Robust Mechanisms for Information Elicitation

1. Robust Mechanisms for Information Elicitation Aviv Zohar & Jeffrey S. Rosenschein The Hebrew University

2. Overview of the talk • Introduction – paying for information. • Mechanisms for information elicitation. • Robust mechanisms. • Multi-agent extensions. • Conclusions.

3. 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.

4. 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…

5. 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.

6. 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.

7. 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.

8. 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!

9. c1 c2 The Model X1 Seller 1 Buyer Ω X2 Seller 2 Variables are governed by a probability distribution px1,x2,…,ω

10. The Requirements from a Proper Mechanism (Single Agent) • Truth-telling: The truth is more profitable than any lie. • Investment: Knowing is better than guessing. • Individual Rationality: There is a positive expected gain from participating.

11. 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:

12. 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) ω2 ω1

13. 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!

14. 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 distribution for which • How do we find the robustness level of a payment scheme? • Find the minimal ε for which a constraint is violated.

15. Robustness of a Payment Scheme • The robustness of one of the truth-telling constraints can be found by solving: • Repeat for every constraint, take the smallest ε. constants variables

16. 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].

17. The full stochastic program: Target function Truth-telling Investment Constraints Individual Rationality variables parameters Possible range of parameters

18. 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.

19. 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.

20. Mechanisms for Multiple Sellers For agents that don’t collude, two main options: • Mechanisms that work in only in equilibrium. • Truth telling is profitable only when everyone else does it. • Other equilibriums may exist. • 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.

21. 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…

22. 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’’

23. 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.

24. 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.