USING LOTTERIES TO APPROXIMATE THE OPTIMAL REVENUEPaul W. Goldberg University of LiverpoolCarmine Ventre Teesside University

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Maximizing the revenue

we_are_the_champions.mp3

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iTunes Revenue = £ 2.97Optimal Revenue = £ 8.00

More revenue!!!

Maximizing the revenue: eliciting “bids”

we_are_the_champions.mp3

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iTunes Revenue = £ 8.00Optimal Revenue = £ 8.00

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Pay-what-you-say (aka 1st price auction) weakness

we_are_the_champions.mp3

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iTunes Revenue = £ 0.03Optimal Revenue = £ 8.00

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1st price1st price

Incentive-compatibility (IC): truthfulness

we_are_the_champions.mp3

v1b1

v2b2

v3b3

is truthful Utility (v1, b2, b3) ≥ Utility (b1, b2, b3) for all b1, b2, b3 def

Utility (b1, b2, b3) = v1– 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

v1b1

pricing

rule

v2b2

v3b3

Utility (b1,b2,b3) + Utility (b1,b2,b3) + Utility (b1,b2,b3)

maximized when bidders bid (v1, v2, v3)

defPricing 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: 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

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 * v1 - p

we_are_the_champions.mp3

v1 v2 v3b1 b2 b3

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

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 (λj, pj) for bidder j• By individual rationality, the lottery must satisfy

L * λj(L, b-j) – pj(L, b-j) ≥ 0in case j has type L

• By truthfulness, the lottery must satisfyH * 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)

Lower bound technique, step 2: setting up a linear system• Requesting an approximation guarantee better than α implies

α * Σj pj(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 bidding H in b λj(L, b-j) + L Σj bidding L in b λj(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)

Lower bound technique, step 3: Carver’s theorem [Carver, 1922]

– (H–L) Σj bidding H in b xj(b-j) + L Σj bidding L in b xj(b-j) > H * nH(b) * (α-1)/α – L * nL(b) * 1/α

for any bid vector b

n = 2 #bidders - 1

m = 2 #bidders

- βiΣj αij xj

Lower bound technique, step 4: finding Carver’s constants (2 bidders)

– (H–L) Σj bidding H in b xj(b-j) + L Σj bidding L in b xj(b-j) > H * nH(b) * (α-1)/α – L * nL(b) * 1/α for any bid vector b

(LL)L x1(L) + L x2(L) > – L * 2 * 1/α

(LH)L x1(H) – (H–L) x2(L) > H * (α-1)/α – L * 1/α

(HL)– (H–L) x1(L) + L x2(H) > H * (α-1)/α – L * 1/α (HH)– (H–L) x1(H) – (H–L) x2(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 x1(L) + L x2(L) + L * 2 * 1/α

L x1(H) – (H–L) x2(L) – H * (α-1)/α + L * 1/α

– (H–L) x1(L) + L x2(H) – H * (α-1)/α + L * 1/α

– (H–L) x1(H) – (H–L) x2(H) – H * 2 * (α-1)/α

weighted sum is non-positive Lottery cannot apx better than α System does not have solutions km+1 ≥ 0

α ≤ (2H-L)/H

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