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'Aquilaguido.proietti@univaq.it

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?