Compact bid languages and core-pricing

Motivation Pricing Suboptimality experimental evaluation conclusion

Compact bid languages and core-pricingin large multi-object auctions

Andor Goetzendorff 1

Martin Bichler 1

Robert Day 2

Pasha Shabalin 1

1Technische Universitat Munchen - Decision Sciences & Systems

2University of Connecticut - Operations and Informations Management

3 September 2014

Goetzendorff, Bichler, Day, Shabalin TUM, UCONN

Compact bid languages and core-pricing 1 / 16


Design of incentive compatible auctions for large markets

VCG prices are not always in the Core → low revenue

Core Pricing (used in spectrum auctions worldwide)

Application of VCG & Core prices suffer from thecomputational hardness of many real-world market designproblems




Design of incentive compatible auctions for large markets

VCG prices are not always in the Core → low revenue

Core Pricing (used in spectrum auctions worldwide)

Application of VCG & Core prices suffer from thecomputational hardness of many real-world market designproblems




pricing rule

Item A

Item B

28

B1

20 B2

14

B3

12 B4

32

32

B5

Pay-As-Bid

VCG

BPOC

Source: Cramton and Day (2009)

Bids

A B ABB1: 28 0 28B2: 0 20 20B3: 14 0 14B4: 0 12 12B5: 0 0 32

VCG Prices

B1: 28− (48− 34) = 14 }= 26

B2: 20− (48− 40) = 12

BPOC Prices

B1: 14 + 3 = 17 }= 32

B2: 12 + 3 = 15




pricing rule

Item A

Item B

28

B1

20 B2

14

B3

12 B4

32

32

B5

Pay-As-Bid

VCG

BPOC


Bids

A B ABB1: 28 0 28B2: 0 20 20B3: 14 0 14B4: 0 12 12B5: 0 0 32

VCG Prices

B1: 28− (48− 34) = 14 }= 26

B2: 20− (48− 40) = 12

BPOC Prices

B1: 14 + 3 = 17 }= 32

B2: 12 + 3 = 15




pricing rule

Item A

Item B

28

B1

20 B2

14

B3

12 B4

32

32

B5

Pay-As-Bid

VCG

BPOC


Bids

A B ABB1: 28 0 28B2: 0 20 20B3: 14 0 14B4: 0 12 12B5: 0 0 32

VCG Prices

B1: 28− (48− 34) = 14 }= 26

B2: 20− (48− 40) = 12

BPOC Prices

B1: 14 + 3 = 17 }= 32

B2: 12 + 3 = 15




pricing rule

Item A

Item B

28

B1

20 B2

14

B3

12 B4

32

32

B5

Pay-As-Bid

VCG

BPOC


Bids

A B ABB1: 28 0 28B2: 0 20 20B3: 14 0 14B4: 0 12 12B5: 0 0 32

VCG Prices

B1: 28− (48− 34) = 14 }= 26

B2: 20− (48− 40) = 12

BPOC Prices

B1: 14 + 3 = 17 }= 32

B2: 12 + 3 = 15




pricing rule

1 W , b∗ ← solve the Winner Determination Problem WD(K );2 foreach k ∈W do3 pvcgk ← compute the VCG price b∗k −

(WD(K )−WD(K−k)

);

4 foreach k ∈W do5 pk ← pvcgk ;

6 while true do7 C ← solve the Core Separation Problem z(p);8 if

∑k pk ≥ z(p) then

9 break;10 else11 θ ← add constraints to Pricing Problem based on C , z(p);12 p ← solve the modified Pricing Problem θ;




pricing rule


(WD(K )−WD(K−k)

);








pricing rule


(WD(K )−WD(K−k)

);








solving the problem optimally

In many combinatorial optimization problems, near-optimalsolutions can be found within minutes for realistic problemsizes.The exact solution is often intractable




solving the problem optimally

Complete enumeration of bids (XOR bidding)

large amounts of bids/items/bidders

Compact bidding languages

concise formulation, domain specific

computationally hard

invidual demand curves

multi-item, multi-unit

economies of scale and scope

Focus on the TV-Ad market, and volume discount auctions




using non-optimal solutions

Issues when using suboptimal solutions

VCG

pvcgk = b∗k − (WD(K )−WD(K−k)) > b∗k

BPOC

similar, causes infeasibilities





TRIM – adjust values after problem solving

avoid infeasibilities by trimming the prices into the appropriateranges





REUSE – dynamic switching of the winning coalition

on every computation of WD:

save the coalition C including all bidsthis allows instant re-computation of WD(C )

if WD(C ) > WD(W ):

switch the winning coalition W to Crecompute VCG pricesrecompute Core constraints





Reusing the found solutions while recreating price vectors

VCG

pvcgk = b∗k −(WD(K )−WD(K−k)

)BPOC

Modify the Pricing Problem to use WD(C ) instead of z(p)




experimental evaluation TRIM & REUSE – attributes

Experimental evaluation of TRIM & REUSE(based on a TV advertisement market, and a volume discount auction market)

Treatment Variables

TV Ads

50 bidders

336 items

50 bid functions

120 units / item

Volume Discount

14 bidders

8 items

14 bid functions

100 units / item





Experimental evaluation of TRIM & REUSE(based on a TV advertisement market, and a volume discount auction market)

Focus variables

Primary metrics

efficiency E , revenue R, duration D

Secondary metrics

ratio: BPOC payments pk to bids bk (core/bid)

ratio: VCG payments pvcgk to bids bk (vcg/bid)

ratio: VCG payments pvcgk to BPOC payments pk (vcg/core)





difficult to compare absolute values

solution: normalization against the optimal computation




experimental evaluation TRIM & REUSE – comparison

Primary attributes

TRIM REUSE Baselineµ µ µ

TV Ads Market LPREfficiency E 0.91 H ◦ 0.93 N ◦ 1.00Revenue R 0.79 N - 0.68 H - -Runtime (minutes) D 95 H - 222 N - -

Volume Discount Auction OPTEfficiency E 0.99 H ◦ 0.99 N ◦ 1.00Revenue R 0.81 N 0.79 H ◦ 0.82Runtime (minutes) D 3 ◦ 3 ◦ 54

H,N: significant difference compared to the competing BPOC algorithm;

◦: significant difference to the baseline




experimental evaluation TRIM & REUSE – comparison

Secondary attributes (Volume Discount Auction)Remember: This is a procurement auction!

TRIM REUSE OPTµ σ µ σ µ σ

bid/core 0.85 0.15 N 0.81 0.15 H 0.82 0.11bid/vcg 0.80 0.18 N 0.72 0.16 H 0.82 0.11core/vcg 0.93 0.13 N 0.90 0.14 H 1.00 0.00

H,N: significant difference compared to the competing BPOC algorithm




Core payments for hard allocation problems

Two approaches to deal with near-optimal solutions:

TRIM – faster, rough price approximationREUSE – slower, good VCG and Core price approximation

→ Core payments can be approximated even with near-optimalsolutions




Core payments for hard allocation problems

Two approaches to deal with near-optimal solutions:

TRIM – faster, rough price approximationREUSE – slower, good VCG and Core price approximation

→ Core payments can be approximated even with near-optimalsolutions



Thank you!

slides: http://bit.ly/tvauction-slidesreference implementation: http://bit.ly/tvauction-project



http://bit.ly/tvauction-slides

http://bit.ly/tvauction-project

Example



WD(K) = max∑j∈J

bjyj (WD)

subject to∑j∈J

dkxij ≤ ci ∀i ∈ I , (1)

dk∑i∈I

rixij ≤ bj ∀k ∈ K , j ∈ Jk , (2)

∑i∈I

wikxij ≤ Myj ∀j ∈ J, (3)

wminj −

∑i∈I

wikxij ≤ M(1− yj) ∀j ∈ J, (4)

∑j∈Jk

yj ≤ 1 ∀k ∈ K , (5)

xij ∈ [0, 1] ∀i ∈ I , j ∈ J, (6)

yj ∈ [0, 1] ∀j ∈ J. (7)



Core Separation Problem

z(pt) = max∑j∈J

bjyj −∑k∈W

(b∗k − ptk)γk (SEPt)

subject to

constraints of WD ,∑j∈Jk

yj ≤ γk ∀k ∈W ,

γk ∈ [0, 1] ∀k ∈W .



Equitable Bidder Pareto-Optimal Problem

θ(ε) = (EBPOt)

min∑k∈W

pk+εm

subject to∑

k∈W\Cτ

pk ≥ z(pτ )−∑

k∈W∩Cτ

pτk ∀τ ≤ t,

pk−m ≤ pvcgk ∀k ∈W ,

pk ≤ b∗k ∀k ∈W ,

pk ≥ pvcgk ∀k ∈W .



Simulation input parameters

Name Parameters Distribution

{µ;σ} or {λ}

I slots 336 -

J bids 50 -

K bidders 50 -

ci slot length {60; 30} Normal

ri reservation prices (in e/s) [1, 2, 5, 10, 50, 75] {1.2} Poisson

dk ad duration {20; 10} Normal

βj bid base price (in e/s) {50; 25} Normal

wmin relj min

∑of campaign priorities (in %) {30; 20} Normal

- correlation of priority to slot reserve price - Linear

- distribution of priorities around the priority/price value - Normal



Education

Compact bid languages and core-pricing