SelfCorrecting SamplingBased Dynamic MultiUnit Auctions

1
Self-Correcting Sampling-BasedDynamic Multi-Unit
Auctions

• Florin Constantin
• Joint work with David Parkes

2
Dynamic Multi-Unit Auctions
Supply
Scen 1 TueWed
Scen S TueWed
Scen S WedThu
Scen 1 WedThu
Sampling-Based

Self-Correcting
Tuesday
Monday
3
Outline

• Model
• Why not optimal policy?
• Solution? Consensus Ironing
• Better solution NowWait
• Results

4
Multi-unit online auctions

• Tickets for sale
• Bidder attributes private
• (arrival, departure, quantity, reward)
• (Tuesday, Friday, 3 tickets, 150)
• Realize need on Tuesday must know by Friday
• (Wednesday, Thursday, 4 tickets, 100)
• Prior distribution for future bidders

5
(No Transcript)
6
Outline

• Model
• Why not optimal policy?
• Solution? Consensus Ironing
• Better solution NowWait
• Results

7
Optimal policy is not truthful
Supply
Winner
5
Tuesday
Monday
Tuesday
Monday
Loser
Winner
Tuesday
Monday
Tuesday
Monday
8
Outline

• Model
• Why not optimal policy?
• Solution? Consensus Ironing
• Better solution NowWait
• Results

9
Self-correct if not monotonic

• Optimal policy
• no algorithm known for computing it
• not truthful
• Output-ironing (Duong Parkes)
• approx optimal policy (Consensus vH)
• self-correct if not truthful
• ok if not optimal

10
Consensus Algorithm (vanHentenryck)
Output-ironing DuongParkes
Supply
5
6
Monday
Tuesday
Offline OPT
Urgent
votes
Offline OPT
Urgent
votes
10
-
4, 6
4
Monday
Monday
9
9
-
-
5, 50
1
6, 50
1
Consensus -
Consensus 4
Tuesday 5, 50 allocated
Tuesday 50 allocated
Ironing CANCELS 5s win better type (6) does
not win
Consensus most voted decision (urgent only)
11
Consensus vH
Active bidders
Active bidders
Scen 1 TueWed
Scen S TueWed

Offline OPT 1
Offline OPT S


Most voted decision U ????? (urgent only)
12
Truthfulness Monotonicity

• Want bidders to be honest
• Good (informed) decision by the seller
• Monotonicity
• If win for bid gt
• Then still win

bid
win
lose
13
Monotonicity

• Necessary and sufficient for truthfulness

bid
lose
win
lose
14
Monotonicity

• Necessary and sufficient for truthfulness

bid
lose
win
lose
15
Partial order on bidders

dominates

16
Monotonicity

If win
17
Monotonicity

Then win
?

If win
18
Departure monotonicity and ironing
Supply
OPT wait until departure Ironing all are
canceled except max patient
Winners
Monday
Tuesday
Winner
Loser
Tuesday
Monday
Tuesday
Monday
19
Outline

• Model
• Why not optimal policy?
• Solution? Consensus Ironing
• Better solution NowWait
• Results

20
Consensus Ironing PD
Consensus vH
Active bidders
Active bidders
Active bidders
Scen S TueWed
Scen 1 TueWed
Scen 1 TueWed
Scen S TueWed


Offline OPT S
Offline OPT 1
Offline OPT 1
Offline OPT S




Most voted decision U ?????(urgent only)
Most voted decision U ?????(urgent only)
Keep if win for all higher ?????
21
Select good properties
Active bidders

• easy to compute
• good policy
• easy to compute ironing
• leads to little ironing

22
Simple Select IgnoDep
23
prob. 1-?
prob. ?
Monday
Tuesday
? a priori probability of not leaving Monday
opp cost11, opp cost25
Supply
NowWait select iff reward opportunity cost
24
Urgent
Active
A
B
Sampled (future)
Offline OPT
25
Outline

• Model
• Why not optimal policy?
• Solution? Consensus Ironing
• Better solution NowWait
• Results

26

Tuesday
Monday

27
Multi-unit demand, exponential
Offline optimum
1
1
Consensus
0.85
Consensus
NowWait
NowWait - Ironing
IgnoDep
IgnoDep - Ironing
Relative efficiency
0.85
Consensus - Ironing
0
28
Myopic monotonicity

• Type ? allocated today
• Type ??
• Monotonicity ? must be allocated
• Myopic monotonicity
• if same decision until yesterday
• then ? must be allocated today
• NowWait, IgnoDep ? Cons Ir ?

29
(No Transcript)
30
Virtual valuations

• Handicap bidders with a priori higher values
• Revenue-optimal static 1-item auction My81
• Highest virtual valuation wins
• Second-price
• Reserve price

31
(No Transcript)
32
Simplicity versus optimality
Bliss point
1 item, no overlap
IgnoDep
Simplicity of ironing
NowWait
Optimal in expectation
value info used
Cardinal A?,B?
Ordinal A ltgt B
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
Trade-off quality vs. monotonicity

• Computing optimal policy may be hard
• Good online policies average-case
• may not allocate for higher bid
• Monotonic policies worst-case
• must allocate for higher bid
• What can be done?

37
Ironing quality vs. monotonicity

• value(ironed ?) value(?) - nonMonot(?)
• Bad if optimal ? highly non-monotone
• Idea ? approx optimal roughly monotone
• value(?) close to value(?)
• nonMonot(?) low
• How to find such ??

38
Consensus algorithm van Hentenryck

• votes(A)0 ?A alloc to bidders departing now
• Sample M future scenarios (bidder worlds)
• For each scenario Scj
• Let Aj be optimal allocation on Scj (offline)
• votes(Aj ? departing now)
• Implement allocation with max votes

39
Reward ironing on Consensus

• Simulate higher rewards on same scenarios
• As reward increases, the decision on a scenario
  can only change once
• overall decision changes scenarios
• Makes bookkeeping manageable
• Subroutine for departure ironing

40
Consensus Algorithm (vanHentenryck)
Supply
Offline optimum
Vote
Scenarios
Consensus
2,2
-
most frequent decision allocate 1
1
1,3
1
1,4
Monday
Tuesday
Wednesday
Important can delay decision until departure