Vijay Vazirani:
Dichotomies in Equilibrium Computation: Markets Provide a Surprise
Abstract:
Equilibrium computation is among the most significant additions to the theory
of algorithms and computational complexity in the last decade  it has its
own character, quite distinct from the computability of optimization problems.
Our contribution to this evolving theory can be summarized in the following
sentence: Natural equilibrium computation problems tend to exhibit striking
dichotomies. The dichotomy for Nash equilibrium, showing a qualitative difference
between 2Nash and k Nash for k > 2, has been known for some
time. We establish a dichotomy for market equilibrium.
For this purpose. we need to define the notion of Leontieffree functions which
help capture the joint utility of a bundle of goods that are substitutes, e.g., bread
and bagels. We note that when goods are complements, e.g., bread and butter,
the classical Leontief function does a splendid job. Surprisingly enough,
for the former case, utility functions had been defined only for special cases
in economics, e.g., CES utility function.
We were led to our notion from the high vantage point provided by an algorithmic
approach to market equilibria.
Note: Joint work with Jugal Garg and Ruta Mehta.
