With a convertible bond, in addition to coupon payments, the bondholder has the right to convert a bond into shares. Meanwhile, if the coupon payments become too expensive the shareholders have the right to end payments by declaring bankruptcy. This makes the convertible bond problem an archetypal non-zero sum Dynkin game. Often in non-zero sum Dynkin games it is assumed that each player would prefer that if the game is stopped, then stopping was done by the opponent. We consider games outside this paradigm, and what it means for the optimal strategies for the players. Joint work with Edward Wang and Gechun Liang