Singleitem dynamic auction interdependent values

1
(No Transcript)
2
Single-item dynamic auction interdependent
values

• One item
• Dynamic (online)
• Bidders have private information (estimate)
• Allow is value to depend on js estimate (i is
  uncertain)

• Main Q which auction makes most money?

3
The bidder model at a (long) glance
4
Dynamic auctions, interdependent values

• Other applications
• Sale of one-of-a-kind item (e.g. Yahoo!)
• Multi-agent AI allocate task to autonomous
  agents
• Related work
• Static auction, interdependent values Economics
• Dynamic auction, private values Parkes et al.,
  Nisan et al.

5
Truthfulness

• aka Incentive Compatibility
• Bidders do not regret reporting honestly even if
  they find out other bidders information
• Truthful reporting ex post Nash equilibrium

6
Main Q
7
Main Q

• Which truthful dynamic
• interdependent-values auction
• makes most money?

8
Simplest private values, static, 1 bidder

• Bids in 0,1 cdf F, pdf f F gt 0
• Price p. Sell if bid ? p. Best p p?

9
Simplest private values, static, 1 bidder

• Bids in 0,1 cdf F, pdf f F gt 0
• Price p. Sell if bid ? p. Best p p?
• Revenue pProb(bid ? p) p(1-F(p))
• p0 revenue 01 p1 revenue 10

10
Simplest private values, static, 1 bidder

• Bids in 0,1 cdf F, pdf f F gt 0
• Price p. Sell if bid ? p. Best p p?
• Revenue pProb(bid ? p) p(1-F(p))
• p0 revenue 01 p1 revenue 10
• Interior max ?/?prevenue 0

11
Simplest private values, static, 1 bidder

• Bids in 0,1 cdf F, pdf f F gt 0
• Price p. Sell if bid ? p. Best p p?
• Revenue pProb(bid ? p) p(1-F(p))
• p0 revenue 01 p1 revenue 10
• Interior max ?/?prevenue 0

Virtual valuation
12
Virtual valuation

• Regular case (many distributions f and F)
• Sell if and only if
• Virtual valuation auction take-it-or-leave-it p

p
Dont sell
Sell
13
Revenue-optimal truthful auction known

• 1 bidder, static, private values
• virtual valuation auction take-it-or-leave-it p

14
Revenue-optimal truthful auction known

• 1 bidder, static, private values
• virtual valuation auction take-it-or-leave-it p
• 2 bidders, static, private values My
• Second price virt val auction with reserve price
  p

15
Revenue-optimal truthful auction known

• 1 bidder, static, private values
• virtual valuation auction take-it-or-leave-it p
• 2 bidders, static, private values My
• Second price virt val auction with reserve price
  p
• 2 bidders, static, interdependent values Br
• Interdep 2nd price virt val auction with reserve
  p

16
Revenue-optimal truthful auction known

• 1 bidder, static, private values
• virtual valuation auction take-it-or-leave-it p
• 2 bidders, static, private values My
• Second price virt val auction with reserve price
  p
• 2 bidders, static, interdependent values Br
• Interdep 2nd price virt val auction with reserve
  p
• 2 bidders, dynamic, interdependent values
• Dont know much about future bidders

17
Truthful ? described by critical estimates
Poster Thu

• Critical estimate
• i wins ? is estimate ? is critical estimate
• If highest value wins, 2s CE e1
• Pay value at CE here, second-highest bid
• CEi cannot depend on own estimate ei
• Hard because other values depend on ei

18
Mixed Integer Program Formulation

• Get discrete revenue-optimal auction via MIP
• Search for best critical estimates at discrete
  points Automated Mechanism Design
• Real and integer variables, linear constraints
• Only viable for small applications
• MIP size exponential in max arity of values and
  critical estimates
• To approximate solution of a MIP is hard
• We used CPLEX, an off-the-shelf solver

19
Instantiation

• vi(ei, e-i) ei0.5avg(e-i), eiU0,1
• (virtual) valuations linear, symmetric
• At depi, seller knows (correct) probability of 3
  arriving p3A

20
Expected virtual-valuation heuristic

• Sell to departing bidder if his v.v. is higher
  than the max expected v.v. of other bidders

21
Expected virtual-valuation heuristic

• Sell to departing bidder if his v.v. is higher
  than the max expected v.v. of other bidders

• Discrete
• Truthful
• Optimal

• Continuous
• Not Truthful
• Not Optimal

22
Results

• Exp-v-v heuristic close to optimal, but not IC
• MIP revenue gt Exp-v-v revenue if uncertainty
• Revenue increases with amount of overlap
• Expected revenue increases as p3A increases
• MIP critical estimates in case of full overlap
  are almost identical to optimal (no obvious bugs)

23
Conclusions
Future work

• MIP formulation for finding revenue-optimal
  auction
• Naïve generalization of optimal static auction
• not truthful
• not optimal, but close

• Values decreasing with time
• Correlation of bidder estimates
• Multiple items