IBM Puzzle [January 09] - Ponder This! (Very Tough)

Source: IBM Research | Ponder This | January 2009 challenges

Consider the following game. You have an opportunity to buy lottery tickets. Each ticket has a value t randomly and independently picked from the continuous and uniform distribution on the interval(0,1). Each ticket costs c (0<c<1). If you don't like the ticket you get you can throw it away and buy another one (at a cost of c). You can do this as often as you want. You can stop at any time and cash in the current ticket at which point the game ends. Your winnings are the value of the ticket you cashed less the cost of all the tickets you bought. You don't have to buy any tickets in which case your winnings are 0. If you adopt the best strategy what are your expected winnings as a function of c?

Suppose we modify the game so that you don't have to throw away any tickets. When you decide to stop buying tickets you can cash in any (but only one) of the tickets you have purchased and the game ends. Now what are your expected winnings (using the best strategy) as a function of c?

Note:
Some of the solvers are overlooking a minor point. For credit please ensure your answer is correct for the entire range of c.