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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