# Tullock Contests and Multiple Equilibria

I am reading this paper on the existence of multiple equilibria in Tullock contests. The authors state on page 3 that "The additional restriction $$(5\alpha_1-\alpha_2)-(\beta_1-\beta_2)>0$$ guarantees a large enough curvature of the best response function to generate asymmetric equilibria $$\{x_i^*=\bar{x},x^*_j=\underline{x}\}$$ and $$\{x_i^*=\underline{x},x^*_j=\bar{x}\}$$" and goes on to define what $$\bar{x},\underline{x}$$ are on page 4. How were they able to derive the condition and definitions?