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
CEO
Sub-unit 1
Sub-unit 2
  ( ,  ,...,  )
0
1
n
………
Sub-unit n
optimize  g (  )
Pay-Fire Incentive
p (  )  {1   
i
i
*
{0 otherwise
-Optimally performing employees are rewarded
-Pay is independent of how other employees
perform
-Assumes that the CEO has complete information
An Organization
t t
0
n
0
i
i

(
t
)

i 1
CEO
 0i (t 0 )
 i (t i )
 i (t i )
Sub-unit 1
Sub-unit 2
 i  ( i (t i ),  i (t i ),  i (t i ))
  ( 0 ,  1 ,...,  n )
 i (t i )
………
Sub-unit n
t i  t i   0i (t 0 )
optimize  g (  )
Own Profit Incentive
n
maximize  g (  )   v i ( *i (t i ))
i 1
p i ( i (t i ))  v i ( i (t i ))   v i ( * j (t j ))
j i
-- 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
p ( (t ))   i .g (( , 
i
i
i
i
*i
)(t ))  Ai
--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
maximize  g (t , x(t ))   v i (t i , x i (t i ))
i 1
p (t , x (t ))  v (t , x (t ))   v (t , x (t ))  h (t )
i
i
i
i
i
i
i
i
i
j i
j
j
j
i
i
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 = te
• Payment to each edge = d G|e  
v e (te , x(t ))  {te e  SP
{0
otherwise
Complexity is O(m *n * log(m))
 d G|e 0
Task Scheduling
•
•
•
•
k tasks
n processors
State of agent i = t i  (t1i ,...t ki )
Goal is to minimize
the completion time of g (t , x(t ))  max i
the set of tasks
(make-span)
• A task need not go to
the agent that does it
the fastest.
t
jx i ( t )
i
j
Min-Work Mechanism
k
g (t , x(t ))   min i t ij
j 1
v (t , x(t ))  {t
i
j
i
j
j  x (t )
i
{0 otherwise
p ij (t , x(t ))  {min t ij'  t ij j  x i (t ) and i'  i
{0 otherwise
Min-Work (contd.)
• Min-Work is truthful
• Nisan and Ronen show it is strongly
truthful
• Min-Work is an n-approximation for makespan
1 k
i i
g (t , opt (t ))  . min t j
n j 1
g (t , x(t ))  n.g (t , opt (t ))
Bounds on approximations
Theorem :
There does not exist a truthful mechanism that implements a
c - approximat ion mechanism for any c  2.
Independen ce :
Let t 1 and t 2 be revelation s and i be an agent. If t1-i  t -2i and
x i (t1 )  x i (t 2 ) then p i (t1 )  p i (t 2 ).
Proof Sketch
T1
1
1
1
1
T2
1
1
1
1
g (t , x(t )) | x (t ) |
2
T1
e
e
1+e
1+e
T2
1
1
1
1
1 2
g (t , opt (t ))  . | x (t ) |  k .e
2
Theorem :
There does not exist a truthful mechanism that implements a
c - approximat ion mechanism for any c  2.
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
Parameters : A real number   1 and s  {1,2}k (selected
uniformly at random)
Input : The revelation s t1 and t 2
Algorithm :
for j  1 to k :
i  s j , i'  3 - i
if t ij   .t i'j
x i  x i {j} , p i    .t i'j
else
x i'  x i' {j} , p i'  
1

.t ij
Theorem :
The randomly - biased min - work mechanism, is a strongly
7
truthful - approximat ion of the make - span problem for
4
4
two agents with  
3
We will first show that the mechanism is truthful.
Weighted VGC Mechanisms
n
maximize  g (t , x(t ))    i .v i (t i , x i (t i ))
i 1
1
p (t , x (t ))  v (t , x (t ))    j .v i (t j , x j (t j ))  hi (t i )
 i j i
i
i
i
i
i
i
i
i
The mechanism is truthful.
Proof Sketch
T1
1
1
1
b
T2
(b+e)
(b+e)
b
1
Opt
1
2
1
2
Rbmw
1
1
rnd
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
Declared times : t ij
Execution times : t'ij
Compensati on to i  {
i
t'
 j
if
jx i ( t )
jx i ( t )
{0
i
t
 j
i
t'
 j
jx i ( t )
otherwise
Bonus to player i  - g(x(t), (t -i , t' ))
Theorem :
The compensati on - bonus mechanism above is a strongly t ruthful
implementa tion of the make - span problem.(? ?)
Generalized Compensation and
Bonus Mechanisms
Declared times : t ij
Execution times : t'ij
ci ( t, t' ) 
i
t'
 j (compensat ion)
jx i ( t )
b i (t, t' )  m i (t -i ,-g(x(t), (t -i , t' )))
Theorem :
The compensati on - bonus mechanism above is a strongly t ruthful
implementa tion of the make - span problem.
---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)
Theorem :
Let x() be an  - approximat ion for make - span. Let m  (x, p) be
the Compensati on and Bonus mechanism based on x(). Then m
is not truthf ul
Proof (by contradict ion) :
Let m be truthful.
Let opt(t) denote the optimal allocation .
Since x(t) is an approximat ion, there exists t such that :
g(opt(t), t)  g(x(t), t)
g(x(t), t)   * g(opt(t), t)
Let t' '  {t
1
j
1
j
{
if j  opt (t )
1
otherwise
Then g(x(t' ' ), t' ' )  g(x(t), t) (since the mechanism is truthful)
Let s be a revelation such that :
s ij  {t ij if j  opt i (t )
{ otherwise
Then g(x(s), s)  g(x(t), t) (?? I dont think this is correct)
But now g(x(s), s)  g(opt(s), s). x(s)  opt(s). The difference is
. Contradict s the approximat ion - ratio.
Bounded Scheduling Problems
The no. of agents n is fixed and there exist b  a  0 such that
for all i, j a  t ij  b.
t ij are rounded up to integer multiples of   f(a,  ).
Horowitz and Sahni(1976 ) show that its a (1   ) - approximat ion
solvable in polynomial time using dynamic programmin g.
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