  Let $K$ be a commutative hypergroup and $\alpha\in \hat{K}$. We show that $K$
is $\alpha$-amenable with the unique $\alpha$-mean $m_\alpha$ if and only if
$m_\alpha\in L^1(K)\cap L^2(K)$ and $\alpha$ is isolated in $\hat{K}$. In
contrast to the case of amenable noncompact locally compact groups, examples of
polynomial hypergroups with unique $\alpha$-means ($\alpha\not=1$) are given.
Further examples emphasize that the $\alpha$-amenability of hypergroups depends
heavily on the asymptotic behavior of Haar measures and characters.
