c ompetitive a uctions 1. w hat will we see today ? were the auctioneer! random algorithms worst...
TRANSCRIPT
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WHAT WILL WE SEE TODAY?
Were the Auctioneer! Random algorithms
Worst case analysis Competitiveness
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OUR PLAYGROUND
Unlimited number of indivisible goods No value for the auctioneer Truthful auctions
Digital goods
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BEFORE WE BEGIN
Normal Auctions (single round sealed bid)
utility vector u bid vector b payment vector p Auction A
Profit is sum of payments
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RANDOM TRUTHFULNESS
Reminder: Truthful auctions are auctions where each bidder maximizes his profit when bids his utility
Random is probability distribution over deterministic auctions
Random Strong Truthfulness One natural approach Our chosen approach A randomized auction is truthful if it can be
described as a probability distribution over deterministic truthful auctions
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BID-INDEPENDENT AUCTIONS
Intuition Masked vector
f a function from masked vectors to prices Every buyer is offered to pay
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AUCTION
Auction 1: Bid-independent Auction: Af(b)
1. ( )
2.if then
2.1. 1 and p
2.1 else x 0 and 0
i i
i i
i i i
i i
t f b
t b
x t
p
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EXAMPLES
Bid vector for buying Lonely-Island new song 4 bets
What have we got? 1-item vickery
For k’th largest bid we get K- item vickery
( ) max( )i if b b
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BID INDEPENDENT -> TRUTHFUL
We are offered T(=20) what should we bid? If U(=15) < T we cant win If U(=30) >= T any bid >= T will win Either way U maximizes bidder’s profit
T U
max profit
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TRUTHFUL -> BID-INDEPENDENT
Theorem : A deterministic auction is truthful if and only if it is equivalent to a deterministic bid-independent auction.
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TRUTHFUL->BID-INDEPENDENT
For bid vector b and bidder i we fix all bids except bi
Lemma1 For each x where i wins he pays same p
Lemma2i wins for x>p (possibly for p)
1 1 1( ,..., , , ,..., )xi i i nb b b x b b
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LEMMA 1 PROOF
Lemma1: i pays p Assume to the contrary
x1,x2 where i pays p1>p2 Than if Ui = x1 i should lie and tell x2
=>In contrast to A’s truthfulness
p2
u2
u1
p1
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LEMMA2:PROOF
Lemma2: for each x>p (and possibly p) x wins
Assume to the contrary w exists
w>p w wins
x exists such that x>p x doesn’t win
if U=x i should lie and say w => In contrast to A’s truthfulness
P wx
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TRUTHFUL->BID-INDEPENDENT
Define
Than for any bid b
For bid b if i in A wins and pays p than also in Af If loses than
p doesn’t exist or bi < p
;if i can win for any x( ) {
;elsei
pf b
fA A
Bid Indepndent is truthful!
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LETS SHAKE THINGS UP
Reminder: Random Auctions Random Truthful Auctions
A randomized bid-independent auction is a probability distribution over bid-independent auctions
=> A randomized auction is truthful iff it is equivalent to a randomized bid-independent auction
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DOT
Deterministic Optimal Threshold single-priced Define opt(b) as the optimum single price
DOT:
Calculates maximum for rest of the group
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WHERE DOT IS OPTIMAL
Bids range from [0$,50$] Bids are i.i.d
DOT optimal for a wide range of problems! For any bounded support i.i.d(without proof)
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WHERE DOT FAILS
For each a bidder : (n/a-1) a-bidders profit for p=a is n-a but for p=1 is n-1 p = 1
For each 1 bidder n/a a-bidders profit for p=1 is n-1 but for p=a is n p = a
Profit is n/a (number of a bidders)
100
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DOT CONCLUSION
Why are we talking worst case? DOT prevails in Bayesian model Loses in worst case When not safe to assume true random source
Competitive outlook is logical
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F-COMPETITIVE FAILURE
Lemma: For any truthful auction Af and any β≥1, there is a bid vector b such that the expected profit of Af on b is less than F(b)/β
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PROOF
2 bidders Define h the smallest value such that
Lets consider the bid {1,H} where H=4βh>1 Profit is at most
For H bidder : For 1 bidder : 1
( ,1); 1b x x
11;Pr[ (1) ]
2h f h
(1 2 )2
Hh
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Set our eyes lower 2-optimal single price bid The optimal bids that sells at least 2 items
Same as f(b) unless there is one bidder with Hugh utility
22( ) max k n kF b kv
2F
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Β-COMPETITIVE
Definition: We say that auction A is β-competitive against F-m if for all bid vectors b, the expected profit of A on b satisfies
( )[ ( )]
mF bE A b
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DETERMINISM SUCKS
Were going to show that no deterministic auction is β competitive
Theorem: Let Af be any symmetric deterministic auction defined by bin-independent function f. Then Af is not competitive. For any m,n there exists a bid vector b of length n such the Af’s profit is at most
Symmetric auction: order of bids doesn’t matter For example, consider F(2). We can find a bid
vector at length 8 such that Af’s profit is at most F(2)/4
( ) ( )m mF b
n
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DETERMINISM SUCKS: PROOF
Lets look at specific m,n at a specific auction Af
Consider bid b where all bids are n or 1
Let f(j) be the price where j bids are n n – 1 – j bid 1
for f(0) > 1 Consider the bids where all bids are 1
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DETERMINISM SUCKS: PROOF k in 0..n-1 the largest integer where f(k) <= 1 We build a bid with
(k+1) n-bids (n – k – 1) 1-bids
1-bidders lose ( f(k+1) > 1) n-bidders win Profit : (k+1)f(k) < k + 1
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DETERMINISM SUCKS: PROOF
For ( )
if k m-1 ( ) and condition holds
A (b) <k+1 m=F *
if k m then ( ) n*(k+1)
A (b) <k+1 *( 1) ( )*
m
m
mf
m
mf
F b
F b n
m
n
F b
mm k F b
n
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CONCLUSION
Why worst case? Not truly random source
How competitive? F is too good
Why random? Because determinism is not good enough
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RANDOM AUCTIONS
Split the bid vector b in two: b’, b’’ Use each part to build auction for the other
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ECCENTRIC MILLIONAIRES EXAMPLE
Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e
b’ b’’
1M
1M+1
1M1
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ECCENTRIC MILLIONAIRES EXAMPLE
Small-time bidders bid small (1) 2 Eccentric millionaires bid h,h+e
b’ b’’
1M
1M+1
1M1M+1
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ECCENTRIC MILLIONAIRES EXAMPLE
F(2) profit is 2h(= 2M) profit is h * Pr[2 high bids are split between
auctions] = h/2(=M/2)
Competitive Ratio of 4
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So, in special cases it has a very good bound In worst case, it is C-competitive C is worse than 4
( )rDSOT
2
2
: there is an absolute constant C, such that for any 0
is (1+ ) competitive again F , with probability at least 1-em C mm
m
Theorem
DSOT
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SCS
Sampling Cost-sharing CostShare-C: if you have k bidders (highest)
which are willing to pay C collectively (bid>C/k). Charge each for C/k
CostShare is truthful For profit is C, else 0 I know exactly how much
I want to make, regardless of bids
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SCS COMPETITIVE
if F’=F’’ profit is at least F’F Auction profit is R = min(F’,F’’) Suppose F’<F’’
b’ cannot achieve F’’ b’’ profit is F’
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COMPETITIVE RATIO
Begins as ¼ Approaches ½
Tight proof Consider 2 high bids h,h+e
But we always throw half Can we improve? Yes, Costshare(rF’) and Costshare(rF’’) Competitive ratio is 4/r
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BOUNDED SUPPLY
If we only have k goods
Than we use k best bidders and run unlimited supply case
Competitive vs
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BOUNDED-SUPPLY TRUTHFULNESS
none of the bidders win at a price lower than the highest ignored bid.
Use k-vickery to get p-v use auction of unlimited supply on winners get auction price p-A
use price max(pv,pA)
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UP TILL NOW
Bid independent is truthful Worst case outlook Our benchmarks: F,T Deterministic is just now good enough competitiveness against F(2)
Examples of random algorithms DOST: C-competitive SCS : 4-competitive
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MULTI-PRICE
F is best single price F(2) comparable to F
What about using T? T is only O(log(n)) better
Mabye other multi-priced?
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MONOTONE FUNCTIONS
F is better than all monotone auctions
Non-monotone example: Hard-coded actions Lets take b* such that half bid 1 and half bid h Lets create function which maximizes profit
Acts as omniscient on b* Poorly on other results Lets generalize
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HARD CODED AUCTIONS
Let b* be out bid specific bid
will maximize profit on b* bad profit on bids that differ in 1
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MONOTONE FUNCTIONS
Basically, if you bid more you will pay less makes sense, for is higher for the lower bidder DOT,DSOT,SCS,
Vickery are monotone
ib
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SUMMARY
Bid independent is truthful Worst case outlook
competitiveness against F(2) use of random auctions
Examples of random algorithms DOST: C-competitive SCS : 4-competitive
F is a good benchmark