We investigate a two-stage competitive model involving multiple contests. In this model, each contest designer chooses two participants from a pool of candidate contestants and determines the biases. Contestants strategically distribute their efforts across various contests within their budget. We first show the existence of a pure strategy Nash equilibrium (PNE) for the contestants, and propose a polynomial-time algorithm to compute an $\epsilon$-approximate PNE. In the scenario where designers simultaneously decide the participants and biases, the subgame perfect equilibrium (SPE) may not exist. Nonetheless, when designers' decisions are made in two substages, the existence of SPE is established. In the scenario where designers can hold multiple contests, we show that the SPE always exists under mild conditions and can be computed efficiently.

翻译：我们研究了一个包含多场竞赛的两阶段竞争模型。在该模型中，每位竞赛设计者从候选参赛者池中选择两名参与者，并确定竞赛偏向。参赛者在预算约束下策略性地将自身努力分配至不同竞赛中。我们首先证明了参赛者纯策略纳什均衡的存在性，并提出了一个多项式时间算法来计算ε-近似纯策略纳什均衡。当设计者同时决定参赛者与偏向时，子博弈完美均衡可能不存在。然而，若设计者的决策分两个子阶段进行，则子博弈完美均衡的存在性得以确立。针对设计者可举办多场竞赛的情形，我们证明在温和条件下子博弈完美均衡始终存在，且可高效求解。