Doris, Laura, and Renee played two card games in which the objectives were : (a) to form pairs by drawing cards, and (b) to escape being left with a singleton.

The players took turns drawing a card from another player’s hand; drawing continued until one player was left with the singleton. When a pair was formed after a drawing, the pair was discarded

After drawing a card from a second player's hand and discarding a pair, some player would be left with no cards at some point; in that event the third player would draw from the second player's hand when it was her turn to draw.

Near the end of each game:

1. Doris held just one card, Laura held just two cards and, Renee held just two cards; the combined hands contained two pairs and a singleton, but no one hand contained a pair.
2. Doris drew a card from Laura's hand and could not form a pair.
3. Laura was to draw a card from Renee's hand, after which Renee was to draw a card from Doris' hand.
4. No player was to hold the same hand twice in any one game.
5. The player who was to be left with the singleton was to be left with it for the first time.

Who escaped being left with the singleton in both games?