I'm not sure I follow your conclusion. Assuming random walk increments are 1 (or less), it would indeed take some time to reach 63 (although you will reach it w

I'm not sure I follow your conclusion. Assuming random walk increments are 1 (or less), it would indeed take some time to reach 63 (although you will reach it with probability 1), but in case the increments are greater (e.g. a 30-50$ dinner), you'll quickly go over $63. The point of "keeping score" is that you can bias the debt process and make it mean-reverting. You're always guaranteed to split costs less than $63 evenly, and can possibly even split a cost of up to $126 in a fair way.