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Using lotteries to approximate the optimal revenue

Using lotteries to approximate the optimal revenue. Paul W. Goldberg University of Liverpool Carmine Ventre Teesside University. iTunes Store. Maximizing the revenue. £ 2.50. £ 2.50. More revenue!!!. w e_are_the_champions.mp3. £ 3.00. i Tunes Revenue = £ 2.97 Optimal Revenue = £ 8.00.

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Using lotteries to approximate the optimal revenue

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  1. Using lotteries to approximate the optimal revenue Paul W. Goldberg University of Liverpool Carmine Ventre Teesside University

  2. iTunes Store

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

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

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

  6. Incentive-compatibility (IC): truthfulness v2 b2 Def: Pricing truthful if all bidders are truthful v1 b1 v3 we_are_the_champions.mp3 b3 pricing rule pricing(b1,b2, b3) def is truthful Utility (v1, b2, b3) ≥ Utility (b1, b2, b3) for all b1, b2, b3 Utility (b1, b2, b3) = v1– if song bought, 0 otherwise

  7. IC: collusion-resistance v2 b2 v1 b1 v3 we_are_the_champions.mp3 b3 pricing rule Pricing collusion-resistant def Utility (b1,b2,b3) + Utility (b1,b2,b3) + Utility (b1,b2,b3) maximized when bidders bid (v1, v2, v3)

  8. 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: v1, v2, v3 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 v1 v2 v3

  9. Pricing “lotteries” Fact: Lotteries truthful iff λi(bi, b-i) ≥ λi(bi’, b-i) iff bi ≥ bi’ and collusion-resistant iff truthful and singular, ie, λi(bi, b-i) = λi(bi, b’-i) for all b-i, b’-i • 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 * v1 - p we_are_the_champions.mp3 v1 v2 v3 b3 b1 b2

  10. 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?

  11. Summary of results

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

  13. Lower bound technique, step 2: setting up a linear system • Requesting an approximation guarantee better than α implies α * Σjpj(b) > OPT(b) = H * nH(b) + L * nL(b) for all bid vectors b • In step 1, we obtained the following upper bounds on the payments: pj(L, b-j) ≤ L * λj(L, b-j) pj(H, b-j) ≤ H – (H–L) * λj(L, b-j) • Then, to get a better than α approximation of OPT the following system of linear inequalities must be satisfied – (H–L) Σj biddingH inbλj(L, b-j) + L Σj biddingL in bλj(L, b-j) > H * nH(b) * (α-1)/α –L * nL(b) * 1/α for any bid vector b xj(b-j) xj(b-j)

  14. Lower bound technique, step 3: Carver’s theorem [Carver, 1922] m = 2 #bidders n = 2 #bidders - 1 – (H–L) Σj biddingH inbxj(b-j) + L Σj biddingL in bxj(b-j) > H * nH(b) * (α-1)/α –L * nL(b) * 1/α for any bid vector b Σjαijxj - βi

  15. Lower bound technique, step 4: finding Carver’s constants (2 bidders) (LL) L x1(L) + L x2(L) > – L * 2 * 1/α – (H–L) x1(H) – (H–L) x2(H) > H * 2 * (α-1)/α (HH) – (H–L) x1(L) + L x2(H) > H * (α-1)/α –L * 1/α (HL) HL (LH) L x1(H) – (H–L) x2(L) > H * (α-1)/α –L * 1/α HH LL LH – (H–L) Σj biddingH inbxj(b-j) + L Σj biddingL in bxj(b-j) > H * nH(b) * (α-1)/α –L * nL(b) * 1/α for any bid vector b

  16. Lower bound: concluding the proof – (H–L) x1(H) – (H–L) x2(H) –H * 2 * (α-1)/α – (H–L) x1(L) + L x2(H) –H * (α-1)/α + L * 1/α L x1(H) – (H–L) x2(L) –H * (α-1)/α + L * 1/α L x1(L) + L x2(L) + L * 2 * 1/α weighted sum is function of α only weighted sum is 0 Lottery cannot apx better than α System does not have solutions km+1 ≥ 0 weighted sum is non-positive α ≤ (2H-L)/H

  17. Conclusions & future research • Take home points • Collusion-resistance = truthfulness, when approximating OPT with lotteries for digital goods • Lotteries much more expressive than universally truthful auctions • New lower bounding technique based on Carver’s result about inconsistent systems of linear inequalities • What next? • Further applications/implications of Carver’s theorem? • Lotteries for settings different than digital goods? E.g., goods with limited supply

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