Transcript iaai06.ppt

Robust Mechanisms for
Information Elicitation
Aviv Zohar & Jeffrey S. Rosenschein
The Hebrew University
Overview of the talk
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Introduction – paying for information.
Mechanisms for information elicitation.
Robust mechanisms.
Multi-agent extensions.
Conclusions.
Purchasing Information From
Strangers
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Information is one of the foundations of
intelligent behavior.
It is often crucial to obtain reliable
information in order to make the right
choices.
We usually purchase information in a
repeated interaction (Buy the same paper
every day).
The reputation of an information source
matters a great deal.
Purchasing Information From
Strangers
But…
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The world is changing.
We are now able to access incredible
amounts of information through the Internet.
(e.g. through web services)
One-shot interaction - no past experience, no
reputation system and no assurance of
reliability.
Can we still purchase reliable information?
Our Approach
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We take a mechanism-design approach:
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Make sure the seller’s best action is to give
correct information.
Create the incentive through payments.
Important assumptions:
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The seller is selfish but not malicious. It is only
interested in its own reward.
The information being sold can be verified
probabilistically.
An Example
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Alice who lives in Jerusalem, wishes to know
the weather in Tel-Aviv.
Bob lives in Tel-Aviv and can go outside to
check the weather.
Getting the information takes some effort.
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A cost of c.
He wants Alice to pay him for his efforts.
Verifying the Information
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Bob is sneaky. He will lie if it helps him.
He may be tempted not to check the
weather to avoid the cost.
Alice needs a way to verify the information
Bob gives her.
She can use the weather in Jerusalem – it is
correlated with the weather in Tel-Aviv.
Still, the weather in Jerusalem may be
different than that in Tel-Aviv.
Conditioned Payments
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Alice can now condition payments to Bob on
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What he tells her about the weather in Tel-Aviv.
The weather in Jerusalem
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Alice publishes the payments in advance.
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Bob knows that Tel-Aviv is usually sunny.
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He can compute the expected payment from saying “sunny”.
His beliefs about probabilities 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!
The Model
1
X1
c1
x1
X2
u , x '1 , x '2
Seller 1
u2 , x '1 , x '2
c2
x2

x'1
Buyer
Seller 2
x' 2
Variables are governed by a probability
distribution px1,x2,…,ω
Ω
The Requirements from a Proper
Mechanism (Single Agent)
1.
Truth-telling: The truth is more profitable than
any lie.
x  x'
2.
p


,x
 u , x   p , x  u , x '

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
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Let’s first assume Pω,x 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:
min
p


,x
,x
 u , x
A little bit about the solutions:
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When can we find a mechanism?
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whenever the verifier can distinguish between
any two events.
Pr( x )
ω2
Pr( x1 )  Pr( x2 )
1
Pr( x2 )
ω1
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What is the optimal cost of a mechanism?
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If any mechanism exists, then there exists a
mechanism with an expected cost of c. (If we
allow negative payments)
Robust Mechanisms
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The problem: We assumed Pω,x is common
knowledge between the seller and buyer.
Adopt a weaker assumption: The buyer has
a probability in mind that is close to that of
the seller.
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px  pˆ x   x
We assume ε is small (according to L∞).
We still want the mechanism to work!
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
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proper for every distribution p  pˆ  
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for which    
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How do we find the robustness level of a
payment scheme?
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Find the minimal ε for which a constraint is
violated.
Robustness of a Payment Scheme
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The robustness of one of the truth-telling
constraints can be found by solving:
min 
 ( pˆ  , x    , x )  (u , x u , x' )  0




,x
0
constants
,x
pˆ  , x    , x  0
variables
   ,x  
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Repeat for every constraint, take the smallest ε.
Finding a Robust solution
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Given an ε, all ε-robust solutions form a
convex set.
Thus, a payment scheme can be found
efficiently.
This is a stochastic programming problem.
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Find a solution to a mathematical program with
uncertainty regarding the constraints.
This particular formulation is due to [Ben-Tal &
Nemirovski].
The full stochastic program:
min
pˆ 


,x
 u , x
Target function
,x
Truth-telling
Investment
Individual
Rationality
p


x  x'
x'
p


,x
,x
 u , x   p , x  u , x '

 u , x  c   p , x  u , x '
,x
,x
p


,x
 u , x  c



,x
0
variables
,x
,x
parameters
p , x  pˆ  , x    , x  0
   ,x  
Constraints
Possible range
of parameters
Robust Mechanisms
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How do we find the most robust solution?
Use binary search.
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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.
Mechanisms for Multiple Sellers
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Collusion between agents is a critical
matter.
If they can share payments and
information, we can treat them as one
agent with multiple sources of information.
An exponential number of constraints is
needed, because the action space of
agents is larger.
Mechanisms for Multiple Sellers
For agents that don’t collude, two main options:
1.
Mechanisms that work in only in equilibrium.
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Truth telling is profitable only when everyone else
does it.
Other equilibriums may exist.
Dominant strategy mechanisms.
2.
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It is always better to tell the truth.
Payments are conditioned on the agent’s own
information only (And the verifier).
Less likely to exist.
Robust Mechanisms for Many
Sellers
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Mechanisms that work in equilibrium are
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
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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?
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We can consider bounded players. Only look some
distance into the belief hierarchy.
We can create finite belief hierarchies via iterated
dominance.
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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.
Conclusions
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Designing information elicitation mechanisms:
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Efficient for one agent.
Can be extended efficiently to robust mechanism.
Complicated for many agent.
Robust extension is unclear in equilibrium.
Collusion makes the design even harder.
Other scenarios we have also looked at:
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Allow the seller access to extra information it does
not sell. Makes the design problem hard.