Lowest Unique Bid

There are non symmetric Nash equilibria. For example with 3 players: (1,1,2) is a silly solution.

There is some published literature on this problem. See for example the following papers and the references therein.

Baek and Bernhardsson, Equilibrium solution to the lowest unique positive integer game

Rapoport et al., Unique bid auctions: Equilibrium solutions and experimental evidence

Ostling et al., Strategic thinking and learning in the field and lab: Evidence from Poisson LUPI lottery games

Houba et al., The Unique-lowest Sealed-bid Auction

Apparently, in general, the Nash equilibria are intractable to describe.