Define a two-player, turn-based climbing game as follows.
Each turn, players have the option to climb or tie a knot at his current position. If the player chooses to climb, there is a 50% chance that he advances his position by $1$, and a 50% that he falls down to the position of his highest knot. If the player chooses to tie a knot, he does so and remains at that position. The winner is the first player to climb to position $n$.
This is a game of complete information, players can see each other's positions and the positions of their knots.
Both players start at position $0$, with knots at position $0$.
Question. What is an optimal strategy for the rope climbing game?
The strategy should be a decision function based on five variables:
$x_1$ and $x_2$, the positions of the respective players
$k_1$ and $k_2$, the positions of the respective players' highest knots
the length of the rope, $n$
We can assume wlog that we're determining the strategy for player 1.
Obviously whenever $x_1=k_1$ the decision should be to climb. For this reason, the strategy is always to climb when $n=1$. ($n=2$ and $n=3$ may be enlightening special cases.)
As $x_1-k_1$ gets larger, climbing gets more risky. However, the smaller $n-x_2$ gets, the more worth it it is to take that risk.