  The $k$-majority game is played with $n$ numbered balls, each coloured with
one of two colours. It is given that there are at least $k$ balls of the
majority colour, where $k$ is a fixed integer greater than $n/2$. On each turn
the player selects two balls to compare, and it is revealed whether they are of
the same colour; the player's aim is to determine a ball of the majority
colour. It has been correctly stated by Aigner that the minimum number of
comparisons necessary to guarantee success is $2(n-k) - B(n-k)$, where $B(m)$
is the weight of the binary expansion of $m$. However his proof contains an
error. We give an alternative proof of this result, which generalizes an
argument of Saks and Werman.
