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

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 subsetof 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 With myopic bidding: D1 = abD2 = ab p(a) = p(b) < 2 p(a) p(b)

7. Example With myopic bidding: p(b) = v1(b|a) = v1(ab) – v1(a) D1 = aD2 = ab p(a) = p(b) = 2+ p(a) p(b)

8. Example With myopic bidding: D1 = a or bD2 = ab p(a) = 8+ p(b) = 2+ p(a) p(b)

9. Example With myopic bidding: the end D1 = a or bD2 = b p(a) = 9 p(b) = 3 p(a) p(b)

10. Example With myopic bidding: the end D1 = a or bD2 = b p(a) = 9 p(b) = 3 Equivalent to theEnglish auction ofGul & Stacchetti (2000) p(a) p(b)

11. Example • A better strategyfor player 2: • As before until p(a)=8, p(b)=2 D1 = a or bD2 = ab p(a) = 8 p(b) = 2 p(a) p(b)

12. Example • A better strategyfor player 2: • As before until p(a)=8, p(b)=2 • Then a “demand reduction”: D1 = a or bD2 = b p(a) = 8 p(b) = 2 p(a) p(b)

13. 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)

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 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.

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

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

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: S1 = {a} , S2 = {b}p(a) = 9 , p(b) = 3is 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 minussum of values of other players in chosen allocation • Truthfulness is a dominant strategy in VCG • Example: S1 = {a} , S2= {b}p1 = 9 , p2 = 2is 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. 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! there exists at least one player with Walrasian price ≠ VCG price the ascending auction with truthful demand reporting terminates in a “jump phase” 

22. The jump phase in the example equilibrium With myopic bidding: ?? D1 = a or bD2 = ab p(a) = 8 p(b) = 2 p(a) p(b)

23. The jump phase in the example equilibrium (hint: need to reach VCG outcome) With myopic bidding: unsuccessful attempt: player 2 (the “big” player) reduces demand (demands only b) D1 = a or bD2 = b p(a) = 8 p(b) = 2 p(a) p(b)

24. The jump phase in the example equilibrium (hint: need to reach VCG outcome) With myopic bidding: 8.5 unsuccessfulattempt: player 2 (the “big” player) reduces demand (demands only b) (sometimes gives wrong incentive to player 1) D1 = a or bD2 = b p(a) = 8 p(b) = 2 p(a) p(b)

25. The jump phase in the example equilibrium (hint: need to reach VCG outcome) With myopic bidding: (turns out that) successfulattempt:player 1 (the “small” player) reduces demand (demands only a) D1 = aD2 = ab p(a) = 8 p(b) = 2 p(a) p(b)

26. The jump phase in the example equilibrium (hint: need to reach VCG outcome) With myopic bidding: (turns out that) successfulattempt:player 1 (the “small” player) reduces demand (demands only a) p(a) = 9 D1 = aD2 = b We reach the VCG outcome. p(b) = 2 p(a) p(b)

27. The jump phase in the example equilibrium (hint: need to reach VCG outcome) With myopic bidding: (turns out that) successfulattempt:player 1 (the “small” player) reduces demand (demands only a) p(a) = 9 D1 = aD2 = b 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) 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?) Dj = a or bDi = 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?) 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?) 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?) 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

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 With equilibrium bidding: D1 = abD2 = ab p(a) = p(b) < 2 p(a) p(b)

39. Example With equilibrium bidding: D1 = aD2 = ab p(a) = p(b) = 2 p(a) p(b)

40. Example With equilibrium bidding: D1 = a or bD2 = ab p(a) = 8 p(b) = 2 p(a) p(b)

41. Example With equilibrium bidding: • 1 asks 2 which item to demand? D1 = a or bD2 = ab p(a) = 8 p(b) = 2 p(a) p(b)

42. Example 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 bD2 = ab p(a) = 8 p(b) = 2 p(a) p(b)

43. Example With equilibrium bidding: • 1 asks 2 which item to demand? • since v2(a) – v2(b) < p(a) – p(b)2 answers ‘demand a’ D1 = aD2 = ab p(a) = 8 p(b) = 2 p(a) p(b)

44. Example With equilibrium bidding: • 1 asks 2 which item to demand? • since v2(a) – v2(b) < p(a) – p(b)2 answers ‘demand a’ D1 = aD2 = 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 With equilibrium bidding: p(a) = 9 D1 = aD2 = b p(b) = 2 p(a) p(b)

46. Example With equilibrium bidding: the end p(a) = 9 D1 = aD2 = b 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) 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.