>That's optimal only if your opponents also act randomly.

By choosing the random strategy you force your opponent to do the same and you have a draw. It is optimal, in this sense.

That's the strategy I use while playing against jsomers, but while playing against chengmi, I always use (seconds+2)%3. (Unfortunately, chengmi's watch is 2 seconds fast (or 1 second slow), so he still ends up winning...)