  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.
