But here you know the content of the "envelope" before you decide whether to switch or not. If the adversary was only allowed one pick I guess I'd only re-draw

But here you know the content of the "envelope" before you decide whether to switch or not. If the adversary was only allowed one pick I guess I'd only re-draw if I had less than 0.5 since statistically you'd be more likely to have a greater number than a weaker one. I'm not sure what the maths are but a simple implementation gave me an average value of about 0.625 using this strategy, so definitely an improvement over a single pick. But I have the feeling that since the adversary can chose the same strategy maybe you should be a little more aggressive and redraw if you are at 0.5? But then the adversary would do the same so I guess the solution would eventually converge to a value a little above 0.6? That'd be my guess at least.