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.