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# Using lotteries to approximate the optimal revenue PowerPoint PPT Presentation

Using lotteries to approximate the optimal revenue. Paul W. GoldbergUniversity 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.

Using lotteries to approximate the optimal revenue

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## Using lotteries to approximate the optimal revenue

Paul W. GoldbergUniversity of Liverpool

Carmine VentreTeesside 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