On Bayes-Nash implementation of combinatorialauctions: structure and efficiency

JTGSS Lecture 8

B. Hajek, with thanks to Vineet Abhishek

Department of Electrical and Computer Engineeringand the Coordinated Science Laboratory

University of Illinois at Urbana-Champaign

1 June 2012

Abstract for lecture 7:

Abstract: An overview of the structure of Bayesian optimalauctions, along the lines of Myerson, but for discrete-valuedvaluations is given. A novel graphical construction of virtualvaluations is provided. In the second part of the presentation, theproblem of characterising revenue optimal auctions forsingle-minded buyers is discussed, meaning each buyer is interestedonly in a specific bundle of items and has a value for the same, andboth the choice of bundle and the value are private information.The talk will also address the question of how efficient the revenueoptimal auction is. (Based on joint work with Vineet Abhishek.Details can be found in [1, 2] )

Outline

Review of revenue optimal allocation in Bayes-Nash setting

On the Efficiency Loss for Revenue Optimal Auctions

Auction to Single-Minded Buyers

Part I: Review of revenue optimal allocation in Bayes-Nashsetting

This follows Myerson [5], although Myerson formulated auctionswith continuous valuations. Here we follow [3] for discretevaluations, but allow multiple winners.

First, focus on a single buyer in an allocation game.

Suppose:

I Winning is worth X to the buyer, where:

P{X = x i} = pi for 1 ≤ i ≤ K where 0 ≤ x1 < . . . < xK .

(If X = x i we say the buyer is type x i , or type i for short.)

I Buyer observes X and reports a bid in {1, . . . ,K}. Buyer istruthful if bid=type.

I qj is the conditional probability of winning, for bid j .

I mj is the payment the buyer makes, for bid j .

Thus, the expected payoff of a type i buyer bidding j is qjx i −mj .

Choice of payments

Incentive Compatibility (IC) constraint:

qjx i −mj ≤ qix i −mi for all i , j ≥ 1

Individual rationality (IR) constraint:

0 ≤ qix i −mi

Geometric interpretation: Let P i = (qi ,mi ) all i , P0 = (0, 0).IC: P j lies above or on slope x i line through P i

IR: P0 lies above or on slope x i line through P i

Move points vertically–can’t meet constraints for these qi values:

2

P

P2

3 q

m

P1

P0q1q q3

Can meet constraints for these qi values. Largest mi ’s shown:

3

q

m

1P

2

3

P

P0

P

1q q2 q

Choice of payments (continued)

Conclude that:

I It is possible to select mi ’s to satisfy the IC constraints if andonly if qi is nondecreasing in i .

I If qi is nondecreasing in i , the maximum choice of mi underthe IC and IR constraints is (with q(x0) = 0):

mi =i∑

j=1

(qj − qj−1)x j .

Maximum (over IC/IR payment rules) expected revenue

E [m(X )] =K∑

i=1

pimi ,

=∑

i ,j :1≤j≤i≤K

(qj − qj−1

)x jpi ,

=K∑

j=1

qj

K∑i=j

x jpi −K∑

i=j+1

x j+1pi

=

K∑j=1

pjqjw j = E [q(X )w(X )],

where w i , x i − (x i+1 − x i )

(∑Kj=i+1 pj

)pi

.

So w i is the value to seller of making a type i buyer a winner.

Geometric interpretation of w i , x i − (x i+1 − x i )(

PKj=i+1 pj)pi .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Cumulative probability

Neg

ativ

e x−

valu

es

Virtual valuations

−x2

−x3

−x4

(1,0)

−x1

p1p2 p3 p4

Single buyer with binary valuation

€

−L

€

−H

€

p(1,0) 

(0,0) 

€

w(H )€

1− p

slope 

€

w(L)slope 

w(H) = H, w(L) = L−pH1−p .

I Rule 1: Always sell: q(H) = q(L) = 1. (m(H) = m(L) = L.)Revenue: E [m(X )] = pw(H) + (1− p)w(L) = L.

I Rule 2: Sell to high: q(H) = 1, q(L) = 0.(m(H) = H,m(L) = 0.) Revenue: E [m(X )] = pw(H) = pH.

Single buyer with binary valuation

€

−L

€

−H

€

p(1,0) 

(0,0) 

€

w(H )€

1− p

slope 

€

w(L)slope 

w(H) = H, w(L) = L−pH1−p .

I Rule 1: Always sell: q(H) = q(L) = 1. (m(H) = m(L) = L.)Revenue: E [m(X )] = pw(H) + (1− p)w(L) = L.

I Rule 2: Sell to high: q(H) = 1, q(L) = 0.(m(H) = H,m(L) = 0.) Revenue: E [m(X )] = pw(H) = pH.

Single buyer with binary valuation

€

−L

€

−H

€

p(1,0) 

(0,0) 

€

w(H )€

1− p

slope 

€

w(L)slope 

w(H) = H, w(L) = L−pH1−p .

I Rule 1: Always sell: q(H) = q(L) = 1. (m(H) = m(L) = L.)Revenue: E [m(X )] = pw(H) + (1− p)w(L) = L.

I Rule 2: Sell to high: q(H) = 1, q(L) = 0.(m(H) = H,m(L) = 0.) Revenue: E [m(X )] = pw(H) = pH.

Single buyer with geometrically distributed valuation

I x i = i for i ≥ 1 (K = +∞)

I pi = θ(1− θ)i−1

I

w i = x i − (x i+1 − x i )

(∑Kj=i+1 pj

)pi

= i − (1− θ)i

θ(1− θ)i−1= i −

(1− θθ

)I Revenue optimal auction: Sell if bid v ≥ i∗

4=⌈

1−θθ

⌉.

I Maximum revenue: i∗(1− θ)i∗−1.

Maximum revenue to seller for a single buyer(Regular case: nondecreasing w ’s)

€

−x1

€

−x 2

€

−x 3

€

−x 4

€

p1

€

p2

€

p3

€

p4(1,0) 

(0,0) 

          slopes are virtual valua2ons 

€

w2

€

w3

€

w4

€

w1

max revenue 

Maximum revenue to seller for a single buyer(Irregular case: switch to monotone w ’s – Myerson’s “rolling”)

€

−x1

€

−x 2

€

−x 3

€

−x 4

€

p1

€

p2

€

p3

€

p4(1,0) 

(0,0) 

€

w

€

w          slopes are virtual valua2ons,           slopes are monotone virtual valua2ons,  

~ ~ 

~ 

~ 

~ €

w1

€

w2

€

w3 = w3

€

w4 = w4

€

w1

€

w2max revenue 

Auction for N buyers having independent private valuations

Switch to N buyers, indexed by subscript n :

I P{Xn = x in} = pi

n for some probabilities pin and

x1n < · · · < xKn

n .

I Bid of buyer n is denoted by vn ∈ {x1n , . . . , x

Knn }.

I Virtual bid for buyer n is denoted by wn(vn).

Suppose winning set of buyers can be any set A with A ∈ A.By single buyer analysis, the maximum revenue (maximum over ICand IR payment rules) for a given allocation mechanism is:1

R(πππ) = E

[N∑

n=1

Qn(XXX )wn(Xn)

]= E

[∑A∈A

πA(XXX )(∑

n∈A

wn(Xn))].

1As long as probability of winning for a buyer is nondecreasing with type.

The optimal allocation for N buyers

The previous slide suggests seller can maximize revenue byallocating a set in

argmaxA∈A

(∑n∈A

wn(vn)

).

I Works fine if for each n, wn(x1n ) ≤ wn(x2

n ) ≤ · · · ≤ wn(xKnn ).

I Otherwise, replace wn’s by monotone virtual valuations(obtained by Myerson’s “rolling procedure”)

I If tie breaking rule is deterministic, or a deterministic functionof some exogenous independent random variable, the paymentof each winner can be the minimum bid the winner wouldhave needed to win.

I If A is lower closed, reserve price for buyer n is xmin{i :ew i

n>0}n .

Geometric interpretation of monotone virtual valuations:

€

w(x1)

€

w(x 2)

€

w(x 3) = w(x 3)

€

−x1

€

−x 2

€

−x 3

€

−x 4

€

p1

€

p2

€

p3

€

p4(1,0) 

(0,0) 

€

w(x 4 ) = w(x 4 )

€

w

€

w          slopes are virtual valua2ons,           slopes are monotone virtual valua2ons,  

€

w(b,x1 )

€

w(b,x 2)

~ 

~ ~ 

~ 

~ 

Part II: On the Efficiency Loss for Revenue OptimalAuctions

Realized welfare vs. maximum welfare

I Realized social welfare of a revenue optimal auction:

N∑n=1

Qn(XXX )Xn or∑A∈A

πA(XXX )(∑

n∈A

Xn

),

I Maximum social welfare (MSW):

MSW(XXX ;A) = E[

maxA∈A

(∑n∈A

Xn

)],

where XXX denotes the vector of random valuations.

Worst case efficiency loss ratio

I The efficiency loss ratio (ELR) for a random vectorXXX ∼ xxx1:N ,ppp1:N of valuations is:

ELR(πππo ,XXX ;A) ,MSW(XXX ;A)− E

∑Nn=1 Qo

n (XXX )Xn

MSW(XXX ;A).

I The worst case ELR η(r ,K ;A) for r ≥ 1 and K ≥ 1, is themaximum ELR over XXX subject to:

Kn ≤ K for all n, and (maxn xKnn )/(minn x1

n ) ≤ r .

Example 1:

I Two i.i.d binary valuation buyers compete for one item, withP{Xn = 2} = 2/3 and P{Xn = 1} = 1/3, for n ∈ {1, 2}.

I The MSW=2 ∗ (1− (1/3)2) + (1/3)2 = 17/9 ≈ 1.89.

I The Vikrey second price auction (efficient) sells to highestbidder at second highest price. Revenue is2 ∗ (2/3)2 + 1− (2/3)2 ≈ 1.45

I Revenue optimal auction sets reserve price to 2, becausewn(1) < 0 (i.e. pH > L) Revenue, and realized welfare, are2 ∗ (1− (1/3)2) = 16/9 ≈ 1.77. Thus, ELR = 1/17 ≈ 0.059.

Example 1:

I Two i.i.d binary valuation buyers compete for one item, withP{Xn = 2} = 2/3 and P{Xn = 1} = 1/3, for n ∈ {1, 2}.

I The MSW=2 ∗ (1− (1/3)2) + (1/3)2 = 17/9 ≈ 1.89.

I The Vikrey second price auction (efficient) sells to highestbidder at second highest price. Revenue is2 ∗ (2/3)2 + 1− (2/3)2 ≈ 1.45

I Revenue optimal auction sets reserve price to 2, becausewn(1) < 0 (i.e. pH > L) Revenue, and realized welfare, are2 ∗ (1− (1/3)2) = 16/9 ≈ 1.77. Thus, ELR = 1/17 ≈ 0.059.

Example 2:

I Two binary valuation buyers compete for one item, withX1 ∈ {5, 10}, X2 ∈ {1, 2}, each taking each value w/p 1/2.

I Efficient allocation always awards item to buyer 1, andMSW=7.5 The price charged can be at most 5.

I An optimal auction sells to buyer 1 if v1 = 10 at price 10,otherwise it sells to buyer 2 at price 1. Revenue is 5.5. Therealized efficiency is 5.75, so the ELR is 1.75/7.5 ≈ 0.23.(Item is always sold. Note that virtual valuations for lowervalues are zero, so seller could also withhold item if both bidsare low.)

Example 2:

I Two binary valuation buyers compete for one item, withX1 ∈ {5, 10}, X2 ∈ {1, 2}, each taking each value w/p 1/2.

I Efficient allocation always awards item to buyer 1, andMSW=7.5 The price charged can be at most 5.

I An optimal auction sells to buyer 1 if v1 = 10 at price 10,otherwise it sells to buyer 2 at price 1. Revenue is 5.5. Therealized efficiency is 5.75, so the ELR is 1.75/7.5 ≈ 0.23.(Item is always sold. Note that virtual valuations for lowervalues are zero, so seller could also withhold item if both bidsare low.)

Example 3:

I Three binary valuation buyers, i.i.d valuations:P{Xn = 1} = P{Xn = 1.6} = 0.5 for n ∈ {1, 2, 3}.

I Each buyer wants the bundle of two items indicated:Buyer 3Buyer 1 Buyer 2

I wn(1.6) = 1.6 and wn(1) = 0.4.

I The revenue optimal auction is not efficient in one case: If thetrue values are (1, 1.6, 1) the efficient choice is to allocate tobuyers 1 and 3, but the revenue optimal auction allocates tobuyer 2.

Example 3:

I Three binary valuation buyers, i.i.d valuations:P{Xn = 1} = P{Xn = 1.6} = 0.5 for n ∈ {1, 2, 3}.

I Each buyer wants the bundle of two items indicated:Buyer 3Buyer 1 Buyer 2

I wn(1.6) = 1.6 and wn(1) = 0.4.

I The revenue optimal auction is not efficient in one case: If thetrue values are (1, 1.6, 1) the efficient choice is to allocate tobuyers 1 and 3, but the revenue optimal auction allocates tobuyer 2.

Example 4:

I Revisit single buyer with P{X = H} = p andP{X = L} = 1− p, where 0 < L < H.

I Suppose pH = L+ then w(L) = 0− and the optimal auctionsets the reserve price to H.

I Then the welfare loss is (1− p)L and MSW=pH + (1− p)L.So the ELR = (r − 1)/(2r − 1), where H/L = r .As r increases to ∞, ELR increases to 1/2.

A bound for binary valuations

Proposition[Abhishek and H.] Suppose A is downward closed.Given any r > 1, the worst case ELR for binary valuedsingle-parameter buyers satisfies:η(r , 2;A) ≤ (r − 1)/(2r − 1) ≤ 1/2.

Proof

The realized social welfare is at least as large as the revenue for arevenue optimal auction, which is at least as large as the revenuefor any IC/IR auction. In particular, use the following two-stepallocation procedure. Compute an efficient allocation (based onthe reported values) and then drop buyers n with Hnpn > Ln fromthe winning set. Then (using qe to denote allocation probabilitiesfor the efficient allocation algorithm used):

Realized welfare =N∑

n=1

(pnqen(Hn)Hn + (1− pn)qe

n(Ln)(wn(Ln))+) ,

and

MSW =N∑

n=1

(pnqen(Hn)Hn + (1− pn)qe

n(Ln)Ln) .

A term by term comparison shows the first of these sums is greaterthan or equal to r

2r−1 times the second of these sums.

Case ELR bounds

N binary valued buyers ELR ≤r − 1

2r − 1≤

1

2

N binary valued i.i.d.. buyers,ELR ≤ min

S

S + N,

r − 1

2r − 1

ffauction of S identical items, S ≤ N

Single item auction, 1 buyer, K discrete values ELR =γ∗

1 + γ∗, γ∗ = (K − 1)

1− r

−1K−1

!

Single item auction, N i.i.d. buyers, K = 2 ELR =

»PNi=0

“r

r−1

”i–−1

≤ 1N+1

Single item auction, N i.i.d. buyers, K > 2

γ1

1 + γ1

≤ ±ELR ≤γ2

1 + γ2

,

γ1 = N

»P∞i=N

1i

“1− 1

r

”i “1− 1

K−1

”i–,

γ2 = N

»P∞i=N

1i

“1− 1

r

”i–

Table: Efficiency loss in revenue optimal auctions - summary of results.

Part III: Auction to Single-Minded Buyers

I Suppose N single-minded buyers compete for bundles ofgoods.

I A single-minded buyer n desires one particular bundle Bn andhas a value Xn for the bundle.

I A key assumption is that both Bn and Xn are privateinformation of the buyer [4], with some known jointdistribution.

I It is supposed that the set of possible values of Xn doesn’tdepend on Bn.

Buyer 7

Buyer 2 Buyer 3Buyer 1

Buyer 4 Buyer 5 Buyer 6

Revenue Optimal Auction for Single-Minded Buyers

Proposition [Abhishek & H.] If the conditional distribution of Xn

given Bn = b is nondecreasing in b, in the hazard rate ordering onprobability distributions, then the auction described earlier is IC,IR, and revenue optimal. In particular, buyers have incentive totruthfully report their desired bundles.

Proof

I The conclusions hold if the wn(x in|Bn = b) are decreasing in b.

I Examining the formula for virtual valuations:

w i = x i − (x i+1 − x i )

(∑Kj=i+1 pj

)pi︸ ︷︷ ︸

reciprocal failure rate

this property follows easily if the w i are monotone (don’tneeded to be rolled).

I The result is proved in general using a new formula formonotone virtual valuations, obtained by convex duality:

w i = argminc<xn

maxz∈[x1,xK ]

(z − c)(1− FX (z))

x i − c,

with the convention that if more than one value of cminimizes the maximum, then the largest such c is selected.

References I

V. Abhishek and B. Hajek.Efficiency loss in revenue optimal auctions.In Proc. IEEE Conf. Decision and Control (full version onarXiv:1005.1121), 2010.

V. Abhishek and B. Hajek.Revenue optimal auction for single-minded buyers.In Proc. IEEE Conf. Decision and Control (full version onarXiv: 1005.1059), 2010.

E. Elkind.Designing and learning optimal finite support auctions.In SODA ’07: Proceedings of the eighteenth annualACM-SIAM symposium on Discrete algorithms, pages736–745, Philadelphia, PA, USA, 2007. Society for Industrialand Applied Mathematics.

References II

D. Lehmann, L. I. Ocallaghan, and Y. Shoham.Truth revelation in approximately efficient combinatorialauctions.J. ACM, 49(5):577–602, 2002.

R. Myerson.Optimal auction design.Mathematics of Operations Research, 6(1):58–73, 1981.