Robust Mechanisms for
Information Elicitation
Aviv Zohar & Jeffrey S. Rosenschein
Overview of the talk
•
•
•
•
•
•
Motivation – how to pay for information
Scoring Rules
Mechanisms for information elicitation
Robust mechanisms for a single agent
Multi-agent extensions
Conclusion
Motivation - How to Pay for
Information
• Alice wishes to know the weather in TelAviv
– This cannot be predicted in advance by
anyone!
• She’s only interested in the chance of rain.
• There are two options:
Paying for Information
• Bob lives in Tel-Aviv.
• He can go outside and check the weather.
• Getting the information costs him some
effort. A cost of c.
• He wants Alice to pay him for his efforts.
Paying for Information
• If Alice pays him c$ no matter what he says he
can just make something up.
• If Alice pays him c1$ for saying rain, and c2$ for
saying clear weather, he will pick the larger
payment every time.
• Conclusion: Alice has to have some way to verify
the information.
• Example: The weather in Jerusalem is
correlated with the weather in Tel-Aviv
• In the real world we often buy unverified
information. We are usually playing a repeated
game.
How to Pay for Information
• Bob knows that Tel-Aviv is usually sunny.
• His beliefs affect the cost-benefit analysis.
• Alice needs to take Bob’s beliefs into
consideration when deciding on payments.
• Does she know what Bob believes?
Usually only approximately!
• Can she find a payment scheme to Bob
that will be robust against small changes
in belief?
Information Elicitation vs.
Preference Elicitation.
• Information Elicitation:
–
–
–
–
Knowledge is changing hands.
The seller only cares about payment.
Not interested in how the knowledge is to be used.
The buyer wants the truth!
• Preference Elicitation:
– Information is just the means to an end: Achieving
some optimal outcome.
– The outcome is the bottom line.
– The mechanism has more freedom of action – can
control outcome as well as payments.
(EXAMPLE: Auctions)
The Direct Revelation Principle
• If any mechanism exists for the problem then
there is a mechanism in which the participants
reveal everything.
Outcome+
Payments
Outcome+
Payments
A1
Mech.
A1
a1
Mech.
A3
a2
A2
A2
a3
A3
Direct Revelation
• Direct revelation can allow us to get over
the problem of learning bob’s beliefs.
• We can ask Bob to reveal everything:
– The information to be sold
– His beliefs about probabilities.
• But…
– We don’t want to reveal everything.
– Information is what Bob sells!
– No trusted third party.
Proper Scoring Rules
• A way to evaluate a probabilistic
prediction.
• For a prediction p, and a final outcome o
we shall pay: S(p,o).
• A proper scoring rule is one in which telling
a more accurate prediction gives a higer
payment:
• Eo~p[S(p,o)] > Eo~p[S(q,o)]
Scoring Rules
• Eo~p[S(p,o)] > Eo~p[S(q,o)]
• There are lots of functions S(.,.) that fulfill
this condition.
• Example:
Logarithmic payments: S(p,i) = log(pi)
In which case:
Ei~ p [S(p, i)]   pi  log( pi )
 pi 
E i ~ p [S(p, i)] - E i ~ p [S(q, i)]   pi  log    0
 qi 
Scoring Rules
• Predictions are given in the form of
probability distributions.
• How do we combine the predictions of two
different experts that have access to
different sources of information?
• We need a model of how their information
interacts.
Our Model
• Seller i owns a random variable Xi that it pays
ci to discover.
• Buyer owns a random variable Ω
• After it learns about values x’ from the sellers
and a value ω, it pays the sellers uiω,x’
• The variables X1,X2,…,Ω are presumably not
independent
• We assume that they are governed by
probability distribution px1,x2,…,ω
• Now we know how to combine information
from several sources. Pr(ω|X1,X2…)
The Model
1
X1
c1
x1
X2
u , x '1 , x '2
Seller 1
u2 , x '1 , x '2
c2
x2

x'1
Seller 2
x' 2
Buyer
Ω
The Requirements from a Proper
Mechanism (Single Agent)
1. Truth-telling: The truth is more profitable than
any lie.
x  x'
p


,x
 u , x   p , x  u , x '

2. Investment: Knowing is better than guessing.
x'
p


,x
 u , x  c   p , x  u , x '
,x
,x
3. Individual Rationality: There is a positive
expected gain from participating.
p


,x
,x
 u , x  c
Finding a Mechanism
• We assume P is known.
• The constraints are all linear in the
payments u.
• We can find a payment scheme using
some LP solver.
• We can optimize the cost too:
min
p


,x
 u , x
,x
• When can we find a good mechanism?
• What is the optimal cost?
The Truth is Enough
• Suppose we have some set of payments
that satisfies the truthfulness constraints:
x  x'
p


,x
 (u , x  u , x ' )  0
• We can scale and shift it
u , x    u , x  
To satisfy the other constraints.
x'
p


,x
 (u , x  u , x ' )  c
,x
 u , x  c
,x
p


,x
The Truthfulness Constraints
• Let’s define:

u x  (u1, x ,..., un , x )

p x  ( p1, x ,..., pn , x )

 
vx, x '  u x  u x '
• Now we get:
x  x'
• And also:
p


,x
 (u , x  u , x ' )  0 
 
px  vx , x '  0
 
px '  vx, x'  0
• We need every pair of vectors px,px’ to be
linearly separated by vx,x’
A Geometric Interpretation of
Truthfulness
ω2
px
 
px  vx, x '  0
Vx,x’
 
px '  vx, x'  0
• Notice that there can be many ways
to select the separating plane
px’
ω1
Existence of a Mechanism
• A mechanism exists if and only if all
vectors px are pair-wise independent.
– One direction is easy: we can’t separate
vectors that are linearly dependent.
– For the other direction: show a working
mechanism: 

px
– Setting u x   does the trick.
px
  
 
p x  (u x  u x ' )  p x  



px
px'
  
px
px'
 
  p x  p x 



px'
 0
px'
Robust Mechanisms
• We return to the case where
P is not

known exactly. px  pˆ x   x
• We assume ε is small (according to L∞).
• certain solutions may be better than others
ω2
px
px’
ω1
Robustness of a Specific Payment
Scheme
• A conservative definition:
A payment scheme u will be considered
ε-robust with regard to distribution
if
it
is
p̂


proper for every distribution p  pˆ  

for which    
• How do we find the robustness level of a
payment scheme?
– Find the minimal ε for which a constraint is
violated.
Robustness of a Payment Scheme
• The robustness of one of the truthfulness
constraints can be found by solving:
min 
 ( pˆ  , x    , x )  (u , x u , x' )  0




,x
0
,x
pˆ  , x    , x  0
   ,x  
• After solving a similar program for every
constraint, take the smallest ε found.
Finding a Robust solution
• Given an ε, all ε-robust solutions form a convex set.
• This is a stochastic programming problem.
– Find a solution to a mathematical program with
uncertainty regarding the constraints.
• The ellipsoid algorithm needs only a separation
oracle in order to optimize over the set of solutions.
• A separation oracle provides a linear separator
between any non-solution and the set of solutions.
• We’ve just seen how to find one! Find a constraint
that the payments + a perturbation violate.
The full stochastic program:
min
p


,x
 u , x
,x
p


x  x'
x'
p


,x
,x
 u , x   p , x  u , x '
 u , x  c   p , x  u , x '
,x
p


,x
 u , x  c



,x
0
,x
,x
p , x  pˆ  , x    , x  0
   ,x  

,x
Robust Mechanisms
• Definition: The robustness level of a problem
p is the largest ε that can be set as the
robustness of a solution.
• How can we find it?
• Use binary search.
– The robustness level is somewhere between 0
and 1.
– Test at any wanted ε in between by trying to
actually find an ε-robust solution.
– Then, update the boundaries according to the
answer.
A Bound for Problem Robustness
• Problem robustness is only is only
bounded by the truthfulness constraints.
– Again, shifting and rescaling takes care of the
other constraints.
• A simple bound can be derived:


  min pˆ x  pˆ x   x, x '   x, x '
tr
*
x, x '

 x, x'
pˆ x  pˆ x '

pˆ x  pˆ x ' 2
ω2

px
εx
ε x’
px’
Mechanisms for Multiple Sellers
• Collusion between agents is a critical
matter.
• If they can move payments and share
information, we can treat them as one
agent with multiple sources of information.
• An exponential number of constraints is
needed.
• Tension within the group may limit their
collusion.
• From here on we assume no collusion is
possible.
Mechanisms for Multiple Sellers
Two main options:
1. Mechanisms that work in only in equilibrium.
– Truth telling is profitable only when everyone
else does it.
– Payments are conditioned on all the information
– Other equilibriums may exist.
2. Dominant strategy mechanisms.
– It is always better to tell the truth.
– Payments are conditioned on the agent’s own
information only (And the verifier).
A Simple Example (2x2x2 )
x1
x2 Pr(x1,x2) Pr(ω=1|x1,x2)
0
0
1
1
0
1
0
1
1/4
1/4
1/4
1/4
0
1- δ
1
δ
Pr(ω|x2=1)=Pr(ω|x2=0)
• Hence no dominant strategy mechanism for
player 2
• But a mechanism in equilibrium exists.
A Simple Example (2x2x2 )
x1
x2 Pr(x1,x2) Pr(ω=1|x1,x2)
0
0
1
1
0
1
0
1
1/4
1/4
1/4
1/4
0
1- δ
1
δ
• The variable Ω is slightly biased to agree with
player 1’s variable.
• We can have a dominant strategy mechanism
for player 1.
Robust Mechanisms for Many Sellers
• Dominant strategy mechanism – Just like the
single agent case. May not always exist.
• Mechanisms that work in equilibrium- problematic.
• An equilibrium is a best response to a best
response.
• A player must believe that its counterpart will play
the equilibrium strategy.
• This only happens if it believes that the other
believes that it will play the equilibrium.
• And so on…
Belief Hierarchies
• Assume player A believes the probability is
p
• player B might conceivably believe it’s p’
• Furthermore it may believe that A believes
it is p’’.
• p’’ may be far from p, and we get further
away with every step.
P’
P
P’’
What can we do?
• We can consider bounded players. Only look
some distance into the belief hierarchy.
• We can create finite belief hierarchies.
– The first player has a dominant strategy.
– The payment to second player depends only on the
first.
– Payment to the third only on the previous two
– Etc.
• Every player considers just beliefs of players
that precede him.
• They do not care about his beliefs. No loops.
Finite belief hierarchies
• Only a single equilibrium.
• Very reasonable that it will be played.
• Such mechanisms might not always exist
• The extreme case:
– All agents have a dominant strategy
mechanism.
Conclusion
• Designing information elicitation mechanisms:
– Easy for one agent.
– Can be extended easily to robust mechanism
– Complicated for many agent.
– Robust extension is unclear in equilibrium.
– Collusion makes the design even harder.