1. 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

2. 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

3. 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

4. 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

5. 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

6. Example a b ab v1 10 4 12 v2 5 14 With myopic bidding: D1 = ab D2 = ab
p(a) = p(b) < 2
p(a)
p(b)

7. Example a b ab v1 10 4 12 v2 5 14 With myopic bidding:
p(b) = v1(b|a) = v1(ab) – v1(a)
D1 = a D2 = ab
p(a) = p(b) = 2+
p(a)
p(b)

8. Example a b ab v1 10 4 12 v2 5 14 With myopic bidding:
D1 = a or b D2 = ab
p(a) = 8+
p(b) = 2+
p(a)
p(b)

9. Example a b ab v1 10 4 12 v2 5 14 With myopic bidding: the end
D1 = a or b D2 = b
p(a) = 9
p(b) = 3
p(a)
p(b)

10. Example a b ab v1 10 4 12 v2 5 14 With myopic bidding: the end
D1 = a or b D2 = b
p(a) = 9
p(b) = 3
Equivalent to the English auction of Gul & Stacchetti (2000)
p(a)
p(b)

11. Example a b ab v1 10 4 12 v2 5 14 A better strategy for player 2:
As before until p(a)=8, p(b)=2
D1 = a or b D2 = ab
p(a) = 8
p(b) = 2
p(a)
p(b)

12. Example a b ab v1 10 4 12 v2 5 14 A better strategy for player 2:
As before until p(a)=8, p(b)=2
Then a “demand reduction”:
D1 = a or b D2 = b
p(a) = 8
p(b) = 2
p(a)
p(b)

13. Situation 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)

14. 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.

15. Related work (1) Collusion also create inefficiencies.
demonstrated many times: a Bayesian-Nash equilibrium in 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.
tion

16. 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
tion

17. 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
tion

18. 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) = xS 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 5 14
S1 = {a} , S2 = {b} p(a) = 9 , p(b) = 3 is a Walrasian equilibrium.

19. 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 minus sum of values of other players in chosen allocation
Truthfulness is a dominant strategy in VCG
Example: a b ab v1 10 4 12 v2 5 14
S1 = {a} , S2 = {b} p1 = 9 , p2 = 2 is VCG’s outcome.

20. 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.

21. there exists at least one player with Walrasian price ≠ VCG price
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 

22. The jump phase in the example
equilibrium a b ab v1 10 4 12 v2 5 14
With myopic bidding: ??
D1 = a or b D2 = ab
p(a) = 8
p(b) = 2
p(a)
p(b)

23. The jump phase in the example
equilibrium a b ab v1 10 4 12 v2 5 14
(hint: need to reach VCG outcome)
With myopic bidding:
unsuccessful attempt: player 2 (the “big” player) reduces demand (demands only b)
D1 = a or b D2 = b
p(a) = 8
p(b) = 2
p(a)
p(b)

24. The jump phase in the example
equilibrium a b ab v1 10 4 12 v2 5 14
(hint: need to reach VCG outcome)
With myopic bidding:
8.5
unsuccessful attempt: player 2 (the “big” player) reduces demand (demands only b)
(sometimes gives wrong incentive to player 1)
D1 = a or b D2 = b
p(a) = 8
p(b) = 2
p(a)
p(b)

25. The jump phase in the example
equilibrium a b ab v1 10 4 12 v2 5 14
(hint: need to reach VCG outcome)
With myopic bidding:
(turns out that) successful attempt: player 1 (the “small” player) reduces demand (demands only a)
D1 = a D2 = ab
p(a) = 8
p(b) = 2
p(a)
p(b)

26. The jump phase in the example
equilibrium a b ab v1 10 4 12 v2 5 14
(hint: need to reach VCG outcome)
With myopic bidding:
(turns out that) successful attempt: player 1 (the “small” player) reduces demand (demands only a)
p(a) = 9
D1 = a D2 = b
We reach the VCG outcome.
p(b) = 2
p(a)
p(b)

27. The jump phase in the example
equilibrium a b ab v1 10 4 12 v2 5 14
(hint: need to reach VCG outcome)
With myopic bidding:
(turns out that) successful attempt: player 1 (the “small” player) reduces demand (demands only a)
p(a) = 9
D1 = a D2 = b
We reach the VCG outcome. But not a Walrasian equilibrium. Player 1 prefers item b over a in these prices (but cannot obtain it in these prices)
p(b) = 2
p(a)
p(b)

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

29. The equilibrium strategy (1st attempt)
If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand
O/W (there are two active players and Di(p) = {{a},{b}} ) then:
i asks the other active player, j, which item to demand

30. The equilibrium strategy (1st attempt)
If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand
O/W (there are two active players and Di(p) = {{a},{b}} ) then:
i asks the other active player, j, which item to demand
j answers ‘item x’: i demands x until p(x)= vi(x), then quits

31. The equilibrium strategy (1st attempt)
If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand
O/W (there are two active players and Di(p) = {{a},{b}} ) then:
i asks the other active player, j, which item to demand
j answers ‘item x’: i demands x until p(x)= vi(x), then quits
j gives invalid answer: i reports true demand from now on

32. The equilibrium strategy (1st attempt)
If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand
O/W (there are two active players and Di(p) = {{a},{b}} ) then:
i asks the other active player, j, which item to demand
j answers ‘item x’: i demands x until p(x)= vi(x), then quits
j gives invalid answer: i reports true demand from now on
If i receives a demand question from another player j then she answers ???

33. The jump phase in general
With equilibrium bidding: (how to reach VCG outcome in general?) a b ab vj ? vi 10 5 14
Dj = a or b Di = ab
p(a)
p(b)
p(a)
p(b)

34. The jump phase in general
With equilibrium bidding: (how to reach VCG outcome in general?) a b ab vj ? vi 10 5 14
vi(a) - vi(b) > p(a) – p(b)
i tells j to ignore a and demand b

35. The jump phase in general
With equilibrium bidding: (how to reach VCG outcome in general?) a b ab vj ? vi 10 5 14
Since Dj = {a} or {b}
vi(a) - vi(b) > p(a) – p(b) = vj(a) - vj(b)
iff
vi(a) + vj(b) > vi(b) + vj(a)
i tells j to ignore a and demand b

36. The jump phase in general
With equilibrium bidding: (how to reach VCG outcome in general?) a b ab vj ? vi 10 5 14
Since Dj = {a} or {b}
vi(a) - vi(b) > p(a) – p(b) = vj(a) - vj(b)
iff
vi(a) + vj(b) > vi(b) + vj(a)
i tells j to ignore a and demand b Sj ab a b  Si

37. The equilibrium strategy
If there are at least three active players or if Di(p) ≠ {{a},{b}} then player i reports true demand
O/W (there are two active players and Di(p) = {{a},{b}} ) then:
i asks the other active player, j, which item to demand
j answers ‘item x’: i demands x until p(x)= vi(x), then quits
j gives invalid answer: i reports true demand from now on
If i receives a demand question from another player j then:
if vi(a) - vi(b) > p(a) – p(b) then i  j : “demand b”
if vi(a) - vi(b) < p(a) – p(b) then i  j : “demand a”
THM: This is ex-post (subgame-perfect) equilibrium

38. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding:
D1 = ab D2 = ab
p(a) = p(b) < 2
p(a)
p(b)

39. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding:
D1 = a D2 = ab
p(a) = p(b) = 2
p(a)
p(b)

40. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding:
D1 = a or b D2 = ab
p(a) = 8
p(b) = 2
p(a)
p(b)

41. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding:
1 asks 2 which item to demand?
D1 = a or b D2 = ab
p(a) = 8
p(b) = 2
p(a)
p(b)

42. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding:
1 asks 2 which item to demand?
since v2(a) – v2(b) < p(a) – p(b) 2 answers ‘demand a’
D1 = a or b D2 = ab
p(a) = 8
p(b) = 2
p(a)
p(b)

43. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding:
1 asks 2 which item to demand?
since v2(a) – v2(b) < p(a) – p(b) 2 answers ‘demand a’
D1 = a D2 = ab
p(a) = 8
p(b) = 2
p(a)
p(b)

44. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding:
1 asks 2 which item to demand?
since v2(a) – v2(b) < p(a) – p(b) 2 answers ‘demand a’
D1 = a D2 = ab
p(a) = 8
p(b) = 2
(for an outsider, the big firm took aggressive action towards the smaller firm)
p(a)
p(b)

45. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding: p(a) = 9
D1 = a D2 = b
p(b) = 2
p(a)
p(b)

46. Example a b ab v1 10 4 12 v2 5 14 With equilibrium bidding: the end
p(a) = 9
D1 = a D2 = b
We reach the VCG outcome. But not a Walrasian equilibrium. Player 1 prefers item a over b in these prices (but cannot obtain it in these prices)
p(b) = 2
p(a)
p(b)

47. 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:
If i follows the strategy she receives her VCG outcome (bundle+price)
If she follows any other strategy and receives some bundle S she pays at least piVCG(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?)

48. 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).

49. 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)

50. 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.

51. 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

52. 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

53. 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

54. 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

55. jump  VCG outcome Easy proof for two players:
Suppose w.l.o.g. the situation in the picture
p(a)
p(b) = v1(b|a)
p(a)
p(b)

56. jump  VCG outcome Easy 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) = v1(b|a)
p(a)
p(b)

57. jump  VCG outcome Easy 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) = v1(b|a)
p(a)
p(b)

58. 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)

59. 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