Setting Lower Bounds on Truthfulness

1
Setting Lower Boundson Truthfulness

• Speaker Michael Schapira (www.cs.huji.ac.il/mike
  sch)
• Joint work with Ahuva Mualem

2
Algorithmic Mechanism Design

• Most work in Algorithmic Mechanism Design
  Nisan-Ronen aspires to implement a dominant
  strategy equilibrium.
• Due to the revelation principle we limit
  ourselves to truthful mechanisms.

3
Non-Utilitarian Social-Choice Functions

• Traditional mechanism design provides us with a
  general technique (namely VCG) for the truthful
  implementation of utilitarian social choice
  functions.
• e.g. social-welfare maximization in combinatorial
  auctions.
• In many economic and computational settings we
  wish to optimize a non-utilitarian social-choice
  function.
• Machine scheduling.
• Congestion minimization.
• Fair allocation of indivisible items.
• Revenue maximization in auctions.

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Non-Utilitarian Social-Choice Functions

• In general, non-utilitarian social choice
  functions cannot be optimally implemented in a
  truthful manner.
• Question How well can these functions be
  truthfully approximated?
• Very few lower bounds on the approximability of
  truthful mechanisms are known. This is especially
  true for multi-parameter settings.

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Setting Lower Bounds on Truthfulness

• We present and discuss general techniques for
  setting lower bounds on the approximability of
  truthful mechanisms for non-utilitarian
  optimization problems.
• Our techniques
• General and simple.
• Apply to multi-parameter settings.
• Make no computational assumptions.
• Apply to randomized mechanisms.

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Related Work

• In their seminal paper Algorithmic Mechanism
  Design, Nisan and Ronen present this new field
  via the machine scheduling problem with unrelated
  machines Lenstra-Tardos-Shmoys. In particular
  they prove a lower bound for deterministic
  truthful mechanisms for this non-utilitarian and
  multi-parameter problem.
• Most work in AMD focused on single parameter
  settings. These settings are pretty well
  understood with regards to truthfulness.
• Some research works addressed the problem of
  proving lower bounds for deterministic,
  polynomial-time, truthful mechanisms
  Lavi-Mualem-Nisan, Dobzinski-Nisan. These works
  make strong assumptions on the mechanisms (e.g.
  IIA, VCG).

7
Related Work

• Bikhchandani-Chatterji-Lavi-Mualem-Nisan-Sen
  study a property maintained by any truthful
  mechanism weak monotonicity. This property will
  play a crucial role in our techniques.

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Machine Scheduling

• We will present our techniques via the scheduling
  problem with unrelated machines. In the
  scheduling problem with unrelated machines we
  have m machines (agents) 1,,m and n tasks 1,,n.
  
• Each machine i has a valuation function vin?R
  (vi(?)0). Every valuation function vi is
  additive (a.k.a. linear), i.e., for every set of
  tasks S, vi(S)Sj?S vi(j).
• The goal is to minimize the makespan. That is, to
  find an allocation of the n tasks to the m
  machines, S1,,Sm , that minimizes the expression
  maxi vi(Si).

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Machine Scheduling

• Nisan and Ronen present a truthful deterministic
  upper bound of m. They also present a lower bound
  of 2-e for the case of two machines (an
  non-truthful FPTAS exists), thus showing that
  their upper bound is tight for this case.
• For the case of two machines Nisan and Ronen
  prove a randomized upper bound of 7/4 (thus
  showing that randomness helps).
• We prove that this result can be generalized to a
  randomized upper bound of 7m/8 will not be
  presented here.
• No lower bound for truthful randomized mechanisms
  for this problem was previously known.

10
Machine Scheduling

• We prove a lower bound of (2-1/m-e) for truthful
  randomized mechanisms. This shows that the
  randomized mechanism of Nisan and Ronen for the
  case of 2 machines is nearly tight.
• This is the first lower bound for truthful
  randomized mechanisms in multi-parameter
  settings.
• Randomness has proven to be useful in designing
  truthful mechanisms for multi-parameter settings
  Nisan-Ronen, Lavi-Swamy, Dobzinski-Nisan-Schapira
  

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Machine Scheduling

• We shall present the proof of our lower bound in
  three steps
• First, we will present a different and simpler
  proof of the 2-e lower bound of Nisan and Ronen
  for deterministic mechanisms.
• We will then use this proof as a building block
  in the proof of the (2-1/m-e) lower bound for
  universally truthful randomized mechanisms.
• We shall then show that this lower bound also
  holds for randomized mechanisms that are truthful
  in expectation (a weaker notion).

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Deterministic Mechanisms

• We shall present an alternative proof to the 2-e
  lower bound of Nisan and Ronen for deterministic
  mechanisms.
• The Nisan-Ronen proof was based on exploiting the
  properties of the payments assigned by truthful
  mechanisms.
• We prove our lower bound for weakly-monotone
  mechanisms. This class of mechanisms is known to
  contain all truthful deterministic mechanisms.

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Truthfulness ? W-MON

• Lemma If f is truthful then pi(v) pi (a,
  v-i ), where f(v) a.
• Proposition (Truthfulness ? W-MON)
• Suppose f (vi , v-i ) a and f (ui , v-i )
  b. Then pi(a, v-i ) - vi (a) gt pi(b, v-i ) -
  vi (b),(otherwise player i would declare ui
  instead of vi).And, pi(b, v-i ) - ui (b) gt
  pi(a, v-i ) - ui (a),(otherwise player i would
  declare ui instead of vi).? vi (a) ui (b)
  ui (a) vi (b).

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Deterministic Mechanisms

• Theorem Any weakly-monotone deterministic
  mechanism cannot obtain an approximation ratio
  better than 2.
• Theorem Any strongly-monotone deterministic
  mechanism cannot obtain an approximation ratio
  better than m.A similar lower bound was
  independently proven by Lavi and Swamy. The proof
  will not be presented here.

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A 2-e Lower Bound
16
A 2-e Lower Bound
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A 2-e Lower Bound
vII
2e vI(1)vI(13) gt vI(13)vI(1) 2 A
contradicition to weak monotonicity
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Randomized Mechanisms

• Two notions of truthfulness for randomized
  mechanisms
• universal truthfulness a probability
  distribution over truthful deterministic
  mechanisms (stronger)
• Truthfulness in expectation truthful behavior
  maximizes the expected profit (weaker)
• Risk-averse bidders might benefit from untruthful
  behavior.
• The outcomes of the random coins must be kept
  secret.

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Universally-Truthful Randomized Mechanisms

• Recall, that a universally truthful randomized
  mechanism is a probability distribution over
  truthful deterministic mechanisms.
• Natural approach Find an instance on which any
  universally truthful randomized mechanism fails
  to obtain the approximation ratio.
• Yaos principle Find a probability distribution
  over instances D on which any truthful
  deterministic mechanism fails to provide the
  approximation factor.

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Universally-Truthful Randomized Mechanisms

• Theorem Any universally-truthful randomized
  mechanism cannot obtain an approximation ratio
  better than 2-1/m.
• We will prove this for the case m2.
• Our proof will present a probability distribution
  D over instances of the problem on which any
  weakly-monotone deterministic mechanism fails to
  give an approximation-ratio better than 2-1/m.

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A 3/2-e Lower Bound
PrD(I0)e PrD(I1) PrD(I2) ½ - e
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A 3/2-e Lower Bound
Let M be a weakly-monotone deterministic
mechanism.
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A 3/2-e Lower Bound
Therefore, the approximation ratio obtained by M
for I1 is at least 2/(1e). Hence, the
approximation ratio of M for P is at least (1-
PrP(I1))1 PrP(I1)2/(1d) gt 3/2-e.
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Trtuthfulness in Expectation

• Any randomized mechanism can be regarded as a
  function from n-tuples of valuations to
  probability distributions over the set of
  feasible alternatives.
• Definition We define the extended valuation
  function V as follows For every valuation
  function v, and for every probability
  distribution P over the set of alternatives
  A,V(v,P) Sa?A P(a)v(a)

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Trtuthfulness in Expectation

• Definition A randomized mechanism M is said to
  be weakly monotone in the extended sense if for
  every machine i, for every vi,vi, and for every
  v-i Let M(vi,v-i)P, and let M(vi,v-i)Q,
  thenV(vi,P0) V(vi,Q) V(vi,P) V(vi,P)
• Proposition Any randomized mechanism that is
  truthful in expectation is weakly-monotone in the
  extended sense.

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Trtuthfulness in Expectation

• Theorem Any randomized mechanism that is weakly
  monotone in the extended sense cannot obtain an
  approximation ratio better than 2-1/m.
• We shall prove this for the case m2.
• Intuition
• A randomized mechanism can be viewed as
  generating, for each task, a probability
  distribution over the machines to which it is
  allocated.
• We show that the extended version of weak
  monotonicity implies certain helpful relations
  between these probability distributions.

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A 3/2-e Lower Bound
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A 3/2-e Lower Bound
Let M be a randomized mechanism that is
weakly monotone in the extended sense, and let
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A 3/2-e Lower Bound

• Definition For every probability distribution R
  over task allocations, let pi,t(R) be the
  probability that machine i is assigned task t
  given R.
• Case 1 There is some machine r such that
  pr,r(P0) lt 1-e2.
• If so, task r is not assigned to machine r with
  probability e2. However, if r does not get r,
  the makespan value is at least 4/e2. while the
  optimal makespan value is 2. The expected
  approximation ratio M obtains for I0 is therefore
  at least e22/e2 2 (QED).

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A 3/2-e Lower Bound

• Case 2 For every machine i1,2 pi,i(P0)
  1-e2.
• Let r be a machine such that pr,3(P0) ½.(such
  a machine exists!). Consider the case r1 (the
  case r2 is similar).
• Lemma If p1,3(P0) ½ then p1,3(P1) ½ e.
• ProofBy weak monotonicity in the extended
  sense
• V(v1,P0) V(v1,P1) V(v1,P0) V(v1,P1)

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A 3/2-e Lower Bound

• That is Sa?A P0(a)v1(a)Sa?A P1(a)v1(a)
  Sa?A P0(a)v1(a)Sa?A P1(a)v1(a)
• Equivalently St p1,t(P0)v1(t) St
  p1,t(P1)v1(t) St p1,t(P0)v1(t) St
  p1,t(P1)v1(t)

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A 3/2-e Lower Bound

• After the removal of identical summands and the
  assignment of values p1,1(P0) p1,3(P1)e
  p1,3(P0)e p1,1(P1)
• Since p1,1(P0) 1-e2 and p1,1(P1) 11-e2
  p1,3(P1)e p1,3(P0)e 1 p1,3(P1) p1,3(P0)
  e ½ e (because p1,3(P0) ½)

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A 3/2-e Lower Bound

• So, if p1,3(P0) ½ then p1,3(P1) ½ e.
• Therefore, with probability of at least (½ - e),
  M does not assign task 3 to 1 for instance I1.
• The approximation ratio of M for I1 is at least
  1(½ e)(2/(1e))(½ - e) gt 3/2-e

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Other Results

• We apply our techniques to the problem of
  workload-minimization (congestion-minimization)
  in the interdomain routing setting
  Feigenbaum-Papadimitriou-Sami-Shenker.
• We discuss and prove several results for notions
  of non-utilitarian fairness (Max-Min fairness
  Rawls, Lavi-Mualem-Nisan, Min-Max fairness,
  envy-minimization Lipton-Markakis-Mossel-Saberi)
  .

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Open Questions
truthfulapproximations
lower bounds
121/2 (Christodoulou-Koustoupias-Vidali) 2-1/m
Machine scheduling
m (Nisan-Ronen) 7m/8
Workload minimization

• n

21/2
Proving lower bounds on polynomial time
truthfulness