We got Bill from Washago, Ontario on the phone to talk about his desire to stop the water from going through his deck onto his Lawnboy. He was thinking about putting some kind of water proof membrane over the joists under the deck boards.
The problem with this is that there is not enough of a slope to really run the water off, and you would collect all kinds of debris that would eventually cause rotting problems.
My favorite solution to this problem is to hang one or more of those corrugated fiberglass panels under the joists and support it on a support it so it slopes down away from the house. You can even put a rain gutter on the end if you wish to take all the water to one place. If you find this unsightly, or you are afraid of bumping into it, simply cover the end with a drop down skirt attached to the edge of the deck to the depth of the "rain catcher". Not only will it be dry below, with this you can flush the cracks between the deck boards clean with a garden hose, and everything will flow forward and out.