Online Mechanism Design
(Randomized Rounding on the Fly)
Piotr Krysta,
Berthold Vöcking,
University of Liverpool, UK
RWTH Aachen University, Germany
Combinatorial Auctions
m indivisible items (goods) given for sale
n potential buyers (bidders), each with a valuation function v(.) for subsets
(bundles) of goods
v(.) may express complex preferences, e.g.:
• complements: v(camera + battery) > v(camera) + v(battery)
• substitutes:
v(Apple iPhone + Samsung Galaxy) <
v(Apple iPhone) + v(Samsung Galaxy)
Goal: Partition m goods among n bidders to maximize the social welfare (SW)
Example:
m=2 bidders {A,B}
n=2 goods {x,y}
v( {x} )
v( {y} )
v( {x,y} )
A
1
5
7
B
3
5
6
Opt SW = 8
Combinatorial Auctions: Applications
Combinatorial auctions have many important applications:
* Government Spectrum Auctions (UK, Germany, Sweden, USA, …)
* Allocation of Airspace System Resources
* Auctions for Truckload Transportation
* Auctioning Bus Routes (London)
Combinatorial Auctions: Problem definition
|U|= m = 8
Combinatorial Auction (CA): n bidders
U = set of m items (goods)
Each e ε U available in b ≥ 1 copies (supply)
Bidder i has valuation f-n:
U
1
2
vi : 2 ® R³0
3
Meaning: For a bundle (subset) S Í U :
vi (S) = money i is willing to pay for S
v4
v
Assume: vi (Æ) = 0, "S Í T : vi (S) £ vi (T)
Allocations:
n
ìï
üï
U n
A = íS = (S1,..., Sn ) Î (2 ) "e Î U : å Si Ç{e} £ bý
ïî
ïþ
i=1
Assume: vi (S) = vi (Si ) (no externalities)
Problem: compute allocation
S Î A maximizing social welfare:
v5
v
v6
v
v
7
v
v(S) = å vi (Si )
n
i=1
b=2
v8
How are bidders represented ? (Demand oracles)
Problem:
The length of bidder’s valuation v(.) is exponential in m.
p0 2
v(.) given by demand oracles Di(Ui, p):
Given item prices p = (pe )eÎU what is utility maximizing
subset Si Í Ui and its valuation v(Si) ?
Utility of bidder i for set Si:
ui (p, Si ) = vi (Si ) - å
eÎSi
Demand oracle is:
• restricted
if Ui
• unrestricted if U
ÌU
i =U
pe
p0 1
Di (Ui , p) :
vi ( {3, 4, 6} ) - 3× p0
p0 3
p0 vv4
p0 v5
v
p0 v6
v
p0 vv7
p0 v8
Truthful mechanisms
( f , q) -- deterministic mechanism for CA:
f :V n ® A
(given bidders' declarations output an alloc.)
qi :V n ® R
(bidder i =1,..., n pays qi )
A mechanism ( f , q) is truthful if for each bidder i, all vi , vi’ and all
declarations v-I of the other bidders except bidder i:
vi ( f (vi , v-i ))- qi (vi , v-i ) ³ vi ( f (vi¢, v-i )) - qi (vi¢, v-i )
v-i = (v1,..., vi-1, vi+1,..., vn )
Randomized mechanism = prob. distribution over deterministic
mechanisms.
It is universally truthful if each of these mechanisms is truthful.
Truthfulness via direct characterization & on-line algs
Achieve truthfulness by serving bidders one by one in a given order, say
i=1,2,…,n, and offering items at fixed (posted) prices:
If Ui Í U set of items offered to bidder i,
i
i
define prices p = (pe )eÎU (indep. from i)
and compute:
* bundle
Si := Di(Ui , pi)
* payment
i
qi = å
eÎSi
pe
(without knowing the valuations of bidders i+1,…,n)
Arrival models:
* random order of arrivals (secretary model)
* arbitrary (adversarial) order of arrivals.
Need u.b.: 1£ vi (S) £ m, "i, S
Goal: find alloc. S in A maximizing the social welfare.
We use standard on-line competitive analysis (CR = competitive ratio)
On-line models: standard definition & some aspects
Competitive ratio CR (of a randomized online algo.):
Σ = set of all arrival sequences of n bidders with valuations for m items
For σ ε Σ: S(σ) = alloc. computed by algo., opt(σ) = opt offline alloc.
CR = sups ÎS v(opt(s )) E[v(S(s ))]
OBSERVE: Adversarial arrival model: If valuations v() are unbounded,
then R cannot be bounded.
REASON: The b bidders arriving last might have huge v()’s, s.t. copies
cannot be given to any bidders that arrive before them.
Thus: assume 1 ≤ vi(S) ≤ μ for every bidder i, S subset U.
Random arrival model: We assume unbounded valuations.
NOTE: Random arrivals used only to extract estimate of the bids’ range.
Our contributions: CAs + Random arrivals model
1. General v(): for any b ≥ 1 we obtain CR
O(m
1/(b+1)
)
(the first online result with
log( bm))
b = o(log(m))
2. General v(), bundles size ≤ d: for any b ≥ 1 we obtain CR
O(d1/b log( bm))
(previous O(d2) only for
b=1)
3.O(log(
XOS v() and
m)) b = 1: we obtain a CR
W(m
1/(b+1)
v(.))
)
(the first online result
1/b for submodular/XOS
O(d )
Previous results:1/(b+1)
-comp. l.b.
-best known u.b.
O(m
)
XOS v(): O(log (m)  log log (m))-apx univ. truthful offline mech. [Dobzinski ‘09]
GeneralO(
v():
m)
-apx truthful in exp. offline mech.
O(b× m1/(b-2)-apx
) (b=1) univ. truthful
[Lavi, Swamy ’05]
[Dobzinski, Nisan, Schapira ‘05]
Our contributions: CAs + Adversarial arrivals model
1. General v(): for any b ≥ 1 we obtain CR
O(m
1/(b+1)
)
(the first online result with
log( m bm))
b = o(log(m))
2. General v(), bundles size ≤ d: for any b ≥ 1 we obtain CR
O(d1/b log( m bm))
(previous O(d2) only for
b=1)
3.O(log(
XOS v() mand
m))b = 1: we obtain a CR
W(m
1/(b+1)
v(.))
)
(the first online result
1/b for submodular/XOS
O(d )
Previous results:1/(b+1)
-comp. l.b.
-best known u.b.
O(m
)
XOS v(): O(log (m)  log log (m))-apx univ. truthful offline mech. [Dobzinski ‘09]
GeneralO(
v():
m)
-apx truthful in exp. offline mech.
O(b× m1/(b-2)-apx
) (b=1) univ. truthful
[Lavi, Swamy ’05]
[Dobzinski, Nisan, Schapira ‘05]
Overselling Multiplicative Price Update (MPU)
Algorithm [inspired by BGN 05, …]
r × p0
sold bundle:
Bidders:
p0
S1 = {1, 3, 4}
r × p0
Bidder 1
r × p0 v
v
Your most
profitable
bundle ?
p0 v
v
p0 v
v
D1 (U1, p) :
v1 ( {1, 3, 4} ) - 3× p0
p0 v
v
b=2
p0 v
Overselling Multiplicative Price Update (MPU)
Algorithm [inspired by BGN 05, …]
r × p0
sold bundle:
Bidders:
r × p0
S2 = {2, 4}
r × p0
Bidder 2
r × r × p0 vv
Your most
profitable
bundle ?
p0 v
v
p0 v
v
D2 (U2 , p) :
v2 ( {2, 4} ) - (p0 + r × p0 )
p0 v
v
b=2
p0 v
Overselling Multiplicative Price Update (MPU)
Algorithm [inspired by BGN 05, …]
r × p0
sold bundle:
Bidders:
r × p0
S3 = {3, 4, 6}
r × r × p0
Bidder 3
r × r × r × p0 v
v
Your most
profitable
bundle ?
p0 v
v
r × p0 v
v
D3 (U3, p) :
v3 ( {3, 4, 6} ) - (r × p0 + r × r × p0 + p0 )
p0 v
v
b=2
p0 v
Warm-up: Overselling Multiplicative Price Update
(MPU) Algorithm [inspired by BGN 05, …]
• Order of bidders 1,2,…,n is arbitrary (adversarial).
0 £ L £ vmax and m ³1 s.t. there is £1 bidder with valuation ³ m L
p0 = L / (4bm)
• 1. For each good e Î U do p1e :=
• 2. For each bidder i =1, 2,..., n do
• 3.
• 4.
Set Si := Di (Ui , pi ),
Update for each good
• NOTE: Bidder i gets set
p0.
for a suitable Ui Í U.
e Î Si : pei+1 := pei × 21/b.
Si and pays pi = å pei
eÎSi
Overselling MPU Algorithm: Analysis
p0 = L / (4bm)
r = 21/b
1£ vi (S) £ m, "i, S
0 £ L £ vmax
S = (S1,..., Sn ) = final alloc.
LEMMA 1. For any U1,...,Un Í U :
* S assigns ≤ sb copies
* v(S) =
å
n
overselling
factor
infeasible
of each item,
v (Si ) ³ bå
i=1 i
eÎU
p - bmp0
*
e
LEMMA 2. For U1 =... =Un =U :
p0 × r sb = m L
s = log(4mbm)+ 2 b
pe* = final price of e
v(S) ³ v(opt) - bå
THEOREM 1. S  infeasible alloc.
if U1 =... =Un =U :
v(S) ³ (3
eÎU
8)× v(opt).
pe*.
b(r -1)-apx
Overselling MPU Algorithm: Analysis
å
pe* - bmp0
å
pe*.
LEMMA 1 (Part II). For any U1,...,Un Í U, v(S) ³ b
*
where pe = final price of e.
LEMMA 2. For U1 =... =Un =U, v(S) ³ v(opt) - b
eÎU
eÎU
THEOREM 1. The algorithm with p0 = L (4bm) outputs an infeasible
alloc. S:
(1) where £ blog(4mbm)+ 2 copies of each item is assigned;
(2) if U1 =... =Un =U , then v(S) ³ (3 8)× v(opt).
å
PROOF: (1) is by LEMMA 1 (P. I). By L. 1 (P. II): b
which with LEMMA 2 gives:
eÎU
pe* £ v(S) + bmp0
v(S) ³ v(opt) - v(S)- bmp0 Û 2v(S) ³ v(opt)- bmp0
By v(opt) ≥ L, we have the following and this implies claim (2):
p0 £ v(opt) (4bm) Û bmp0 £ v(opt) 4
☐
Overselling MPU Algorithm: Analysis
LEMMA 1 (Part I). For any U1,...,Un Í U, alloc. S assigns ≤ sb copies
of each item, where
s = log(4mbm)+ 2 b.
PROOF: Consider e ε U. Suppose, after some step,
éêbs - 2ùú ³ b × log(4mbm) copies of e assigned to bidders.
Þ
price of e ≥
p0 × 2 log(4 mbm) ³ m L
After this step, the algorithm might give further copies of e to bidders
whose maximum valuation exceeds μL.
By definition of μ, L there is ≤ 1 such bidder that receives ≤ 1 copy of e.
Hence, at most éêbs -1ùú £ bs copies of e assigned.
☐
Overselling MPU Algorithm: Analysis
LEMMA 2. For U1 =... =Un =U,
v(S) ³ v(opt) - b × å pe*.
eÎU
PROOF: T
= (T1,...,Tn ) = feasible allocation (allocates ≤ b of each item)
Algo. uses demand oracle:
vi (Si ) - å
eÎSi
vi (Si ) ³ vi (Ti ) - å
(*)
eÎTi
By using
pei ³ vi (Ti ) - å
eÎTi
pei , so
pei
"e,i : pe* ³ pei and summing (*) for all bidders we obtain
n
n
i=1
i=1
n
v(S) = å vi (Si ) ³ å vi (Ti ) - å å pe* ³ v(T ) - b å pe*
i=1 eÎTi
eÎU
(last “≥” follows because T allocates ≤ b copies of each item)
Taking T = opt implies the claim.
☐
Overselling MPU Algorithm with Oblivious RR
Larger price update factor r
 “more feasible” solution + worse approximation
Smaller
r
 helps “learn” correct prices, but, produces in-feasible solution.
Idea: Achieve feasibility and good approximation by defining appropriate
sets Ui for demand oracles, and using RR.
Idea: Provisionally assign bundles Si of virtual copies to bidders following
MPU algorithm
 learn correct prices
Number of virtual copies ≤ b  log(μbm)
(LEMMA 1)
Oblivious randomized rounding (RR)  used to decide (with small Pr = q)
which provisional bundles Si become final bundles.
Overselling MPU Algo. with Oblivious Randomized
Rounding
r × p0
final
bundle:bundle:
provisional
R
S1 = {1, 3, 4}
Bidders:
p0
r × p0
Bidder 1
r × p0 v
v
Your most
profitable
bundle ?
v
p0 v
v
 YES! (Pr=q)
D1 (U1, p) :
v1 ( {1, 3, 4} ) - 3× p0
p0 v
p0 v
v
b=2
p0 v
Overselling MPU Algo. with Oblivious Randomized
Rounding
r × p0
final
bundle:bundle:
provisional
R
S2 = {2, 4}
Bidders:
r × p0
r × p0
Bidder 2
r × r × p0 vv
Your most
profitable
bundle ?
v
p0 v
v
 YES! (Pr=q)
D2 (U2 , p) :
v2 ( {2, 4} ) - (p0 + r × p0 )
p0 v
p0 v
v
b=2
p0 v
Overselling MPU Algo. with Oblivious Randomized
Rounding
r × p0
final
bundle:bundle:
provisional
R
Æ 6}
!!!
S3 = {3,
Bidders:
r × p0
r × r × p0
Bidder 3
r × r × p0
Your most
profitable
bundle ?
v
r × p0 v
v
 NO! (Pr=1-q)
D3 (U3, p) :
v3 ( {3, 6} ) - (r × p0 + p0 )
p0 v
p0 v
v
b=2
p0 v
MPU Algorithm with Oblivious RR
• Order of bidders 1,2,…,n is arbitrary (adversarial).
0 £ L £ vmax and m ³1 s.t. there is £1 bidder with valuation ³ m L
q Î [0,1] -- given probability
p0 = L / (4bm)
• 1. For each good e Î U do p1e := p0 ,
• 2. For each bidder i =1, 2,..., n do
be1 := b.
Si := Di (Ui , pi ), for Ui = {e Î U : bei > 0}.
• 3.
Set
• 4.
Update for each good
• 5.
With prob.
• 6.
Update for each good
e Î Si : pei+1 := pei × 21/b.
q set Ri := Si else Ri := Æ.
e Î Ri : bei+1 := bei -1.
MPU Algorithm with Oblivious RR: remarks
•
•
•
•
•
•
1. For each good e Î U do p1e := p0 , be1 := b.
2. For each bidder i =1, 2,..., n do
i
i
3.
Set Si := Di (Ui , p ), for Ui = {e Î U : be > 0}.
i+1
i
1/b
4.
Update for each good e Î Si : pe := pe × 2 .
5.
With prob. q set Ri := Si else Ri := Æ.
i+1
i
6.
Update for each good e Î Ri : be := be -1.
(a)
(b)
(c)
(d)
(e)
å
pei.
The algorithm outputs allocation R; payment for Ri is
eÎRi
Def. of Ui in line 3. ensures that R is feasible!
If q=0, then the provisional alloc. S is same as MPU algo. with Ui=U.
If q=0, then the output alloc. R is empty.
With prob. 1-q the algo. increases prices of e in Si but does not sell
Si (and thus “learns” the correct prices).
(f) If q>0, then LEMMA 1 holds, but LEMMA 2 doesn’t!
(g) We will show a stochastic version of LEMMA 2 to imply O(1/q)-apx.
Overselling MPU Algo. with RR: Analysis
S = (S1,..., Sn ) = prov. alloc.
R = (R1,..., Rn ) = final alloc.
p0 = L / (4bm)
Recall the previous analysis:
LEMMA 1. For any U1,...,Un Í U :
* S assigns ≤ sb copies of each item,
* v(S) =
å
n
v (Si ) ³ bå
i=1 i
eÎU
p - bmp0
*
e
LEMMA 2. For U1 =... =Un =U :
Si := Di (Ui , pi )
vi (Si ) ³ vi (Ti ) - å
opt bundle for i
Always holds
s = log(4mbm)+ 2 b
pe* = final price of e
v(S) ³ v(opt) - bå
eÎU
pe*.
Not always holds!!!
eÎTi
pei
Problem: Ti Ë Ui
A stochastic LEMMA 2’ for CAs with d-bundles
LEMMA 2’. Consider CA with |bundles| ≤ d, and let
q-1 = 2e× d1/b × s
Then for any
s = log(4mbm)
i =1, 2,..., n and any bundle T Í U, T £ d :
E[vi (T ÇUi )] ³ (1 2)× vi (T).
and
E[v(S)] ³ (1 8)× v(opt) and E[v(R)] ³ (q 8)× v(opt).
æ sb ö b æ esb öb æ b öb 1
THEOREM Pr[e
2. The
MPU
algorithm
and
ÏU
£ ç oblivious
÷ × q with
÷ × ç RR
÷ q£ as above is
i ]= ç
1b
è b ø è 2ed sb ø 2d
è1/b b ø
O(d × log(mbm))-competitive
for CA with |bundles| ≤ d and multiplicity b.
Summary and further questions?
We design the first online (universally truthful) mechanisms achieving
competitiveness for any supply b ≥ 1.
New technique: we combine the online allocation of bidders with the
concept of oblivious randomized rounding.
Our mechanisms are simple and intuitive: each bidder’s demand oracle
is queried only once, …
We achieve competitive ratios close to or even beating the best known
approx. factors for the corresponding offline setting.
Question: The main open problem is to design similar deterministic
mechanisms.
Thanks!
Questions ?
Problem definition: Submodular and XOS valuations
We also consider special valuations

v : 2 ® R³0
U
Submodular (decreasing marginal utilities):
"e, S Í T : v(T Èe)- v(T ) £ v(S Èe)- v(S)
 XOS (fractionally subadditive):
"e Î S : å
k:eÎTk
a k ³ 1, a k Î [0,1]
(sets Tk Í S are fractional cover of S)
Þ v(S) £ å a k × v(Tk )
k
FACT: If v() if submodular then it is XOS.
PROOFS
Overselling MPU Algorithm: Analysis
v(S) ³ b × å pe* - bmp0
LEMMA 1 (Part II). For any U1,...,Un Í U,
where v(S) =
eÎU
n
*
and
p
v
(S
)
åi i
e = final price of e.
i=1
r = 21 b and
lei = no. copies of e alloc. to bidders before bidder i
PROOF: Let
le* = len+1 = total alloc. of e to all bidders
As bidders are individually rational: vi (Si ) ³
le* -1
å
eÎSi
pei, hence:
r -1
v(S) ³ å å p =å å p0 × r =p0 å å r = p0 å
i=1 eÎSi
i=1 eÎSi
eÎU k=0
eÎU r -1
n
n
i
e
Now
lei
k
le*
p = p0 r and 1 (r -1) =1 (21 b -1) ³ b imply the claim.☐
*
e
le*
MPU Algo. with RR: stochastic LEMMA 2’
LEMMA 2’. If for any
(**)
then
i =1, 2,..., n
and any
T Í U, T £ d,
E[vi (T ÇUi )] ³ (1 2)× vi (T).
E[v(S)] ³ (1 8)× v(opt) and E[v(R)] ³ (q 8)× v(opt).
T = (T1,...,Tn )  feasible opt alloc.
PROOF: Fix bidder i and
By Si := Di (Ui , pi ) for any coin flips of the algorithm
vi (Si ) ³ vi (Ti ÇUi ) - å
eÎTi ÇUi
pei
Þ E[vi (Si )] ³ E[vi (Ti ÇUi )]- å
eÎTi ÇUi
Þ E[vi (Si )] ³ (1 2)× vi (Ti ) - å
eÎTi ÇUi
by (**)
E[ pei ]
E[ pei ]
(***)
MPU Algo. with RR: stochastic LEMMA 2’
LEMMA 2’. If for any
i =1, 2,..., n
(**)
then
and any
T Í U, T £ d,
E[vi (T ÇUi )] ³ (1 2)× vi (T).
E[v(S)] ³ (1 8)× v(opt) and E[v(R)] ³ (q 8)× v(opt).
PROOF: Sum (***) for all bidders:
n
n
i=1
i=1
n
2E[v(S)] = 2å E[vi (Si )] ³ å vi (Ti ) - 2å å E[ pe* ] ³
i=1 eÎTi
³ v(T ) - 2b å E[ pe* ]
eÎU
Þ 2E[v(S)] ³ v(opt) - 2b å E[ pe* ]
eÎU
(i)
MPU Algo. with RR: stochastic LEMMA 2’
LEMMA 2’. If for any
i =1, 2,..., n
T Í U, T £ d,
E[vi (T ÇUi )] ³ (1 2)× vi (T).
(**)
then
and any
E[v(S)] ³ (1 8)× v(opt) and E[v(R)] ³ (q 8)× v(opt).
PROOF:
By LEMMA 1: b
å
Þ bå
eÎU
eÎU
2E[v(S)] ³ v(opt) - 2b å E[ pe* ]
p £ v(S) + bmp0
*
e
E[ pe* ] £ E[v(S)]+ bmp0
eÎU
p0 £ v(opt) (4bm)
Þ 2E[v(S)] ³ v(opt) - 2E[v(S)]- 2bmp0 ³
³ v(opt) - 2E[v(S)]- v(opt) 2 Û E[v(S)] ³ v(opt) 8
Now: E[v(Ri)]=qE[v(Si)] as Pr[Ri=Si]=q, so E[v(R)]=qE[v(S)], and finally
E[v(R)] ≥ q  v(opt)/8. ☐
(i)
Proving (**) for d-bundles
LEMMA. Consider CA with |bundles| ≤ d ≥ 1, and let
s = log(4mbm)+ 2 b
q-1 = 2e× d1/b × s
Then for any
i =1, 2,..., n and any bundle T Í U of at most d items:
(**)
E[vi (T ÇUi )] ³ (1 2)× vi (T).
PROOF: Fix bidder i. By LEMMA 1, e Î U is in £ l = b× log(4mbm)+ 2
of the provisional bundles S1,..., Si-1, and each of them becomes final
with prob. q = b (2e× d1/b × l ).
Consider e Î T and note that e Î Ui if e was sold £ b -1 times, i.e.,
at most b-1 of its provisional bundles became final.
b
b
1
Thus the prob. that e Ï Ui is: æ l ö b æ el ö æ b ö
£
ç
÷× q £ ç ÷ ×ç
1b ÷
è b ø è 2ed l ø 2d
By T £ d and union bound
è b ø
☐
Pr[T ÇU = T] ³1 2 Þ E[v (T ÇU )] ³ (1 2)× v (T).
i
i
i
i
END