Transcript Truthful and Near-Optimal Mechanism Design via Linear Programming Ron Lavi California Institute of Technology Joint work with Chaitanya Swamy.

Truthful and Near-Optimal
Mechanism Design via Linear
Programming
Ron Lavi
California Institute of Technology
Joint work with Chaitanya Swamy
Overview of the Talk
• The model of Combinatorial Auctions
– Definition, motivation, challenges and goals,
previous results.
• Our results
– Plus a word on the “big picture”.
• Intuition to our construction and proofs
Combinatorial Auctions
• m indivisible non-identical items for sale
• n bidders compete for subsets of these items
• Each bidder i has a valuation for each set of items:
vi(S) = value that i assigns to acquiring the set S
– vi is non-decreasing (“free disposal”)
– vi () = 0
• The multi-unit case: B>1 copies of each item; no player
desires more than one copy of each item
• Objective: Find a partition of the items (S1…Sn) that
maximizes the social welfare: i vi (Si)
Example
s1
• Each player wants a sourcesink path, for some value.
• Each edge is an item. We need
to allocate items to players.
• Each edge can be allocated to
at most one player.
t2
s2
V1=10
V2=4
t1
Example
s1
• Each player wants a sourcesink path, for some value.
• Each edge is an item. We need
to allocate items to players.
• Each edge can be allocated to
at most one B players.
In the multi-unit case
t2
s2
V1=10
V2=4
t1
Motivation
• Abstracts complex resource allocation problems in
systems with distributed ownership (scheduling,
allocation of network resources).
• Real Applications (e.g. the FCC spectrum auction).
Strategic issues
The classic model:
V1(·)
v2 (·)
·
·
·
·
·
·
vn (·)
ALG
·
S1
S2
·
·
·
·
·
Sn
A game-theoretic view:
• Bidders aim to maximize their own utility: vi(Si) – price.
– Thus a player may manipulate the alg. -- declare a false vi (·).
• Wish to produce an approximately optimal outcome with respect
to the true value functions.
– Thus want to create an incentive to report truthfully.
Mechanism Design and Truthfulness
A mechanism:
V1(·)
v2 (·)
·
·
·
S1 , P1
S2 , P2
·
·
·
·
·
·
·
·
·
vn (·)
ALG
Mechanism
Sn , Pn
A truthful mechanism: No matter what the other players declare,
player i will maximize his utility by reporting truthfully.
Mechanism Design and Truthfulness
A mechanism:
V1(·)
v2 (·)
·
·
·
S1 , P1
S2 , P2
·
·
·
·
·
·
·
·
·
vn (·)
ALG
Mechanism
Sn , Pn
A truthful mechanism: No matter what the other players declare,
player i will maximize his utility by reporting truthfully.
Theorem [Vickrey-Clarke-Groves, the 70’s] : If the algorithm finds
the exact optimal welfare then there exist truthful prices.
Mechanism Design and Truthfulness
A mechanism:
V1(·)
v2 (·)
·
·
·
S1 , P1
S2 , P2
·
·
·
·
·
·
·
·
·
vn (·)
ALG
Mechanism
Sn , Pn
A truthful mechanism: No matter what the other players declare,
player i will maximize his utility by reporting truthfully.
Theorem [Vickrey-Clarke-Groves, the 70’s] : If the algorithm finds
the exact optimal welfare then there exist truthful prices.
• Unfortunately finding the exact optimum is computationally hard.
Complexity Issues
• Communication: input is exponential (in m).
– No algorithm can approximate better than m1/(B+1) with
polynomial communication [Nisan; Nisan and Segal; Dobzinski
and Schapira]
• Computation:
– It is NP-hard to approximate better than m1/(B+1), even for
short valuations [Lehmann, O'Callaghan, Shoham; Bartal, Gonen,
Nisan]
• There exist polynomial time O(m1/(B+1))-approximations
– In particular when B=Ω(log m) there exists a (1+ε)approximation
We seek truthful and computationally feasible
mechanisms.
In other words, are there other ways to embed
truthfulness into a given algorithm?
Previous attempts for resolution
• The “single minded” case :
– √m approx. when B=1
[Lehmann, O'Callaghan, Shoham]
– (1+ε)-approx. when B=Ω(log m) [Archer, Papadimitriou, Talwar, Tardos]
– O(m1/B) -approx. for any B
[Briest, Krysta, Vocking]
• For general valuations:
– O(B·m1/B-2) for B>3
[Bartal, Gonen, Nisan]
– O(√m) for B=1
[Dobzinski, Nisan, Schapira]
• Bundling equilibria in VCG to reduce communication (essentially a
negative result).
[Holzman, Kfir-Dahav, Monderer, Tennenholtz]
• No result for the general case; a large gap from the best
approximability results for the non-single minded case.
Our results
Main construction: Given any alg. for general CA that also bounds the
integrality gap of the LP relaxation, one can construct a randomized,
truthful in expectation, mechanism that has the same approx. ratio.
Immediate Applications: strategic mechanisms with approximation
guarantees that match the best known non-strategic ones:
– A strategic O(m1/B+1) approx. for general valuations and any B.
– If B=Ω(log m) this yields a (1+ε)-approx. mechanism.
– This technique applies to other “packing domains”, for example
multi-parameter knapsack problems.
• By moving from deterministic to randomized mechanisms, we
completely close the strategic -- non-strategic gap for general CAs.
Truthfulness in expectation
Truthfulness in expectation [Archer and Tardos] :
No matter what the other players declare, player i will
maximize his expected utility by reporting truthfully.
• A worst case notion (the distribution is created by the
mechanism, not assumed on the input).
• A player need not assume anything about the rationality of
others.
• This implicitly implies, however, that a player is risk-neutral.
– Thus weaker than deterministic truthfulness.
An aside – a more general view
• Does deterministic truthfulness can yield such results?
– For B=1, any deterministic mechanism that is also IIA cannot
obtain a reasonable approximation
[Lavi, Mu’alem, Nisan]
• Other GT notions might yield distribution-free/worst-case results?
– “Rationalizable strategies” for single-item first price auctions
[Dekel and Wolinsky, Battigalli and Siniscalchi]
– “Set-Nash” for online auctions [Lavi and Nisan]
– “Implementation in undominated strategies” for single-value
combinatorial auctions [Babaioff, Lavi, Pavlov]
– What else?
More on VCG
Truthfulness :  vi, v-i, v’i :
vi(f(vi, v-i)) – pi(vi, v-i) > vi(f(v’i , v-i)) – pi (v’i, v-i)
Theorem [Vickrey-Clarke-Groves] : If the algorithm finds the
exact optimal welfare then there exist truthful prices.
The prices: If (s1,…,sn) is the optimal allocation according to the
reported types v=(v1,…,vn), set prices to pi(v) = -Σj≠ivj(sj) + hi(v-i)
Proof: Suppose a player says v’i and the chosen allocation is
(s’1,…,s’n). His utility is
vi(s’i) - pi(v’i, v-i) = vi(s’i) + Σj≠ivj(s’j) < vi(si) + Σj≠ivj(sj)
= vi(si) - pi(vi, v-i)
i.e. telling his true value would weakly improve his utility.
The fractional case
• xi,s is the fraction of bundle S that player i gets.
vi ( xi )  S xi,S  vi (S )
• The fractional case is easy to solve by an LP:
max i vi ( xi )
s.t.


S
xi , S  1
i , S : jS
xi , S  B
 playeri
 it em j
• Thus we can use VCG for this case.
The fractional case
• xi,s is the fraction of bundle S that player i gets.
vi ( xi )  S xi,S  vi (S )
• The fractional case is easy to solve by an LP:
1
max i vi ( xi )
c
s.t.


S
xi , S  1
i , S : jS
xi , S  B
 playeri
 it em j
• Thus we can use VCG in this case as well.
For every c>1
The fractional case
• xi,s is the fraction of bundle S that player i gets.
vi ( xi )  S xi,S  vi (S )
• The fractional case is easy to solve by an LP:
xi
max i vi ( )
c
s.t.


S
xi , S  1
i , S : jS
xi , S  B
 playeri
 it em j
• Thus we can use VCG in this case as well.
For every c>1
More on solving the LP
• “Short” valuations (the LP is succinctly describable)
– We have a (one shot) truthful in expectation mechanism.
– For example k-minded players. The first strategic
mechanism for this case.
• General valuations: the LP is efficiently solvable with a
“demand oracle”
[Blumrosen-Nisan]
– We have an iterative mechanism; truthfulness in
expectation is ex-post Nash equilibrium.
– The first strategic mechanism with polynomial
communication, computation, and tight approximation
bounds.
A randomized truthful integral mechanism
Construction:
1. Compute a fractional solution x* (optimal w.r.t. the declared
values).
2. Decompose x*/c = λ1·x1+…+ λL·xL where {xl}l are the integral
solutions, and λ1 +…+ λL = 1.
The main technical construction.
Works if c is the integrality gap, and
if furthermore we have an
algorithm that “verifies” this.
For this we extend a technique of
[Carr and Vempala].
A randomized truthful integral mechanism
Construction:
1. Compute a fractional solution x* (optimal w.r.t. the declared
values).
2. Decompose x*/c = λ1·x1+…+ λL·xL where {xl}l are the integral
solutions, and λ1 +…+ λL = 1.
3. Choose xl with probability λ1 and set the expected price to be the
VCG price in the fractional setting.
Claim: This is truthful in expectation
A randomized truthful integral mechanism
Construction:
1. Compute a fractional solution x* (optimal w.r.t. the declared values).
2. Decompose x*/c = λ1·x1+…+ λL·xL where {xl}l are the integral solutions,
and λ1 +…+ λL = 1.
3. Choose xl with probability λ1 and set the expected price to be the VCG price
in the fractional setting.
Claim: This is truthful in expectation
Proof: Suppose that vi  y* and v’i  z* . We have:
vi(y*/c) – pi(vi, v-i) > vi(z*/c) – pi (v’i, v-i)
As the fractional
mechanism is
truthful
A randomized truthful integral mechanism
Construction:
1. Compute a fractional solution x* (optimal w.r.t. the declared values).
2. Decompose x*/c = λ1·x1+…+ λL·xL where {xl}l are the integral solutions,
and λ1 +…+ λL = 1.
3. Choose xl with probability λ1 and set the expected price to be the VCG price
in the fractional setting.
Claim: This is truthful in expectation
Proof: Suppose that vi  y* and v’i  z* . We have:
vi(y*/c) – pi(vi, v-i) > vi(z*/c) – pi (v’i, v-i)
[λy1·vi(x1)+…+ λyL·vi(xL)] – pi(vi, v-i) >
[λz1·vi(x1)+…+ λzL·vi(xL)] – pi (v’i, v-i)
By the
decomposition
A randomized truthful integral mechanism
Construction:
1. Compute a fractional solution x* (optimal w.r.t. the declared values).
2. Decompose x*/c = λ1·x1+…+ λL·xL where {xl}l are the integral solutions,
and λ1 +…+ λL = 1.
3. Choose xl with probability λ1 and set the expected price to be the VCG price
in the fractional setting.
Claim: This is truthful in expectation
Proof: Suppose that vi  y* and v’i  z* . We have:
vi(y*/c) – pi(vi, v-i) > vi(z*/c) – pi (v’i, v-i)
[λy1·vi(x1)+…+ λyL·vi(xL)] – pi(vi, v-i) >
[λz1·vi(x1)+…+ λzL·vi(xL)] – pi (v’i, v-i)
By construction
E[ vi(f(vi, v-i)) – pi(vi, v-i) ] > E[ vi(f(v’i , v-i)) – pi (v’i, v-i) ]
The decomposition (1)
Claim: Given a c-approx. algorithm to the optimal fractional
solution, one can decompose any fractional point x*/c to a
convex combination of integral points, i.e. x*/c = λ1·x1+…+ λL·xL
(where xl is integral), in polynomial time.
Remark: The alg. should “work” for any weights {wi,s}
Method (based on [Carr and Vempala]):
min l l
s.t.
l
*

x

x
l l i,S i,S / c (i, S )

l
l
l  0
1
The decomposition (1)
Claim: Given a c-approx. algorithm to the optimal fractional
solution, one can decompose any fractional point x*/c to a
convex combination of integral points, i.e. x*/c = λ1·x1+…+ λL·xL
(where xl is integral), in polynomial time.
Remark: The alg. should “work” for any weights {wi,s}
Method (based on [Carr and Vempala]):
min l l
max
s.t.
s.t.
l
*

x

x
l l i,S i,S / c (i, S )

l
l
l  0
1
x wi,s
xz
1
*
x
 wi , S  z

i,S i,S
c
l
w

x
i,S i,S i,S  z  1
l
z  0 ; wi , S unconstrained
The decomposition (2)
Observation: If (wi,s , z) is feasible then
1
*
x

i
, S  wi , S  z  1
i ,S
c
max
1
*
x
 wi , S  z

i,S i,S
c
s.t.
l
w

x
i,S i,S i,S  z  1
l
z  0 ; wi , S unconstrained
The decomposition (2)
Observation: If (wi,s , z) is feasible then
1
*
x

i
, S  wi , S  z  1
i ,S
c
Proof: Suppose o/w.
(1/c) Σi,s x*i,s · wi,s > 1 - z
max
1
*
x
 wi , S  z

i,S i,S
c
s.t.
l
w

x
i,S i,S i,S  z  1
l
z  0 ; wi , S unconstrained
The decomposition (2)
Observation: If (wi,s , z) is feasible then
1
*
x

i
, S  wi , S  z  1
i ,S
c
Proof: Suppose o/w. Using A, find xl s.t.
Σi,s wi,s · xli,s > (1/c) Σi,s x*i,s · wi,s > 1 - z
contradicting feasibility.
max
1
*
x
 wi , S  z

i,S i,S
c
s.t.
l
w

x
i,S i,S i,S  z  1
l
z  0 ; wi , S unconstrained
The decomposition (2)
Observation: If (wi,s , z) is feasible then
1
*
x

i
, S  wi , S  z  1
i ,S
c
Proof: Suppose o/w. Using A, find xl s.t.
Σi,s wi,s · xli,s > (1/c) Σi,s x*i,s · wi,s > 1 - z
contradicting feasibility.
Implications:
1. The optimal solution is 1, as we need.
max
1
*
x
 wi , S  z

i,S i,S
c
s.t.
l
w

x
i,S i,S i,S  z  1
l
z  0 ; wi , S unconstrained
The decomposition (2)
Observation: If (wi,s , z) is feasible then
1
*
x

i
, S  wi , S  z  1
i ,S
c
Proof: Suppose o/w. Using A, find xl s.t.
Σi,s wi,s · xli,s > (1/c) Σi,s x*i,s · wi,s > 1 - z
contradicting feasibility.
max
1
*
x
 wi , S  z

i,S i,S
c
s.t.
l
w

x
i,S i,S i,S  z  1
l
z  0 ; wi , S unconstrained
Implications:
1. The optimal solution is 1, as we need.
2. We can use the ellipsoid method to find it in polynomial time:
• A separation oracle is implemented as above.
• This yields a dual program of polynomial size. Its dual will
give us the convex decomposition.
Summary
• Studied the clash between computational and game-theoretic
considerations.
• For a variety of domains, we give a technique to embed
truthfulness in existing algorithmic methods, via randomization
and Linear Programming.
• Our technique closes the existing large approximation gaps in
the literature, providing several new and tight results.
– CAs, multi-parameter knapsack problems, Routing and flow problems.
• Still open:
– Deterministic truthfulness in CAs.
– Truthfulness for special cases of CAs (e.g. sub-modularity of value
functions).
– Other methods for truthful constructions?