1) The document discusses using compact bidding languages and core pricing in large multi-object auctions to address computational hardness. 2) It proposes two approaches, TRIM and REUSE, to allow application of VCG and core pricing using non-optimal solutions from compact bidding languages in near real-time. 3) An experimental evaluation shows that TRIM and REUSE maintain high efficiency and revenue compared to optimal benchmarks, while significantly reducing computation time.

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