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Online Ascending Auctions for Gradually Expiring Items
Ron Lavi and Noam Nisan
SISL/IST, Caltech Hebrew University
The Model (I)
• M identical items that “expire” at different times.• Players arrive over time, and desire one item
between their arrival time and their deadline.
. . .1 2 3 4
Items:
Expiration times:
Player 1arrival time
deadline
Player 2
Player 3
Time 1
Time 2
• Player i has value vi for receiving a desired item.
• Players are selfish:– All information (arrival time, deadline, value) is private, known
only to the player.– Each player acts in order to maximize his own utility: value -
price.
• Our goal is to maximize the sum of (true) values of players that receive an item (the “social welfare”).
• Applications:– In economic settings e.g. transportation tickets– In computational settings e.g. bandwidth allocation
The Model (II)
Algorithmic Status
• Well studied - equivalent to scheduling of unit jobs.
• Offline optimal allocation is poly-time computable(has a matroid structure).
• Lower bound of 1.618 for online approximation. [Hajek]
• Online greedy is a 2 - approximation:
greedy: at time t, allocate item t to the player with highest value.
– This assumes obedient players that simply reveal theirprivate information.
Truthfulness and its difficulties • A popular approach: truthful auctions.
– Motivating the player to reveal his true parameters.
– Strong argument of dominant strategy: no matter what others do, the truth will maximize “my” utility.
– Many recent positive examples for truthful auctions.
• Unfortunately, we show that:
Theorem: Any deterministic truthful auction for our allocation problem cannot obtain an approximation ratio better than M.
– A simple truthful M - approximation exists.
How to approach this difficulty?• Relax the equilibrium notion to Bayesian - Nash:
– Not a worst-case analysis. Requires strong distributional assumptions.
• Add assumptions about player types. E.g. assume values in [vmin , vmax]. Then a randomized truthful 2 log(vmax - vmin) approximation exists (a special case of [BSZ]).
– vs. a deterministic 2 - approximation without any assumptions when truthfulness is dropped.
• Our approach:– New, relaxed, notion of equilibrium.
– Worst - case analysis. No distributional assumptions.
– No additional assumptions about player types.
Outline for rest of the talk
• Describe two ascending auctions:– Their algorithmic properties– Intuition to an equilibrium notion that fits well
• Describe a new notion of equilibrium– discuss its properties
• Main theorem– Intuition for the proof
The Online Iterative Auction• Maintain temporary prices and owners for each item (initialized to
0).• At each time unit t=1,2… :
• Repeat:Some player that doesn’t currently own an item temporarily takes an item, and increases the price by .
Until no losing player wishes to make a new bid.
• Allocate item t to its current owner for the listed price - . Keep prices and temporary owners for next time unit.
• This is an adaptation of the Iterative Auction of [DGS].
Example
1 2Item
Temp. winner
Temp. price
-- --
0 0
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
=1
Example
1 2Item
Temp. winner
Temp. price
I --
1 0
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
(phase 1)
=1
Example
1 2Item
Temp. winner
Temp. price
I II
1 1
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
(phase 2)
=1
Example
1 2Item
Temp. winner
Temp. price
III II
2 1
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
(phase 3)
=1
Example
1 2Item
Temp. winner
Temp. price
I II
3 1
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
(phase 4)
=1
Example
1 2Item
Temp. winner
Temp. price
I II
3 1
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
(phase 4)
Player I did not bid for the item with lowest price.
=1
Example
1 2Item
Temp. winner
Temp. price
I II
3 1
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
Result:
Player I wins item 1 and pays 2.
If no new player will arrive, player II will win item 2 for a price of 0.
But, player II might not win at all if a new high valued player will now arrive.
=1
Players’ behaviors (the offline case)
DFN([DGS]): A player is myopic if he always bids on the item with lowest price among those he desires.
THM([DGS],[GS]): Assume all players arrive at time 1:• When all players are myopic then the online
iterative auction finds the optimal allocation*.• When all other players are myopic, player i will
maximize* his utility by behaving myopically.
* up to a difference of about .
• A tight block B S: |B|=d and jB d(j) < d.
• Tight blocks must be prefixes of S, thus contained one in the other.
• Special focus in the minimal tight block f.
Basic structure of allocations
d1 2 . . . M
S = the optimal allocation
j’’ jj’
• A tight block B S: |B|=d and jB d(j) < d.
• Tight blocks must be prefixes of S, thus contained one in the other.
• Special focus in the minimal tight block f.
• Every j in f can be located first.
• Therefore, its “social cost” is the value of the highest unallocated player.
Basic structure of allocations
d
f
Highest un-allocated player determines VCG price of all players in f
1 2 . . . M
S = the optimal allocation
j’’ jj’
i*
The offline iterative auction with myopic players finds the optimal allocation
• All prices in f are equal (because of the structure of swaps):
– p(j’) < p(j’’) since j’ is myopic
– p(j’’) < p(j’) since j’’ is myopic and has far-enough deadline.
• Prices will continue to go up exactly until v(i*).
d
f
Highest un-allocated player determines VCG price of all players in f
1 2 . . . M
S = the optimal allocation
j’’ jj’
i*
• In the online case, non myopic behaviors might perform better.
• E.g. bidding more aggressively for the current item makes sense if one anticipates that many competitive players will arrive later on.
DFN: A player is semi - myopic if he bids on some item with price lower than his value.
THM: If all players are semi - myopic then the online iterative auction obtains a 3 - approximation.
Players’ behaviors (the online case)
The Sequential Japanese Auction
• Item t is sold at time t using a classic Japanese auction:– The auctioneer starts raising a price.
– Each player decides whether to drop or to stay as the price ascends.
– We allow to observe how many players remain at each moment.
– The price halts when only one player remains. This player wins and pays the price that was reached(up to some tie breaking adjustment rule).
Example
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
What if players I and II decide not to participate at all in the auction for item 1?
Player III will win item 1.
Player I will certainly not win anything.
Player II might win item 2, but for a price of 3.
Example continued
Player I: v=3, d=2
Player II: v=5, d=2
Player III: v=2, d=1
Suppose players I and II decide to stay until the price reaches their value, or until there remain two players in the auction (including themselves):
2At price=2, player III will drop.
Immediately afterwards, both I and II drops.
So either I or II wins and pays 2.
Price
Players’ behaviors (the offline case)Surprisingly, a notion of myopic behavior leads to the optimal allocation here as well:
DFN: A player is myopic if, at any time t, he drops exactly:– when the price reaches his value, or– when d - t other players remain (where d is his
deadline).
THM: If all players arrive at time 1, and are all myopic, then the Sequential Japanese Auction finds the optimal allocation.
Proof• p*=value of highest unallocated player i*, |f|=d
• Price < p* implies that no one from f drops:
– At least d+1 players still remain (all f + i*)
– Price is still low.
• At price = p* all remaining unallocated players drop, and after them all remaining players of S.
• Players of f start to drop only after all others have dropped.
winner of item 1 = optimal item 1 winner.
• Continue inductively.
p*
Price
• In the online case, again, bidding more aggressively for the current item makes sense if one anticipates that many competitive players will arrive later on.
DFN: A player is semi - myopic if, at any time t, he drops:– not earlier than d-t other players remain, and– not later than when the price reaches his value.
THM: If all players are semi - myopic then the Sequential Japanese Auction obtains a 3 - approximation.
Players’ behaviors (the online case)
Summary of auctions
OnlineIterative
SequentialJapanese
Myopic behavior Semi-myopic behavior
bid for the item with the lowest price
bid for some item with price < value
Drop when(i) price reaches value or(ii) Exactly d-t other players remain
Drop in between (i) price reaches value and(ii) d-t other players remain
Proving the approximation
Lemma: Any semi - myopic algorithm obtainsa 3 - approximation.
Lemma: When players are semi - myopic then both our auctions are semi - myopic algorithms.
MyopicGreedyAllocate to bidder with highest value
Allocate according to current best allocation
Semi - myopic
Allocate to someone with value > value of the winner of item t in a current best allocation ( = an optimal allocation of items t,…,M among the active players at time t ).
Semi - Myopic Algorithms
Set - Nash Equilibrium• The above intuition implies that we do not expect a player to
follow a specific strategy. Instead, we define a set of “recommended strategies” Ri for player i.
DFN: The strategy sets R1 … Rn are in Set – Nash
equilibrium if a best response to every s-i R-i exists in Ri
• Comment 1: If | Ri |=1, then equivalent to regular Nash.
• Comment 2: Best response to mixed strategies might be outside Ri
– stronger definitions can require that too.
• Comment 3: Only interesting if you can say something about the outcome when everyone plays in Ri
• Comment 4: Naturally generalizes to games with incomplete information without a Bayesian prior: Ri(ti)
Stronger set notions
>Set domination:
Player i’s strategies
Strategies of other players
Ri
R-i
(coordinate-wise)
Stronger set notions
>Set mixed Nash:
Player i’s strategies
Strategies of other players
Ri
R-i
Eπ( )
Eπ( )
Stronger set notions
RiSet-Nash:
Player i’s strategies
Strategies of other players
Ri
R-i
MAX( )
MAX
Main Theorem: The Online Iterative Auction and the Sequential Japanese Auction Set - Nash implementa 3 - approximation of the welfare.
I.e., both auctions have Set - Nash equilibrium that are all semi - myopic, hence guarantee a 3 - approximation.
• All the recommended strategies are not dominated.
• The recommended strategies contain best responses even if the strategies of the others are from a much larger set.
• The recommended strategies do not necessarily contain b.r. to mixed recommended strategies -- We think this is an interesting open problem.
Proof structure
Basic building block:
Semi Myopic Mechanism
Recommended Strategies thatare in Set - Nash
Sequential Japanese:
Semi Myopic Mechanism
“Ignorable extension”
Online Iterative:
“Ignorable extension”
Semi Myopic Mechanism
Reminder: with myopic players, the ascending auctions compute ft and VCG prices
Reminder: basic structure of allocations
d
ft
Highest un-allocated player determines VCG
price of all players in ft
t t+1 . . . M
St = the optimal allocation
j’ j
i*
Semi Myopic MechanismsStrategy space. Extended direct revelation:
{ arrival time, value, “false” deadline, “true” deadline }
(Similar in spirit to “2nd chance mechanisms” [NR])
Allocation rule. Compute St according to “pretend deadlines”:
– Allocate item t to some player in ft .
Payment Rule.– For any player i, let ct(i) be his VCG price for entering St .
– Set temporary prices
– The winner i pays maxt’<t pt’(i)
“false” deadline If this has not passed.“true” deadline Otherwise.
“pretend deadline” =
ct(i) If ift .[0, ct(i) ] If i St - ft .0 Otherwise
pt(i) =
Set - Nash in Semi Myopic Mechanisms
Recommended strategies: declare true arrival time, value, and true “true deadline”, and any “false deadline” < true deadline.
Lemma 1: When all players play recommended strategies then the allocation rule of a semi myopic mechanism is a semi myopic algorithm.
Lemma 2: These recommended strategies form a Set -Nash Equilibrium.
Semi Myopic Mechanism Ascending Auctions
Recommended strategies for the Online Iterative Auction:play myopically with a fake deadline until it has passed, and myopically with the real deadline afterwards.
Lemma: The Semi Myopic Mechanism is embedded in the Online Iterative Auction.
Proof sketch: need to show that the requirements of the semi-myopic mechanism hold:
– winners belong to ft
– prices are VCG
Already know these from the offline analysis
Summary• We study an online setting with “gradually expiring items”.
• We first saw that truthful auctions cannot perform well.
• We then explored a new approach to this difficulty.– Worst case, no additional assumptions on players.
• Analyzed two adaptations to classical ascending auctions.– Both obtain a 3 - approximation under a large family of selfish
behaviors.
• Introduced the notion of “Set - Nash equilibrium”.– Both our auctions have Set - Nash equilibrium that guarantees a 3 -
approximation of the social welfare.
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