1 of 6

# Halve or diminish, and race to unity! v2

This is follow-up question to Halve or diminish, and race to unity!.

Alice and Bob are playing a game. In the beginning, they randomly choose an positive integer between $$2$$ to $$n$$ where $$n$$ is a finite number. In a move, a player, can:

• either decrease the number on the board by $$1$$ (i.e., replace $$n$$ by $$n-1$$), or
• halve the number, and round it up to the next integer if a fraction is obtained (i.e., replace $$n$$ by $$\left\lceil \frac{n}{2}\right\rceil$$).

As always, Alice moves first, and then they make moves in turns. The first person to get to $$1$$ wins.

Assuming both players play perfectly, who has the better chance to win and what is that chance?