Truthful Mechanisms for One-parameter Agents

Aaron Archer, Eva Tardos

Presented by: Ittai Abraham

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Truthful Mechanisms for One-parameter Agents IntroductionIntroduction Terminology and notation Related work Characterization of truthful mechanisms Examples:

– Scheduling to minimize makespan– LP– Uncapacitated facility location

Lower bounds

3

Introduction: Q||Cmax

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

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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

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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 dominant strategy

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Truthful Mechanisms for One-parameter Agents IntroductionIntroduction Terminology and notation Related work Characterization of truthful mechanisms Examples:

– Scheduling to minimize makespan– LP– Uncapacitated facility location

Lower bounds

7

Terminology and notation

M agents, represented by index set I Each agent i has a private value ti Each agent reports a bid bi

t1 t2 ti tm

Mechanism

1 2 i m

b1b2

bi bm

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Terminology and notation O is the set of allowable outcomes Output is a function o:mO Payment is a function p:m m Mechanism is a pair <o,p>

Mechanism

t1 t2 ti tm

1 2 i m

b1b2

bi bm

op1

p2pi

pm

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Terminology and notation

Each outcome assigns work w:Om

Each Agent i wants to maximize her profit

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

pi (b)- ti wi (o(b))

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Main questions

Characterization: What output functions admit truthful payment schemes ?

Mechanism design: For an output function that admits a truthful payment scheme, what is the payment scheme ?

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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

12

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

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Truthful Mechanisms for One-parameter Agents IntroductionIntroduction Terminology and notation Related work Characterization of truthful mechanisms Examples:

– Scheduling to minimize makespan– LP– Uncapacitated facility location

Lower bounds

14

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

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Pictorial proof of Theorem 1 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 !

A B

x y

li(x)

li(y)

Cost(x) when t=yCost(x) when t=y

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Characterization of truthful mechanisms Theorem 2: A decreasing output

function o(b) admits a truthful payment scheme if and only if it is of the form:

For example if b=y then p=c-A

y

li(y)A

duubowbbowbb i

b

iiiiii

i

)),(()),(()(h0

i

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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 ))

i

i i

ii

b

iiiiiiiiiii

b b

iiii

tbi

iiii

i

iiii

duubowbbowbbpbbp

dudu

ubodwudu

du

ubdp

db

bbodwt

db

bbdpt

0

0 0

)),(()),(()0,(),(

)),((),(

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

ti

li(t)

l

li(l) G

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Pictorial proof of Theorem 2 (if) Bidding truthfully gives –T Bidding higher gives –H.

ti

li(t)

h

li(h)G

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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

Where the hi are arbitrary functions

duubowbbowbb i

b

iiiiii

i

)),(()),(()(h0

i

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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 and we can choose it to be

duuwi ),b(0

i-

ib

iiiiiiiii duubowbbowbbbp )),(()),((),(

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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

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Truthful Mechanisms for One-parameter Agents IntroductionIntroduction Terminology and notation Related work Characterization of truthful mechanisms Examples:

– Scheduling to minimize makespan– LP– Uncapacitated facility location

Lower bounds

24

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

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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 !

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From Scheduling to Bin Packing and fractional relaxations 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 TCmax 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

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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

i

l l

j

k kij bpbpTij11

/1/,max:

i

l l

j

kk

ijij

LB

b

pbpT

1

1

1,maxminmax

jji

j

k k

ji

l l

pbTj

pbTj

)(

1

)(

1

/:

/:

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Remember we need decreasing allocations 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

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Randomized allocations for truthful mechanisms 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 load on agent i

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Randomized Rounding

Start with the greedy valid fractional 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

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Theorem and proof of 3 approximation Theorem: allocation admits a truthful

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

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Proof of decreasing expected work load and voluntary participation The expected load on bin i is the load on the

greedy fractional assignment: TLB/bi Suppose some machine claims she is slower

replacing bi with bi where >1 Clearly TLBT’LB but T’LBTLB (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)

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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

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More examples

LP Uncapacitated facility location Scheduling to minimize sum of

completion times Max flow

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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

37

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

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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

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Summery

For the model where profiti=pi(x)-ti*li(x) gave a full characterization of allocation functions that admit truthful payment schemes

For scheduling related machines to minimize makespan, shown a 3-approx truthful mechanism

Can be used for other combinatorial allocations (max flow, facility location, …)

Can be used to prove lower bounds