1. Motivation Pricing Suboptimality experimental evaluation conclusion
Compact bid languages and core-pricing
in large multi-object auctions
Andor Goetzendor 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
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 1 / 16

2. 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 suer from the
computational hardness of many real-world market design
problems
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 2 / 16

3. 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 suer from the
computational hardness of many real-world market design
problems
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 2 / 16

4. Motivation Pricing Suboptimality experimental evaluation conclusion
pricing rule
B3
20 B2
12 B4
Item A
Item B
B1
28
14
32
32
B5
Pay-As-Bid
VCG
BPOC
Source: Cramton and Day (2009)
Bids
A B AB
B1: 28 0 28
B2: 0 20 20
B3: 14 0 14
B4: 0 12 12
B5: 0 0 32
VCG Prices
B1: 28 (48 34) = 14 o
= 26
B2: 20 (48 40) = 12
BPOC Prices
B1: 14 + 3 = 17 o
= 32
B2: 12 + 3 = 15
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16

5. Motivation Pricing Suboptimality experimental evaluation conclusion
pricing rule
B3
20 B2
12 B4
Item A
Item B
B1
28
14
32
32
B5
Pay-As-Bid
VCG
BPOC
Source: Cramton and Day (2009)
Bids
A B AB
B1: 28 0 28
B2: 0 20 20
B3: 14 0 14
B4: 0 12 12
B5: 0 0 32
VCG Prices
B1: 28 (48 34) = 14 o
= 26
B2: 20 (48 40) = 12
BPOC Prices
B1: 14 + 3 = 17 o
= 32
B2: 12 + 3 = 15
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16

6. Motivation Pricing Suboptimality experimental evaluation conclusion
pricing rule
B3
20 B2
12 B4
Item A
Item B
B1
28
14
32
32
B5
Pay-As-Bid
VCG
BPOC
Source: Cramton and Day (2009)
Bids
A B AB
B1: 28 0 28
B2: 0 20 20
B3: 14 0 14
B4: 0 12 12
B5: 0 0 32
VCG Prices
B1: 28 (48 34) = 14 o
= 26
B2: 20 (48 40) = 12
BPOC Prices
B1: 14 + 3 = 17 o
= 32
B2: 12 + 3 = 15
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16

7. Motivation Pricing Suboptimality experimental evaluation conclusion
pricing rule
B3
20 B2
12 B4
Item A
Item B
B1
28
14
32
32
B5
Pay-As-Bid
VCG
BPOC
Source: Cramton and Day (2009)
Bids
A B AB
B1: 28 0 28
B2: 0 20 20
B3: 14 0 14
B4: 0 12 12
B5: 0 0 32
VCG Prices
B1: 28 (48 34) = 14 o
= 26
B2: 20 (48 40) = 12
BPOC Prices
B1: 14 + 3 = 17 o
= 32
B2: 12 + 3 = 15
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 3 / 16

8. Motivation Pricing Suboptimality experimental evaluation conclusion
pricing rule
1 W; b solve the Winner Determination Problem WD(K);
2 foreach k 2 W do
3 pvcg
k compute the VCG price b
k
WD(K) WD(Kk )
;
4 foreach k 2 W do
5 pk pvcg
k ;
6 while true do
7 C P
solve the Core Separation Problem z(p);
8 if
k pk z(p) then
9 break;
10 else
11 add constraints to Pricing Problem based on C, z(p);
12 p solve the modi

9. ed Pricing Problem ;
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 4 / 16

10. Motivation Pricing Suboptimality experimental evaluation conclusion
pricing rule
1 W; b solve the Winner Determination Problem WD(K);
2 foreach k 2 W do
3 pvcg
k compute the VCG price b
k
WD(K) WD(Kk )
;
4 foreach k 2 W do
5 pk pvcg
k ;
6 while true do
7 C P
solve the Core Separation Problem z(p);
8 if
k pk z(p) then
9 break;
10 else
11 add constraints to Pricing Problem based on C, z(p);
12 p solve the modi

11. ed Pricing Problem ;
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 4 / 16

12. Motivation Pricing Suboptimality experimental evaluation conclusion
pricing rule
1 W; b solve the Winner Determination Problem WD(K);
2 foreach k 2 W do
3 pvcg
k compute the VCG price b
k
WD(K) WD(Kk )
;
4 foreach k 2 W do
5 pk pvcg
k ;
6 while true do
7 C P
solve the Core Separation Problem z(p);
8 if
k pk z(p) then
9 break;
10 else
11 add constraints to Pricing Problem based on C, z(p);
12 p solve the modi

13. ed Pricing Problem ;
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 4 / 16

14. Motivation Pricing Suboptimality experimental evaluation conclusion
solving the problem optimally
In many combinatorial optimization problems, near-optimal
solutions can be found within minutes for realistic problem
sizes.
The exact solution is often intractable
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 5 / 16

15. 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 speci

16. c
computationally hard
invidual demand curves
multi-item, multi-unit
economies of scale and scope
Focus on the TV-Ad market, and volume discount auctions
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 6 / 16

17. Motivation Pricing Suboptimality experimental evaluation conclusion
using non-optimal solutions
Issues when using suboptimal solutions
VCG
pvcg
k = b
k (WD(K) WD(Kk )) b
k
BPOC
similar, causes infeasibilities
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 7 / 16

18. Motivation Pricing Suboptimality experimental evaluation conclusion
using non-optimal solutions
TRIM { adjust values after problem solving
avoid infeasibilities by trimming the prices into the appropriate
ranges
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 8 / 16

19. 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 bids
this allows instant re-computation of WD(C)
if WD(C) WD(W):
switch the winning coalition W to C
recompute VCG prices
recompute Core constraints
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 9 / 16

20. Motivation Pricing Suboptimality experimental evaluation conclusion
using non-optimal solutions
Reusing the found solutions while recreating price vectors
VCG
pvcg
= b
WD(K) )
k k WD(Kk BPOC
Modify the Pricing Problem to use WD(C) instead of z(p)
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 10 / 16

21. 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
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 11 / 16

22. 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
eciency E, revenue R, duration D
Secondary metrics
ratio: BPOC payments pk to bids bk (core/bid)
ratio: VCG payments pvcg
k to bids bk (vcg/bid)
ratio: VCG payments pvcg
k to BPOC payments pk (vcg/core)
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 12 / 16

23. Motivation Pricing Suboptimality experimental evaluation conclusion
experimental evaluation TRIM REUSE { attributes
dicult to compare absolute values
solution: normalization against the optimal computation
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 13 / 16

24. Motivation Pricing Suboptimality experimental evaluation conclusion
experimental evaluation TRIM REUSE { comparison
Primary attributes
TRIM REUSE Baseline
TV Ads Market LPR
Eciency E 0.91 H 0.93 N 1.00
Revenue R 0.79 N - 0.68 H - -
Runtime (minutes) D 95 H - 222 N - -
Volume Discount Auction OPT
Eciency E 0.99 H 0.99 N 1.00
Revenue R 0.81 N 0.79 H 0.82
Runtime (minutes) D 3 3 54
H;N: signi

25. cant dierence compared to the competing BPOC algorithm;
: signi

26. cant dierence to the baseline
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 14 / 16

27. Motivation Pricing Suboptimality experimental evaluation conclusion
experimental evaluation TRIM REUSE { comparison
Secondary attributes (Volume Discount Auction)
Remember: This is a procurement auction!
TRIM REUSE OPT
bid/core 0.85 0.15 N 0.81 0.15 H 0.82 0.11
bid/vcg 0.80 0.18 N 0.72 0.16 H 0.82 0.11
core/vcg 0.93 0.13 N 0.90 0.14 H 1.00 0.00
H;N: signi

28. cant dierence compared to the competing BPOC algorithm
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 15 / 16

29. Motivation Pricing Suboptimality experimental evaluation conclusion
Core payments for hard allocation problems
Two approaches to deal with near-optimal solutions:
TRIM { faster, rough price approximation
REUSE { slower, good VCG and Core price approximation
! Core payments can be approximated even with near-optimal
solutions
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 16 / 16

30. Motivation Pricing Suboptimality experimental evaluation conclusion
Core payments for hard allocation problems
Two approaches to deal with near-optimal solutions:
TRIM { faster, rough price approximation
REUSE { slower, good VCG and Core price approximation
! Core payments can be approximated even with near-optimal
solutions
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 16 / 16

31. Thank you!
slides: http://bit.ly/tvauction-slides
reference implementation: http://bit.ly/tvauction-project
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 17 / 16

32. Example
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 18 / 16

33. WD(K) = max
X
j2J
bjyj (WD)
subject to
X
j2J
dkxij ci 8i 2 I ; (1)
dk
X
i2I
ri xij bj 8k 2 K; j 2 Jk ; (2)
X
i2I
wikxij Myj 8j 2 J; (3)
wmin
j
X
i2I
wikxij M(1 yj ) 8j 2 J; (4)
X
j2Jk
yj 1 8k 2 K; (5)
xij 2 [0; 1] 8i 2 I ; j 2 J; (6)
yj 2 [0; 1] 8j 2 J: (7)
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 19 / 16

34. Core Separation Problem
z(pt ) = max
X
j2J
bjyj
X
k2W
k ptk
(b
)
k (SEPt )
subject to
constraints of WD ;
X
j2Jk
yj
k 8k 2 W;
k 2 [0; 1] 8k 2 W:
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 20 / 16

35. Equitable Bidder Pareto-Optimal Problem
() = (EBPOt )
min
X
k2W
pk+m
subject to
X
k2WnC
pk z(p )
X
k2WC
p
k 8 t;
pkm pvcg
k 8k 2 W;
pk b
k 8k 2 W;
pk pvcg
k 8k 2 W:
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 21 / 16

36. Simulation input parameters
Name Parameters Distribution
f; g or fg
I slots 336 -
J bids 50 -
K bidders 50 -
ci slot length f60; 30g Normal
ri reservation prices (in e/s) [1, 2, 5, 10, 50, 75] f1.2g Poisson
dk ad duration f20; 10g Normal

37. j bid base P
price (in e/s) f50; 25g Normal
wmin rel
j min
of campaign priorities (in %) f30; 20g Normal
- correlation of priority to slot reserve price - Linear
- distribution of priorities around the priority/price value - Normal
Goetzendor, Bichler, Day, Shabalin TUM, UCONN
Compact bid languages and core-pricing 22 / 16