Attainable best guarantee for the accuracy of $k$-medians clustering in $[0,1]$

The scalar $k$-medians clustering problem is considered in the context of a two-player zero-sum game. The set of strategies of the first player coincides with a family of fixed-length samples from the interval $[0,1]$. The strategies of the second player are all possible partitions of an arbitrary sample of a given length into a given number of clusters. The quality of the clustering is evaluated by the payoff function equal to the sum of deviations of the elements from the centers of clusters nearest to them. It is easy to see that the game has no value except for rare cases. For arbitrary positive integers $n$ and $k$, we establish an upper bound $0.5n/(2k-1)$ for the lower value of the game and prove its attainability for $k>1$ and sufficiently large $n=n(k)$. Thus, we show that a clustering of an arbitrary sample of length $n$ can be constructed by the $k$ medians method so that the payoff does not exceed the obtained bound, and the bound is attainable for an arbitrary number of clusters and for sufficiently long samples. These results are applicable in combinatorial optimization in the proof of polynomial solvability of subclasses of intractable extremal problems.