Robust Mechanisms for Information Elicitation Aviv Zohar & Jeffrey S. Rosenschein The Hebrew...
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Transcript of Robust Mechanisms for Information Elicitation Aviv Zohar & Jeffrey S. Rosenschein The Hebrew...
Robust Mechanisms for Information Elicitation
Aviv Zohar & Jeffrey S. RosenscheinThe Hebrew University
Overview of the talk
Introduction – paying for information.
Mechanisms for information elicitation.
Robust mechanisms. Multi-agent extensions. Conclusions.
Purchasing Information From Strangers
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
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?
But…
Our Approach
We take a mechanism-design approach: Make sure the seller’s best action is to give
correct information. Create the incentive through payments.
Important assumptions: 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 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. A cost of c.
He wants Alice to pay him for his efforts.
Verifying the Information 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
Alice can now condition payments to Bob on What he tells her about the weather in Tel-Aviv. The weather in Jerusalem
Alice publishes the payments in advance.
Bob knows that Tel-Aviv is usually sunny. 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
X1
Seller 1
Buyer
Ω
X2
Seller 2
1'x
2'x
c1
1x
c2
2x
2',', 21 xxu
1',', 21 xxu
Variables are governed by a probability distribution px1,x2,…,ω
The Requirements from a Proper Mechanism (Single Agent)
1. Truth-telling: The truth is more profitable than any lie.
2. Investment: Knowing is better than guessing.
3. Individual Rationality: There is a positive expected gain from participating.
',,,,' xxxx upupxx
x
xxx
xx upcupx,
',,,
,,'
cupx
xx ,
,,
Finding a Mechanism
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:
x
xx up,
,,min
A little bit about the solutions:
When can we find a mechanism? whenever the verifier can distinguish
between any two events.
What is the optimal cost of a mechanism? If any mechanism exists, then there exists a
mechanism with an expected cost of c. (If we allow negative payments)
)Pr()Pr( 21 xx ω1
ω2)Pr( 1x
)Pr( 2x
Robust Mechanisms
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.
We assume ε is small (according to L∞). We still want the mechanism to work!
xxx pp
ˆ
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 proper for every distributionfor which
How do we find the robustness level of a payment scheme? Find the minimal ε for which a constraint
is violated.
p̂
pp ˆ
Robustness of a Payment Scheme
min
0,
, x
x
x,
0ˆ ,, xxp
0)()ˆ( ',,,, xxxx uup
The robustness of one of the truth-telling constraints can be found by solving:
Repeat for every constraint, take the smallest ε.
variables
constants
Finding a Robust solution
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. 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:
0,
, x
x
x,
0ˆ ,,, xxx pp
x
xx up,
,,ˆmin
',,,,' xxxx upupxx
x
xxx
xx upcupx,
',,,
,,'
cupx
xx ,
,,
Target function
Constraints
Possible range of parameters
parameters
variables
Truth-telling
Investment
Individual Rationality
Robust Mechanisms
How do we find the most robust solution?
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.
Mechanisms for Multiple Sellers
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.
Truth telling is profitable only when everyone else does it.
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). Less likely to exist.
Robust Mechanisms for Many Sellers
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
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 via iterated dominance. 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
Designing information elicitation mechanisms: 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: Allow the seller access to extra information it does
not sell. Makes the design problem hard.