  Cooperative games provide a framework for fair and stable profit allocation
in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are
such solution concepts that characterize stability of cooperation. In this
paper, we study the algorithmic issues on the least-core and nucleolus of
threshold cardinality matching games (TCMG). A TCMG is defined on a graph
$G=(V,E)$ and a threshold $T$, in which the player set is $V$ and the profit of
a coalition $S\subseteq V$ is 1 if the size of a maximum matching in $G[S]$
meets or exceeds $T$, and 0 otherwise. We first show that for a TCMG, the
problems of computing least-core value, finding and verifying least-core payoff
are all polynomial time solvable. We also provide a general characterization of
the least core for a large class of TCMG. Next, based on Gallai-Edmonds
Decomposition in matching theory, we give a concise formulation of the
nucleolus for a typical case of TCMG which the threshold $T$ equals $1$. When
the threshold $T$ is relevant to the input size, we prove that the nucleolus
can be obtained in polynomial time in bipartite graphs and graphs with a
perfect matching.
