Side-Communication Improves Efficiency of Ascending Auctions: The Two-Items Case

Side-Communication Improves Efficiency of Ascending Auctions:

The Two-Items Case

Ron Lavi

Industrial Engineering and Management

Technion – Israel Institute of Technology

Sigal Oren

Computer Science

Cornell University

Motivation

• Ascending auctions: Auctioneer gradually increases item prices in response to bidders’ demand reports.

– Popular over the Internet, in governmental auctions, even in experimental computerized systems.

• However, more collusion opportunities,

– since bidding process is longer

– since bids can be used to as signaling

Motivation

Real examples:

• Netherlands' 3G Telecom Auction: a bidder firm threatened legal action if another firm continued to bid (Klemperer ’02)

• FCC auctions: bids included single dollar quantities, probably to coordinate to lower competition (Cramton & Schwartz ’00)

How do players use communication to increase utility?

– they aim to collude and reduce prices, but how?

– few and partial theoretical models exist

Motivation

Real examples:

• Netherlands' 3G Telecom Auction: a bidder firm threatened legal action if another firm continued to bid (Klemperer ’02)

• FCC auctions: bids included single dollar quantities, probably to coordinate to lower competition (Cramton & Schwartz ’00)

Are these phenomena good or bad?

– in both cases the rules were changed to prevent such events

– common argument: less competition => less efficiency

The Model• Basic setup:

– two non-identical items, {a,b}

– players have private valuations for every subset of items

– items are substitutes: vi(ab) < vi(a) + vi(b)

– players’ utilities are quasi-linear (= value minus price)

– Seller’s goal is social efficiency: maximizing the sum of players’ values for the items they receive

• Ascending auctions: extensive-form game. In each node,

– players report their demanded set

– if no over-demand: each player receives a demanded set, pays sum of prices of items in this set, game ends

– otherwise: prices of over-demanded items increase

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = p(b) < 2

With myopic bidding:

D1 = abD2 = ab

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = p(b) = 2+ D1 = aD2 = ab

With myopic bidding:

p(b) = v1(b|a) = v1(ab) – v1(a)

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8+D1 = a or bD2 = ab

With myopic bidding:

p(b) = 2+

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 9D1 = a or bD2 = b

With myopic bidding:

p(b) = 3

the end

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 9D1 = a or bD2 = b

With myopic bidding:

p(b) = 3

the end

Equivalent to theEnglish auction ofGul & Stacchetti (2000)

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8D1 = a or bD2 = ab

A better strategyfor player 2:

• As before until p(a)=8, p(b)=2

p(b) = 2

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8D1 = a or bD2 = b

A better strategyfor player 2:

• As before until p(a)=8, p(b)=2

• Then a “demand reduction”:

p(b) = 2

Situation without side-communication

• Without side-communication:

– truthful demand reporting is not an ex-post equilibrium

– no efficient ex-post equilibrium exists even for two items(with at least four item: Gul & Stacchetti ’00)

– An inefficient Bayesian-Nash equilibrium exists(Goeree & Lien ’09)

This work: with side-communication

Main Result: with one bit of allowed communication per-bidder, there exists an efficient ex-post subgame perfect equilibrium.

Conceptually,

• Myopic bidding sometimes creates bubble prices

– the bidder firm who threatened legal action may be right

• Our strategy prevents such bubbles. In its general form:

– initially bidders bid myopically

– at a well-defined point they perform a demand reduction, whose exact nature is determined by a single message.

• This fits the appearance of real-life collusion, but guarantees optimal social efficiency.

Related work (1)

• Collusion also create inefficiencies.

– demonstrated many times: a Bayesian-Nash equilibriumin which players exploit probabilistic knowledge to “agree” on too-low prices (even without side-communication)

– For example in Brusco and Lopomo (’02);Albano, Germano and Lovo (’06); Zheng (’06)

• We show how a certain form of limited side-communication may be the answer.

Related work (2)

• Other ways to reach the efficient outcome via indirect mechanisms:

– Ausubel (2006) - using multiple price trajectories

– Parkes (1999)

– Ausubel and Milgrom (2002)

– ….

• We add another possible way: ascending prices, using anonymous item prices, but with side communication.

using non-anonymous bundle prices

Rest of talk

• Some technical background

• More details on the problematic aspects of myopic bidding

– Crucial to understanding the proposed equilibrium strategies

• Description of the proposed equilibrium strategies

• Few proofs

• Summary

Some technical background (1)

• The “demand” of player i in prices p is:

Di(p) = argmax S{a,b} vi(S) – p(S) ( where p(S) = xS p(x) )

• “Walrasian equilibrium”: allocation S1,…,Sn and prices p(a),p(b) such that (1) Si Di(p) ; (2) i Si = {a,b}

• Example:

a b ab

v1 10 4 12

v2 10 5 14

S1 = {a} , S2 = {b}

p(a) = 9 , p(b) = 3

is a Walrasian equilibrium.

Some technical background (2)• VCG is the following direct mechanism (= players report values):

– Items are allocated to maximize social efficiency (according to reported types).

– Player i pays the “damage” she causes to the other players:

sum of values of optimal allocation without i minussum of values of other players in chosen allocation

– Truthfulness is a dominant strategy in VCG

• Example:a b a

b

v1 10 4 12

v2 10 5 14

S1 = {a} , S2 = {b}

p1 = 9 , p2 = 2

is VCG’s outcome.

The necessity of a demand reduction

• By a revenue-equivalence argument: in any efficient ex-post equilibrium, prices must be VCG prices.

– Thus our strategy must always reach VCG prices

• Myopic bidding results in a Walrasian equilibrium

• Gul & Stacchetti: Walrasian prices are larger than VCG prices

• Therefore a demand reduction is necessary.

• We pin-point a simple way to do a correct demand reduction via side-communication.

Myopic bidding

• In our example, with myopic bidding, prices exceeded VCG prices in a “jump phase” where:

– only two players i,j have non-empty demand

– player j demands {a,b} and player i demands {{a},{b}}

Lemma: For any valuations v1,...,vn,

• Proof (quite technical) implies that before the jump phase, prices are lower than VCG prices myopic players know exactly when prices cross VCG prices!

the ascending auction with truthful demand reporting

terminates in a “jump phase”

there exists at least one player with Walrasian

price ≠ VCG price

The jump phase in the example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8

With myopic bidding:

p(b) = 2

equilibrium??

D1 = a or bD2 = ab

The jump phase in the example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8

With myopic bidding:

p(b) = 2

equilibrium

unsuccessful attempt: player 2 (the “big” player) reduces demand (demands only b)

(hint: need to reach VCG outcome)

D1 = a or bD2 = b

The jump phase in the example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8

With myopic bidding:

p(b) = 2

equilibrium

unsuccessful attempt: player 2 (the “big” player) reduces demand (demands only b)

(hint: need to reach VCG outcome)

(sometimes gives wrong incentive to player 1)

8.5

D1 = a or bD2 = b

The jump phase in the example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8

With myopic bidding:

p(b) = 2

equilibrium

(turns out that) successful attempt: player 1 (the “small” player) reduces demand (demands only a)

(hint: need to reach VCG outcome)

D1 = aD2 = ab

The jump phase in the example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 9

With myopic bidding:

p(b) = 2

equilibrium

(turns out that) successful attempt: player 1 (the “small” player) reduces demand (demands only a)

(hint: need to reach VCG outcome)

We reach the VCG outcome.D1 = aD2 = b

The jump phase in the example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 9D1 = aD2 = b

With myopic bidding:

p(b) = 2

equilibrium

(turns out that) successful attempt: player 1 (the “small” player) reduces demand (demands only a)

(hint: need to reach VCG outcome)

We reach the VCG outcome.But not a Walrasian equilibrium. Player 1 prefers item b over ain these prices (but cannotobtain it in these prices)

The equilibrium strategy (1st attempt)

1. If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand

The equilibrium strategy (1st attempt)

1. If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand

2. O/W (there are two active players and Di(p) = {{a},{b}} ) then:

a) i asks the other active player, j, which item to demand

The equilibrium strategy (1st attempt)

1. If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand

2. O/W (there are two active players and Di(p) = {{a},{b}} ) then:

a) i asks the other active player, j, which item to demand

b) j answers ‘item x’: i demands x until p(x)= vi(x), then quits

The equilibrium strategy (1st attempt)

1. If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand

2. O/W (there are two active players and Di(p) = {{a},{b}} ) then:

a) i asks the other active player, j, which item to demand

b) j answers ‘item x’: i demands x until p(x)= vi(x), then quits

c) j gives invalid answer: i reports true demand from now on

The equilibrium strategy (1st attempt)

1. If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand

2. O/W (there are two active players and Di(p) = {{a},{b}} ) then:

a) i asks the other active player, j, which item to demand

b) j answers ‘item x’: i demands x until p(x)= vi(x), then quits

c) j gives invalid answer: i reports true demand from now on

3. If i receives a demand question from another player j then she answers ???

The jump phase in general

a b ab

vj ? ? ?

vi 10 5 14

p(a) p(b)

p(a)Dj = a or b

Di = ab

With equilibrium bidding: (how to reach VCG outcome in general?)

p(b)

The jump phase in generalWith equilibrium bidding: (how to reach VCG outcome in general?)

vi(a) - vi(b) > p(a) – p(b)

a b ab

vj ? ? ?

vi 10 5 14

i tells j to ignore a and demand b

The jump phase in generalWith equilibrium bidding: (how to reach VCG outcome in general?)

vi(a) - vi(b) > p(a) – p(b) = vj(a) - vj(b)

iff

vi(a) + vj(b) > vi(b) + vj(a)

Since Dj = {a} or {b}

a b ab

vj ? ? ?

vi 10 5 14

i tells j to ignore a and demand b

The jump phase in generalWith equilibrium bidding: (how to reach VCG outcome in general?)

vi(a) - vi(b) > p(a) – p(b) = vj(a) - vj(b)

iff

vi(a) + vj(b) > vi(b) + vj(a)

Since Dj = {a} or {b}

a b ab

vj ? ? ?

vi 10 5 14

i tells j to ignore a and demand b

Sjab a b

Si b a ab

The equilibrium strategy

1. If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand

2. O/W (there are two active players and Di(p) = {{a},{b}} ) then:

a) i asks the other active player, j, which item to demand

b) j answers ‘item x’: i demands x until p(x)= vi(x), then quits

c) j gives invalid answer: i reports true demand from now on

3. If i receives a demand question from another player j then:

1. if vi(a) - vi(b) > p(a) – p(b) then i j : “demand b”

2. if vi(a) - vi(b) < p(a) – p(b) then i j : “demand a”

THM: This is ex-post (subgame-perfect) equilibrium

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = p(b) < 2

With equilibrium bidding:

D1 = abD2 = ab

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = p(b) = 2 D1 = aD2 = ab

With equilibrium bidding:

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8D1 = a or bD2 = ab

p(b) = 2

With equilibrium bidding:

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8D1 = a or bD2 = ab

p(b) = 2

With equilibrium bidding:

• 1 asks 2 which item to demand?

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8D1 = a or bD2 = ab

p(b) = 2

With equilibrium bidding:

• 1 asks 2 which item to demand?

• since v2(a) – v2(b) < p(a) – p(b)

2 answers ‘demand a’

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8D1 = a D2 = ab

p(b) = 2

With equilibrium bidding:

• 1 asks 2 which item to demand?

• since v2(a) – v2(b) < p(a) – p(b)

2 answers ‘demand a’

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 8D1 = a D2 = ab

p(b) = 2

With equilibrium bidding:

• 1 asks 2 which item to demand?

• since v2(a) – v2(b) < p(a) – p(b)

2 answers ‘demand a’

(for an outsider, the big firm took aggressive action towards the smaller firm)

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 9D1 = aD2 = b

p(b) = 2

With equilibrium bidding:

Example

a b ab

v1 10 4 12

v2 10 5 14

p(a) p(b)

p(a) = 9D1 = aD2 = b

p(b) = 2

We reach the VCG outcome.But not a Walrasian equilibrium. Player 1 prefers item a over bin these prices (but cannotobtain it in these prices)

With equilibrium bidding:

the end

Proof – general structure

standard argument: suppose all other players play the strategy.To show strategy is best response for i, it is sufficient to show:

1. If i follows the strategy she receives her VCG outcome (bundle+price)

2. If she follows any other strategy and receives some bundle S she pays at least pi

VCG(S) – the VCG payment if she would declare a value vi* that will lead her to receive S.

Remarks• 2nd requirement is important; shows why players cannot coordinate

arbitrary allocations• For subgame-perfection, we show this for all starting prices• It is the unique efficient equilibrium. (open: unique equilibrium?)

Proof idea for 2nd requirement

an example case: player i receives {a,b} (using some strategy), efficient allocation without i gives both items to player j.

Need to show: i’s payment is at least vj(ab).

Proof idea for 2nd requirement

an example case: player i receives {a,b} (using some strategy), efficient allocation without i gives both items to player j.

Need to show: i’s payment is at least vj(ab).

Case I: player j did not jump during the course of the auction.

p(a) > vj(a) , p(b) > vj(b) p(a)+p(b) > vj(a) + vj(b) > vj(ab)

Proof idea for 2nd requirement

an example case: player i receives {a,b} (using some strategy), efficient allocation without i gives both items to player j.

Need to show: i’s payment is at least vj(ab).

Case I: player j did not jump during the course of the auction.

p(a) > vj(a) , p(b) > vj(b) p(a)+p(b) > vj(a) + vj(b) > vj(ab)

Case II: player j jumps, and i communicates “demand a”.

p(b) > vj(b|a) , p(a) > vj(a)

p(a) + p(b) > vj(b|a) + vj(b) = vj(ab)

• Proof of other cases uses similar arguments.

Proof idea for 1st requirement

• need to show: if all players follow the strategy VCG outcome

Proof structure:

• no jump VCG payments = Walrasian payments

from the analysis of myopic bidding

Proof idea for 1st requirement

• need to show: if all players follow the strategy VCG outcome

Proof structure:

• no jump VCG payments = Walrasian payments and

no jump auction ends in Walrasian outcome

since by G&S myopic bidding ends in Walrasian outcome

Proof idea for 1st requirement

• need to show: if all players follow the strategy VCG outcome

Proof structure:

• no jump VCG payments = Walrasian payments and

no jump auction ends in Walrasian outcome

no jump VCG outcome

Proof idea for 1st requirement

• need to show: if all players follow the strategy VCG outcome

Proof structure:

• no jump VCG payments = Walrasian payments and

no jump auction ends in Walrasian outcome

• jump VCG outcome

no jump VCG outcome

jump VCG outcomeEasy proof for two players:

• Suppose w.l.o.g. the situation in the picture

p(a) p(b)

p(a)

p(b) = v1(b|a)

jump VCG outcomeEasy proof for two players:

• Suppose w.l.o.g. the situation in the picture

• If 2 wins both items:

– v2(a|b) > v1(a) efficient to give a,b to 2

– 2 pays v1(a) + v1(b|a) = v1(ab) = 2’s VCG price

p(a) p(b)

p(a)

p(b) = v1(b|a)

jump VCG outcomeEasy proof for two players:

• Suppose w.l.o.g. the situation in the picture

• If 2 wins both items:

– v2(a|b) > v1(a) efficient to give a,b to 2

– 2 pays v1(a) + v1(b|a) = v1(ab) = 2’s VCG price

• If 1 receives a, 2 receives b:

– again efficient

– 2 pays v1(b|a)

– 1 pays v2(a|b)

p(a) p(b)

p(a)

p(b) = v1(b|a)

Summary (1)

• Study side-communication in a two-item ascending auction

– Show an ex-post efficient equilibrium with limited signaling

– With no signaling: inefficient Bayesian equilibria exist, no efficient ex-post equilibrium exists.

• Conceptual contribution:

– Bidders’ side: explain how signaling is used strategically

– In equilibrium: first bid myopically, then a “demand reduction”. This behavior was observed in reality

– Auctioneer side: Perhaps no-need to disallow cheap talk?

– Side-communication in sealed-bid auctions was studied many times, e.g. Matthews and Postlewaite (1989)

Summary (2)Possible criticism:

• Equilibrium selection: bidders may still choose an inefficient Bayesian equilibrium

– If bidders hate ex-post regret this is solved

– Bayesian equilibrium is more sensitive to problematic assumptions, so less stable

• Allowing communication simply “looks bad”

– Embed strategy in the auction

– Give price discount at the end (as in Mishra and Parkes ‘07)

Extensions and future research:

• more items (we have partial results; similar conclusions)

• non-substitutes items (revenue?)

• budgets, non-quasi-linear utilities