An Optimal Lower Bound for Anonymous Scheduling Mechanisms

1
An Optimal Lower Bound for Anonymous Scheduling
Mechanisms

• Shahar Dobzinski
• Joint work with Itai Ashlagi and Ron Lavi

2
Unrelated Machines Scheduling

• n jobs to be assigned to m selfish machines
• Each machine i needs tij time units to complete
  job j, and incurs a cost of tij.
• Private Information.
• Goal assign jobs to machines to minimize the
  maximal load (minimize the makespan).
• We use payments to motivate truthfulness.

3
The Nisan-Ronen Conjecture

• Nisan and Ronen a simple mechanism gives an
  upper bound of m, and there is a lower bound of
  2.
• Ignoring computational issues.
• The 99 paper that introduced Algorithmic
  Mechanism Design!
• Conjecture Nisan-Ronen The lower bound of m.

4
Previous Work

• Many efforts to prove or disprove the conjecture.
• Christodoulou, Koutsoupias, and Vidali 07 an
  improved lower bound (about 2.41, and then 2.61).
• A huge gap between the upper bound and the lower
  bound.
• Mualem and Schapira 07 and Christodoulou et al
  07 give a lower bound of 2 for randomized and
  fractional mechanisms.
• Dobzinski and Sundararajan 08, and Christodoulo,
  Koutsoupias, and Vidali 08 characterize the 2
  machines case.

5
Previous Work Special Cases

• Lavi and Swamy 07 prove that in the two values
  case (low and high jobs) there are constant
  approximation mechanisms.
• Dhangwatnotai, Dobzinski, Dughmi, and Roughgarden
  08 show that if the machines are related there
  is a PTAS.
• The problem was introduced by Archer and Tardos
  01.
• Is the Nisan-Ronen conjecture true?

6
Anonymity

• We provide the first concrete evidence that the
  Nisan-Ronen conjecture is true.
• A lower bound of m for anonymous mechanisms.
• A mechanism f is anonymous if the names of the
  machines do not matter.
• Two machines that switch cost vectors also switch
  their assignments.
• Very weak notion of anonymity.

7
Why Anonymity?

• Thats what we can prove ?
• Very natural from an algorithmic perspective.
• Powerful even from a mechanism design
  perspective.
• Related machines Dhangwatnotai-Dobzinski-Dughmi-R
  oughgarden
• 2 values Lavi-Swamy
• Recent interest in the AGT community.
  Daskalakis-Papadimtriou
• We will talk about fractional mechanisms later
• First evidence that the Nisan-Ronen conjecture is
  true.
• First lower bound for a large class that is super
  constant.
• At least, the algorithm is strange.
• Still, for revenue maximization in digital goods
  naming the players helps! Aggarwal-Fiat-Goldberg-
  Hartline-Immorlica-Sudan
• But in a single parameter setting.

8
Weak Monotonicity

• Definition an allocation function f is weakly
  monotone if for every ti, ti, t-i suppose that
  machine i is allocated S in f(ti, t-i), and that
  it is allocated T in f(ti,t-i). Then,
• ti(T) ti(T) ti(S) ti(S)
• Reminder Fix t-i.Each bundle has an associated
  payment (independent of ti). The machine is
  allocated the bundle that maximizes its profit.
• ? Every truthful mechanism is weakly monotone.
• Interpretation of WMON the profit from taking T
  must increase more than the profit from taking S.
• Easy corollary If machine i is allocated S, and
  lowers its cost for all jobs in S while raising
  its cost for all jobs not in S, then machine i
  still receives S.
• Well also use WMON in more delicate ways.

9
The Main Result

• Theorem Every anonymous mechanism that provides
  a finite approx ratio must allocate as follows
• Thus it provides an approx ratio no better than
  m.
• Intuition this is how VCG allocates the jobs.

t1e gt tm gt gt t2 gt t1
10
Outline of the Proof

• Proof is by induction on the number of jobs.
• In this talk only 3 machines and 3 jobs, hence a
  lower bound of 3.
• An easy base case, and 5 induction steps.
• Steps are modular.
• More or less
• Lots of omissions and shameless cheating,
  sometimes in technical details.

11
The Base Case One Job, 3 Machines
Towardsa contradiction
WMON
WMON
t3gt t2 gt t1
A contradictionto the anonymityof the mechanism!
12
Induction Steps
t3gt t2 gt t1 gtgt a gtgt d
13
Step 1

• Informally The cost of J3 is very small so we
  can ignore this job.
• By the induction hypothesis it must allocate both
  big jobs to M1.
• More formally, we fix the costs of J3 and define
  a mechanism on J1 and J2. The induction
  hypothesis applies to this mechanism.

14
Induction Steps
d
t3gt t2 gt t1 gtgt a gtgt d
15
Step 2
WMON
Towards contradiction
WMON
The mechanism does notprovide a finite approx
ratio!
16
Induction Steps
t3
t3gt t2 gt t1 gtgt a gtgt d
17
Step 3
Step 3(a)
Step 3(b)
18
Step 3(a)
WMON
19
Step 3
Step 3(a)
Step 3(b)
20
Step 3(b)
Given (no proof)
Lemma 1 (M1 gets atleast one bigjob)
Lemma 2 (One big, 2small M1 getseverything)
Proof of 3(b)
WMON
By Lemma 1, towards a contradiction
A contradiction to Lemma 2
21
Induction Steps
t2
t3gt t2 gt t1 gtgt a gtgt d
22
Step 4
I.e., machine 2 gets nothing in such ordered
instances.
The 2nd machine got a job. A contradiction.
WMON
Towardscontradiction
23
Induction Steps
t1
t3gt t2 gt t1 gtgt a gtgt d
24
Step 5
WMON
A contradiction tothe previous step! (M1 should
get everything)
Towards a contradiction
(A similar argument if M1 is allocated two jobs)
25
Summary

• We showed that anonymous mechanisms provide only
  a trivial approximation ratio.
• First evidence that the Nisan-Ronen conjecture is
  indeed correct
• Might help in proving a lower bound on all
  mechanisms anonymity is without loss of
  generality for fractional mechanisms.

26
Tool Induced Mechanisms

• Suppose f is a mechanism for n jobs and m
  machines.
• Define f (a mechanism for (n-1) jobs and m
  machines) fix identical costs for the nth job.
  Allocate in f the first n-1 jobs as in f.

f
f
27
Induced Mechanisms (cont.)

• Proposition f is truthful ? f is truthful.
• Proof
• f satisfies weak monotonicity. To finish, we will
  prove that f satisfies weak monotonicity too.
• Suppose that machine i gets S in f(ti,t-i), and T
  in f(ti,t-i). f satisfies weak monotonicity
• ti(T) - ti(T) ti(S) - ti(S)
• So in f we have that
• ti(T \ n) - ti(T \ n) ti(S \ n) - ti(S
  \ n)