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?



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