Position Auctions with Bidder-Specific Minimum Prices

1
Position Auctions with Bidder-Specific Minimum
Prices

• Eyal Even-Dar Google
• Jon Feldman Google
• Yishay Mansour Tel-Aviv Univ., Google
• S. Muthukrishnan Google

2
Sponsored Search

• Monetization of search results
• Search engine needs to balance
• advertiser efficiency
• user experience
• revenue

3
First Model

• Advertisers submit per-click bids b(i)
• Position effect q(1) gt q(2) gt gt q(k).
• Separability Prclick(i,j) p(i) q(j)
• Rank ads by b(i) p(i)
• Under separability, maximizes efficiency
• GSP charge next price b(i1) p(i1)/p(i)
• VCG charge effect on others efficiency

4
Envy-free equilibria

• GSP not truthful, but admits an efficient
  envy-free equilibrium with same outcome as VCG.
  EOS, V, AGM
• Envy-free equilibrium Every bidder would
  rather have her own (position, price) than any
  other available.
• Stronger than Nash Equilibria in GSP

5
Reserve Prices

• Pays for SEs loss in quality
• Boosts revenue (in undersold auctions)
• In many cases, reserve prices should be
  bidder-specific
• Both Google and Yahoo use them
• AdWords FAQ Minimum prices are based on the
  quality and relevance of the keyword, its ad, and
  associated landing page.

6
Our work

• How do bidder-specific reserve prices affect
  GSP?
• - GSP equilibria no longer efficient
• - Envy locality no longer holds
• - Despite this, GSP with bidder-specific reserve
  prices still has an envy-free equilibrium. Main
  result

7
Warm-up VCG

• How do bidder-specific reserve prices affect the
  equilibria of VCG?
• Naïve application of reserve prices breaks
  truthfulness

Position effects q(1)1, q(2)1/2, q(3)1/4
Bidder Value Reserve price
A 1.50 0
B 1.25 1.00
C 0.50 0
D 0.25 0
VCG price/click for bidder B Position 1 1.5/2
.5/4 .25/4 15/16 Position 2 (.5/4 .25/4)
/ (1/2) 3/8
8
Fixes to VCG with reserve prices

• Virtual values
• For price(i), use maxb(j),R(i) instead of
  b(j).
• Efficient, truthful, not envy-free.
• Minimum price offset
• Subtract R(i) from bids b(i) b(i) R(i),
  then run VCG.
• Truthful (easy), efficient in v(i) v(i) R(i)

9
GSP with bidder-spec. reserves

• Bad news Not necessarily efficient

Bidder Value Reserve price
A 1.00 0
B 0.68 0.67
Position effects q(1) 1, q(2) 1/2
Bidder A Profit at 1st pos 1 (1.00 -
b(B)) lt 0.33 Profit at 2nd pos ½
(1.00 0) 0.50
Bidder A will underbid bidder B in any
equilibrium.
10
GSP with bidder-spec. reserves

• Bad news no envy locality
• simple example in paper locally high reserve
  prices, bargain at the bottom.
• thus, global argument is needed to show
  envy-freeness

11
GSP with bidder-spec. reserves

• Good news
• Theorem The GSP auction with arbitrary
  bidder-specific minimum prices admits an
  envy-free equilibrium

12
GSP with bidder-spec. reserves

• Proof setup
• Slot prices define bipartite best response
  graph modeling envy
• Matching in graph that hits all slots implies
  equilibrium assignment
• Tâtonnement process to raise prices
• Maintain matching on slot prefix (Halls thm)
• Grow prefix by increasing prices
• Prove if not all slots in matching, can proceed

13
Revenue

• Theorem
• Let Pvcg(j) price at pos. j under VCG without
  reserve prices
• Let Pres(j) envy-free price at pos. j under
  GSP with reserve prices
• Then, assuming all bidders have v(i) gt R(i), we
  have
• Pres(j) Pvcg(j)

14
Conclusions

• Bidder-specific reserve prices are important
  tools used by search engines.
• In VCG, naïve application can break
  truthfulness, but there are fixes
• In GSP, reserve prices can hurt efficiency, only
  help revenue, complicate bidder dynamics, but
  equilibrium still exists.

15
Future work

• Relationship of VCG variants, GSP equilibrium?
• Equilibrium discovery?
• Position-specific reserve prices?
• Gonen, Vassilvitskii, tomorrow
• Minimum quality score (ctr)?