(Guest post by Vijay Vazirani.)

On

Theory Day on Nov 30, 2012
Vijay Vazirani gave a talk:

*New (Practical) Complementary Pivot Algorithms for Market Equilibrium*.

He was inspired by the reaction to the talk to write a guest blog

which I present here!

**Where is the Youth of TCS?**
by Vijay Vazirani

I have always been impressed by the researchers of our community, especially the young researchers -- highly competent, motivated, creative, open-minded ... and yet cool! So it has been disconcerting to note that over the last couple of years, each time I have met Mihalis Yannakakis, I have lamented over the lack of progress on some fundamental problems, and each time the same thought has crossed my mind, ``Where is the youth of TCS? Will us old folks have to keep doing all the work?''

Is the problem lack of information? I decided to test this hypothesis during my talk at NYTD. To my dismay, I found out that there is a lot of confusion out there! By a show of hands, about 90% of the audience said they believed that Nash Equilibrium is PPAD-complete and 3% believed that it is FIXP-complete! I would be doing a disservice to the community by not setting things right, hence this blog post.

First a quick primer on PPAD and FIXP, and then the questions. Ever since Nimrod Megiddo, 1988, observed that proving Nash Equilibrium NP-complete is tantamount to proving NP = co-NP, we have known that the intractability of equilibrium problems will not be established via the usual complexity classes. Two brilliant pieces of work gave the complexity classes of PPAD (Papadimitriou, 1990) and FIXP (Etessami and Yannakakis, 2007), and they have sufficed so far. A problem in PPAD must have rational solutions and this class fully characterizes the complexity of 2-Nash, which has rational equilibria if both payoff matrices have rational entries. On the other hand, 3-Nash, which may have only irrational equilibria, is PPAD-hard; however, its epsilon-relaxation is PPAD-complete. That leaves the question, ``Exactly how hard is 3-Nash?''

Now it turns out that 3-Nash always has an equilibrium consisting of algebraic numbers. So one may wonder if there is an algebraic extension of PPAD that captures the complexity of 3-Nash, perhaps in the style of Adler and Beling, 1994, who considered an extension of linear programs in which parameters could be set to algebraic numbers rather than simply rationals. The class FIXP accomplishes precisely this: it captures the complexity of finding a fixed point of a function that uses the standard algebraic operations and max. Furthermore, Etessami and Yannakakis prove that 3-Nash is FIXP-complete. The classes PPAD and FIXP appear to be quite disparate: whereas the first is contained in NP INTERSECT co-NP, the second lies somewhere between P and PSPACE (and closer to the harder end of PSPACE, according to Yannakakis).

Now the questions (I am sure there are more):

- Computing an equilibrium for an Arrow-Debreu market under separable, piecewise-linear concave utilities is PPAD-complete (there is always a rational equilibrium). On the other hand, if the utility functions are non-separable, equilibrium consists of algebraic numbers. Is this problem FIXP-complete? What about the special case of Leontief utilities? If the answer to the latter question is ``yes,'' we will have an interesting demarcation with Fisher markets under Lenotief utilities, since they admit a convex program.

- An Arrow-Debreu market with CES utilities has algebraic equilibrium if the exponents in the CES utility functions are rational. Is computing its equilibrium FIXP-complete? Again, its epsilon-relaxation is PPAD-complete.

- A linear Fisher or Arrow-Debreu market with piecewise-linear concave production has rational equilibria if each firm uses only one raw good in its production, and computing it is PPAD-complete. If firms use two or more raw goods, equilibria are algebraic numbers. Is this problem FIXP-complete?