[Paper Review] Complexity Theory, Game Theory, and Economics

Tim Roughgarden|arXiv (Cornell University)|Jan 1, 2018

Economic theories and models3 citations

TL;DR

This paper presents a mini-course linking complexity theory, game theory, and economics, demonstrating how computational complexity reveals barriers in finding equilibria in games and markets, while also inspiring new complexity-theoretic insights. It introduces two lecture series—Solar Lectures on equilibrium computation complexity and Lunar Lectures on economic applications—offering foundational tools without requiring prior game theory knowledge.

ABSTRACT

This document collects the lecture notes from my mini-course Complexity Theory, Game Theory, and Economics, taught at the Bellairs Research Institute of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th McGill Invitational Workshop on Computational Complexity. The goal of this mini-course is twofold: (i) to explain how complexity theory has helped illuminate several barriers in economics and game theory; and (ii) to illustrate how game-theoretic questions have led to new and interesting complexity theory, including recent several breakthroughs. It consists of two five-lecture sequences: the Solar Lectures, focusing on the communication and computational complexity of computing equilibria; and the Lunar Lectures, focusing on applications of complexity theory in game theory and economics. No background in game theory is assumed.

Motivation & Objective

• To clarify fundamental computational barriers in finding equilibria in games and markets using complexity theory.
• To show how game-theoretic problems have driven advances in complexity theory, including recent breakthroughs.
• To provide accessible lecture notes that bridge complexity theory with economic and game-theoretic reasoning.
• To establish a foundation for researchers without prior game theory background to engage with modern complexity-theoretic insights in economics.
• To explore the interplay between communication complexity and equilibrium computation in strategic settings.

Proposed method

• Leveraging the Solar Lectures to analyze communication and computational complexity in equilibrium computation, particularly in games with bounded rationality.
• Using the Lunar Lectures to apply complexity-theoretic tools to economic problems such as market equilibria and mechanism design.
• Introducing foundational complexity classes (e.g., PPAD, FIXP) to formalize the difficulty of computing equilibria.
• Employing reductions and completeness results to demonstrate hardness of equilibrium computation problems.
• Presenting case studies on exchange economies and auctions to illustrate complexity-theoretic insights in economic mechanisms.
• Using informal yet rigorous lecture notes to make advanced concepts accessible to researchers without prior game theory training.

Experimental results

Research questions

• RQ1 What are the fundamental computational barriers to computing Nash equilibria in games, and how do complexity classes like PPAD formalize these barriers?
• RQ2 How does communication complexity constrain the efficiency of algorithms that compute equilibria in strategic environments?
• RQ3 In what ways do game-theoretic questions lead to new developments in complexity theory, such as new completeness results?
• RQ4 How can complexity-theoretic tools be applied to understand the computational feasibility of market equilibria in exchange economies?
• RQ5 What insights does complexity theory provide into the design and analysis of mechanisms in auction theory and mechanism design?

Key findings

• The existence of Nash equilibria in finite games is guaranteed by Brouwer’s fixed-point theorem, but computing such equilibria is PPAD-complete, indicating inherent computational difficulty.
• Communication complexity lower bounds demonstrate that even in simple two-player games, computing equilibria may require exponentially many bits of communication in the worst case.
• The complexity class FIXP captures the difficulty of computing equilibria in exchange economies with continuous preferences, showing that such problems are computationally harder than PPAD.
• Game-theoretic problems such as finding market equilibria in certain economies have led to new completeness results in complexity theory, expanding the scope of FIXP and related classes.
• The interplay between complexity theory and game theory reveals that many economic mechanisms are computationally infeasible to implement exactly, even when they are theoretically optimal.
• The lecture series demonstrates that complexity-theoretic insights are essential for understanding the practical limitations of equilibrium computation in real-world economic and strategic settings.