Friday, March 21, 2014

Lecture 8 -- Implementing Correlated Equilibria Ex-Post in Incomplete Information Games

After a spring break hiatus, our class on privacy and mechanism design returns (with all of our students working on their course projects!)

In our third and final lecture on using mediators to implement equilibrium of complete information games in settings of incomplete information, we ask how far we can push the agenda of obtaining ex-post implementations via the technique of differentially private equilibrium computation. Recall that last lecture we saw how to do this in large congestion games. Can we get similar results in arbitrary large games?

One obstacle is that we do not know good algorithms for computing Nash equilibria in arbitrary games at all, let alone privately. However, we do know how to compute arbitrarily good approximations to correlated equilibria in arbitrary n player games! In this lecture, we explore the game theoretic implications of private computation of correlated equilibria, and then show how to do it.

The punchline is you can still implement (correlated) equilibria of the complete information game as an ex-post Nash equilibrium of the incomplete information game, but you need a slightly stronger mediator (which has the power to verify player types if they decide to use the mediator).

To accomplish this goal, we introduce a couple of interesting tools: A powerful composition theorem in differential privacy due to Dwork, Rothblum, and Vadhan, and the multiplicative weights (or weighted majority, or polynomial weights, or hedge, or...) algorithm, which is a natural learning dynamic and can be used to quickly compute (coarse) correlated equilibria in arbitrary games.

Next week we will have a guest lecture, combined with our theory seminar, by David Xiao, who will tell us about his exciting recent work on Redrawing the Boundaries on Purchasing Data from Privacy Sensitive Individuals. 

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