Imagine two robots on the number line starting at position $0$. If you choose one of the robots it goes forward one step with prob $p$ and backwards one step with prob $1-p$ with $p>1/2$. You want to get either one of the robots to the point 10 with minimum total cost. The cost is the sum of the square of the number of steps each robot has taken.
At each step you choose to move one of the two robots. How do you best play the game to minimise the expected cost?
The obvious strategy of always picking the robot closest to 10 has the the problem that more steps might have been taken by that robot, so increasing the number of steps by one will be more expensive than moving the robot that is further away. That is because the cost is the sum of the square of the number of steps taken by each robot.
For example, say robot 1 has moved 5 steps left and 5 right and robot 2 has moved one step to -1. Moving robot 2 increases the cost by $2^2-1^1=3$ whereas moving robot 1 increases the cost by $11^2-10^2=21$ so it might be better to move the robot that was further from the destination.
UPDATE
Changed walk to be biased to ensure finite expected time to the destination.