Two at a Time Against the Dealer

A boardwalk hustler shuffles a full 52-card deck and peels off the top two cards over and over, 26 pairs in all. Every red–red pair goes into your pile, every black–black pair into hers, and every mixed pair goes into the trash. Whoever ends the deck with more pairs collects $\$10$ from the other; equal piles push. How much should you be willing to pay for the right to play?