The Complexity of Tullock Contests

Despite the extensive literature on Tullock contests, computational results for the general model with heterogeneous contestants remain scarce. This paper studies the algorithmic complexity of computing a pure Nash equilibrium (PNE) in such general Tullock contests. We find that the elasticity parameters {r_i}, which govern the returns to scale of contestants’ production functions, play a decisive role in the problem’s complexity.

Our core conceptual insight is that the computational hardness is determined specifically by the number of contestants with medium elasticity (r_i in (1, 2]). This is illustrated by a complete set of algorithmic results under two parameter regimes. In the efficient regime, when the number of contestants with medium elasticity is logarithmically bounded by the total number of contestants (O(log n)), we provide an algorithm that determines the existence of a PNE and computes an epsilon-PNE in polynomial time in both n and log(1/epsilon) whenever it exists. This result generalizes classical findings for concave (r_i <= 1) and convex (r_i > 2) cases, establishing computational tractability for a broader class of mixed-elasticity contests.

In contrast, we show that when the number of medium-elasticity contestants exceeds Omega(log n), determining the existence of PNEs is NP-complete and it is impossible for any algorithm to compute an epsilon-PNE within running time Poly(n, log(1/epsilon)). We then design a Fully Polynomial-Time Approximation Scheme (FPTAS) that computes an epsilon-PNE in Poly(n, 1/epsilon), guaranteeing efficient approximations for hard instances.