> if we follow this strategy we’ll win a little bit less than one-third of the time.
That is not possible. If N players use the same strategy then by symmetry each of them must win on average 1 time in N. So against two other players we should win exactly one third of the time, not "a little bit less" than one third of the time.
Also, if all players use the same strategy then it doesn't matter what the strategy actually is. The only requirement is that it have some probabilistic element, otherwise all players would choose the same number every time.
> if we follow this strategy we’ll win a little bit less than one-third of the time.
That is not possible. If N players use the same strategy then by symmetry each of them must win on average 1 time in N. So against two other players we should win exactly one third of the time, not "a little bit less" than one third of the time.
Also, if all players use the same strategy then it doesn't matter what the strategy actually is. The only requirement is that it have some probabilistic element, otherwise all players would choose the same number every time.