Redraw or keep

Idea. Against an opponent who uses threshold $t$, keeping a number $x$ wins with probability $F(x)$, while redrawing wins with a fixed probability $W$. You keep exactly when $F(x)\ge W$. In equilibrium the threshold where the two are equal must be $t$ itself.

Step 1: the opponent's final number. With probability $t$ they redraw and get a fresh uniform number; otherwise they keep a number that is uniform on $[t,1]$. So for $y\in [0,1]$, $$F(y)=ty+\max (0,y-t).$$

Step 2: redrawing. A fresh uniform number $U$ wins with probability $$W={\int }_0^{1}F(y)dy=\frac{t}{2}+\frac{(1-t{)}^2}{2}.$$

Step 3: indifference at $t$. Keeping exactly $t$ wins with probability $F(t)=t^2$. Setting $t^2=W$: $$2t^2=t+1-2t+t^2,$$ that is, $t^2+t-1=0$.

Step 4. The root in $[0,1]$ is $$t=\frac{\sqrt{5}-1}{2}\approx 0.618,$$ the golden ratio's reciprocal. Since $F$ is increasing, keeping is better exactly above $t$, so the threshold rule is indeed a best reply.

Intuition. Without an opponent, to maximise your expected number, the threshold would be $\frac{1}{2}$. Here only beating the opponent counts, and the opponent's final number is skewed upwards by their own redraws, so a middling number is worth less and you redraw more.