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 Tel- Aviv –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 c 1 $ for saying rain, and c 2 $ 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. A1 Mech. A3 A2 Outcome+ Payments a1 Mech. a3 a2 Outcome+ Payments A1A3 A2

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: E o~p [S(p,o)] > E o~p [S(q,o)]

Scoring Rules E o~p [S(p,o)] > E o~p [S(q,o)] There are lots of functions S(.,.) that fulfill this condition. Example: Logarithmic payments: S(p,i) = log(p i ) In which case:

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 X i that it pays c i 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 u i ω,x’ The variables X 1,X 2,…,Ω are presumably not independent We assume that they are governed by probability distribution p x1,x2,…,ω Now we know how to combine information from several sources. Pr(ω|X 1, X 2 …)

The Model X1X1 Seller 1 Buyer Ω X2X2 Seller 2 c1c1 c2c2

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.

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: 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: We can scale and shift it To satisfy the other constraints.

The Truthfulness Constraints Let’s define: Now we get: And also: We need every pair of vectors p x,p x’ to be linearly separated by v x,x’

A Geometric Interpretation of Truthfulness V x,x’ p x’ pxpx Notice that there can be many ways to select the separating plane ω1ω1 ω2ω2

Existence of a Mechanism A mechanism exists if and only if all vectors p x 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: –Setting does the trick.

Robust Mechanisms We return to the case where P is not known exactly. We assume ε is small (according to L ∞ ). certain solutions may be better than others p x’ pxpx ω1ω1 ω2ω2

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.

Robustness of a Payment Scheme The robustness of one of the truthfulness constraints can be found by solving: 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:

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: p x’ pxpx ω2ω2 ε x’ ε xε x

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 ) Pr(ω=1|x 1,x 2 )Pr(x 1,x 2 )x2x2 x1x1 01/ δ1/ δ 11 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 ) Pr(ω=1|x 1,x 2 )Pr(x 1,x 2 )x2x2 x1x1 01/ δ1/ δ 11 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.