Using lotteries to approximate the optimal revenue
This presentation is the property of its rightful owner.
Sponsored Links
1 / 17

Using lotteries to approximate the optimal revenue PowerPoint PPT Presentation


  • 103 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Using lotteries to approximate the optimal revenue

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Using lotteries to approximate the optimal revenue

Paul W. GoldbergUniversity of Liverpool

Carmine VentreTeesside University


iTunes Store


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?


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) ≥ 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


  • Login