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 exists under mild conditions and can be computed efficiently.

翻译：我们研究了一个包含多个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赛者，并确定偏向参数。参赛者在预算约束下，策略性地将努力分配到不同竞赛中。我们首先证明了参赛者存在纯策略纳什均衡（PNE），并提出了一种多项式时间算法来计算$\epsilon$-近似PNE。在竞赛设计者同时决定参赛者和偏向的情景中，子博弈完美均衡（SPE）可能不存在。然而，当设计者的决策分为两个子阶段时，我们确立了SPE的存在性。在竞赛设计者可以举办多个竞赛的情景中，我们证明了SPE在温和条件下存在，并且能够被高效计算。