= Quadratic-cost two-player proportional contest
{title2=$b_i=\dfrac{\sqrt{v_i}}{\sqrt{v_1}+\sqrt{v_2}}\,\dfrac{(v_1v_2)^{1/4}}{\sqrt2}$}
With values $v_i>0$ and effort costs $b_i^2$, the unique pure <Nash equilibrium> has the efforts displayed above. Interior <first-order conditions> give $v_i b_j/(b_1+b_2)^2=2b_i$, hence $b_1/b_2=\sqrt{v_1/v_2}$ and $(b_1+b_2)^2=\sqrt{v_1v_2}/2$. Against a positive rival effort, each payoff is a <strictly concave function>, so these conditions identify global <best responses>. On the axes, lowering a positive uncontested effort or adding a sufficiently small effort at the zero tie rules out equilibrium.
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