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