

[ubij(tij) ≥ eij] = rbij

Thus, setting entry fees optimally on all items for all bidders, we obtain entry feerevenue

∑i

∑j r

bij = rb. The revenue obtained from these separate EA auctions on

each item is upper bounded by the revenue obtained from separate Myerson auctionson each item, giving rb ≤ SMyer, as desired.

15

In total we have,

SURPLUS ≤ TAIL + CORE ≤ (rb) + (2rb + 2EA) ≤ 3 · SMyer + 2 · EA

as desired.

UNDER

We rearrange UNDER to be in terms of xM : the expost allocation of the Mmechanism.

∑i

Eti∼Di

[∑j

πMij (ti)tij Prv−i∼D−i

[bij(tij) ≤ Bb

ij(v−i)]]

=∑i

Eti∼Di

[∑j

Et−i∼D−i[xMij (ti, t−i)

]tij(1− πbij(tij)

)]

=∑j

Et∼D

[∑i

xMij (t)tij(1− πbij(tij)

)]≤∑j

Et∼D[maxitij(1− πbij(tij)

)]≤∑j

c · Revj(Myer)

=c · SMyer

by the “type-loss trade off” property of the A-auction, as desired.

5 Type-loss trade off of common mechanisms

Throughout this section, we discuss the revenue of the posted price mechanismRev(PP ) (abbreviated PP ). In this mechanism, we globally announce some fixedprice and allocate to any bidder willing to pay it. Thus, we allocate if any typeexceeds the posted price, and the optimal posted price mechanism achieves revenue

PP = maxrr Prt∼D

[r ≤ max

iti

]Theorem 5.1. In a second-price auction, we have

ti Prt−i∼D−i

[ti ≤ max

j 6=itj

]≤ PP

16

for all bidders i and all possible types ti of bidder i.

Proof. We see

ti Prt−i∼D−i

[ti ≤ max

j 6=itj

]≤max

rr Prt−i∼D−i

[r ≤ max

j 6=itj

]≤max

rr Prt∼D

[r ≤ maxiti]

=PP

Corollary 5.2. In a multi-item auction setting where bidder’s utilities are additive,the revenue achieved by the revenue-optimal mechanism OPT is upper bounded

Rev(OPT ) ≤ 6Rev(SMyer) + 2Rev(EV CG)

Proof. VCG is a sealed-bid deterministic mechanism. From Theorem 5.1, we havethat

maxi

(ti(1− πbi (ti)

))= max

i

(ti Prv−i∼D−i

[ti ≤ max

j 6=ivj

])≤ PP

for all t ∈ T . And thus,

Et∼D[maxi

(ti(1− πbi (ti)

))]≤ PP ≤ Rev(Myer)

So, we can apply Theorem 4.1 with c = 1 to obtain the desired result.

The EVCG mechanism is much simpler than the BVCG mechanism of Yao andCai cite. In BVCG, separate VCG auctions are run on each item. The outcomes ofthese auctions are observed by all, and then the auctioneer offers each bidder i accessto his winnings if he pays both the VCG payments AND an entry fee ei(v−i) that is afunction of how all the other bidders played the auction. This entry fee function maybe quite complex to compute in practice. In EVCG, however, a FIXED entry fee ei ischarged at the start of the auction. Once paid, those who entered simply participatein separate VCG auctions. Even the “ghost bidders” here are not necessary. The onlyplace they were used in the analysis was to ensure that bidders did not deviate fromthe regular A-auction bid equilibrium and that their interim utility was not affectedby which bidders enter. However, in VCG, bidders will always play truthfully, and wecan show that interim utility (i.e. type − expected next highest bid) can only increasewhen a subset of bidders participate. This EVCG auction (together with SMyer) is

17

the simplest mechanism thus far proven to be approximately revenue optimal in themulti-item setting.

Theorem 5.3. In a first-price auction, under any bid equilibrium b, we have

Et∼D[maxi

(ti Prv−i∼D−i

[bi(ti) ≤ Bb

i (v−i)])]≤ 4PP

Proof. We note that

ti Prt−i∼D−i

[bi(ti) ≤ Bb

i (t−i)]

=ti

(1− Pr

t−i∼D−i

[bi(ti) > Bb

i (t−i)])

≤ti − (ti − bi(ti)) Prt−i∼D−i

[bi(ti) > Bb

i (t−i)]

=ti − ubi(ti)

Therefore,

Et∼D[maxi

(ti Prv−i∼D−i

[bi(ti) ≤ Bb

i (v−i)])]≤ Et∼D

[maxi

(ti − ubi(ti)

)]and we want to show

Et∼D[maxi

(ti − ubi(ti)

)]≤ 4PP

We use the following lemma

Lemma 5.4. Let F : R → [0, 1] be a function (in practice a CDF) and t ∈ R. Letu(t) = maxb(t− b)F (b), that is the area of the largest box that lies underneath the Fcurve with bottom right corner at (t, 0). Let a(t) = maxr≤t r(1 − F (r)), that is thearea of the largest box that lies above the F curve with top left corner at (0, 1). Then,√

u(t) +√a(t) ≥

√t

t0

1F

u(t)

a(t)

18

Note, we specifically consider the maximal above box with right border left of t,but considering all above boxes gives us a true, weaker statement. This is to ensurea ≤ t.

Proof. Say u(t) ≤ k1. Then, we must have F (x) ≤ k1t−x for all x ∈ [0, t]. Similarly,

saying a(t) ≤ k2, we must have F (x) ≥ 1− k2x

for all x ∈ [0, t]. And so, we must havek1t−x ≥

(1− k2

x

). From the first order condition, we see k1

t−x −(1− k2

x

)is minimized at

∂

∂x

(k1t− x

−(

1− k2x

))= − k1

(t− x)2+k2x2

= 0

∴ x√k1 = (t− x)

√k2, ∴ x∗ =

t√k2√

k1 +√k2

and so

minx

k1t− x

−(

1− k2x

)=

k1t− x∗

+k2x∗− 1

=k1t

(1

1−√k2√

k1+√k2

)+k2t

(1√k2√

k1+√k2

)− 1

=k1t·√k1 +

√k2√

k1+k2t·√k1 +

√k2√

k2− 1

=(√k1 +

√k2)

2

t− 1

And so, there is a gap between these upper and lower bound curves iff√k1+√k2 ≥

√t,

as desired.

Applying this lemma with F corresponding to the CDF of Bbi (t−i) over the ran-

domness of t−i ∼ D−i, we obtain

ubi(ti) ≥ (√ti −√ai)

2 ≥ ti − 2√aiti

where ai = maxr≤ti rPrt−i∼D−i [r < Bbi (t−i)]. We see

ai = maxr≤ti

r Prt−i∼D−i

[r < Bbi (t−i)]

≤ maxr≤ti

r Prt−i∼D−i

[r < maxj 6=i

tj]

≤ maxrr Prt∼D

[r < maxiti]

= PP

19

Thus,

Et∼D[maxi

(ti − ubi(ti)

)]≤Et∼D

[maxi

(2√aiti)]

≤2√PP · Et∼D

[√maxiti

]

We finish with the following lemma:

Lemma 5.5.

Et∼D[√

maxiti

]≤ 2√PP

Proof. We see

Pr

[√maxiti ≤ x

]= Pr

[maxiti ≤ x2

]≥ max

(1− PP

x2, 0

)since PP ≥ x2 Prt∼D [x2 ≤ maxi ti]. And so,

Et∼D[√

maxiti

]=

∫ ∞0

(1− Pr

[√maxiti ≤ x

])dx

≤∫ ∞0

(1−max

(1− PP

x2, 0

))dx

=

∫ √PP0

1dx+

∫ ∞√PP

PP

x2dx

= 2√PP

Thus,

Et∼D[maxi

(ti − ubi(ti)

)]≤ 2√PP · Et∼D

[√maxiti

]≤ 4PP

as desired.

20

Corollary 5.6. In a multi-item auction setting where bidders’ utilities are additive,if bidder’s type distributions lead to a unique equilibrium in the entry fee first priceauction (EFP), then the revenue achieved by the revenue-optimal mechanism M isupper bounded by

Rev(M) ≤ 9Rev(SMyer) + 2Rev(EFP ).

Proof. First price is a deterministic mechanism and, by assumption, EFP has a uniqueequilibrium strategy for these bidders. Thus, from Theorem 5.3, we have that

Et∼D[maxi

(ti(1− πi(bi(ti))))]

= Et∼D[maxi

(ti Prv−i∼D−i

[bi(ti) ≤ Bi(v−i)]

)]≤ 4PP

≤ 4Rev(Myer)

So, we can apply Theorem 4.1 with c = 4 to obtain the desired result.

Corollary 5.7. In a multi-item auction setting where bidder’s utilities are additive,if bidder’s type distributions lead to a unique equilibrium in the entry fee first priceauction (EFP), then the revenue achieved by the revenue-optimal mechanism M isupper bounded

Rev(M) ≤ 18e

e− 1Rev(SFP ) + 2Rev(EFP )

where SFP is the mechanism where each item is sold in a separate reserve first priceauction. That is, for each item, each bidder is informed of a specific fixed value theirbid must exceed in order to be considered for the auction.

Proof. This is simply due to the single-dimensional Price of Anarchy for AuctionRevenue result of Hartline [2]. It states that in a single-item auction

Rev(reserve first-price) ≥ e− 1

2eRev(Myer)

This directly implies SFP ≥ e−12eSMyer as these mechanisms involve running sepa-

rate reserve first-price and Myerson auctions respectively on each of the items. There-fore, the desired result is a direct consequence of Corollary 5.6.

This is the first multi-item revenue approximation result thus far proven forwinner-pays-bid mechanisms. Even the assumption of unique EFP equilibrium isnot that strong. Here, discuss stuff about how unique fp equilibrium almost alwaysoccurs, with proper citation. Additionally, start to discuss how, when we implement,this assumption will not be necessary as we will use online-learning for the auctioneer

21

that converges towards whatever equilibrium the bidder’s choose to play.

Even though these mechanisms are winner pays bid, EFP does not achieve credi-bility in the formal sense defined in [6]. Whenever a ghost bidder wins, the auctioneeralways has incentive to deviate and allocate the item to a real bidder. discuss formaldefinition, why it doesn’t apply to EFP because of ghost bidders, and how all-paywill work

We will be able to apply our arguments for first-price type-loss trade off to theall-pay setting using a technique introduced by Hartline, Hoy and Taggart [2]. It is ameans of viewing any mechanism as a winner-pays-bid mechanism. Representing theinterim utility in terms of interim allocation and interim payment, we have

ui(ti) = maxb

(πi(b)ti − pi(b)) = maxbπi(b)

(ti −

pi(b)

πi(b)

)We define βi(b) = pi(b)

πi(b)to be the equivalent bid for action b. In the first price auction,

this is the identity function. Now, we can view the interim utility of any auctionas a box optimization problem: the largest box with bottom-right corner at (ti, 0)and with top left corner on the parametric curve (βi(b), πi(b)). We want to applyLemma 5.4 on this curve, but first, we want to make a stronger claim. We note thatβi(b) is not necessarily monotonically increasing in b. This means we could have twobids b1 < b2 for which βi(b1) > βi(b2) and πi(b1) < πi(b2). Such b1 could never bea solution to the box optimization problem. We can make a slightly stronger claimabout interim utility with the introduction of the “equivalent threshold bid” curve.First, we define the “equivalent bid set” Si(x) = {b|βi(b) = x}. Then, we define theequivalent threshold bid

τi(x) =

{maxb∈Si(x) πi(b) if Si(x) 6= ∅0 otherwise

This is essentially an ironing of the parametric curve (βi(b), πi(b)) so that it is mono-tonically increasing.3 graphs: bid CDF, parametric curve, equiv thresh curveEven after this ironing, we still have ui(ti) = maxx τi(x)(ti−x). The b that maximizesπi(b)(ti − βi(b)) is surely the largest b with equivalent bid = βi(b). Otherwise, therewould exist a larger bid with identical (ti − βi(b)) but with greater allocation. Wereplicate our arguments for first-price with this curve.

Theorem 5.8. Let A be a single-item sealed bid auction such that, for every bidderi, the equivalent threshold bid curve satisfies the following:

maxrr(1− τi(r)) ≤ c · PP

22

Then, we have

Et∼D[maxi

(ti(1− πi(bi(ti))))]≤ 4√cPP

Proof. We note that

ti(1− πi(bi(ti))) ≤ ti − (πi(bi(ti))ti + pi(bi(ti))) = ti − ui

Therefore,

Et∼D[maxi

(ti(1− πi(bi(ti))))]≤ Et∼D

[maxi

(ti − ui)]

and we want to showEt∼D

[maxi

(ti − ui)]≤ 4√cPP

Applying Lemma 5.4 with F corresponding to τi, we obtain

ui ≥ (√ti −√ai)

2 ≥ ti − 2√aiti

where

ai = maxr≤ti

r(1− τi(r))

≤ maxrr(1− τi(r))

≤ c · PP

Thus,

Et∼D[maxi

(ti − ui)]

≤Et∼D[maxi

(2√aiti)]

≤2√c · PP · Et∼D

[√maxiti

]≤4√cPP

by Lemma 5.5, as desired.

Theorem 5.9. For any collection of bidders participating in an all-pay auction, wehave

maxrr(1− τi(r)) ≤ 4 · PP

23

Proof. In the all-pay auction, we have interim payment pi(b) = b. So, we haveequivalent bid βi(b) = b

πi(b). First, we show

maxrr(1− τi(r)) ≤ 4 ·max

rr(1− πi(r))

If we let ai = maxr r(1− πi(r)), then r ≤ ai1−πi(r) for all r. Therefore,

βi(r) =r

πi(r)≤ aiπi(r)(1− πi(r))

So, the parametric curve (βi(r), πi(r)) lies to the left of the parametric curve(

aiz(1−z) , z

).

In turn,(

aiz(1−z) , z

)lies to the left of parametric curve(

max

(ai

z(1− z),

2ai1− z

), z

).

We will now prove that for r > 4ai, τi(r) ≥ 1− 2air

. Recall that,

τi(x) = max πi(r) s.t. βi(r) = x

≥ max z s.t. max

(ai

z(1− z),

2ai1− z

)= x

≥ 1− 2aix.

Thus,

(1− τi(r)) ≤

{1 r ≤ 4ai2air

r > 4ai

Hence, maxr r(1− τi(r)) ≤ 4ai = 4 ·maxr r(1− πi(r)).

maxrr(1− πi(r)) = max

rr Prt−i∼D−i

[r ≤ Bi(v−i)]

≤ maxrr Prt−i∼D−i

[r < maxj 6=i

tj]

≤ maxrr Prt∼D

[r < maxiti]

= PP

24

since bi(ti) ≤ ti in any all-pay equilibrium strategy. We conclude that

maxrr(1− τi(r)) ≤ 4 · PP.

The following corollaries directly follow by applying Theorem 5.8 and Theorem5.9 in Theorem 4.1.

Corollary 5.10. In a multi-item auction setting where bidders’ utilities are additive,if bidders’ type distributions lead to a unique equilibrium in the entry fee all-payauction (EAP), then the revenue achieved by the revenue-optimal mechanism M isupper bounded by

Rev(M) ≤ 13Rev(SMyer) + 2Rev(EAP ).

Corollary 5.11. In a multi-item auction setting where bidders’ utilities are additive,if bidders’ type distributions lead to a unique equilibrium in the entry fee all-payauction (EAP), then the revenue achieved by the revenue-optimal mechanism M isupper bounded by

Rev(M) ≤ 26e

e− 1Rev(SFP ) + 2Rev(EAP ).

We know that SFP is a credible mechanism [6]. The entry-fee all-pay auction(EAP) is also credible. This is because the entry-fee paid by the bidders is a sunkcost and once they pay the entry-fee, they participate in separate all-pay auctionsfor each item. It was proved in [6] that single-dimensional all-pay auction is credible.Therefore, all we need to show is that the auctioneer does not have any incentiveto deviate when the item is allocated to a ghost bidder. This is true because in anall-pay auction, the auctioneer is indifferent to who the item is allocated to, underthe assumption that the item to be sold is time-sensitive and unallocated item (itemallocated to a ghost bidder) cannot be resold. Thus, the above corollary shows thatthe revenue of the optimal mechanism can be approximately achieved by a crediblemechanism.

Following the result in [2] which states that revenue of an all-pay mechanism with2-duplicates, i.e., at least two bidders have values drawn from each distribution, givesa 4e/(e − 1) approximation to the revenue of the optimal mechanism, we get thefollowing result.

25

Corollary 5.12. In a multi-item auction setting where bidders’ utilities are additive,if bidders’ type distributions are such that at least two bidders have values drawn fromeach distribution, and lead to a unique equilibrium in the entry fee all-pay auction(EAP), then the revenue achieved by the revenue-optimal mechanism M is upperbounded

Rev(M) ≤ 52e

e− 1Rev(SAP ) + 2Rev(EAP )

where SAP is the mechanism where each item is sold in a separate all-pay auction.

6 Implementation

Discuss online learning things here

7 Additional Corollaries

Discuss welfare loss of fp here

References

[1] Yang Cai, Nikhil R. Devanur, S. Matthew Weinberg (2018). “A Duality-BasedUnified Approach to Bayesian Mechanism Design”, arXiv:1812.01577 [cs.GT]

[2] Jason Hartline, Darrell Hoy, Sam Taggart (2019). “Price of Anarchy for AuctionRevenue”, arXiv:1404.5943 [cs.GT]

[3] Moshe Babaioff, Nicole Immorlica, Brendan Lucier, and S. Matthew Wein-berg (2014). “A Simple and Approximately Optimal Mechanism for an AdditiveBuyer”. In the 55th Annual IEEE Symposium on Foundations of Computer Sci-ence (FOCS)

[4] Tim Roughgarden, Vasilis Syrgkanis, Eva Tardos (2016). “The Price of Anarchyin Auctions”, arXiv:1607.07684 [cs.GT]

[5] Bernard Lebrun, Uniqueness of the equilibrium in first-price auctions,Games and Economic Behavior, Volume 55, Issue 1, 2006, Pages131-151, ISSN 0899-8256, https://doi.org/10.1016/j.geb.2005.01.006.(http://www.sciencedirect.com/science/article/pii/S0899825605000540)

[6] Mohammad Akbarpour and Shengwu Li. “Credible Mechanisms” EC (2018).

26

[7] Aviad Rubinstein and S. Matthew Weinberg. Simple mechanisms for a subadditivebuyer and applications to revenue monotonicity. In Proceedings of the SixteenthACM Conference on Economics and Computation, EC 15, Portland, OR, USA,June 15-19, 2015, pages 377394, 2015. 1.1, 1.3.3, 2, 1

[8] Constantinos Daskalakis and S. Matthew Weinberg. Symmetries and OptimalMulti- Dimensional Mechanism Design. In the 13th ACM Conference on ElectronicCommerce (EC), 2012. 2, 1

A. Discrete to Continuous

appendix package? In this appendix, we showref theorem from earlier rather than writing new theorem

Theorem 7.1. Consider a multi-item auction setting with type space T ⊆ Rn×m,continuous atomless valuation distribution D, and additive bidders. Say that each Tijis bounded: Tij ⊆ [0, H].

For all i and each v−i ∈ T−i, let Rv−i0 , R

v−i1 , · · · , Rv−i

m be a “preference” partition ofthe type space Ti. That is, for each item j and each v−i, there exists a non-decreasingutility function Uv−ij : Tij → R ≥ 0 such that, for all j 6= 0

ti ∈ Rv−ij ⇔Uv−ij (tij) ≥ Uv−ik (tik) ∀k 6= j

Uv−ij (tij) > Uv−ik (tik) ∀k < j

Uv−ij (tij) > 0

andti ∈ Rv−i

0 ⇔ Uv−ij (tij) = 0 ∀j

That is, the region containing ti is determined by the item j which ti most prefers,breaking ties lexicographically. The 0 region contains all ti with 0 utility on all items.

Let πM represent a feasible interim allocation. That is, for all i, j, there ex-ists some mapping xMij : Rn×m → [0, 1] so that πMij (ti) = Et−i∼D−i

[xMij (ti, t−i)

]and∑

i xMij (t) ≤ 1 for all items j and t ∈ T . Let OPT be the revenue-optimal BIC mecha-

nism with values drawn from D. Then, the expected revenue of OPT is upper boundedby

Rev(OPT ) ≤∑i

Eti∼Di

[∑j

πMij (ti)

(tij · Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]+ ϕ̃∗ij(tij) · Pr

v−i∼D−i

[ti ∈ Rv−i

j

])]

27

where ϕ̃∗ij(tij) = max(ϕ̃ij(tij), 0) and ϕ̃ij(tij) represents Myerson’s continuous ironedvirtual value for the distribution Dij.

Our proof closely parallels arguments presented in [1].

Proof. Our approach will be to consider a discretization of D: D+,ε. We define D+,εij

to first sample tij ∼ Dij and then output t+,εij = ε2 ·dtij/ε2e. We see that D+,εij will have

finite support, as the support of Dij is bounded ∈ [0, H]. Our approach is as follows.Due to the coupling of samples from D and D+,ε, we will be able to show that therevenue-optimal mechanism OPT for values drawn from D achieves approximatelythe same revenue as the revenue optimal mechanism OPT+,ε for values drawn fromD+,ε: RevD(OPT ) ≈ RevD

+,ε

(OPT+,ε). Since D+,ε has finite support, we will be ableto apply theorem 2.1 bounding the revenue of OPT+,ε by its virtual welfare. Then,one last argument on the coupled distributions will give that the virtual welfare upperbound is approximately the desired result.

We make use of the following theorem.

Theorem 7.2. [[7], [8]] Let M↓ be any BIC mechanism for additive bidders withvalues drawn from distribution D↓. For all i, let D↓i and D↑i be any two distributionswith coupled samples t↓i and t↑i such that t↑i · xi ≥ t↓i · xi for all feasible allocationsx ∈ F . If δi = t↑i − t

↓i , then for any ε > 0, there exists a BIC mechanism M↑ such

that

RevD↑(M↑) ≥ (1− ε)

(RevD

↓(M↓)− V AL(δ)

ε

)where V AL(δ) denotes the expected welfare of the VCG allocation when bidder is typeis drawn according to the random variable δi.

Using this theorem, we will be able to bound the gap between RevD(OPT ) and

RevD+,ε

(OPT+,ε). We introduce D−,εij , defined similarly to D+,εij , that first samples

tij ∼ Dij and then outputs t−,εij = ε2 · (dtij/ε2e − 1). Also, define OPT−,ε to be therevenue-optimal mechanism for values drawn from D−,ε. We will apply Theorem 7.2with D as D↓ and D+,ε as D↑ and OPT as M↓ as well as with D−,ε as D↓ andD as D↑ and OPT−,ε as M↓. In both cases, due to the coupling, we will have thenecessary t↑i ·xi ≥ t↓i ·xi for all x. Additionally, we will have δij ≤ ε2 for all i, j. Thus,V AL(δ) ≤ mε2 as the welfare contribution of any one item is at most ε2 for types δi.So, applying this theorem in these two settings gives, for some mechanisms M↑

1 ,M↑2 ,

RevD+,ε

(OPT+,ε) ≥ RevD+,ε

(M↑1 ) ≥ (1− ε)(RevD(OPT )−mε)

RevD(OPT ) ≥ RevD(M↑2 ) ≥ (1− ε)(RevD−,ε(OPT−,ε)−mε)

28

Lastly, note that

RevD+,ε

(OPT+,ε) = RevD−,ε

(OPT−,ε) +mε

as every buyer values every item at exactly ε more in D+,ε versus D−,ε. For everyBIC mechanism with values drawn from D−,ε, there is an analogous mechanism forvalues D+,ε in which every bidders payment increases by exactly ε times the numberof items they are expost allocated. Thus, we have

RevD(OPT ) ∈

[(1− ε)(RevD+,ε

(OPT+,ε)− 2mε),RevD

+,ε

(OPT+,ε)

1− ε+mε

]

so there is a typo in the Cai paper which gives[(1− ε)RevD−,ε(OPT−,ε)−mε, Rev

D+,ε

(OPT+,ε)

1− ε+

mε

1− ε

]

I went to the original RW paper to see what the real theorem stated and therewas this extra η term i didn’t understand. Anyways, we should double check this:https://arxiv.org/pdf/1501.07637.pdf Theorem 5.2 Page A20

So, as ε → 0, we achieve discrete type distributions D+,ε for which there existsmechanisms OPT+,ε with revenue arbitrarily close to RevD(OPT ). Since we have

discretized the type space, we can apply Theorem 2.1 and bound RevD+,ε

(OPT+,ε)

≤∑i

∑ti∈T+

i

∑j

fD+,ε

i (ti)πOPT+,ε

ij (ti)

(tij · Pr

v−i∼D+,ε−i

[ti 6∈ Rv−i

j

]+ ϕ̃+

ij(tij) · Prv−i∼D+,ε

−i

[ti ∈ Rv−i

j

])

where T+ is the discretized type space, fD+,ε

i is the PMF of D+,εi , ϕ̃+

ij(tij) is Myerson’s

discrete ironed virtual value for the distribution D+,εij , and the R

v−ij s are a preference

partition of the continuous type space T , which will also be a preference partition onthe discrete subset T+.

To use this theorem, we must show that preference partitions are always upwards-closed partitions, which is true due to the following argument.

Let tij, t′ij ∈ Tij with t′ij > tij. Say for some type vector ti = (ti1, · · · , tij, · · · , tim)

we have ti ∈ Rv−ij in a preference partitionRv−i . We want to show t′i = (ti1, · · · , t′ij, · · · , tim) ∈

Rv−ij . We see

29

ti ∈ Rv−ij ⇒Uv−ij (tij) ≥ Uv−ik (tik) ∀k 6= j

Uv−ij (tij) > Uv−ik (tik) ∀k < j

Uv−ij (tij) > 0

⇒Uv−ij (t′ij) ≥ Uv−ik (tik) ∀k 6= j

Uv−ij (t′ij) > Uv−ik (tik) ∀k < j

Uv−ij (t′ij) > 0

⇒ t′i ∈ Rv−ij

since Uv−ij is non-decreasing, as desired.

Now, returning to the virtual value term, instead of sampling directly from thediscrete distribution, we can instead think about sampling ti ∼ Di from the continuousdistribution and then letting t+i be the rounded discrete type in terms of ti. That is,t+ij = ε2 · dtij/ε2e. Is this notation confusing? Is it necessary to write t+i (ti) We have

∑i

∑t+i ∈T

+i

∑j

fD+,ε

i (t+i )πOPT+,ε

ij (t+i )

(t+ij · Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]+ ϕ̃+

ij(t+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

])

=∑i

∫ti∈Ti

∑j

fDi (ti)πOPT+,ε

ij (t+i )

(t+ij · Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]+ ϕ̃+

ij(t+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

])dti

where fDi is the PDF of Di. First, we define πMij (ti) = πOPT+,ε

ij (t+i ). We see that thissatisfies the desired properties of a feasible interim allocation as OPT+,ε is a feasiblemechanism and sampling ti from the continuous distribution and then rounding isidentical to sampling from the discrete distribution. We rewrite

30

∑i

∫ti∈Ti

∑j

fDi (ti)πMij (ti)

(t+ij · Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]+ ϕ̃+

ij(t+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

])dti

=∑i

Eti∼Di

[∑j

πMij (ti)

(t+ij · Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]+ ϕ̃+

ij(t+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

])]

=∑i

∑j

Eti∼Di[πMij (ti)t

+ij Prv−i∼D−i

[t+i 6∈ R

v−ij

]]+∑i

∑j

Eti∼Di[πMij (ti)ϕ̃

+ij(t

+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

]]

We see

Eti∼Di[πMij (ti)t

+ij Prv−i∼D−i

[t+i 6∈ R

v−ij

]]= Eti∼Di

[πMij (ti)(t

+ij − tij) Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]]+ Eti∼Di

[πMij (ti)tij

(Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]− Pr

v−i∼D−i

[ti 6∈ Rv−i

j

])]+ Eti∼Di

[πMij (ti)tij Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]

We can bound Eti∼Di[πMij (ti)(t

+ij − tij) Prv−i∼D−i

[t+i 6∈ R

v−ij

]]≤ ε2 as πMij (ti) ≤ 1,

t+ij − tij ≤ ε2, and Prv−i∼D−i[t+i 6∈ R

v−ij

]≤ 1. Similarly, we bound

Eti∼Di[πMij (ti)tij

(Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]− Pr

v−i∼D−i

[ti 6∈ Rv−i

j

])]≤ H · Eti∼Di

[Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]− Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]= H · Eti,v−i∼D

[1[t+i 6∈ R

v−ij

]− 1

[ti 6∈ Rv−i

j

]]≤ H · Eti,v−i∼D

[1[ti ∈ Rv−i

j ∧ t+i 6∈ Rv−ij

]]= H · Ev−i∼D−i

[Prti∼Di

[ti ∈ Rv−i

j ∧ t+i 6∈ Rv−ij

]]= H · Ev−i∼D−i

[∑k 6=j

Prti∼Di

[ti ∈ Rv−i

j ∧ t+i ∈ Rv−ik

]]

31

In order to have ti ∈ Rv−ij and t+i ∈ R

v−ik , we must have Uv−ij (tij) ≥ Uv−ik (tik) and

Uv−ij (t+ij) < Uv−ik (t+ik) in the event j < k. Similarly, we must have Uv−ij (tij) > Uv−ik (tik)

and Uv−ij (t+ij) ≤ Uv−ik (t+ik) in the event j > k. We assume WLOG j < k as the argu-

mentation is symmetric in both cases.

We think about the two-dimensional plane Tij×Tik of possible values (tij, tik). Wecan view the discretization (t+ij, t

+ik) as a division of this plane into a grid of squares of

side length ε2. Here, (t+ij, t+ik) represents the upper corner of whichever square (tij, tik)

belongs to. In order to have Uv−ij (tij) ≥ Uv−ik (tik) and Uv−ij (t+ij) < Uv−ik (t+ik), we must

have the “indifference curve” pass through the square containing (tij, tik). However,only so many squares will contain a piece of this indifference curve, enabling us tobound the probability of such an event.

Formally, we define the indifference curve I to be the set of points{(Ij, Ik) ∈ Tij × Tik

∣∣∣∣∣ ∃u s.t. Uv−ij (Ij) = Uv−ik (Ik) = u

∧ Ij = min{tj|Uv−ij (tj) = u

}or Ik = max {tk|Uv−ik (tk) = u}

}

Adjust this definition to account for the fact that U could be discontinuous. Thisshould just be the border between the two regions. The only property we need of thisis that it goes up and right, which is certainly true due to the upwards-closedness.The second part of this definition deals with the case when there exists a whole in-terval of values in Tij with utility u and a whole interval of values in Tik with utilityu. In such a case, the set of points in the Tij × Tik plane with equal utility wouldcontain an entire rectangle. However, all the points in the rectangle would belong toregion j due to the lexicographic tie-breaking. Thus, the actual border between theregions would trace along the edge of the rectangle, which is what is described in thesecond condition.

Now, we can index the grid of squares as an ordered pair (x, y) where x and yare integers in the range [1, H/ε2]. Square (x, y) contains the points ((x− 1)ε2, xε2]×((y − 1)ε2, yε2]. I claim that, for any two squares (x1, y1), (x2, y2) containing piecesof the indifference curve, we must have x1 + y1 6= x2 + y2. Assume for the sakeof contradiction that x1 + y1 = x2 + y2 and WLOG x1 < x2, y1 > y2. Say wehad (Ij,1, Ik,1) in square (x1, y1) with Uv−ij (Ij,1) = Uv−ik (Ik,1) = u1 and (Ij,2, Ik,2)in square (x2, y2) with Uv−ij (Ij,2) = Uv−ik (Ik,2) = u2. We cannot have u1 > u2 asthat would imply Uv−ij (Ij,1) > Uv−ij (Ij,2). However, Ij,1 < Ij,2 since x1 < x2 andUv−ij is non-decreasing, a contradiction. Similarly, we cannot have u1 < u2 as thatwould imply Uv−ik (Ik,1) < Uv−ik (Ik,2). However, Ik,1 > Ik,2 since y1 > y2 and Uv−ik

is non-decreasing, a contradiction. Thus, we must have u1 = u2. But, this implies

32

Ij,2 6= min{tj|Uv−ij (tj) = u2

}since Ij,1 < Ij,2 and Ik,2 6= max {tk|Uv−ik (tk) = u2} since

Ik,1 > Ik,2, a contradiction.

Thus, the indifference curve can only pass through one square along the diagonalof squares (x, y) satisfying x + y = u. Since x, y are integers in the range [1, H/ε2],we have x+ y ∈ [2, 2H/ε2]. Therefore, the indifference curve passes through at most2H/ε2 − 1 squares.

Then, since Di is an atomless distribution, the probability density function of Dij

is bounded for every j. Is this correct or can atomless have infinite density at places.Even if it can asymptote, I suspect some limit argument work here. Thus, there issome finite P that upper bounds the PDF of the joint distribution Dij×Dik for everypair j, k. So, the probability of (tij, tik) belonging to any specific square is at mostP (ε2)2. Therefore, the probability that (tij, tik) belongs to a square containing a pieceof the indifference curve is ≤ 2HPε2. So,

Eti∼Di[πMij (ti)tij

(Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]− Pr

v−i∼D−i

[ti 6∈ Rv−i

j

])]≤ H · Ev−i∼D−i

[∑k 6=j

Prti∼Di

[ti ∈ Rv−i

j ∧ t+i ∈ Rv−ik

]]

≤ H · Ev−i∼D−i

[∑k 6=j

2HPε2

]≤ 2mH2Pε2

So, in total, we bound

Eti∼Di[πMij (ti)t

+ij Prv−i∼D−i

[t+i 6∈ R

v−ij

]]≤Eti∼Di

[πMij (ti)tij Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]+ ε2(2mH2P + 1)

Similarly, we decompose the other term

33

Eti∼Di[πMij (ti)ϕ̃

+ij(t

+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

]]= Eti∼Di

[πMij (ti)(ϕ̃

+ij(t

+ij)− ϕ̃ij(t+ij)) Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]]+ Eti∼Di

[πMij (ti)ϕ̃ij(t

+ij)

(Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]− Pr

v−i∼D−i

[ti 6∈ Rv−i

j

])]+ Eti∼Di

[πMij (ti)ϕ̃ij(t

+ij) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]

where ϕ̃ij represents Myerson’s continuous ironed virtual value for the distributionDij. In [1], they prove that the discrete virtual value converges to the continuousvirtual value for increasingly fine discretizations (Observation 9), limε→0 ϕ

+ij(v) =

ϕij(v) for all v. We can easily extend this argument to ironed virtual values as ϕ+ij

converges to ϕij at all points. So,

limε→0

ϕ̃+ij(v) = lim

ε→0maxv≤t

ϕ+ij(v) = max

v≤tϕij(v) = ϕ̃ij(v)

Thus,

Eti∼Di[πMij (ti)(ϕ̃

+ij(t

+ij)− ϕ̃ij(t+ij)) Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]]= oε(1)

and from our previous argument

Eti∼Di[πMij (ti)ϕ̃ij(t

+ij)

(Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]− Pr

v−i∼D−i

[ti 6∈ Rv−i

j

])]≤ 2mH2Pε2

as ϕ̃ij(t+ij) ≤ t+ij. Thus, all that remains to show is that we can replace the ϕ̃ij(t

+ij)

term in Eti∼Di[πMij (ti)ϕ̃ij(t

+ij) Prv−i∼D−i

[ti 6∈ Rv−i

j

]]with a ϕ̃ij(tij). Here, we make use

of the relaxation of virtual value to positive virtual value: ϕ̃∗ij(tij) = max(ϕ̃ij(tij), 0).Clearly, this upper bounds the virtual value. It will give us a weaker result, but stilla meaningful bound. We have

34

Eti∼Di[πMij (ti)ϕ̃ij(t

+ij) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]≤Eti∼Di

[πMij (ti)ϕ̃

∗ij(t

+ij) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]=Eti∼Di

[πMij (ti)(ϕ̃

∗ij(t

+ij)− ϕ̃∗ij(tij)) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]+Eti∼Di

[πMij (ti)ϕ̃

∗ij(tij) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]We have

Eti∼Di[πMij (ti)(ϕ̃

∗ij(t

+ij)− ϕ̃∗ij(tij)) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]≤ Eti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(tij)

]≤ Eti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij)

]where t−ij = ε2 · (dtij/ε2e − 1). This is true since the IRONED virtual value function isnon-decreasing. We can view the discretization as a breaking up of Tij into segmentsof length ε2. ϕ̃∗ij(t

+ij) − ϕ̃∗ij(t−ij) will be the difference in the endpoints of the interval

containing tij. We break into cases:

Eti∼Di[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij)

]= Eti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij)|ϕ̃∗ij(t+ij)− ϕ̃∗ij(t−ij) ≤ ε

]· Prti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij) ≤ ε

]+ Eti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij)|ϕ̃∗ij(t+ij)− ϕ̃∗ij(t−ij) > ε

]· Prti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij) > ε

]≤ ε · 1 +H · Pr

ti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij) > ε

]Here, we make use of the fact that ϕ̃∗ij is a non-decreasing function with range ⊆ [0, H]as ϕ̃ij(tij) ≤ tij. Due to this fact, there can only be at most H/ε intervals withendpoints differing by at least ε. Again making use of the fact that Dij is atomlessand there is some finite upper bound P on its density function, we can argue thatthe probability of tij ∼ Dij belonging to any specific interval is ≤ Pε2. So,

Prti∼Di

[ϕ̃∗ij(t

+ij)− ϕ̃∗ij(t−ij) > ε

]≤ (H/ε) · Pε2 = HPε

and so,

Eti∼Di[πMij (ti)(ϕ̃

∗ij(t

+ij)− ϕ̃∗ij(tij)) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]≤ (H2P + 1)ε

35

Putting all the pieces together, we have

Eti∼Di[πMij (ti)ϕ̃

+ij(t

+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

]]≤ Eti∼Di

[πMij (ti)ϕ̃

∗ij(tij) Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]]+oε(1)

and

∑i

Eti∼Di

[∑j

πMij (ti)

(t+ij · Pr

v−i∼D−i

[t+i 6∈ R

v−ij

]+ ϕ̃+

ij(t+ij) · Pr

v−i∼D−i

[t+i ∈ R

v−ij

])]

≤mn · oε(1) +∑i

Eti∼Di

[∑j

πMij (ti)

(tij · Pr

v−i∼D−i

[ti 6∈ Rv−i

j

]+ ϕ̃∗ij(tij) · Pr

v−i∼D−i

[ti ∈ Rv−i

j

])]

giving the desired upper bound as ε→ 0.

36