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Mechanism Design
Overview
• Incentives in teams (T. Groves (1973))
• Algorithmic mechanism design (Nisan and Ronen (2000))
- Shortest Path
- Task Scheduling
Framework
• Something needs to be done with the help of n agents• Is there a way of inducing them to do it (might lack
knowledge or control)• The way if it exists is called a “mechanism”• Assumption1 : The agents are rational• Assumption2 : The agents are independent (no
communication)• The mechanism is said to be truthful if there is no
incentive for an agent to lie• A lie is defined as something the agent could do so that
the goal is not achieved.
An Organization
),...,,( 10 n
CEO
Sub-unit 1 Sub-unit 2 Sub-unit n………
)(goptimize
Pay-Fire Incentive
otherwise {0
1{)( * iip
-Optimally performing employees are rewarded
-Pay is independent of how other employees perform
-Assumes that the CEO has complete information
An Organization
),...,,( 10 n )(goptimize
CEO
Sub-unit 1 Sub-unit 2 Sub-unit n………)( ii t)( ii t )( ii t
n
i
ii ttt1
00 )(
)( 00 ti
)( 00 ttt iii ))(),(),(( iiiiiii ttt
Own Profit Incentive
n
i
iii tvg1
* ))(()(maximize
ij
jjiiiiiii tvtvtp ))(())(())(( *
-- Payment to player i is independent of the decisions of the others
--But it is dependent on the messages
--Why is there no advantage in lying ?
Profit Sharing
iii
iiii Atgtp )))(,((.))(( *
--It is hard to remove message dependence without losing truthfulness
--Truthful mechanism – Nobody has incentive to lie
--Strongly truthful mechanism – Truth telling is the only dominant strategy
--Dominant strategy – No unilateral incentive to deviate
Direct Revelation Mechanisms
• The message strategy space and state space (t) are the same
• m(x(t),p(t))
• x(t) is a set of feasible outputs given t
• p(t) is a vector of payments to the agents
• g(t,x(t)) is the function to optimize
• m’(x’(t),p’(t)) is a c-approximation for m(x(t),p(t)) if g(t,x’(t))<= c . g(t,x(t))
VGC mechanisms
• VGC (Vickrey-Groves-Clarke)• VGC mechanisms are truthful• x(t) is feasible iff it maximizes g (so that we concern ourselves
with providing the correct incentive structure.)
n
i
iiii txtvtxtg1
))(,())(,(maximize
)())(,())(,())(,( ii
ij
jjjiiiiiiiii thtxtvtxtvtxtp
Shortest Path
• Each edge is an agent• People want to send
messages to other people• People are at vertices• Goal is to minimize cost• Each edge has a cost =• Payment to each edge =
et
0|| eGeG dd
otherwise 0{
{))(,( SPettxtv eee
Complexity is O(m *n * log(m))
Task Scheduling
• k tasks• n processors• State of agent i = • Goal is to minimize
the completion time of the set of tasks (make-span)
• A task need not go to the agent that does it the fastest.
),...( 1ik
ii ttt
)(
max))(,(txj
ij
i
i
ttxtg
Min-Work Mechanism
ij
k
j
ittxtg
1
min))(,(
otherwise {0
' and )( {min))(,( ' iitxjtttxtp iij
ij
ij
otherwise {0
)( {))(,( txjttxtv iij
ij
Min-Work (contd.)
• Min-Work is truthful• Nisan and Ronen show it is strongly
truthful• Min-Work is an n-approximation for make-
span
ij
k
j
itn
topttg
1
min.1
))(,(
))(,(.))(,( topttgntxtg
Bounds on approximations
2.cany for mechanismion approximat-c
a implements that mechanism truthfulaexist not does There
:Theorem
).()( then )()(
and t tIf agent.an be i and srevelation be tand Let t
:ceIndependen
2121
i-2
i-121
tptptxtx iiii
Proof Sketch
T1 T2
1 1
1 1
1 1
1 1
T1 T2
e 1
e 1
1+e 1
1+e 1
|)(|))(,( 2 txtxtg ektxtopttg .|)(|.2
1))(,( 2
2.cany for mechanismion approximat-c
a implements that mechanism truthfulaexist not does There
:Theorem
Randomized Mechanisms
• A probability distribution over a family of mechanisms that share the same set of strategies and outputs
• Optimize the G=E(g)
• Payments etc. are defined as expectations over payments
Randomly-Biased Min-Work
.t1
p , {j}x x
else
.t p , {j}xx
t. tif
i-3 i' ,s i
:k to1 jfor
:Algorithm
tand tsrevelation The :Input
random)at uniformly
(selected {1,2}s and 1number realA :Parameters
ij
i'i'i'
i'j
iii
i'j
ij
j
21
k
3
4 with agents two
for problemspan -make theofion approximat-4
7 truthful
strongly a is mechanism,work -min biased-randomly The
:Theorem
We will first show that the mechanism is truthful.
Weighted VGC Mechanisms
n
i
iiiii txtvtxtg
1
))(,(.))(,(maximize
)())(,(.1
))(,())(,( ii
ij
jjjij
i
iiiiiiii thtxtvtxtvtxtp
The mechanism is truthful.
Proof Sketch
T1 T2 Opt Rbmw
1 (b+e) 1 1
1 (b+e) 2 1
1 b 1 rnd
b 1 2 rnd
g(t,opt(t))=1 + b + e = 1 + 4/3 = 7/3 , 7/4 * g(t,opt(t))=49/12
g(t,rbmw(t))=1/4((1 + 1 + 1 + b) + (1 + 1 + b) + (1 + 1 + 1) + (b+1))
=1/4(9+3b)
=13/4 = 3.25 <= 49/12
Mechanisms with Verification
• Assumption: Agents actions can be verified
• Routing, Task scheduling etc.
• Check the effect of such a simplifying assumption both on mechanism design and computation
Make-span with Verification
?)problem.(?span -make theoftion implementa
ruthfulstrongly t a is above mechanism bonus-oncompensati The
:Theorem
))t',(tg(x(t),- iplayer toBonus
otherwise {0
t't if t'{ i on toCompensati
t':timesExecution
t: timesDeclared
i-
)(xj
ij
)(xj
ij
)(xj
ij
ij
ij
iii
ttt
Generalized Compensation and Bonus Mechanisms
problem.span -make theoftion implementa
ruthfulstrongly t a is above mechanism bonus-oncompensati The
:Theorem
)))t',(t,-g(x(t),(tm )t'(t,b
ion)(compensat t' )t't,(c
t':timesExecution
t: timesDeclared
i-i-ii
)(xj
ij
i
ij
ij
i
t
---Participation and Bonus Constraints
Computational Problems
• Exponential-time allocation algorithm• Approximations tend to violate truthfulness
(will discuss a theorem from Nisan and Ronen)
• If the no. of agents are fixed, and declarations are bounded a truthful polynomial time approximation mechanism exists. (Computing the exact solution is NP-hard)
t)g(opt(t), * t)g(x(t),
t)g(x(t),t)g(opt(t),
:such that t exists thereion,approximatan is x(t)Since
.allocation optimal thedenote opt(t)Let
truthful.be mLet
:ion)contradict(by Proof
ulnot truthf is
mThen on x(). based mechanism Bonus andon Compensati the
be p)(x,mLet span.-makefor ion approximat-an be Let x()
:Theorem
ratio.-ionapproximat thesContradict .
is difference The opt(s). x(s)s).g(opt(s), s)g(x(s), nowBut
correct) is thisdont think I (?? t)g(x(t),s)g(x(s),Then
otherwise {
)(optj if t{s
:such that revelation a be sLet
truthful)is mechanism the(since t)g(x(t),)'t'),'g(x(t'Then
otherwise {
)(optj if t{'Let t'
iij
ij
11j
1j
t
t
Bounded Scheduling Problems
g.programmin dynamic using timepolynomialin solvable
ionapproximat-)(1 a its that show )Sahni(1976 and Horowitz
).f(a, of multiplesinteger toup rounded are t
b.ta ji, allfor
such that 0abexist thereand fixed isn agents of no. The
ij
ij
Rounding Mechanism
• Compensation using actual times
• Bonus using rounded times.
• All revelations that are rounded up to the same value as the true revelations are dominant strategies.
Extensions
• Repeated games
• e-dominant strategies
• Partial verification