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### Using lotteries to approximate the optimal revenue

Paul W. Goldberg University of Liverpool

Carmine Ventre Teesside University

Maximizing the revenue

£ 2.50

£ 2.50

More revenue!!!

we_are_the_champions.mp3

£ 3.00

iTunes Revenue = £ 2.97

Optimal Revenue = £ 8.00

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

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

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

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)

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”

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

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?

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)

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)

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

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

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

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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