Each of n players simultaneously choose a positive integer, and one of the players who chose [the least number of [the numbers chosen the fewest times of [the numbers chosen at least once]]] is selected at random and that player wins.


For n=3, the symmetric Nash equilibrium is the player chooses m with probability 1/(2^m).


What is the symmetric Nash equilibrium for n=4? Is it known for general n?



Lowest Unique Bid
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](http://arxiv.org/pdf/1001.1065v1)


Rapoport et al., [Unique bid auctions: Equilibrium solutions and experimental evidence](http://mpra.ub.uni-muenchen.de/4185/)


Ostling et al., [Strategic thinking and learning in the field and lab: Evidence from Poisson LUPI lottery games](http://en.scientificcommons.org/23204285)


Houba et al., [The Unique-lowest Sealed-bid Auction](http://dare.ubvu.vu.nl//handle/1871/12947)


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


