A A. ONLINE SUPPLEMENTAL MATERIALS FOR A B THEORY ...home.uchicago.edu/hickmanbr/uploads/HowEfficient... · Appendix A. ONLINE SUPPLEMENTAL MATERIALS FOR How Efﬁcient are Decentralized

HOW EFFICIENT ARE DECENTRALIZED AUCTION PLATFORMS? 1

Appendix A. ONLINE SUPPLEMENTAL MATERIALS FORHow Efficient are Decentralized Auction Platforms?

by Aaron L. Bodoh-Creed, Jörn Boehnke, and Brent R. Hickman

THEORY: TECHNICAL PROOFS APPENDIX

Due to the length of the technical proofs, we have divided our appendices into sec-tions. Appendix A.1 provides some useful results from the prior literature and provessome miscellaneous results from this paper. Appendix A.2 provides the proof of theexistence of an SCE of the limit model (Proposition 2.4). Appendix A.3 describes thetechnical details of the relationship between the large finite games and the limit model(Proposition 6.3). Each section is proceeded by a brief summary of the arguments con-tained therein.

A.1. Preliminary Results. Before beginning our main arguments, we define the Kol-mogorov metric for random variables over R, where F and G are the CDFs of the randomvariables.

dK(F, G) = supx‖F(x)− G(x)‖

Note that when either G or F is continuous (i.e., the underlying measure is atomless),then dK metricizes the weak-∗ topology.

Remark 1. Consider a random variable X : Ω → R with CDF F(x). For N i.i.d. realizations,x1, ..., xN, drawn from F, denote the N realization empirical CDF as FN(x). Then from theGlivenko-Cantelli theorem we have

supx∈R

‖FN(x)− F(x)‖ → 0 almost surely as N → ∞

Now we provide a quick proof of our claim that the equilibria of static auctions canbe translated directly into equilibria of the dynamic auction market that uses the samepayment rule.

Proposition 6.2. Suppose that if δ = 0 there exists an equilibrium σ =(θ, O

). Then we

can define the equilibrium σ = (θ,O) when δ > 0 as

θ(v, CNt , FN

V,t, GNR,t) = θ(v− δEN

t

[VN(v, CN

t+1, FNV,t+1, GN

R,t+1|σ)]

, CNt , FN

V,t, GNR,t)

O(v, CNt , FN

V,t, GNR,t) = O(v− δEN

t

[VN(v, CN

t+1, FNV,t+1, GN

R,t+1|σ)]

, CNt , FN

V,t, GNR,t

Proof. For parsimony, we have supressed all of the arguments for the functions χN and

ρN other than the bid and all of the arguments of VN. First consider the problem facing a

2 HOW EFFICIENT ARE DECENTRALIZED AUCTION PLATFORMS?

buyer with value v who has entered the market in the static (δ = 0) game and is choosingthe optimal bid:

b(v) = argmaxb≥0

χN(b)v− ρN(b)− κ (48)

Now consider the problem of a bidder with value v with a time discount factor δ > 0facing the same auction mechanism:

O(v) = argmaxb≥0

χN(b)v− ρN(b) +(

1− χN(b))

δE[VN]− κ.

A few algebraic rearrangements yields:

O(v) = argmaxb≥0

χN(b)[v− δE

[VN]]− ρN(b) + δE

[VN]− κ.

Since E[VN] and κ are independent of the choice of b, an equivalent problem is:

O(v) = argmaxb≥0

χN(b)[v− δE

[VN]]− ρN(b). (49)

Clearly any solution to problem 49 for a buyer with value v is also a solution to problem

48 for a buyer with value v = v− δE[VN] and vice versa.

In the static game the buyer will choose to enter the market and bid if and only if:

χN(b)v− ρN(b)− κ ≥ 0 (50)

The analogous equation for the dynamic game is:

χN(b)v− ρN(b) +(

1− χN(b))

δE[VN]− κ ≥ δE

[VN]

After some algebra, we find that this is equivalent to:

χN(b)[v− δE

[VN]]− ρN(b)− κ ≥ 0 (51)

Again, a comparison of equations 50 and 51 implies that an agent with value v enters

the dynamic market if and only if an agent with value v = v− δE[VN] enters the static

market in this period.

A.2. Proofs from Section 2. The highlight of this section is an equilibrium existenceresult using a fixed point argument. Our argument consists of several parts. First weargue that the state variables, (C, FV , GR), live in a compact space that we denote Γ. Wemust show that the best responses to the state variables and the transition operator forthe state variables are continuous and closed with respect to Γ.


As part of our argument we must prove that the bid strategies we consider have, statedinformally, both a lower and an upper bound on their derivatives so that the induceddistributions of bids do not admit atoms. Stated formally we need to show that the bestresponse function is closed and continuous over the space CM[0, 1|ϕ], ϕ ∈ (0, 1), thatcontains all continuous, strictly increasing mappings from [0, 1] into [0, 1] that satisfy

f (v)− f (v′) ∈[

ϕ(v− v′),v− v′

ϕ

], ∀v > v′.

The set CM[0, 1|ϕ] is equicontinuous and bounded over a compact domain. From theArzelá-Ascoli theorem, CM[0, 1|ϕ] is compact. The strategy space we need to considercan be defined as σ = (e, β) ∈ Ξ = [0, 1] × CM[0, 1|ϕ]. Once we show that our bestreponse and state transition operators are closed and continuous, a straightforward fixedpoint argument applies.

Our proof characterizes two functions. First, we define the best response dynamics,

BR (C, FV , GR, σ) = σ =(

e, β)

, that describes how each type of buyer responds given

the belief that (C, FV , GR) is stationary and all other agents use the strategy σ. We showthat BR is continuous in (C, FV , GR, σ) and closed within Ξ.

The function T(C, FV , GR|σ) = (C, FV , Gr) describes the transitions of the aggregatestates. We show that T is continuous in (C, FV , GR, σ) for (C, FV , GR) ∈ Γ and σ ∈ Ξ, andT (·|σ) : Γ→ Γ as long as β ∈ CM[0, 1|ϕ]. Once we have proven T and BR are continuous,closed operators over compact spaces, we define the total operator L : Γ× Ξ → Γ× Ξwhere

L(C, FV , GR, σ) = (C, FV , σ)

BR (σ, C, FV , GR, σ) = σ

T(C, FV , GR|σ) = (C, FV)

A straightforward application of Schauder’s fixed point theorem to the operator L givesus existence of a stationary equilibrium. However, proving T and BR are continuousand closed operators over compact spaces requires many small steps.

Several of our results require that FV and GR admit bounded PDFs, fV and gr, except

at r = 0.45 For notational purposes, we let Q [0, q] denote the space of measures over[0, 1] that admit pdfs bounded from above by q > 0.

A.2.1. Continuity of Best Responses. In this section we prove that under certain conditionson the aggregate state, the best responses are continuous as required and lie in CM[0, 1|ϕ]for some choice of ϕ ∈ (0, 1). Our first result is that the measure of entering agents and

45This may fail to be true if low value bidders accumulate in the market by entering even though theyhave arbitrarily low probabilities of consummating a trade.


the distribution of bids are continuous in the underlying economy. To prove this, weneed the following intermediate result. We add the “open neighborhood” qualifier tothe statement of Lemma A.1 so that our result applies to empirical measures that areclose (in the weak-* sense) to a nonatomic measure with a bounded PDF. We exclusivelywork with measures with bounded PDFs in Section A.2, but Section A.3 requires us towork with empirical measures near the SCE steady-state distributions.

Lemma A.1. Consider any increasing function f ∈ CM[0, 1|ϕ], ϕ ∈ (0, 1), and let Z be anatomless CDF over R that admits a pdf that is bounded from above by M. Then Y(s) =

Z( f−1(s)) is uniformly continuous in f ∈ CM[0, 1|ϕ], ϕ ∈ (0, 1) and an open neighborhood ofZ.

Proof. Consider f , f ∈ CM[0, 1|ϕ]. For any x and γ, υ ∈ R++ we can write

f (x + γ/υ) > f (x) > f (x− γ/υ)

so for any y we have f−1(y) ∈ [ f−1(y)− γ/υ, f−1(y) + γ/υ], which implies

Z(

f−1(y))∈[

Z(

f−1(y)− γ/υ)

, Z(

f−1(y) + γ/υ)]

Therefore ∥∥∥Z(

f−1(y))− Z

(f−1(y)

)∥∥∥ <2Mγ

υ

Since this bound holds uniformly over x, our result regarding continuity with respect tof is proven. Continuity with respect to Z is immediate.

Lemma A.1 yields the following result.

Lemma A.2. The distribution of bids, GB and the measure of entering buyers C are continuousin C, FV , and σ = (e, β) provided that the entry cutoff point e < 1, FV admits a bounded pdf,and β ∈ CM[0, 1|ϕ], ϕ ∈ (0, 1).

Proof. The distribution of entering buyers is

FEV(x) =

FV(x)− FV(e)1− FV(e)

for x ≥ e, 0 otherwise

and the measure of entering buyers is C = C [1− FV(e)]. Note that FEV and C are contin-

uous in (C, FV). The distributions of bids can be described as

GB(b) = FEV(x)(β−1(b))

Using Lemma A.1, we find that GB is continuous in β. Finally, if FV is atomless, thenFV(e) varies continuously in e, so (C, GB) is also continuous in e.


Throughout this paper we maintain an assumption on the bidder arrival process

π(k; λ) that E[K2] < ∞. Recall that a bidder’s beliefs about the number of other agentsassigned to her auction are

πM(k; λ) = π(k; λ)(k + 1)

E[K]with E[K] = C

Since C is continuous in C, FV , and σ, πM(k; λ) is continuous with respect to thesevariables as well. The probability of winning the good (from a bidder’s perspective) is

χ(b) =∞

∑k=0

πM(k; λ) ∗ GR(b) ∗ GB(b)k

Lemma A.3. Assume GB is atomless. Then χ is continuous in λ, GR, GB, and b.

Proof. χ(b) is clearly continuous in λ, GR, and GB. χ(·) is continuous in b if GB isatomless.

This result on the continuity of the bids combined with Assumption 2.3 yields thefollowing result.

Lemma A.4. BR (C, FV , GR, σ) =(

e, β)

is continuous in (C, FV , GR) and σ if FV admits a

bounded PDF.

Proof. Given the continuity of GB and λ with respect to (C, FV , GR) and σ, Assumption

2.3 immediately yields continuity of β with respect to (C, FV , GR).Let V(vi, C, GB, GR|σ) denote the value function generated by best responding to (C, GB, GR)

given a value of vi. Theorem 1 of Pavan et al. [2014] implies that V is almost everywheredifferentiable and the derivative, where it exists, takes the value

∂V(v, C, GB, GR|σ)∂v

=∞

∑τ=t

δτ−t(1− χ(β(v)))τ−tχ(β(v)) (52)

Since the strategies are best responses, the probability of sale must be positive for alltypes that enter. This implies the sum that equation 52 is strictly bounded between 0and 1. From the continuity of χ, we know that V is continuous in (C, GB, GR, σ)

e can be defined implictly using

V(e, C, GB, GR|σ) = 0 (53)

Since V is continuous in v, has a strictly positive lower bound on its derivative, and iscontinuous in (C, GB, GR) and σ, the solution to equations 53 (i.e., e) are also continuousin these variables.


A.2.2. Continuity of Aggregate States. Now we turn to proving that the evolution of theaggregate states is continuous as required. Before doing so, let us describe the operatorT(C, FV , GR|σ) in more depth. We iterate the states by using steady-state relations that,in equilibrium, are consistent with the aggregate state. We will need to use the following

function, which the reader is encouraged to think of as an unnormalized version of fV

j(v) =µtV(v)

χ[β(v)

] if e ≤ v

and j(v) = 0 if e > v. Having defined j, we can define C as

C =∫ 1

0j(s)ds

The distributions of buyer types and reservation prices are

fV(v) =j(v)C

GR = GR

There is, at this point, no assurance that fV is bounded (or even well-defined) since χ

could have arbitrarily low values for some entrants. We rule this problem out with thefollowing lemma.

Lemma A.5. χ(b) ≥ κ > 0.

Proof. For a buyer to find it optimal to enter given a positive entry cost, his expectedpayoff from entering must at least cover the cost of entry. This implies:

χv− ρ ≥ κ

Therefore

χ = PrTransaction ≥ κ

v≥ κ

where the last inequality follows from the fact that v ∈ [0, 1].

Lemma A.5 implies that

j(v) ≤ µtV(v)κ

for entrants. These relations give us

C ∈ [µ, µ/κ]

fV(v) ∈[

0,tV(v)

κ

]Therefore we can restrict attention to fV ∈ Q [0, q] where q is the maximum of tV(·)/κ. In

other words, fV admits a bounded PDF. Furthermore, fV is continuous in χ as required.


Finally, C, FV and GR inherit continuity with respect to λ, GR, GB, and σ from thecontinuity of χ.

We can now prove the existence of a stationary equilibrium of the continuum model.

Proposition 2.4. A stationary competitive equilibrium exists, and a positive mass ofbuyers choosing to enter the market if κ is not too large.

Proof. Lemmas A.1 through A.5 in conjunction with the continuity provided by Assump-tion 2.3 prove that the dynamics of our limit model imply L is a continuous mappingfrom Ξ× Γ into itself. Given the continuity of the mapping and the compactness of thespaces, Schauder’s fixed point theorem implies that there exists a fixed point that definesa stationary equilibrium of our model.

To see that it is optimal for some buyers to enter for κ sufficiently small, first assumethere is an equilibrium where none of the buyers enter (i.e., e = 1). Suppose a buyer witha value of v = 1 deviated to choosing Enter and bid b = 0. Given there is an atom ofreservation prices at r = 0, there is a positive probability that the buyer is matched intoan auction with a reservation price of 0. Since there are no competing bidders, the buyerwins the good at a price of 0. Since the expected payoff of this deviation is GR(0)v, if κ

is less than this value it cannot be an equilibrium for no buyers to choose Enter.

A.3. Proofs from Section 6.2. Our approximation result requires two steps. First, wemust show that the limit game has a utility structure that is close to the utility structureof a sufficiently large finite game. In particular, we must show that the correlation insale price and winning probability across auctions vanishes as N → ∞. Second, wemust show that these facts imply that with high probability there are no deviations thatyield a significant improvement in the utility of any agent. We conduct each task inseparate sections. Throughout the sections we focus on an SCE strategy of the limitgame (σ, C, FV).

A.3.1. Convergence of Utility. First note that if an agent chooses Out, then his utility is0 regardless of the number of other agents. For the remainder of the section we willassume the agent in question chooses Enter. The utility of a bidder in the current periodof the N-agent game given the bidder enters and bids b is

χN(b)vi − ρN(b)− κ

If we can show that

χN(b) → χ(b) (54)

ρN(b) → ρ(b)


uniformly over b when we hold b, C, FV , and GR fixed, then we will have shown that theutility function in the N agent game converges to the utility functions of the limit game.

We first show that the probability of buyers being unmatched vanishes as CN increases.

Lemma A.6. We have

Pr (A particular buyer is unmatched) = O(

1CN

)Proof. Let Dl denote the number of bidders matched to auction l. For any buyers to beunmatched, the total “demand” for bidders from sellers must fall short of the supply,

CN, which means that i bidders are not matched if and only if

SN

∑l=1

Dl = CN − i

Since any bidder is equally likely to be amongst the unmatched buyers, conditional on ibidders being unmatched, the probability that a particular bidder is unmatched is i/CN.The total probability a particular bidder is unmatched is

Pr (A particular buyer is unmatched) =CN

∑i=1

iCN Pr

(SN

∑l=1

Dl = CN − i

)

Using assumption 2.1, we can approximate the probaility mass function of the sum ofthe Dl using a normal distribution probability density function. This lets us write:

CN

∑i=1

iCN Pr

[SN

∑l=1

Dl = CN − i

]' 1CN

CN

∑i=1

i√SNVar[K]

ψ

[−i√

SNVar[K]

]

≤ 1CN

∫ ∞

0xψ(x)dx = O

(1CN

)where ψ is the standard normal PDF.

The asymptotics results required to prove equation 54 holds is complicated by the factthat in the finite game the buyers are sampled without replacement when assigned toauctions. Denote the number of auctions generated in the N agent game as AN, andAN → ∞ almost surely as N → ∞. Intuition suggests that as N → ∞ the auctionsbecome asymptotically independent, and we prove this in the following lemma.


Lemma A.7. Supose that all buyers follow some SCE of the limit game, σ = (e, β). For anyε, γ > 0 we can choose N∗ such that for any N > N∗ and any (CN, FN

V , GR) we have

P

(1CN

∥∥∥∥∥ CN

∑i=1

[xN

i (b)− χN(b)]∥∥∥∥∥ > ε

)< 1− γ (55)

P

(1CN

∥∥∥∥∥ CN

∑i=1

[pN

i (b)− ρN(b)]∥∥∥∥∥ > ε

)< 1− γ

supb∈[0,1]

∥∥∥χN(b)− χ(b)∥∥∥ = O

(1√CN

)(56)

supb∈[0,1]

∥∥∥ρN(b)− ρ(b)∥∥∥ = O

(1√CN

)The choice of ε and ρ can be chosen uniformly over (CN, FN

V , GR).

Proof. We provide a proof for Equations 55 and 56, but essentially identical argumentssuffice for the other results. For notational cleanliness, we will provide a proof of theprobability of large positive deviations, but the analogous result for large negative devi-ations is essentially identical.

From Chebyshev’s inequality we have

P( 1CN

CN

∑i=1

(xN

i (b)− χN(b))

> ε)≤ 1

ε2 Var

(1CN

CN

∑i=1

(xN

i (b)− χN(b)))

(57)

=1ε2

1

(CN)2

[ CN

∑i=1

Var(

xNi (b)− χN(b)

)+ (58)

2CN

∑i=2

CN

∑j=1

cov(

xNi (b)− χN(b), xN

j (b)− χN(b)) ]

Since xNi (b) is bounded, we know

1

(CN)2

CN

∑i=1

Var(

xNi (b)− χN(b)

)= O(C−1

N ) (59)

We now bound the covariance term. If the agents assigned to the auctions were gener-ated through a process of sampling with replacement (SWR), then

cov(


j (b)− χN(b))= 0

The only difference between sampling with and without replacement is an event whereinone of the following occurs:


(1) The bidder is unmatched(2) The two auctions share a common bidder

Lemma A.6 provides a uniform upper bound on the probability of event (1). Let ES

denote the event the auctions share a bidder in a SWR regime, which allows us to write

cov(


j (b)− χN(b))< Pr ES+

1CN

where the second term captures the probability of a buyer being unmatched.Fix two auctions with m and n bidders. The probability the auctions do not share a

bidder is46 (1− 1CN

)(1− 2CN

)...(

1− m + n− 1CN

)>

(1− m + n

CN

)m+n

The probability the auctions share an agent is no more than

∑m,n:m+n≤CN

π(m; λ)π(n; λ)

[1−

(1− m + n

CN

)m+n]

A binomial expansion yields:(1− m + n

CN

)m+n= 1− (m + n)2

CN + o(C−1

N

)Therefore

∑m,n:m+n≤CN

π(m; λ)π(n; λ)

[1−

(1− m + n

CN

)m+n]

= ∑m,n:m+n≤CN

π(m; λ)π(n; λ)(m + n)2

CN + o(

1CN

)

= ∑m,n:m+n≤CN

π(m; λ)π(n; λ)m2 + 2mn + n2

CN + o(

1CN

)

≤ 2E[K2] + 2E[K]2

CN + o(

1CN

)= O

(1CN

)46If we think of drawing the m + n buyers in order, the second buyer chosen cannot have the same

identity as the first (first term), the the third buyer chosen must not have the same identity as either of thefirst two (second term), etc.


Putting all of these results together, we have that

PrES = O(

1CN

)Pulling our argument together we have

1ε2

1

(CN)2

CN

∑i=2

i−1

∑j=1

cov(


j (b)− χN(b))= O

(1CN

)

Referring back to equation 57, we then have

P( 1CN

CN

∑i=1

(xN

i (b)− χN(b))> ε)= O

(1CN

)

The final step is proving that χN and χ are close to one another. Note that the dif-

ference between these two functions is generated by the fact that χN is generated byconstructing the set of auctions with a sampling without replacement process and thatbuyers can be unmatched, whereas the process generating χ insures that all buyers arematched into auctions with opponent types generated by a sampling with replacementprocess. Our argument regarding the correction required to account for these differencesimplies ∥∥∥χN(b)− χ(b)

∥∥∥ = O(

1CN

)The uniformity with respect to b follows from standard arguments based on the separa-bility of the reals and both functions being monotone in b. The uniformity with respect

to (CN, FNV , GR) follows from the fact that the proof is completely independent of these

variables.

The following lemma implies that the within-period utility of the game with a finitenumber of agents converges to the within-period utility function of the continuum limitgame.

Lemma A.8. Supose that all agents follow some SCE of the limit game, (e, β). For any ε > 0

we can choose N∗ such that for any N > N∗ and any (CN, FNV , GN

R ) we have

supb,v∈[0,1]

∥∥∥χN(b)vi − ρN(b)− (χ(b)vi − ρ(b))∥∥∥ < ε (60)

Proof. The result follows from Lemma A.7.


A.3.2. Mean Field Lemma. The goal of this subsection is to prove that the evolution of thefinite game approaches the deterministic evolution of the limit game as N → ∞. We startby showing the initial distribution of agent types in the finite and limit game convergesin this limit.

Lemma A.9. The empirical distribution of types and reserve prices in period 0 of the N-agentgame converges weakly to FV and GR with a convergence rate of O(N−0.5).

Proof. Follows from Remark 1.

We use time indices for the variables in the next proposition to make the evolution

of the aggregate variables clear. Let QN(CNt , FN

V,t, FNC,t|σ) = (CN

t+1, FNV,t+1, FN

C,t+1) denote

the aggregate state iterator in the N-agent game, where (CNt+1, FN

V,t+1, FNC,t+1) is a random

variable.

Lemma A.10. Consider a stationary SCE strategy σ = (e, β) and aggregate state (C, FV , GR).

For any η, γ > 0, we can choose N∗ such that for all N > N∗ and(

CNt , FN

V,t, GNR,t

)such that∥∥∥CN

t − C∥∥∥+ ∥∥∥FN

V,t − FV

∥∥∥+ ∥∥∥GNR,t − GR

∥∥∥ <η

2

we have the following with probability at least 1− γ∥∥∥CNt+1 − C

∥∥∥+ ∥∥∥FNV,t+1 − FV

∥∥∥+ ∥∥∥GNR,t+1 − GR

∥∥∥ < η (61)

where QN

(CN

t , FNV,t, GN

R,t|σ−i, σ′i

)=(

CNt+1, FN

V,t+1, GNR,t+1

).

Proof. As an initial note, when any single agent deviates from σ in the limit game, nochange in the aggregate variables occurs. When a single agent deviates in the finite game,

it causes a change of at most (CN)−1. Since (CN)−1 → ∞ if e < 1 in any equilibrium ofthe finite game, these deviations do not affect the convergence arguments presentedbelow.

First, we have that GNR,t+1 → GR by Remark 1. What remains is to show that

CNt+1N →

Ct+1 and FNV,t+1 → FV,t+1. Focusing on the new potential entrants, there are dNµe agents

added to the game with types (v1, ..., vdNµe) drawn from TV . dNµeN → µ as N → ∞ and

Remark 1 imply

1dNµe

dNµe

∑i=1

1 v ≥ vi ≥ e → TV(v) uniformly over v almost surely

This means that the only thing we need to show is that the number and type distributionof agents continuing onto the next period in the finite game converges to the analogous


measure and distribution in the limit game as N → ∞. Lemma A.7 implies the following,where the convergence is in probability and uniform over v

1CN

t+1

CNt

∑i=1

1 vi ≥ e (1− xi(β(v)) →∫ 1

01 vi ≥ e (1− χ(β(v))dFV,t(s)

1CN

t+1

CNt

∑i=1

1 v ≥ vi ≥ e (1− xi(β(v)) →∫ 1

01 v ≥ vi ≥ e (1− χ(β(v))dFV,t(s)

Bringing these results together, we have(

CNt+1, FN

V,t+1, GNR,t+1

)→ (Ct+1, FV,t+1, GR) in

probability as N → ∞, which is equivalent to our desired result.

Iterating Lemma A.10 immediately gives us the following:

Corollary A.11. Consider a stationary SCE strategy σ and aggregate state (C, FV , GR). For any

∆, γ > 0, we can choose η > 0 and N∗ such that for all N > N∗ and(

CNt , FN

V,t, GNR,t

)such that∥∥∥CN

t − C∥∥∥+ ∥∥∥FN

V,t − FV

∥∥∥+ ∥∥∥GNR,t − GR

∥∥∥ < η

we have for all t′ ∈ t, .., t + τ with probability at least 1− γ∥∥∥CNt′ − C

∥∥∥+ ∥∥∥FNV,t′ − FV

∥∥∥+ ∥∥∥GNR,t′ − GR

∥∥∥ < ∆ (62)

where for k ∈ 0, .., τ− 1 we define QN

(Ct+k, FN

V,t+k, GNR,t+k|σ−i, σ′i

)=(

CNt+k+1, FN

V,t+k+1, GNR,t+k+1

).

A.4. No Profitable Deviations. In this subsection, we finally prove our approximationresult. We start by proving that the limit model has a continuous per-period utilityfunction.

Lemma A.12. χ(b)v − ρ(b) − κ is continuous with respect to (C, FV , GR) and β when FV

admits a PDF that is bounded from above

Proof. χ(b) was proven to be continuous in Lemma A.3. We can write

ρ(b) = πM(1; λ)∫ b

0u GR(du) +

∞

∑k=2

π(k; λ)∫ b

0

∫ b

0maxu, tGR(du)Gk−1

B (dt)

Lemma A.1 implies that GB is continuous in (FV , GR) and β under the conditions of ourlemma. Since the integrands are continuous and GB and GR are continuous, then theresulting integral is continuous.

These results, together with the convergence of the utility functions, yields the follow-ing result on the convergence of value functions.


Lemma A.13. Consider a stationary SCE strategy σ and aggregate state (C, FV , GR). For any

ε, γ > 0 we can choose η > 0 and N∗ such that for all N > N∗ and(

CN0 , FN

V,0, GNR,0

)such that

‖C0 − C‖+∥∥∥FN

V,0 − FV

∥∥∥+ ∥∥∥GNR,0 − GR

∥∥∥ < η (63)

we have with probability at least 1− γ

For all v,∥∥∥VN

(v, CN

0 , FNV,0, GN

R,0|σ)− V(v, C, FV , GR|σ)

∥∥∥ < ε

Proof. First note that if v < e, then we are done since the buyer never enters the marketand receives the same payoff in either game. For the duration we assume that v ≥ e.

Let EN0 [xt], etc. refer to an agent’s expectation in period 0 about an event that occurs

in period t of the finite game. We can write the value functions as

VN(

v, CN0 , FN

V,0, GNR,0|σ

)=

∞

∑t=0

δtEN0

[xt v− pt − κ|CN

0 , FNV,0, GN

R,0, σ]

V(v, C, FV , GR|σ) =∞

∑t=0

δt (χv− ρ− κ)

Choose T such that δT < ε3 , and note:

∞

∑t=T

δt [(xtv− pt − κ)] < (1− δ)δTv <ε

3

From hereon, we consider only the first T periods.

Lemma A.8 implies that for any sample path of(

CNt , FN

V,t, GNR,t

)∞

t=0we can choose

an N∗ sufficiently large so that for all t ∈ 0, .., T

supb,v∈[0,1]

∥∥∥ENt

[xN

t (b)v− pNt (b)|CN

t , FNV,t, GN

R,t

]− (χ(b)v− ρ(b))

∥∥∥ < ε (64)

where χ and ρ are conditioned on (CNt , FN

V,t, GNR,t). Lemma A.11 implies that for any

∆, γ > 0 we can choose η sufficiently small and N sufficiently large such that∥∥∥CNt+τ − C

∥∥∥+ ∥∥∥FNV,t+τ − FV

∥∥∥+ ∥∥∥GNR,t+τ − GR

∥∥∥ ≤ ∆ (65)

for all τ ≤ T with probability at least 1− γ.From Lemma A.12, if Equation 65 holds and ∆ is sufficiently small, we have for all

t ∈ 0, .., T we have:∥∥∥E0

[xv− p− κ|CN

t , FNV,t, GN

R,t, σ]− (χ(b)v− ρ(b))

∥∥∥ <ε

3Twhere χ and ρ are conditioned on the steady state aggregate variables. Note that thisresult holds uniformly over v and the closed neighborhood defined by Equation 65.


Finally, in the complementary event that the sample path of(

CNt , FN

V,t, GNR,t

)∞

t=0is not

close to (C, FV , GR) (i.e., Equation 65 fails to hold for ∆ sufficiently small) we have:∥∥∥EN0

[(xv− p− κ) |CN

t , FNV,t, GN

R,t, σ]− E0 [(xv− p− κ) |C, FV , GR, σ]

∥∥∥ < 1

Therefore, for ∆ (and hence η) sufficiently small and γ < ε3(T+1) , we have uniformly over

v ∥∥∥VN(v, CNt , FN

V,t, GNR,t|σ)− V(v, C, FV , GR|σ)

∥∥∥ < T ∗ ε

3T+

ε

3+ (T + 1)γ < ε

where the first error term refers to errors that occur in the approximation when Equation65 holds, the second term includes errors accruing in periods after T, and the final termis the expected error from the event when equation CloseEqn fails to hold.

Now we prove our main approximation results.

Proposition 6.3. Consider a SCE (σ, C, FV , GR) where e(C, FV , GR) < 1. For any ε > 0 wecan choose η > 0 and N∗ such that for all N > N∗ and γ > 0, σ is an ε− BNE strategy

if the state in the first period of the N-agent game,(

CN0 , FN

V,0, GNR,0

), satisfies∥∥∥CN

0 − C∥∥∥+ ∥∥∥FN

V,0 − FV

∥∥∥+ ∥∥∥GNR,0 − GR

∥∥∥ < η

Proof. From the one-step deviation principle, it suffices to consider a deviation by a singleagent in a single period. Without loss of generality, let us assume the deviation occursin period 0 by a bidder that chooses Enter. Lemma A.10 implies that for any η, γ > 0we can choose η sufficiently small that with probability 1− γ∥∥∥CN

1 − C∥∥∥+ ∥∥∥FN

V,1 − FV

∥∥∥+ ∥∥∥GNR,1 − GR

∥∥∥ < η (66)

even if a single agent deviates from the SCE strategy in period 1. Lemma A.13 impliesthat for any ε > 0 we can choose η, γ > 0 sufficiently small and N sufficiently large thatwith probability 1− γ∥∥∥VN

(v, CN

1 , FNV,1, GN

R,1

∣∣∣ σ)− V(v, C, FV , GR|σ)

∥∥∥ <ε

4(67)

for(

CN1 , FN

V,1, GNR,1

)that satisfy equation 66. Equation 67 implies that the effect of the

current deviation on future periods is small.

Let βdev(v) denote the optimal deviation for type v at(

CN0 , FN

V,0, GNR,0

). From Lemma

A.2 and Assumption 2.3 we have that for any δ ≥ 0 and all v that we can choose η

sufficiently small that

‖βdev(v)− β(v)‖ < δυ (68)


From Lemma A.8 we have for N sufficiently large:

supb,v∈[0,1]

∥∥∥EN0

[x(b)v− p(b)

∣∣∣CN0 , FN

V,0, GNR,0

]− E0

[x(b)v− p(b)

∣∣∣CN0 , FN

V,0, GNR,0

]∥∥∥ <ε

6

Combining this with Lemma A.12 yields for η sufficiently small and N sufficiently large

∥∥∥EN0

[x(βdev(v))v− p(βdev(v))|C1, FN

V,0, GNR,0

]− E0 [x(b)v− p(b) |C, FV , GR ]

∥∥∥ <ε

4(69)

Equations 67 and 69 and the fact that the SCE strategy is weakly optimal in the limitgame given the SCE aggregate values of C, FV , and GR we get

VN(

v, CN0 , FN

V,0, GNR,0

∣∣∣ σ)+ ε ≥ EN

0 [(x(βdev(v))vi − p(βdev(v))− κ) +

(1− x(βdev(v)))δVN(

vi, CN1 , FN

V,1, GNR,1|σ

) ∣∣∣σ, CN0 , FN

V,0, GNR,0

]

which implies that following the SCE strategy is an ε-BNE for η > 0 sufficiently small.


Appendix B. ONLINE SUPPLEMENTAL MATERIALS FORHow Efficient are Decentralized Auction Platforms?

by Aaron L. Bodoh-Creed, Jörn Boehnke, and Brent R. Hickman

EMPIRICS: ADDITIONAL DETAILS AND FIGURES

B.1. GMM Standard Errors. In this section we relate our stage I estimator to well-knownGMM theory, and we present our calculations for standard errors in stages I and II. Let

α = (λ1, λ2, αb2, . . . , αbIb−1, αr1, . . . , αrIr−1)> denote a column vector of all stage-I free

parameters, let I1 ≡ 2 + Ib + 1 + Ir + 2 denote its dimension, and let wl = (kl, yl, rl)

denote the observables for the lth auction, l = 1, . . . , L. Moreover, let m(wt; α) denote our

(3L× 1)-dimensional moment condition function, for which the nth component is

mn(wt; α) =

π(kn|rn; λ, αb)− L1

(kt = kn

) K(

rt−rnhR

)∑L

u=1K(

rt−ruhR

) , n ∈ 1, . . . , L

H(yn; λ, αb)− 1(yt ≤ yn−L), n ∈ L + 1, . . . , 2L, andGR(rn; αr)− 1(rt ≤ rn−2L), n ∈ 2L + 1, . . . , 3L,

and satisfying E [m(wt; α)] = 0. Note that a consistent estimator of the asymptotic

variance of m(wt; α), denoted Σ, a (3L × 3L)-dimensional matrix, is given by Σ =

L−1 ∑Lt=1 m(wt; α)m(wt; α)>. The analogous empirical moment conditions are mL(α) =

L−1 ∑Lt=1 m(wt; α), and note that our estimators (27) and (28) are equivalent to finding

the standard GMM estimator α = arg minα LmL(α)>mL(α).47 Finally, let M denote the

(3L× I1)-dimensional matrix of (estimated) first derivatives, where the (i, j)th element is

defined as Mij ≡ L−1 ∑Lt=1

∂mi(wt;α)∂θj

, where

∂mi(wt; α)

∂θj=

∂π(ki|ri;λ,αb)∂λj

i = 1, . . . , L, j = 1, 2,∂H(yi−L;λ,αb)

∂λji = L + 1, . . . , 2L, j = 1, 2,

0 i = 2L + 1, . . . , 3L, j = 1, 2,

∂π(ki|ri;λ,αb)∂αb,j−2

i = 1, . . . , L, j = 3, . . . , Kb + 5,∂H(yi−L;λ,αb)

∂αb,j−2i = L + 1, . . . , 2L, j = 3, . . . , Kb + 5,

0 i = 2L + 1, . . . , 3L, j = 3, . . . , Kb + 5,

0 i = 1, . . . , 2L, j = Kb + 6, . . . , Kb + Kr + 8,∂GR(ri−2L;αr)

∂αr,j−Kb−5i = 2L + 1, . . . , 3L, j = Kb + 6, . . . , Kb + Kr + 8.

47Note that throughout this section we take the choice of Ib, Ir, and knot vectors nb and nr as fixed, andwe adopt the identity matrix for weighting in the GMM objective function.


With these definitions, it follows from well-known econometric theory (see [Hayashi,

2000, Chapter 7]) that α is asymptotically normal, converging at rate√

L, and a consistentestimator of its (I1 × I1)-dimensional asymptotic variance-covariance matrix is

avar(α) = L−1(

M>M)−1

M>ΣM(

M>M)−1

.

B.1.1. Stage II Standard Errors. Given this result, a straightforward approach to comput-

ing standard errors for stage II objects—including χ, ρ, κ, V(v), β, β, and FV , all functions

of α—is the delta method. Let υ(α) = (υ1(α), . . . , υd(α))> denote an I2-dimensional,

smooth, vector-valued function of the Stage I free parameters, and let Υ(α) denote its

I2 × I1-dimensional partial derivative matrix, where the (i, j)th element is given by

Υij(α) =∂υi(α)

∂θj, i = 1, . . . , I2, j = 1, . . . , I1.

Since α is approximately distributed as multivariate normal with mean α and varianceavar(α), and linear transformations of normal random variables are normal, and sincesmooth functions are approximately locally linear, it follows that

υ(α)− υ(α) ∼ N(

0, Υ(α)avar(α)Υ(α)>)

.

Therefore, the variance of υ(α) can be consistently estimated by Υ(α)avar(α)Υ(α)>. Inprinciple, this method could be used for both parameters (such as κ) and functionals(such as tV), if we define υ as a vector of functional values on a grid of d domain points.

B.2. Additional Figures. Figure 9 provides intuition about the nature of the B-splinefunctions we use to fit the variables of our model. The top panel of the figure displaysthe observed distribution of the highest losing bids (H(y)) as well as the underlyingparent distribution of bids (GB(b; αb)). The center panel provides the distribution ofthe reserve prices. Since both the top and center panels describe observable variables,we have included the empirical CDFs as well. A comparison between the empiricalCDFs and the B-spline fit shows a very close correspondence. Finally, the bottom paneldescribes the distribution values that we infer from the observables. To illustrate theunderlying components of our B-spline functions, we have included the locations of theknots and the basis functions in the plot as well. The knot locations are described by thevertical lines extending below the top of each panel. The families of basis functions aredrawn at the bottom of each panel.


80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

B (EQUILIBRIUM BIDS)

CD

Fs

H(y;λ,αb)H(y)GB(b;αb)KNOTSBASIS FUNCTIONS

20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R (SELLER RESERVE PRICES)

CD

Fs

GR(r;αr)GR(r)KNOTSBASIS FUNCTIONS

100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V (PRIVATE VALUES, δ=0.8871)

CD

F

FV (v;αv)KNOTSBASIS FUNCTIONS

Figure 9. Stage I Estimates

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