# Simple and Near-Optimal Auctions Tim Roughgarden (Stanford)

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Simple and Near-Optimal Auctions Tim Roughgarden (Stanford)

2 Motivation Optimal auction design: what's the point? One primary reason: suggests auction formats likely be useful in practice. Exhibit A: single-item Vickrey auction.  maximizes welfare (ex post) [Vickrey 61]  with suitable reserve price, maximizes expected revenue with i.i.d. bidder valuations [Meyerson 81 ]

3 The Dark Side Issue: in more complex settings, optimal auction can say little about how to really solve problem. Example #1: single-item auction, independent but non-identical bidders. to maximize revenue:  winner = use highest "virtual bid"  charge winner its "threshold bid" Example #2: combinatorial auctions (max welfare)  VCG hard to implement, provable approximations are mostly theoretical

4 Alternative Approach Standard Approach: solve for optimal auction over huge set, hope optimal solution is reasonable Alternative: optimize only over "plausibly implementable" auctions. Sanity Check: want performance of optimal restricted auction close to that of optimal (unrestricted) auction.  if so, have theoretically justified and potentially practically useful solution

5 Plan for Talk Part I: simple near-optimal auctions for revenue maximization in single-parameter problems.  VCG + monopoly reserves almost as good as OPT  [Hartline/Roughgarden EC 09]  follow-ups: [Chakraborty et al. WINE 10], [Chawla et al STOC 10], [Yan SODA 11] Part II: equilibrium welfare guarantees for combinatorial auctions with "item bidding"  mechanism = parallel second-price auctions  [Bhawalkar/Roughgarden SODA 11]

Simple versus Optimal Auctions (Hartline/Roughgarden EC 2009)

7 Single-Parameter Problems Example: single-minded combinatorial auctions set of m different goods each bidder i wants a known bundle S i of goods feasible subsets = bidders with disjoint bundles Downward-closed environment: n bidders, independent valuations from F 1,...,F n public collection of feasible sets of bidders subsets of feasible sets must be again feasible

8 Simple vs. Optimal Auctions Question: is there a simple auction that's almost as good as Myerson's optimal auction?

9 Simple vs. Optimal Auctions Question: is there a simple auction that's almost as good as Myerson's optimal auction? Theorem: [Hartline/Roughgarden EC 09] Yes! Expected revenue of VCG + "monopoly reserve prices" is at least 50% of OPT. assumes F i 's have "monotone hazard rates"  better bounds hold under stronger assumptions  single-item auction, anonymous reserve: can get between 25-50% (open: determine tight bound)

10 VCG + Monopoly Reserves Input: general downward-closed environment, unknown v i 's drawn from known MHR F i 's set reserve r i = argmax p p(1 - F i (p)) [each i] delete all bidders with v i < r i run VCG: maximize sum of v i 's (i.e., welfare) of remaining bidders, subject to feasibility price of winner i =max{VCG price, r i }

11 A Simple Lemma Lemma: Let F be an MHR distribution with monopoly price r (so ϕ(r) = 0). For every v ≥ r: r + ϕ(v) ≥ v. Proof: We have r + ϕ(v) = r + v - 1/h(v)[defn of ϕ] ≥ r + v - 1/h(r)[MHR, v ≥ r] = v.[ϕ(r) = 0]

Welfare Guarantees in Combinatorial Auctions with Item Bidding (Bhawalkar/Roughgarden SODA 2011)

13 Combinatorial Auctions n bidders, m heterogeneous goods bidder i has private valuation v i (S) for each subset S of goods [≈ 2 m parameters]  assume nondecreasing with v i (φ) = 0 quasi-linear utility: player i wants to maximize v i (S i ) - payment allocation = partition of goods amongst bidders welfare of allocation S 1,S 2,...,S n : ∑ i v i (S i )  goal is to allocate goods to maximize this quantity

14 Item Bidding Note: if m (i.e., # of goods) is not tiny, then direct revelation is absurd. Goal: good welfare with much smaller bid space.  "truthfulness" now obviously impossible Item Bidding: each bidder submits one bid per good. Each good sold via an independent second-price auction. only solicit m parameters per bidder this auction is already being used! (via eBay)

15 Item-Bidding Example Example: two players (1 & 2), two goods (A & B). Player #1: v 1 (A) = 1, v 1 (B) = 2, v 1 (AB) = 2. Player #2: v 2 (A) = 2, v 2 (B) = 1, v 2 (AB) = 2. OPT welfare = 4. A full-information Nash equilibrium:  #1 bids 1 on A, 0 on B  #2 bids 0 on A, 1 on B which has welfare only 2.

16 The Price of Anarchy Definition: price of anarchy (POA) of a game (w.r.t. some objective function): Well-studied goal: when is the POA small?  benefit of centralized control --- or, a hypothetical perfect (VCG) implementation --- is small  note POA depends on choice of equilibrium concept  only recently studied much in auctions/mechanisms optimal obj fn value equilibrium objective fn value the closer to 1 the better

17 Complement-Free Bidders Fact: need to restrict valuations to get interesting worst-case guarantees. Complement-Free Bidder: v i (S ∪ T) ≤ v i (S) + v i (T) for all subsets S,T of goods.

18 Complement-Free Bidders Fact: need to restrict valuations to get interesting worst-case guarantees. Complement-Free Bidder: v i (S ∪ T) ≤ v i (S) + v i (T) for all subsets S,T of goods. Fact : "bluffing" can yield zero-welfare equilibria.  consider valuations = 1 and 0, bids = 0 and 1 No Overbidding: for all i and S, ∑ jєS b ij ≤ v i (S).  our bounds degrade gracefully as this is relaxed

19 Summary of Results Theorems [Bhawalkar/Roughgarden SODA 11]: worst-case POA of pure Nash equilibria = 2  when such equilibria exist  extends [Christodoulou/Kovacs/Shapira ICALP 08] worse-case POA of Bayes-Nash eq ≤ 2 ln m  also, strictly bigger than 2  also hold for mixed, (coarse) correlated Nash eq  assumes independent private valuations  with correlated valuations, worst-case POA is polynomial in m.

20 XOS Valuations To illustrate ideas: prove POA of pure Nash eq is ≤ 2 for subclass of complement-free valuations. XOS valuation: there exist vectors a 1,a 2,...,a k, indexed by goods, such that for all S: v(S) = max l { ∑ jєS a lj } every XOS valuation is complement-free every submodular valuation is XOS

21 Pure POA = 2 for XOS Fix XOS valuations v 1,v 2,...,v n. Let O 1,O 2,...,O n be an optimal allocation. Let b 1,b 2,...,b n be a pure Nash equilibrium with no overbidding.

22 Pure POA = 2 for XOS Fix XOS valuations v 1,v 2,...,v n. Let O 1,O 2,...,O n be an optimal allocation. Let b 1,b 2,...,b n be a pure Nash equilibrium with no overbidding. Fix player i. Let a i satisfy v i (O i ) = ∑ jєS a ij. By the Nash equilibrium condition [u i denotes payoff]: u i (b) ≥ u i (a i ;b- i ) ≥ ∑ jєOi [a ij - max l≠i b lj ] ≥ v i (O i ) - ∑ jєOi max l b lj

23 Pure POA = 2 for XOS (cont'd) So far: u i (b) ≥ v i (O i ) - ∑ jєOi max l b lj Summing over i: Nash eq welfare ≥ ∑ i v i (O i ) - ∑ i ∑ jєOi max l b lj

24 Pure POA = 2 for XOS (cont'd) So far: u i (b) ≥ v i (O i ) - ∑ jєOi max l b lj Summing over i: Nash eq welfare ≥ ∑ i v i (O i ) - ∑ i ∑ jєOi max l b lj = OPT welfare - ∑ i ∑ jєSi max l b lj = OPT welfare - ∑ i ∑ jєSi b ij

25 Pure POA = 2 for XOS (cont'd) So far: u i (b) ≥ v i (O i ) - ∑ jєOi max l b lj Summing over i: Nash eq welfare ≥ ∑ i v i (O i ) - ∑ i ∑ jєOi max l b lj = OPT welfare - ∑ i ∑ jєSi max l b lj = OPT welfare - ∑ i ∑ jєSi b ij ≥ OPT welfare - ∑ i v i (S i ) no over- bidding

26 Pure POA = 2 for XOS (cont'd) So far: u i (b) ≥ v i (O i ) - ∑ jєOi max l b lj Summing over i: Nash eq welfare ≥ ∑ i v i (O i ) - ∑ i ∑ jєOi max l b lj = OPT welfare - ∑ i ∑ jєSi max l b lj = OPT welfare - ∑ i ∑ jєSi b ij ≥ OPT welfare - ∑ i v i (S i ) So: Nash eq welfare ≥ (OPT welfare)/2. no over- bidding

27 The Complement-Free Case Lemma 1: for every complement-free valuation v and subset S, there is a bid vector a i such that: ∑ jєS a ij ≥ v i (S)/(ln m) ∑ jєT a ij ≤ v i (T) for every subset T of goods Proof idea: modify primal-dual set cover algorithm and analysis. Lemma 2: implies POA bound of 2 ln m.

28 Relation to Smoothness Key Proof Step: invoke Nash equilibrium condition once per player, with "canonical deviation". deviation is independent of b- i Fact: this conforms to the "smoothness paradigm" of [Roughgarden STOC 09] Corollary: POA bound of 2 (ln m) extends automatically to mixed Nash + correlated equilibria, no-regret learners, and Bayes-Nash equilibria with independent private valuations.

29 Correlated Valuations Lower Bound: for Bayes-Nash equilibria with correlated valuations, POA can be m ¼.  even for submodular valuations approach #1: direct construction  based on random planted matchings approach #2: [ongoing with Noam Nisan] communication complexity  good POA bounds imply good poly-size "sketches" for the valuations  [Balcan/Harvey STOC 11] such sketches do not exist

30 Open Problems is independent Bayes-Nash POA = O(1)? what about for independent 1st-price auctions?  [Hassidim/Kaplan/Mansour/Nisan EC 11] when do smoothness bounds extend to POA of Bayes-Nash equilibria with correlated types? optimal trade-offs between auction performance and "auction simplicity"  e.g., in terms of size of bid space, number of "tunable parameters", etc.

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