We consider the game as described in http://www.datagenetics.com/blog/november12011/ . Each person rolls a dice and the person who gets 6 on the face can start and the other keeps waiting. If the completion of your move results in your token landing at the foot of a ladder, you instantly climb to the top of that ladder. Correspondingly, if your move lands you on the head of a snake, you are forced to slide down the snake to an earlier square. There are no consequences to landing on the top of a ladder or the tail of a snake. The change compared to that in the link above is that if the person reaches the same spot as where the other person is at already, the person who was at that position earlier gets out and has to restart the game (again waiting for a 6 face). Thus, the plays of the two players become dependent.
With this variant, what is the probability for the person starting first to win?
One approach is to write the complete Markov transition probability but it will have joint state of players making it $100^2$ states which is not straightforward analytically (can be potentially done by computer in which $10^4\times 10^4$ matrix can be handled). Is there any other approach?