Algorithmic Mechanism Design: an Introduction

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• 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
• guido.proietti_at_univaq.it

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Results obtained so far

• 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

Centralized algorithm private-edge mechanism
SP O(mn log n) O(mn log n) (VCG)
MST O(m ?(m,n)) O(m ?(m,n)) (VCG)
SPT O(mn log n) O(mn log n) (one-parameter)
In all these basic examples, the underlying
optimization problem is polytime computablebut
what does it happen if this is not the case?
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Single-minded combinatorial auction
t1 20
SCF the set X?F with the highest total value
r120
t215
r216
t36
r37
the mechanism decides the set of winners and
the corresponding payments
Each player wants a specific bundle of
objects ti value player i is willing to pay
for her bundle ri value player i offers for her
bundle
F X?1,,n winners in X are compatible
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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,j?X, with i?j,
  Si?Sj? (and so Si is allocated to buyer i)
• Buyer is valuation of X?F
• vi(ti,X) ti if i?X (and so Si is allocated to
  buyer i), 0 otherwise
• SCF (to maximize) Total value of X ?i?X ti

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CA problem single-minded case (2)

• 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

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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)

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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
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The VCG-mechanism

• Mltg,pgt
• g(r) arg maxX?F ?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?
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Hardness of the CA problem
Theorem Approximating the CA problem within a
factor better than m1/2-? is NP-hard, for any
fixed ?gt0 (recall m is the number of items).
proof
Reduction from the maximum independent set problem
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Maximum Independent Set (MIS) problem

• Input
• a graph G(V,E) of n nodes and m edges
• Solution
• U?V, such that no two vertices in U are joined by
  an edge
• Measure
• Cardinality of U

Theorem (J. Håstad, 2002)
Approximating the MIS problem within a factor
better than n1-? is NP-hard, for any fixed ?gt0.
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The reduction from MIS to CA
Let be given an instance G(V,E) of the MIS pb
then, we build an instance of the CA pb in which
a
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input graph
each edge is an object
1
G(V,E)
d
each node i is a buyer with Si set of edges
incident to i
b
c
f
e
3
4
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CA instance S1a,b,c,d, S2a, S3b,e,m,
S4c,e,f,g, S5d,f,h,l, S6m, S7g,h,i,
S8i,l
h
g
l
m
7
i
8
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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!
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
and since mO(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-?.
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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 PNP.
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?
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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.
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A problem is binary demand (BD) if
Reminder

• ais type is a single parameter ti??
• ais 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
? The CA problem is clearly BD a buyer is either
selected or not in the solution!
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Reminder (2)

• An algorithm g for a maximization BD problem is
  monotone if
• ? agent ai, and for every r-i(r1,,ri-1,ri1,,rN
  ), wi(g(r-i,ri)) is of the form

1
ri
?i(r-i)
?i(r-i)???? threshold
payment from ai is pi(r) ?i(r-i)
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Our new goal

• To design a (truthful) OP and BD mechanism
  Mltg,pgt satisfying
• g is monotone
• 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)
• g and p are computable (efficiently) in
  polynomial time

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A greedy ?m-approximation algorithm

• reorder (and rename) the bids such that
• W ? ? X ? ?
• for i1 to n do
• if Si?W? then X ? X?i W ? W?Si
• return X

r1/?S1 ? r2/?S2 ? ? rn/?Sn
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Monotonicity of g
Theorem The algorithm g is monotone
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?
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Computing the payments
we have to compute for each selected bidder i
her threshold value

• How much can bidder i decrease her bid until she
  is non-selected?

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Computing the payment pi
Consider the greedy order without i
r1/?S1 ? ? ri/?Si ? ?
rn/?Sn
index j
Use the greedy algorithm to find the smallest
index jgti (if any) such that 1. j is
selected 2. Sj?Si??
pi 0 if j doesnt 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?
pi rj ?Si/?Sj otherwise
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The approximation bound on g
Let OPT be an optimal solution for the CA
problem, and let X be the solution computed by
the algorithm g, then
?i?OPT ri ? ?m ?i?X ri
?i?X
proof
let OPTij?OPT j ?i and Sj?Si??
Observe that ?i?X OPTiOPT indeed, any player j
selected in OPT must either have a non-empty
intersection with at least a player iltj selected
in X, or j is selected in X as well (because of
the greedy approach)
?
rj ? ?m ri
?i?X
Then it suffices to prove that
j?OPTi
crucial observation for greedy order we have
ri ?Sj
?j?OPTi
rj ?
?Si
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proof (contd.)
then, ?i?X
?
?
ri
?Sj
rj ?
? ?m ri
?Si
j?OPTi
j?OPTi
CauchySchwarz inequality
we can bound
?
?
Sj
?Sj
? ?OPTi
? ?Si?m
j?OPTi
j?OPTi
Si
m
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CauchySchwarz inequality
1/2
1/2
in our case
xj1
n OPTi
for j1,,OPTi
yj?Sj
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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!

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• Thanks for your attention!
• Questions?