USING LOTTERIES TO APPROXIMATE THE OPTIMAL REVENUE Paul W. GoldbergUniversity of Liverpool Carmine...
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Transcript of USING LOTTERIES TO APPROXIMATE THE OPTIMAL REVENUE Paul W. GoldbergUniversity of Liverpool Carmine...
USING LOTTERIES TO APPROXIMATE THE OPTIMAL REVENUEPaul W. Goldberg University of Liverpool
Carmine Ventre Teesside University
Maximizing the revenue
we_are_the_champions.mp3
£ 2.50
£ 2.50
£ 3.00
iTunes Revenue = £ 2.97Optimal Revenue = £ 8.00
More revenue!!!
Maximizing the revenue: eliciting “bids”
we_are_the_champions.mp3
£2.50
£ 2.50
£ 3.00
iTunes Revenue = £ 8.00Optimal Revenue = £ 8.00
£ 2.50
£ 2.50
£ 3.00£ 2.50£ 2.50
£ 3.00Promoted!?
Pay-what-you-say (aka 1st price auction) weakness
we_are_the_champions.mp3
£2.50
£ 2.50
£ 3.00
iTunes Revenue = £ 0.03Optimal Revenue = £ 8.00
£ 0.01
£ 0.01
£ 0.01Fired!1st pric
e
1st pric
e
1st pric
e
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?
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 * 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