algorithmic mechanism design: an introduction
DESCRIPTION
Algorithmic Mechanism Design: an Introduction Approximate (one-parameter) mechanisms, with an application to combinatorial auctions Guido Proietti Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica University of L'Aquila [email protected]. Results obtained so far. - PowerPoint PPT PresentationTRANSCRIPT
Algorithmic Mechanism Design: an Introduction
Approximate (one-parameter) mechanisms, with an application to combinatorial auctions
Guido ProiettiDipartimento di Ingegneria e Scienze dell'Informazione e
Matematica
University of L'[email protected]
Results obtained so far
Centralized algorithm
private-edge mechanism
SP O(m+n log n)
O(m+n log n) (VCG)
MST O(m (m,n))
O(m (m,n)) (VCG)
SPTO(m+n log
n)O(m+n log n) (one-
parameter)In all these basic examples, the underlying optimization problem is polytime computable…but what does it happen if this is not the case?
Two classes of truthful mechanisms:•VCG-mechanisms: arbitrary valuation functions and types, but only utilitarian problems•OP-mechanisms: arbitrary social-choice function, but only one-parameter types and workloaded monotonically non-increasing valuation functions
Single-minded combinatorial auction
t1
=20
t2=15
t3=6
SCF: the set XF with the highest total value
the mechanism decidesthe set of winners and thecorresponding payments
Each player wants a specific bundle of objectsti: value player i is willing to pay for
her bundleri: value player i offers for her bundleF={ X{1,…,n} : winners in X
are compatible}
r1=20
r2=16
r3=7
Combinatorial Auction (CA) problem – single-minded case
Input: n buyers, m indivisible objects each buyer i:
wants a subset Si of the objects has a value ti for Si (or any superset of Si), while she is not interested in
any other allocation not containing all the items in Si (single-minded case); basically, ti is the maximum amount buyer i is willing to pay for Si
Solution: X{1,…,n}, such that for every i,jX, with ij, SiSj= (and so Si is
allocated to buyer i) Buyer i’s valuation of XF:
vi(ti,X)= ti if iX (and so Si is allocated to buyer i), 0 otherwise
SCF (to maximize): Total value of X: iX ti
Each buyer makes a payment to the system pi(X) as a consequence of the selected output X; as usual, payments are used by the system to incentive players to be collaborative.
Then, for each feasible outcome X, the utility of player i (in terms of the common currency) coming from outcome X will be:
ui(ti,X) = pi(X) + vi(ti,X) = pi(X) + ti
CA problem – single-minded case (2)
Designing a mechanism for the CA game
Each buyer is selfish Only buyer i knows ti (while Si is public) We want to compute an optimal solution w.r.t.
the true values (we will see this is a hard task) We do it by designing a mechanism that:
Asks each buyer to report her value ri
Computes a solution using an output algorithm g(r) Receives payments pi from buyer i using some
payment function p (depending on the computed solution)
How to design a truthful mechanism for the
problem?Notice that:
the (true) total value of a feasible solution X is:
i vi(ti,X)
… and so the problem is utilitarian!
VCG-mechanisms (should) apply
The VCG-mechanism M=<g,p>:
g(r) = arg maxXF j vj(rj,X)
pi = -j≠i vj(rj,g(r-i)) +j≠i vj(rj,g(r))
g(r) has to compute an optimal solution…
…but can we do that?
Theorem: Approximating the CA problem within a factor better than m1/2- is NP-hard, for any fixed >0 (recall m is the number of items).
Hardness of the CA problem
proof
Reduction from the maximum independent set problem
Maximum Independent Set (MIS) problem
Input: a graph G=(V,E) of n
nodes and m edges Solution:
UV, such that no two vertices in U are joined by an edge
Measure: Cardinality of U
Approximating the MIS problem within a factor better than n1- is NP-hard, for any fixed >0.
Theorem (J. Håstad, 2002)
The reduction from MIS to CA
Then, it is easy to see that the CA instance has a solution of total value k if and only if there is an IS of size k
G=(V,E)each edge is an objecteach node i is a buyer with
Si: set of edges incident to i
…and since m=O(n2), if we could find an approximate solution for CA of ratio better (i.e., less) than m1/2- , then we would find an IS with a ratio better than n1-.
Let be given an instance G=(V,E) of the MIS pb; then, we build an instance of the CA pb in which:
CA instance: S1={a,b,c,d}, S2={a}, S3={b,e,m}, S4={c,e,f,g}, S5={d,f,h,l}, S6={m}, S7={g,h,i}, S8={i,l}
12
3 4 5
67 8
a
b c df
g h
e
i
lm
input graph
Observation: the obtained CA instance is quite special: each object is contended by only two players, and any two players contend at most one object!
How to design a truthful mechanism for the
problem?
So, the CA problem is utilitarian, and we could in principle apply a VCG-mechanism, but the solution that should be returned by its algorithm is not computable in polynomial time, unless P=NP.The question is: If we want to keep on to guarantee the truthfulness of the VCG-mechanism, can we provide in polynomial time a reasonable approximate solution for the SCF?
A general negative result
For many natural mechanism design minimization problems (and the CA problem is one of them), any truthful VCG-mechanism is either optimal, or it produces results which are arbitrarily far from the optimal (this means, truthfulness will bring the system to compute an inadequate solution!)
What can we do for the CA problem?
…fortunately, the problem is one-parameter, and we now show that a corresponding one-parameter mechanism will produce a reasonable result.
A problem is binary demand (BD) if1. ai‘s type is a single parameter ti
2. ai‘s valuation is of the form:
vi(ti,o)= ti wi(o),
wi(o){0,1} workload for ai in o
When wi(o)=1 we say that ai is selected in o
Reminder
The CA problem is clearly BD: a buyer is either selected or not in the solution!
An algorithm g for a maximization BD problem is monotone if
agent ai, and for every r-i=(r1,…,ri-1,ri+1,…,rN), wi(g(r-i,ri)) is of the form:
1
Өi(r-i) ri
Өi(r-i){+}: threshold
payment from ai is:pi(r)= Өi(r-i)
Reminder (2)
Our new goal
To design a (truthful) OP and BD mechanism M=<g,p> satisfying:
1. g is monotone2. Solution returned by g is a “good”
solution, i.e., a provably approximate solution (we will actually show a O(m)-approximate solution, which is tight)
3. g and p are computable (efficiently) in polynomial time
A greedy m-approximation algorithm
1. reorder (and rename) the bids such that
2. W ; X 3. for i=1 to n do
if SiW= then X X{i}; W W{Si}
4. return X
r1/|S1| r2/|S2| … rn/|Sn|
Theorem: The algorithm g is monotone
Monotonicity of g
proof
It suffices to prove that, for any selected agent i, we have that i is still selected when she raises her bid.
In fact, increasing ri can only move bidder i up in the greedy order, making it easier to win for her.
Homework: it is easy to see that the running time of g is polynomial in n and m. What is your faster implementation for g?
How much can bidder i decrease her bid until she is
non-selected?
Computing the payments
…we have to compute for each selected bidder i her threshold value
Computing the payment pi
r1/|S1| … ri/|Si| … rn/|Sn|
Consider the greedy order without i
index jUse the greedy algorithm to findthe smallest index j>i (if any) such that:
1. j is selected2. SjSi
pi= rj |Si|/|Sj| otherwise
pi= 0 if j doesn’t exist
Homework: it is easy to see that each payment can be computed in O(mn) time, and so we need a total of O(mn2) time for all the payments. Can you provide a faster implementation?
Let OPT be an optimal solution for the CA problem, and let X be the solution computed by the algorithm g, then
The approximation bound on g
iX
iOPT ri m iX ri
proof let OPTi={jOPT : j i and SjSi}
Observe that iX OPTi=OPT; indeed, any player j selected in OPT must either have a non-empty intersection with at least a player i<j selected in X, or j is selected in X as well (because of the greedy approach)
Then it suffices to prove that:
jOPTi
rj m ri
crucial observationfor greedy order we have
ri |Sj|
iX
jOPTi|Si|rj
proof (contd.)
we can boundCauchy–Schwarz inequality
then, iX
jOPTi
rj jOPTi
ri
|Si||Sj|
jOPTi
|Sj| |OPTi| jOPTi
|Sj|
≤|Si|≤ m
m ri
|Si|m
Cauchy–Schwarz inequality
yj=|Sj|xj=1
n= |OPTi| for j=1,…,|OPTi|
…in our case…
1/2 1/2
Conclusions
We have introduced a simple type of combinatorial auction, the single-minded one, for which it is computationally hard to find an optimal solution (i.e., a best possible allocation of objects)
In a corresponding strategic setting in which types are private, the problem is both utilitarian and one-parameter, but VCG-mechanisms cannot be used since they will return an arbitrarily bad allocation!
On the other hand, it is not hard to design an OP-mechanism, which is instead satisfactory: we showed a straigthforward greedy monotone algorithm returning an O(m)-approximate solution, which is tight!
Thanks for your attention!Questions?