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: -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) (i.e., Poly(n,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. -Hard Regime: In contrast, we show when the number of medium elasticity contestants exceed 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.

翻译：尽管关于塔洛克竞赛的文献十分丰富，但针对具有异质参与者的通用模型的计算结果仍然稀缺。本文研究了在此类通用塔洛克竞赛中计算纯纳什均衡的计算复杂性。我们发现，决定参与者生产函数规模报酬的弹性参数 {r_i} 在问题的复杂性中起着决定性作用。我们的核心概念性见解是，计算难度具体由具有中等弹性（r_i ∈ (1, 2]）的参与者数量决定。这通过两种参数体系下的一套完整算法结果得以阐明：- 高效体系：当中等弹性参与者的数量以参与者总数的对数形式有界（O(log n)）时，我们提供了一种算法，该算法能在多项式时间（即 Poly(n,log(1/epsilon))）内判定纯纳什均衡的存在性，并在其存在时计算一个 epsilon-近似纯纳什均衡。这一结果推广了针对凹性（r_i <= 1）和凸性（r_i > 2）情形的经典发现，为更广泛的混合弹性竞赛类别建立了计算易处理性。- 困难体系：相反，我们证明当中等弹性参与者的数量超过 Omega(log n) 时，判定纯纳什均衡的存在性是 NP 完全问题，并且任何算法都无法在 Poly(n,log(1/epsilon)) 的运行时间内计算出一个 epsilon-近似纯纳什均衡。随后，我们设计了一个完全多项式时间近似方案，该方案能在 Poly(n,1/epsilon) 时间内计算出一个 epsilon-近似纯纳什均衡，从而保证了针对困难实例的高效近似。