Truthful and Near-Optimal Mechanism Design via Linear Programming

1
Truthful and Near-Optimal Mechanism Design via
Linear Programming

• Ron Lavi
• California Institute of Technology
• Joint work with Chaitanya Swamy

2
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

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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 Bgt1 copies of each item no
  player desires more than one copy of each item
• Objective Find a partition of the items (S1Sn)
  that maximizes the social welfare ?i vi (Si)

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Example
t2
s1

• Each player wants a source-sink 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.

s2
t1
V110 V24
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Example
t2
s1

• Each player wants a source-sink 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.

s2
t1
V110 V24
In the multi-unit case
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Motivation

• Abstracts complex resource allocation problems in
  systems with distributed ownership (scheduling,
  allocation of network resources).
• Real Applications (e.g. the FCC spectrum
  auction).

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Strategic Issues and Truthfulness

• Study rational bidders that aim to maximize
  vi(Si) price
• A mechanism is M (A, p1 , p2 , ? , pn ), where
  A is an algorithm, and pi V ? R is the
  payment function of player i.
• We would like a truthful mechanism, i.e. ? vi,
  v-i, vi
• vi(A(vi, v-i)) pi(vi, v-i) gt vi(A(vi ,
  v-i)) pi (vi, v-i)
• Theorem Vickrey-Clarke-Groves, the 70s If
  the algorithm finds the exact optimal welfare
  then there exist truthful prices.
• This is true for any problem domain.
• Unfortunately, since finding the exact optimum is
  computationally hard, we cannot use this.

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Strategic issues
The classic model
V1()
S1
v2 ()
S2
ALG




vn ()
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.

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Mechanism Design and Truthfulness
A mechanism
V1()
S1 , P1
ALG
v2 ()
S2 , P2




Mechanism
vn ()
Sn , Pn

• A truthful mechanism No matter what the other
  players declare, player i will maximize his
  utility by reporting truthfully.

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Mechanism Design and Truthfulness
A mechanism
V1()
S1 , P1
ALG
v2 ()
S2 , P2




Mechanism
vn ()
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 70s If
  the algorithm finds the exact optimal welfare
  then there exist truthful prices.

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Mechanism Design and Truthfulness
A mechanism
V1()
S1 , P1
ALG
v2 ()
S2 , P2




Mechanism
vn ()
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 70s If
  the algorithm finds the exact optimal welfare
  then there exist truthful prices.
• Unfortunately finding the exact optimum is
  computationally hard.

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Complexity Issues

• Communication input is exponential (in m).
• No algorithm can approximate better than m1/(B1)
  with polynomial communication Nisan Nisan and
  Segal Dobzinski and Schapira
• Computation
• It is NP-hard to approximate better than
  m1/(B1), even for short valuations Lehmann,
  O'Callaghan, Shoham Bartal, Gonen, Nisan
• There exist polynomial time O(m1/(B1))-approximat
  ions
• In particular when BO(log m) there exists a
  (1e)-approximation

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• We seek truthful and computationally feasible
  mechanisms.
• In other words, are there other ways to embed
  truthfulness into a given algorithm?

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Previous attempts for resolution

• The single minded case
• vm approx. when B1 Lehmann, O'Callaghan,
  Shoham
• (1e)-approx. when BO(log m) Archer,
  Papadimitriou, Talwar, Tardos
• O(m1/B) -approx. for any B Briest, Krysta,
  Vocking
• For general valuations
• O(Bm1/B-2) for Bgt3 Bartal, Gonen, Nisan
• O(vm) for B1 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.

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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/B1) approx. for general
  valuations and any B.
• If BO(log m) this yields a (1e)-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.

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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.

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An aside a more general view

• Does deterministic truthfulness can yield such
  results?
• For B1, any deterministic mechanism that is also
  IIA cannot obtain a reasonable approximation
  Lavi, Mualem, Nisan
• Other GT notions might yield distribution-free/wor
  st-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?

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More on VCG

• Truthfulness ? vi, v-i, vi vi(f(vi, v-i))
  pi(vi, v-i) gt vi(f(vi , v-i)) pi (vi,
  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) -Sj?ivj(sj)
  hi(v-i)
• Proof Suppose a player says vi and the chosen
  allocation is (s1,,sn). His utility isi.e.
  telling his true value would weakly improve his
  utility.

vi(si) - pi(vi, v-i) vi(si) Sj?ivj(sj) lt
vi(si) Sj?ivj(sj) vi(si) - pi(vi,
v-i)
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The fractional case

• xi,s is the fraction of bundle S that player i
  gets.
• The fractional case is easy to solve by an LP
• Thus we can use VCG for this case.

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The fractional case

• xi,s is the fraction of bundle S that player i
  gets.
• The fractional case is easy to solve by an LP
• Thus we can use VCG in this case as well.

For every cgt1
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The fractional case

• xi,s is the fraction of bundle S that player i
  gets.
• The fractional case is easy to solve by an LP
• Thus we can use VCG in this case as well.

For every cgt1
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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.

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A randomized truthful integral mechanism

• Construction
• Compute a fractional solution x (optimal w.r.t.
  the declared values).
• Decompose x/c ?1x1 ?LxL where xll are
  the integral solutions, and ?1 ?L 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.
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A randomized truthful integral mechanism

• Construction
• Compute a fractional solution x (optimal w.r.t.
  the declared values).
• Decompose x/c ?1x1 ?LxL where xll are
  the integral solutions, and ?1 ?L 1.
• Choose xl with probability ?1 and set the
  expected price to be the VCG price in the
  fractional setting.
• Claim This is truthful in expectation

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A randomized truthful integral mechanism

• Construction
• Compute a fractional solution x (optimal w.r.t.
  the declared values).
• Decompose x/c ?1x1 ?LxL where xll are
  the integral solutions, and ?1 ?L 1.
• Choose xl with probability ?1 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 vi ? z . We
  have
• vi(y/c) pi(vi, v-i) gt vi(z/c) pi (vi,
  v-i)

As the fractional mechanism is truthful
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A randomized truthful integral mechanism

• Construction
• Compute a fractional solution x (optimal w.r.t.
  the declared values).
• Decompose x/c ?1x1 ?LxL where xll are
  the integral solutions, and ?1 ?L 1.
• Choose xl with probability ?1 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 vi ? z . We
  have
• vi(y/c) pi(vi, v-i) gt vi(z/c) pi (vi,
  v-i)
• ?y1vi(x1) ?yLvi(xL) pi(vi, v-i) gt
  ?z1vi(x1) ?zLvi(xL) pi (vi, v-i)

By the decomposition
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A randomized truthful integral mechanism

• Construction
• Compute a fractional solution x (optimal w.r.t.
  the declared values).
• Decompose x/c ?1x1 ?LxL where xll are
  the integral solutions, and ?1 ?L 1.
• Choose xl with probability ?1 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 vi ? z . We
  have
• vi(y/c) pi(vi, v-i) gt vi(z/c) pi (vi,
  v-i)
• ?y1vi(x1) ?yLvi(xL) pi(vi, v-i) gt
  ?z1vi(x1) ?zLvi(xL) pi (vi, v-i)
• E vi(f(vi, v-i)) pi(vi, v-i) gt E vi(f(vi ,
  v-i)) pi (vi, v-i)

By construction
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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 ?1x1 ?LxL
  (where xl is integral), in polynomial time.
• Remark The alg. should work for any weights
  wi,s
• Method (based on Carr and Vempala)

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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 ?1x1 ?LxL
  (where xl is integral), in polynomial time.
• Remark The alg. should work for any weights
  wi,s
• Method (based on Carr and Vempala)

x wi,s
x z
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The decomposition (2)

• Observation If (wi,s , z) is feasible then

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The decomposition (2)

• Observation If (wi,s , z) is feasible then
• Proof Suppose o/w.

(1/c) Si,s xi,s wi,s gt
1 - z
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The decomposition (2)

• Observation If (wi,s , z) is feasible then
• Proof Suppose o/w. Using A, find xl s.t.
• contradicting feasibility.

Si,s wi,s xli,s gt (1/c) Si,s xi,s wi,s gt 1
- z
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The decomposition (2)

• Observation If (wi,s , z) is feasible then
• Proof Suppose o/w. Using A, find xl s.t.
• contradicting feasibility.
• Implications
• The optimal solution is 1, as we need.

Si,s wi,s xli,s gt (1/c) Si,s xi,s wi,s gt 1
- z
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The decomposition (2)

• Observation If (wi,s , z) is feasible then
• Proof Suppose o/w. Using A, find xl s.t.
• contradicting feasibility.
• Implications
• The optimal solution is 1, as we need.
• 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.

Si,s wi,s xli,s gt (1/c) Si,s xi,s wi,s gt 1
- z
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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?