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```Truthful Mechanisms for
One-parameter Agents
Aaron Archer, Eva Tardos
Presented by: Ittai Abraham
Truthful Mechanisms for
One-parameter Agents





Introduction
Terminology and notation
Related work
Characterization of truthful mechanisms
Examples:
– Scheduling to minimize makespan
– LP
– Uncapacitated facility location

Lower bounds
2
Introduction: Q||Cmax

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Scheduling jobs on related parallel machines
to minimize makespan
Each job j has processing requirement pj
Each machine i runs in speed si
If job j is scheduled on machine i it takes pj/si
Goal: allocate jobs so that last job finishes as
early as possible (makespan)
Its NP-complete, and there is a known PTAS
3
Introduction: the Q||Cmax game

Each machine i is a distinct economic agent
which incurs a cost proportional to the total
time it spends processing
 Only machine i knows its true speed si
 Our mechanism :
– Asks each machine to report its speed
– Allocates jobs using some output function o
– Hands payments pi to each machine i using some
payment function p
4
Introduction: mechanism design
of the Q||Cmax game

We would like our mechanism to:
– Cause truth-telling to be a (weakly)
dominant strategy
– Reach a (near) optimal allocation
– Use polynomial resources
– Never give truth tellers negative profits
– Pay as little as possible

For the PTAS allocation there is no payment scheme that
causes profit-interested agents to be truthful
A 3-approximation allocation combined with a payment scheme
(both polynomial and shown later) cause truth-telling to be a
5
dominant strategy

Truthful Mechanisms for
One-parameter Agents





Introduction
Terminology and notation
Related work
Characterization of truthful mechanisms
Examples:
– Scheduling to minimize makespan
– LP
– Uncapacitated facility location

Lower bounds
6
Terminology and notation
M agents, represented by index set I
 Each agent i has a private value ti
 Each agent reports a bid bi

Mechanism
b1
1
t1
b2
2
t2
bi
bm
i
ti
m
tm
7
Terminology and notation
O is the set of allowable outcomes
 Output is a function o:mO
 Payment is a function p:m m
 Mechanism is a pair <o,p>

p1
b1
1
t1
b2 p2
2
t2
o
Mechanism
pm
bm
pi bi
i
ti
m
tm
8
Terminology and notation
Each outcome assigns work w:Om
 Each Agent i wants to maximize her profit

pi (b)- ti wi (o(b))

A mechanism is truthful if truth-telling is a
dominant strategy. Formally fix any b-i then
for all bi profiti(b-i,ti)profiti(b-i,bi)
 An output function o admits a truthful
payment scheme if there exists a payment
scheme p such that:
Mechanism <o,p> is truthful
9
Main questions
Characterization: What output
schemes ?
 Mechanism design: For an output
function that admits a truthful payment
scheme, what is the payment scheme ?

10
Truthful Mechanisms for
One-parameter Agents





Introduction
Terminology and notation
Related work
Characterization of truthful mechanisms
Examples:
– Scheduling to minimize makespan
– LP
– Uncapacitated facility location

Lower bounds
11
Related Work

Vickery-Clarke-Groves Mechanism
maximizes the sum of the agent valuations
(social welfare)
 Algorithmic mechanism design. (Ronen,
Nisan) focus on scheduling unrelated
machines through a Vickery auction for each
job (reach m-approximation, 2 is best known)
 Algorithms for rational agents. (Ronen)
characterize all truthful 0-1 load functions
12
Truthful Mechanisms for
One-parameter Agents





Introduction
Terminology and notation
Related work
Characterization of truthful mechanisms
Examples:
– Scheduling to minimize makespan
– LP
– Uncapacitated facility location

Lower bounds
13
Characterization of truthful
mechanisms
Definition: for a given b-i the load on
agent i is: li(x)= wi (o(b-i,x))
 Definition:The output function o(b) is
decreasing if for all b-i and for all i: li is
decreasing
 Theorem 1: The output function o(b)
admits a truthful payment scheme only if
it is decreasing

14
Pictorial proof of Theorem 1
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profiti=payi-costi: payi=pi (b) costi=ti wi (o(b))
If ti=y then costi(x)+A+B=costi(y)
If ti=x then costi(x)+A=costi(y)
So pi (y)- pi (x) is at least A+B and at most A
But B is Positive !
li(y)
li(x)
A
B
Cost(x) when t=y
Cost(x) when t=y
x
y
15
Characterization of truthful
mechanisms

Theorem 2: A decreasing output
function o(b) admits a truthful payment
scheme if and only if itbis of the form:
i
h i (bi )  bi wi (o(bi , bi ))   wi (o(bi , u )) du
0
For example if b=y then
p=c-A
A
li(y)
y
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Proof of Theorem 2 (only if)

Suppose li(x) is differentiable, so for all b-i and
for all ti the point ti is a maximum of pi (b-i,x)- ti
wi(o(b-i,x ))
 dpi (bi , bi )
dwi o((bi , bi )) 
ti : 
 ti
0

dbi
dbi

 bi ti
bi
bi
dpi (bi , u )
dwi (o(bi , u ))
du
0 du du  0 u
du
bi
pi (bi , bi )  pi (bi ,0)  bi wi (o(bi , bi ))   wi (o(bi , u )) du
0
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Pictorial proof of Theorem 2 (if)
Profit is pi (b)- ti wi (o(b))
 Bidding truthfully gives –T
 Bidding lower gives –L

li(l)
G
li(t)
l
ti
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Pictorial proof of Theorem 2 (if)
Bidding truthfully gives –T
 Bidding higher gives –H.

li(t)
li(h)
G
ti
h
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Characterization so far

The output function o(b) admits a
truthful payment scheme if only if it is
decreasing. In this case the mechanism
is truthful if and only if the payments
pi(b-i,bi) are of the form b
i
h i (bi )  bi wi (o(bi , bi ))   wi (o(bi , u))du

0
Where the hi are arbitrary functions
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Characterization: Voluntary
participation

A mechanism satisfies the voluntary
participation condition if agents who bid
truthfully never incur a loss
 Need to set hi(b-i) to be at least as large as the
integral 0 to ti for all ti
 Theorem 3: A decreasing output function
o(b) admits a truthful payment scheme that
satisfies the voluntary participation condition if
and only if w (b , u )du   and we can choose
i
-i

it to be
0

pi (bi , bi )  bi wi (o(bi , bi ))   wi (o(bi , u )) du 21
bi
Generalization of Vickery auction

The Vickery auction is a special case
were agents bid their costs load is 0 or
1 and the lowest bidder pays the
amount of the second lowest bid
Critical value
22
Truthful Mechanisms for
One-parameter Agents





Introduction
Terminology and notation
Related work
Characterization of truthful mechanisms
Examples:
– Scheduling to minimize makespan
– LP
– Uncapacitated facility location

Lower bounds
23
Scheduling jobs on related parallel
machines to minimize makespan

Each job j has processing requirement pj
 Each machine i runs in speed si , so ti =1/ si
 If job j is scheduled on machine i it takes pj/si
 Output function: allocate jobs to minimize
makespan

wi (o(b)) is the sum of the pj assigned to
machine i
 So we need a decreasing allocation function
24
Known allocations are not
decreasing

The PTAS of Hochbaum and Shmoys uses
rounding and dynamic programming,
announcing a slightly slower speed may
cause receiving a different set of jobs and the
load could increase because of rounding
 The greedy is not decreasing: two machines
of almost equal speeds and jobs 2,1+,1+.
First, fast machine gets job 2 then, slow
machine gets both 1+ jobs – so slower gets
more work !
25
From Scheduling to Bin Packing
and fractional relaxations
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Equivalent to bin packing with uneven bins
Let Cmax be the optimal makespan
Given a guess T at the value of Cmax , create
bins of size T/bi for each machine i
TCmax iff exists an assignment of jobs s.t.
each bin is at least as large as the total size
of jobs assigned to it
Get lower bound by relaxing the requirement
and allowing fractional assignments of jobs
26
Valid fractional assignments

A fractional assignment is valid if
– Each bin is at least as large as the total size of all
fractional jobs assigned to it
– Every bin receiving a piece of a job is large
enough to contain the entire job

The smallest T for which there exists a valid
assignment is a lower bound for Cmax
 Given such a T the greedy algorithm finds the
allocation: Assign the largest unassigned job
to the largest bin that is not full yet
 Number bin and jobs from largest to smallest
b1  …  bm and p1  …  pn
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Finding the lower bound of valid
fractional allocation with greedy


When is greedy valid ? For every job j, let i(j) denote
the last bin that is as large as job j
Greedy is valid iff for all j, the total capacity of the first
i(j) bins is at least as large as the total size of the first
j jobs


ji : T  max p j bi , k 1 pk / l 11 / bl
j
j :  T / b   p


pk 

j : T / b  p



k 1
TLB  max min max  p j bi , i

i
j
1 



l 1 bl 


i( j)
l 1
i( j)
j
i
j
k 1
l
k
j
28
Remember we need decreasing
allocations
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Lemma: Sizing bins as TLB, greedy yields a
valid fractional assignment s.t. each bin
contains some full jobs and at most two
partial jobs
So round each split job to the faster machine
and we get a 2-approximation
But suppose pjbi is a bottleneck and job j
exactly finishes bin i
For bi+ TLB gets bigger so job j+1 gets
partially in bin i increasing the load on bin i
Seems difficult to overcome deterministically 29
Randomized allocations for
truthful mechanisms
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What does it mean for a randomized
allocation to be truthful ?
Agents aim to maximize their expected profit
Truth telling a dominant strategy for agent i if
bidding ti maximizes her expected profit
regardless of what other agents bid
A mechanism is truthful if for all agents truth
telling is a dominant strategy
So now interpret wi (o(b)) as the expected
30
Randomized Rounding
assignment given by TLB
 Randomly assign the partial jobs in the
following way:
 Job j is assigned to machine i with
probability equal to the proportion of j
that is fractionally assigned to bin i

31
Theorem and proof of 3
approximation
payment scheme satisfying voluntary
participation and deterministically yields
a polytime 3-approximation mechanism
for Q||Cmax
 3-approximation follows from valid
allocation and at most two partial jobs

32
Proof of decreasing expected work

greedy fractional assignment: TLB/bi
 Suppose some machine claims she is slower
replacing bi with bi where >1
 Clearly TLBT’LB but T’LBTLB (check) so
TLB/bi  T’LB/bi thus allocation is decreasing
 If a machine is very slow it will not receive
any job (The T without it divided by bi is
smaller than any job)
33
Computing payments in
polynomial time
For a given b-i the load on agent i is:
li(x)= wi (o(b-i,x))= TLB (x)/x
 TLB (x)is either (1) constant, (2) or of the
form cx, (3) or of the form c/(d+1/x)
 Breakpoint occur only when x coincides
with another agent’s bid or when the
two terms in TLB (x) cross ((3) to (2))
 The number of intervals is polynomial

34
More examples
LP
 Uncapacitated facility location
 Scheduling to minimize sum of
completion times
 Max flow

35
Truthful Mechanisms for
One-parameter Agents





Introduction
Terminology and notation
Related work
Characterization of truthful mechanisms
Examples:
– Scheduling to minimize makespan
– LP
– Uncapacitated facility location

Lower bounds
36
Lower bounds
Scheduling on machines with speeds to
minimize weighted sum of completion
times
 Theorem: No truthful mechanism for
Q||wjCj can achieve an approximation
ratio better than 2/3, even with just two
jobs and two machines

37
Lower bounds

Proof sketch:
– Job 1 has weight and processing requirement 1
– Job 2 has weight w and processing req. p>1
– Machine 1 runs at speed 1, machine 2 at speed s

Set pw<1 so that opt will be non-monotone
 OPT:
– For small s both jobs will be on 1, then split, then
swap, then both on 2. Not monotone

A decreasing allocation:
– For s<<1 both on 1 and for s>>1 both on 2
38
Summery

For the model where profiti=pi(x)-ti*li(x) gave a
full characterization of allocation functions