Online Ascending Auctions for Gradually Expiring Items

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


(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


(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


(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


(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


(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


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


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


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


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


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)


>Set mixed Nash:



Ri

R-i

Eπ( )

Eπ( )


RiSet-Nash:



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:


“Ignorable extension”

Online Iterative:

“Ignorable extension”


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.

Documents

Online Ascending Auctions for Gradually Expiring Items