I currently own an Optoma HD70 and am very happy with it, apart from the fact that it's hardly useable in daylight. Especially for playing games which are a bit dark-ish like Splinter Cell, the display is just not dark enough to enjoy the game. When it's darkening up a little, it's a LOT better.
So now I'm thinking about buying a new projector with a lot more ANSI (mine has 1000 if I'm not mistaken). What would I need to be able to game during the day too?
Very hard question to answer.
Basically, If the daylight/sunshine hits the projection area, your stuffed. The average UK daylight can be over 50,000 Lumens. However, depending on angles and directions of light, you can effectively project onto certain surfaces. As Gary mentioned, there are several Glass Beaded or High Gain screens which can help in these scenario's.
I've had many demo's and tests on many projectors in daylight situations and when outside, it's impossible but, with a little shade, you can get away with using aprox' 4000 Ansi Lumen pj's but you do require special optics and the shade.
I have a 2800 Ansi Lumen pj here which works well in daylight, without the curtains closed but with the compromise of lower quality video processing. You can find pj's like this from £700 to £1000+.
If you look into the 3000 lumen pj's with high quality video processing, they start around £3000+. Some cheaper options can be found but there are no High Def ones at that price.
One option that many are going for at the moment is buying a cheap 1:1 manual screen, used as a roller/blackout blind. Perfect for instant blackout.
Hope this helps.