The Complexity of Tullock Contests
Yu He, Fan Yao, Yang Yu, Xiaoyun Qiu, Minming Li, Haifeng Xu
TL;DR
This work analyzes the algorithmic complexity of computing pure Nash equilibria in heterogeneous Tullock contests, revealing a sharp elasticity-driven boundary: when all players have small ($r_i\le1$) or large ($r_i>2$) elasticity, PNEs are computable in polynomial time; but if a polynomial number of players have medium elasticity ($1<r_i\le2$), determining PNE existence becomes NP-complete. To address the intractable case, the authors design a Fully Polynomial-Time Approximation Scheme (FPTAS) that computes an $\epsilon$-PNE (and hence an $(L\epsilon)$-Nash equilibrium) when a PNE exists, with a provable runtime of $O(\frac{n^6}{\epsilon^2}\log^2 n)$. They implement these methods in a Python module and validate performance across settings, offering practical tools for equilibrium analysis in large-scale, heterogeneous contests such as blockchain mining. The results provide a principled bridge between equilibrium computation in asymmetric contests and computational complexity, highlighting how production elasticity shapes tractability and suggesting directions for efficient approximation in complex multi-agent settings.
Abstract
This paper studies the algorithmic complexity for computing the Nash equilibrium in the extensively studied Tullock Contest game. The (possibly heterogeneous) elasticity parameter $r_i$ determines whether a contestant $i$'s cost function is convex or concave. Our core conceptual insight is that the domains of $r_i$ governs the complexity for solving Tullock contents. This is illustrated by our following complete set of algorithmic results. - When at most $O(\log n)$ number of contestants have $r_i \in (1,2]$, we show polynomial-time algorithms to determine the existence of a pure Nash equilibrium (PNE), and compute a PNE whenever it exists. Notably, this result is proved under three different regimes, each via quite different techniques: (1) all $r_i \leq 1$, (2) all $r_i > 2$, and (3) a mixed situation with at most $O(\log n)$ number of $r_i \in (1, 2]$. - When polynomially many contestants have $r_i \in (1,2]$, we prove that determining the existence of a PNE is NP-complete. In response to this hardness result, we design a Fully Polynomial-Time Approximation Scheme (FPTAS) that finds an $ε$-PNE when an exact PNE exists.