You'll win ¼ of the time so you should pay ¼ of a dollar 
or 25 cents to break even in the long run.  But here is 
the work:


Let y be the amount to pay

    Coin     Winnings     Probability    Expectation
   tosses    or losses     of this        per game  
                X            P(X)         X·P(X)
   ---------------------------------------------
     HH       $1-y            ¼         (1-y)(¼)   
     HT         -y            ¼            -y(¼) 
     TH         -y            ¼            -y(¼)
     TT         -y            ¼            -y(¼)

         Sum of expectations = 0
(1-y)(¼) - y(¼) - y(¼) -y(¼) = 0

Multiply through by 4

  (1-y) - y - y - y = 0
  1 - y - y - y - y = 0
             1 - 4y = 0
                -4y = -1
                  y = ¼ = .25 = 25 cents

 Edwin