Auctioning one item

1. Auctioning one item Tuomas Sandholm Computer Science Department Carnegie Mellon University

2. Auctions • Methods for allocating goods, tasks, resources... • Participants: auctioneer, bidders • Enforced agreement between auctioneer & winning bidder(s) • Easily implementable e.g. over the Internet • Many existing Internet auction sites • Auction (selling item(s)): One seller, multiple buyers • E.g. selling a bull on eBay • Reverse auction (buying item(s)): One buyer, multiple sellers • E.g. procurement • We will discuss the theory in the context of auctions, but same theory applies to reverse auctions • at least in 1-item settings

3. Auction settings • Private value : value of the good depends only on the agent’s own preferences • E.g. cake which is not resold or showed off • Common value : agent’s value of an item determined entirely by others’ values • E.g. treasury bills • Correlated value : agent’s value of an item depends partly on its own preferences & partly on others’ values for it • E.g. auctioning a transportation task when bidders can handle it or reauction it to others

4. Auction protocols: English(first-price open-cry = ascending) • Protocol: Each bidder is free to raise his bid. When no bidder is willing to raise, the auction ends, and the highest bidder wins the item at the price of his bid • Strategy: Series of bids as a function of agent’s private value, his prior estimates of others’ valuations, and past bids • Best strategy: In private value auctions, bidder’s ex post equilibrium strategy is to always bid a small amount more than current highest bid, and stop when his private value price is reached • No counterspeculation, but long bidding process • Variations • In correlated value auctions, auctioneer often increases price at a constant rate or as he thinks is appropriate • Open-exit: Bidder has to openly declare exit without re-entering possibility => More info to other bidders about the agent’s valuation

5. Auction protocols: First-price sealed-bid • Protocol: Each bidder submits one bid without knowing others’ bids. The highest bidder wins the item at the price of his bid • Single round of bidding • Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations • Best strategy: No dominant strategy in general • Strategic underbidding & counterspeculation • Can determine Nash equilibrium strategies via common knowledge assumptions about the probability distributions from which valuations are drawn

6. Strategic underbidding in first-price sealed-bid auction Example 1 N risk-neutral bidders Common knowledge that their values are drawn independently, uniformly in [0, vmax] Claim: In symmetric Nash equilibrium, each bidder i bids bi = b(vi) = vi (N-1) / N Proof. First divide all bids by vmax so bids were in effect drawn from [0,1]. We show that an arbitrary agent, agent 1, is motivated to bid b1 = b(v1) = v1 (N-1) / N given that others bid b(vi) = vi (N-1) / N Prob{b1 is highest bid} = Pr{b1 > b2} … Pr{b1 > bN} = Pr{b1 > v2 (N-1)/N} … Pr{b1 > vN (N-1)/N} = Pr{b1 > v2 (N-1)/N)}N-1 = Pr{b1 N / (N-1) > v2}N-1 = (b1 N / (N-1))N-1 E[u1|b1] = (v1-b1) Prob{b1 is highest bid} = (v1-b1) (b1 N / (N-1))N-1 dE[u1|b1] / db1 = (N/(N-1))N-1 (-N b1N-1 + v1 (N-1) b1N-2) = 0 <=> N b1N-1 = v1 (N-1) b1N-2 | divide both sides by b1N-2  0 N b1= v1 (N-1) <=> b1= v1 (N-1) / N ■

7. Strategic underbidding in first-price sealed-bid auction… • Example 2 • 2 risk-neutral bidders: A and B • A knows that B’s value is 0 or 100 with equal probability • A’s value of 400 is common knowledge • In Nash equilibrium, B bids either 0 or 100, and A bids 100 +  (winning more important than low price)

8. Auction protocols: Dutch(descending) • Protocol: Auctioneer continuously lowers the price until a bidder takes the item at the current price • Strategically equivalent to first-price sealed-bid protocol in all auction settings • Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations • Best strategy: No dominant strategy in general • Lying (down-biasing bids) & counterspeculation • Possible to determine Nash equilibrium strategies via common knowledge assumptions regarding the probability distributions of others’ values • Requires multiple rounds of posting current price • Dutch flower market, Ontario tobacco auction, Filene’s basement, Waldenbooks

9. Dutch (Aalsmeer) flower auction

10. Auction protocols: Vickrey(= second-price sealed bid) • Protocol: Each bidder submits one bid without knowing (!) others’ bids. Highest bidder wins item at 2nd highest price • Strategy: Bid as a function of agent’s private value & his prior estimates of others’ valuations • Best strategy: In private value auctions with risk neutral bidders, weakly dominant strategy is to bid one’s true valuation. (Also, in this setting, outcome is the same as in English auction.) • No counterspeculation • Independent of others’ bidding plans, operating environments, capabilities... • Single round of bidding • Widely advocated for computational multiagent systems • Old [Vickrey 1961], but not widely used among humans • Revelation principle --- proxy bidder agents on www.ebay.com, www.webauction.com, www.onsale.com

11. Vickrey auction is a special case of Clarke tax mechanism • Who pays? • The bidder who takes the item away from the others (makes the others worse off) • Others pay nothing • How much does the winner pay? • The declared value that the good would have had for the others had the winner stayed home = second highest bid

12. Results for private value auctions

13. Results for private value auctions • Dutch strategically equivalent to first-price sealed-bid • Risk neutral agents => Vickrey outcome same as English • All four protocols allocate item efficiently • (assuming no reservation price for the auctioneer) • English & Vickrey have truthful (ex post or weakly dominant) strategies => no effort wasted in counterspeculation • Which of the four auction mechanisms gives highest expected revenue to the seller? • Assuming valuations are drawn iid & agents are risk-neutral • The four mechanisms have equal expected revenue!

14. More generally: revenue equivalence [version from Nisan’s review book chapter in the book Algorithmic Game Theory] • Note: • General allocation space • No additional i.i.d. assumptions • Assumes quasilinearity

15. 

16. Revenue equivalence holds also for non-private values settings (settings with signals) as long as the setting is symmetric [see, e.g., Krishna: “Auction Theory”]

17. Revenue equivalence ceases to hold if agents are not risk-neutral • Risk averse bidders: • Dutch, first-price sealed-bid ≥ Vickrey, English • Reason: in the former two auctions, the bidder can “insure” himself by bidding more than a risk-neutral bidder. In contrast, truthful bidding is still a best strategy in the Vickery and English auctions • Risk averse auctioneer: • Dutch, first-price sealed-bid ≤ Vickrey, English • Reason: in the latter two, the seller gets the 2nd-highest valuation, while in the former he only gets the expectation of the 2nd-highest valuation

18. Revenue equivalence ceases to hold if agents have budget constraints • Prop. In Vickrey auction, the dominant strategy is: bidi = min{vi, budgeti} • Thm. In 1st-price auction, if there is an equilibrium of the form bidi=min{b(vi), budgeti}, then the expected revenue is higher than in the Vickrey auction

19. Revenue equivalence might not hold between 1st and 2nd-price auctions if distributions are asymmetric • Example where 2nd-price auction yields higher expected revenue: • Bidder 1’s valuation is 2; Bidder 2’s valuation is 0 or 2 with equal probability • Expected revenue under 2nd-price auction is 2 * ½ = 1 • In 1st-price auction, • Bidder 1 can guarantee expected payoff 1 by bidding ε; bidding above 1 would yield expected payoff <1; so he will not bid more than 1 • If Bidder 2 bids 1+ε he gets expected payoff ½ (2 – (1+ε)) = ½ - ε. So her ex ante expected payoff is at least ½ • Since the sum of the bidders’ payoffs is at least 1.5, the auctioneer’s expected revenue can be at most 1/2 • There are also settings where the 1st-price auction yields more expected revenue than the 2nd-price auction • E.g., Bidder 1’s valuation is 10, Bidder 2’s valuation is uniform [0,1] • For another example, see Chapter 4.3 of the book “Auction Theory”, by Krishna, Academic Press, 2002 • Q: How do these reconcile with the revenue equivalence theorem? • A: They have different social choice functions f, and/or some bidder’s expected payment when he draws his lowest type differs

20. [Based on lecture notes from Jason Hartline, Tim Roughgarden, David Parkes, Shaddin Dughmi, the AGT book, and the Myerson paper.] Optimal (=expected revenue maximizing) auction design

21. “Efficient” vs “optimal” auctions • Earlier: efficient auctions • Yields allocation that maximizes social surplus – sum of valuations of bidders • Vickrey auctions, VCG are efficient • … assuming rational bidders – both are IC, but if bidders are dumb, all allocation guarantees are off • Efficient auctions can have very bad revenue properties • Easy worst-case $0 revenue examples for second-price and VCG • Now: optimal auctions • Maximize seller’s profit (payments minus cost) ... • … in expectation given prior --> Bayesian Optimal • … for any agents’ valuations --> worst-case

22. Do better mechanisms exist? I. • 2 bidders with valuations v1,v2 ~ U[0,1] • Want: expected revenue of Vickrey auction • Given valuations, revenue is min(v1,v2) • Pr[vi < x] = x, so Pr[vi > x] = 1-x • Pr[min(v1,v2)>x] = Pr[v1 > x & v2 > x] = (1-x)2 • E[min(v1,v2)] = Pr[min(v1,v2) > x]dx=(1-x)2dx = 1/3

23. Do better mechanisms exist? II. • Same setting, but now we’ll run a Vickrey auction with reserve price r. • Sell item iff a bid is greater than r • Revenue, conditional on some bidder winning, is now max(min(v1,v2), r) • Want: expected revenue when r = 1/2 • C1: v1 < 1/2 & v2 < 1/2 --> 0 revenue • C2: v1 > 1/2 & v2 > 1/2 --> revenue E[min(v1,v2)] = 2/3 • Expectation calculation similar to last slide, v1,v2 ~ U[1/2,1] • C3: vi > 1/2 & v3-i < 1/2 --> revenue r • C1, C2 have probability 1/4, C3 has probability 1/2 • 1/4*0 + 1/4*2/3 + 1/2*1/2 = 5/12 • Vickrey auction with reserve r=1/2 yields revenue 5/12 > 1/3, the expected revenue from standard Vickrey auction!

24. Bayesian optimal auctions • Prior knowledge about valuations can increase expected revenue • Our setting: • N agents, 1 item • Valuations v1, v2, …, vNdrawn independently • Not identical distributions F1, F2, …, FN over [0,hi] • Cost function c(x) to producing an allocation x • E.g. c(x) = 0 if the item is allocated, else c(x) = ∞ • Myerson [Math of OR 1981] gives a profit-maximizing (in expectation) auction

25. Myerson’s optimal auction, I. • Given Fi, the CDF for vi is Fi(z) = Pr[vi < z] • Then the PDF for vi is fi(z) = dFi(z)/dz • Virtual valuation function given vi: • ϕi(vi) = vi – (1-Fi(vi)) / fi(vi) • Then the virtual surplus of allocation x is: • Σ ϕi(vi)xi – c(x) • (We show later that expected virtual surplus = expected profit of the mechanism.)

26. Myerson’s optimal auction, II. • Myerson’s auction: • Translate bids bi to virtual bids bi’ = ϕi(bi) • Run VCG on b’ --> allocation x’, prices p’ • Remap the prices: pi = ϕi-1(pi’) • Output allocation x=x’ and prices p • Note 1: Myerson’s auction maximizes virtual (social) surplus – if bidders act truthfully • VCG is efficient and is run on virtual valuations • Note 2: Myerson does not necessarily allocate the item! All virtual valuations can be negative

27. Myerson’s optimal auction, III. • Theorem: Myerson’s auction is the revenue-maximizing sales mechanism for private-valuations, single-item settings. • Proof sketch in two steps:maxx,pprofit -----2-----> maxxvirtual surpluss.t. feasiblity s.t. feasiblityIC -----1-----> monotonicity IR

28. Myerson’s optimal auction: Truthful? Theorem: A mechanism is truthful in expectation iff the allocation rule xi(v) is monotone non-decreasing in v and pi(v) = vixi(v) - xi(z,v-i)dz • Monotone: “if the winner bids more, does she still win?” • For deterministic allocation rules, the payment above is just the minimum value at which an agent would’ve won the item. Lemma: If Fi is regular then the virtual surplus maximization on virtual valuations ϕiis monotone non-decreasingin valuation vi • Know: surplus maximization is monotone in agents’ valuations v (see [Nisan et al. AGT]) • So virtual surplus maximization is monotone in virtual valuation • Lemma restricts to Fi regular -> virtual valuation functions ϕi monotone • Myerson’s paper has an “ironing” procedure for non-regular distributions, which gives a principled way to move to the“nearest” regular distribution – take concave closure • Convex closure of revenue curve • Virtual surplus max monotone in virtual valuations which are monotone in valuation --> virtual surplus max monotone in valuations --> truthfulness, via Myerson’s VCG call maximizing virtual surplus + theorem above

29. Lemma and proof from Hartline & Roughgarden Myerson’s optimal auction: Profit max? Theorem: The expected profit of any truthful mechanism is equal to its expected virtual surplus. Proof: Expected profit is E[Σpi(bi)] = ΣE[pi(bi)] = ΣE[ϕi(bi)xi(bi)] = E[Σϕi(bi)xi(bi)], by linearity of expectation & valuations drawn independently.

30. Example optimal auction • Setting: 2 bidders, i.i.d. valuations – so F1 = F2 = F and ϕ1 = ϕ2 =ϕ • Bidder 1 wins when ϕ(b1) ≥ max(ϕ(b2),0) • Pays: p1 = inf{b | ϕ(b) > ϕ(b2) & ϕ(b) > 0}= inf{b | b > b2 & b > ϕ-1(0)} • Bidder 2 wins when ϕ(b2) ≥ max(ϕ(b1),0) • Pays: p2 = inf{b | b > b1 & b > ϕ-1(0)} Theorem: The optimal single-item auction for N bidders with i.i.d. valuations is the Vickrey auction with reserve r=ϕ-1(0). • From before: 2 bidders, vi ~ U[0,1]. Can compute r=ϕ-1(0). • F(z) = Pr[x < z] = z • f(z) = dF(z)/dz = d(z)/dz = 1 • ϕ(vi) = vi – (1-F(vi))/f(vi)= vi– 1 + vi= 2vi– 1 • ϕ-1(0): 0 = 2r – 1 --> r = ½ • So, for the earlier example, setting r =1/2 is actually optimal

31. Results for non-private value auctions

32. Results for non-private value auctions • Dutch strategically equivalent to first-price sealed-bid • Vickrey not equivalent to English • Winner’s curse • Common value auctions: • Agent should lie (bid low) even in Vickrey & English Revelation to proxy bidders? • Model: • Signals can be correlated (joint distribution not a product distribution) • Affiliated signals: if a subset of the signals X1…XN is large, then it is more likely that the rest of them are large • Symmetric model means • signals Xi drawn from same interval, and • vi(X) = u(Xi,X-i), where u is symmetric in the last n-1 components • Thrm (revenue non-equivalence ). Consider symmetric model with at least 2 bidders. Let signals be affiliated. Expected revenues: English ≥ Vickrey ≥ Dutch = first-price sealed bid

33. Results for non-private value auctions... signal vector • Symmetric equilibria may be inefficient • Bidder with highest signal wins, but might not have highest valuation • Def. Single-crossing property: • Thm. [see Krishna, “Auction Theory”] Consider symmetric model with affiliated signals. Suppose single-crossing property holds. Then 2nd-price, English, and 1st-price auction all have efficient symmetric equilibria • In English auction, efficient equilibrium exists also for 2 asymmetric bidders, but not necessarily for more than 2 • Without single crossing property, no efficient mechanism might exist in asymmetric setting • If single-crossing holds, truth-telling is an efficient ex post equilibrium in VCG mechanism • Winner pays vi(smallest xi that would have won, x-i)

34. Results for non-private value auctions... Revenue ranking (aka linkage) principle • Informally: “The greater the linkage between a bidder’s own information and how he perceives others will bid, the greater the expected price paid upon winning” • Let W(z,x) denote the expected price paid by bidder 1 if he is the winning bidder when he receives signal x but bids as if his signal were z, i.e., bids ß(z) • Let W’(z,x) be the partial derivative of W(z,x) wrt x • Thm. Let A and B be two auctions in which the highest bidder wins and only he pays a positive amount. Suppose each has a symmetric and increasing equilibrium s.t. • for all x, W’A(x,x) ≥ W’B(x,x), and • WA(0,0) = WB(0,0) = 0. • then A’s expected revenue is at least as large as B’s • Can be applied to show that seller is best off revealing all his information

35. Results for non-private value auctions... • Common knowledge that auctioneer has private info • Q: What info should the auctioneer release ? • A: in symmetric setting with affiliated signals, in 1st-price and 2nd-price auction, auctioneer is best off releasing all of it • “No news is worst news” • Mitigates the winner’s curse • A: among asymmetric bidders, information revelation can decrease revenue in 2nd-price auction • [see Krishna “Auction Theory” p. 115 for an example]

36. Results for non-private value auctions... • Asymmetric info among bidders • E.g. 1: auctioning pennies in class • E.g. 2: first-price sealed-bid common value auction with bidders A, B, C, D • A & B have same good info. C has this & extra signal. D has poor but independent info • A & B should not bid; D should sometimes • => “Bid less if more bidders or your info is worse” • Most important in sealed-bid auctions & Dutch

37. Results for non-private value auctions... • Impossibility of efficiency if at least one agent has a multi-dimensional signal • Reason: • if bidder A’s valuation only depends on the sum of his signals, it is impossible to elicit his signals, and • bidder 2’s valuation might depend on one of A’s signals, so for efficiency it might be necessary to elicit A’s signals • This impossibility holds even if single-crossing is satisfied • See examples in Krishna book p. 244-245

38. Other 1-item auction topics

39. Vulnerability to bidder collusion[even in private-value auctions] • v1 = 20, vi = 18 for others • Collusive agreement for English: e.g. 1 bids 6, others bid 5. Self-enforcing • Collusive agreement for Vickrey: e.g. 1 bids 20, others bid 5. Self-enforcing • In first-price sealed-bid or Dutch, if 1 bids below 18, others are motivated to break the collusion agreement • Need to identify coalition parties

40. Vulnerability to shills • Only a problem in non-private-value settings • English & all-pay auction protocols are vulnerable • Classic analyses ignore the possibility of shills • Vickrey, first-price sealed-bid, and Dutch are not vulnerable

41. Vulnerability to a lying auctioneer • Truthful auctioneer classically assumed • In Vickrey auction, auctioneer can overstate 2nd highest bid to the winning bidder in order to increase revenue • Bid verification mechanisms, e.g. cryptographic signatures • Trusted 3rd party auction servers (reveal highest bid to seller after closing) • In English, first-price sealed-bid, Dutch, and all-pay, auctioneer cannot lie because bids are public

42. Auctioneer’s other possibilities • Bidding • Seller may bid more than his reservation price because truth-telling is not dominant for the seller even in the English or Vickrey protocol (because his bid may be 2nd highest & determine the price) => seller may inefficiently get the item • In symmetric private-value auctions, the revenue maximizing auction is a Vickrey auction with a reserve price that is set like this • (This is a special case of the Myerson auction [Myerson 81]) • I.e., optimal auctions are not Pareto efficient (not surprising in light of Myerson-Satterthwaite theorem) • Setting a minimum price • Refusing to sell after the auction has ended

43. Undesirable private information revelation • Agents’ strategic marginal cost information revealed because truthful bidding is a dominant strategy in Vickrey (and English) • Observed problems with subcontractors • First-price sealed-bid & Dutch may not reveal this info as accurately • Lying • No dominant strategy • Bidding decisions depend on beliefs about others

44. Untruthful bidding with local uncertainty even in Vickrey • Uncertainty (inherent or from computation limitations) • Many real-world parties are risk averse • Computational agents take on owners’ preferences • Thm [Sandholm ICMAS-96]. It is not the case that in a private value Vickrey auction with uncertainty about an agent’s own valuation, it is a risk averse agent’s best (dominant or equilibrium) strategy to bid its expected value • Higher expected utility e.g. by bidding low

45. Wasteful counterspeculation Thrm [Sandholm ICMAS-96, IJEC-00]. In a private value Vickrey auction with uncertainty about an agent’s own valuation, a risk neutral agent’s best (deliberation or information gathering) action can depend on others. E.g. two bidders (1 and 2) bid for a good. v1 uniform between 0 and 1; v2 deterministic, 0 ≤ v2 ≤ 0.5 Agent 1 bids 0.5 and gets item at price v2: Say agent 1 has the choice of paying c to find out v1. Then agent 1 will bid v1 and get the item iff v1 ≥ v2 (no loss possibility, but c invested)

46. Sniping = bidding very late in the auction in the hopes that other bidders do not have time to respond Especially an issue in electronic auctions with network lag and lossy communication links

47. [from Roth & Ockenfels]

48. Sniping… [from Roth & Ockenfels]

50. Conclusions on 1-item auctions • Nontrivial, but often analyzable with reasonable effort • Important to understand merits & limitations • Unintuitive mechanisms may have better properties • Vickrey auction induces truth-telling & avoids counterspeculation (in limited settings) • Myerson auction best for revenue • Choice of a good auction protocol depends on the setting in which the protocol is used, and the objective