USING LOTTERIES TO APPROXIMATE THE OPTIMAL REVENUE Paul W. GoldbergUniversity of Liverpool Carmine VentreTeesside University

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Maximizing the revenue we_are_the_champions.mp3 £ 2.50 £ 3.00 iTunes Revenue = £ 2.97 Optimal Revenue = £ 8.00 More revenue!!!

Maximizing the revenue: eliciting “bids” we_are_the_champions.mp3 £2.50 £ 3.00 iTunes Revenue = £ 8.00 Optimal Revenue = £ 8.00 £ 2.50 £ 3.00 £ 2.50 £ 3.00 Promoted!?

Pay-what-you-say (aka 1 st price auction) weakness we_are_the_champions.mp3 £2.50 £ 3.00 iTunes Revenue = £ 0.03 Optimal Revenue = £ 8.00 £ 0.01 Fired! 1 st price

Incentive-compatibility (IC): truthfulness we_are_the_champions.mp3 v1v1 v1v1 b1b1 b1b1 v2v2 v2v2 b2b2 b2b2 v3v3 v3v3 b3b3 b3b3 is truthful Utility (v 1, b 2, b 3 ) ≥ Utility (b 1, b 2, b 3 ) for all b 1, b 2, b 3 def Utility (b 1, b 2, b 3 ) = v 1 – if song bought, 0 otherwise pricing(b 1, b 2, b 3 ) Def: Pricing truthful if all bidders are truthful pricing rule

IC: collusion-resistance we_are_the_champions.mp3 v1v1 v1v1 b1b1 b1b1 pricing rule v2v2 v2v2 b2b2 b2b2 v3v3 v3v3 b3b3 b3b3 Utility (b 1,b 2,b 3 ) + Utility (b 1,b 2,b 3 ) + Utility (b 1,b 2,b 3 ) maximized when bidders bid (v 1, v 2, v 3 ) def Pricing collusion- resistant

Designing “good” IC pricing rules We want to design IC pricing rules that approximate the optimal revenue as much as possible Not hard to see that “individually rational” deterministic pricing rules can only guarantee bad approximations Example: v 1, v 2, v 3 in {L,H}, L < H – aka, binary domain If bid vector is (L,L,L) then a bidder has to be charged at most L Bid vector (H,L,L): opt=H+2L, revenue=3L, apx ratio ≈ H/L v1v1 v1v1 v2v2 v2v2 v3v3 v3v3

Pricing “lotteries” We propose to price lotteries akin to [Briest et al, SODA10] Pay something for a chance to win the song A lottery has two components: Price p Win probability λ Risk-neutral bidders: Utility ( ) = λ * v 1 - p we_are_the_champions.mp3 v1v1 v1v1 v2v2 v2v2 v3v3 v3v3 b1b1 b1b1 b2b2 b2b2 b3b3 b3b3 Fact: Lotteries truthful iff λ i (b i, b -i ) ≥ λ i (b i ’, b -i ) iff b i ≥ b i ’ and collusion-resistant iff truthful and singular, ie, λ i (b i, b -i ) = λ i (b i, b’ -i ) for all b -i, b’ -i Fact: Lotteries truthful iff λ i (b i, b -i ) ≥ λ i (b i ’, b -i ) iff b i ≥ b i ’ and collusion-resistant iff truthful and singular, ie, λ i (b i, b -i ) = λ i (b i, b’ -i ) for all b -i, b’ -i

Lotteries for binary domains {L,H} Let us consider the following lottery: λ(L) = ½, priced at L/2 λ(H) = 1, priced at H/2 Properties collusion-resistant truthful since monotone non-decreasing singular (offer depends only on the bidder’s bid) anonymous (no bidder id used) approximation guarantee: ½ Tweaking the probabilities we can achieve an approximation guarantee of (2H-L)/H Can a truthful lottery do any better?

Summary of results

Lower bound technique, step 1: Upper bounding the payments Take any truthful lottery (λ j, p j ) for bidder j By individual rationality, the lottery must satisfy L * λ j (L, b -j ) – p j (L, b -j ) ≥ 0 in case j has type L By truthfulness, the lottery must satisfy H * λ j (H, b -j ) – p j (H, b -j ) ≥ H * λ j (L, b -j ) – p j (L, b -j ) in case j has type H We then have the following upper bounds on the payments p j (L, b -j ) ≤ L * λ j (L, b -j ) p j (H, b -j ) ≤ H * λ j (H, b -j ) – H * λ j (L, b -j ) + p j (L, b -j ) ≤ H – (H–L) * λ j (L, b -j )

Lower bound technique, step 2: setting up a linear system Requesting an approximation guarantee better than α implies α * Σ j p j (b) > OPT(b) = H * n H (b) + L * n L (b) for all bid vectors b In step 1, we obtained the following upper bounds on the payments: p j (L, b -j ) ≤ L * λ j (L, b -j ) p j (H, b -j ) ≤ H – (H–L) * λ j (L, b -j ) Then, to get a better than α approximation of OPT the following system of linear inequalities must be satisfied – (H–L) Σ j bidding H in b λ j (L, b -j ) + L Σ j bidding L in b λ j (L, b -j ) > H * n H (b) * (α-1)/α – L * n L (b) * 1/α for any bid vector b x j (b -j )

Lower bound technique, step 3: Carver’s theorem [Carver, 1922] – (H–L) Σ j bidding H in b x j (b -j ) + L Σ j bidding L in b x j (b -j ) > H * n H (b) * (α-1)/α – L * n L (b) * 1/α for any bid vector b n = 2 #bidders - 1 m = 2 #bidders - β i Σ j α ij x j

Lower bound technique, step 4: finding Carver’s constants (2 bidders) – (H–L) Σ j bidding H in b x j (b -j ) + L Σ j bidding L in b x j (b -j ) > H * n H (b) * (α-1)/α – L * n L (b) * 1/α for any bid vector b (LL) L x 1 (L) + L x 2 (L) > – L * 2 * 1/α (LH) L x 1 (H) – (H–L) x 2 (L) > H * (α-1)/α – L * 1/α (HL) – (H–L) x 1 (L) + L x 2 (H) > H * (α-1)/α – L * 1/α (HH)– (H–L) x 1 (H) – (H–L) x 2 (H) > H * 2 * (α-1)/α HH HL LH LL

weighted sum is function of α only weighted sum is 0 Lower bound: concluding the proof L x 1 (L) + L x 2 (L) + L * 2 * 1/α L x 1 (H) – (H–L) x 2 (L) – H * (α-1)/α + L * 1/α – (H–L) x 1 (L) + L x 2 (H) – H * (α-1)/α + L * 1/α – (H–L) x 1 (H) – (H–L) x 2 (H) – H * 2 * (α-1)/α weighted sum is non-positive Lottery cannot apx better than αSystem does not have solutionsk m+1 ≥ 0 α ≤ (2H-L)/H

Extension to any finite domain Upper bound For a domain of size d, we can easily get a d-approximation of OPT with a collusion-resistant lottery Straightforward generalization of the upper bound for {L,H} domain Lower bound Given a 3-value domain {L,M,H}, we get by using the previous argument a bound of (3HM−HL−M 2 )/(HM)… … which is not very nice and scarcely informative By working on the system built in step 2, we can prove that for any d and ε>0, there exist d values such that no truthful lottery can approximate OPT better than d-ε over the domain given by those values (Non-trivial) generalization of the lower bound for binary domains

Lotteries for domains [1,H] Easy to come up with an anonymous collusion-resistant lottery guaranteeing a ln(H)+1 approximation of OPT Bidder bidding b wins with probability ln(e*b)/ln(e*H) Matching Lower bound can be proved for any truthful lottery over [1,H] Proof uses… … a technique designed for universally truthful auctions by [Goldberg et al, GEB 2006]… … and a bijection between truthful lotteries and universally truthful auctions E.g., the ½-approximating lottery for {L,H} can be viewed as a uniform distribution over two simple auctions charging H and L, respectively

Related literature Focus for “competitive auctions” [Goldberg et al, GEB 2006] is on F (2) rather than OPT, as OPT is “impossible” to approximate This research can then also be seen as the study of the implications of the knowledge of the domain on the approximation of OPT Moreover, lotteries are a more natural interpretation of universally truthful auctions In certain cases (ie, when Cumulative Distribution Functions do not have Probability density functions) lotteries are far more expressive than randomized auctions

Conclusions & future research We show that there is no separation between collusion- resistant and truthful lotteries for digital goods, when approximating OPT = We introduce a new lower bounding technique based on Carver’s result about inconsistent systems of linear inequalities The technique of [Goldberg et al, GEB 2006] does not allow to prove asymptotic bounds in terms of the size of the domains Further applications/implications of Carver’s theorem? Lotteries for settings different than digital goods? E.g., goods with limited supply