Favorite Theorems: Equilibrium

The past decade has seen an explosion of work tying together theoretical computer science and micro-economics. Much has come in the area of market design, where tools and ideas from the algorithms community apply nicely in developing auctions that achieve some close to idea outcome.

In complexity, the most exciting work studied the problem of finding equilibriums of games. How does one find a Nash Equilibrium when each player's payoffs are given by a matrix. In a 2-player zero-sum game (my gain is your loss), one can easily find equilibriums using linear programming. But if we have unrelated payoffs, computing the equilibrium can be much more difficult.

• The Complexity of Computing a Nash Equilibrium by Constantinos Daskalakis, Paul Goldberg and Christos Papadimitriou
• Settling the Complexity of Two-Player Nash Equilibrium by Xi Chen and Xiaotie Deng

Daskalakis et al. showed that computing the Nash Equilibrium between three players in PPAD-complete and Chen and Deng extended their work to 2-player games. PPAD represents a search problem, solvable if P = NP though not believed to be NP-hard. 

You don't have to know the formal definition of PPAD to appreciate this work. PPAD corresponds to a discrete version of the Brouwer fixed-point theorem, the critical tool in proving that Nash Equilibriums exist. One proof of the fixed-point theorem uses Sperner's Lemma, also PPAD-complete. The New York Times has a nice example of using Sperner's Lemma to fairly divide up the rent among three friends who share an apartment with different sized rooms.

Paul Goldberg wrote a nice survey on these and related results, such as finding equilibria in markets.