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
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 2 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 2 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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;
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 4 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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;
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 4 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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;
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 4 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 5 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 6 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
using non-optimal solutions
Issues when using suboptimal solutions
VCG
pvcgk = b∗k − (WD(K )−WD(K−k)) > b∗k
BPOC
similar, causes infeasibilities
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 7 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
using non-optimal solutions
TRIM – adjust values after problem solving
avoid infeasibilities by trimming the prices into the appropriateranges
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 8 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
using non-optimal solutions
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 9 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
using non-optimal solutions
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)
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 10 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 11 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
experimental evaluation TRIM & REUSE – attributes
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)
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 12 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
experimental evaluation TRIM & REUSE – attributes
difficult to compare absolute values
solution: normalization against the optimal computation
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 13 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 14 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 15 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 16 / 16
Motivation Pricing Suboptimality experimental evaluation conclusion
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 16 / 16
Thank you!
slides: http://bit.ly/tvauction-slidesreference implementation: http://bit.ly/tvauction-project
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 17 / 16
Example
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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)
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 19 / 16
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 .
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 20 / 16
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 .
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 21 / 16
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
Goetzendorff, Bichler, Day, Shabalin TUM, UCONN
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