  In cooperative game theory, the primary focus is the equitable allocation of
payoffs or costs among agents. However, in the practical applications of
cooperative games, accurately representing games is challenging. In such cases,
using an allocation method sensitive to small perturbations in the game can
lead to various problems, including dissatisfaction among agents and the
potential for manipulation by agents seeking to maximize their own benefits.
Therefore, the allocation method must be robust against game perturbations.
  In this study, we explore optimization games, in which the value of the
characteristic function is provided as the optimal value of an optimization
problem. To assess the robustness of the allocation methods, we use the
Lipschitz constant, which quantifies the extent of change in the allocation
vector in response to a unit perturbation in the weight vector of the
underlying problem. Thereafter, we provide an algorithm for the matching game
that returns an allocation belonging to the
$\left(\frac{1}{2}-\epsilon\right)$-approximate core with Lipschitz constant
$O(\epsilon^{-1})$. Additionally, we provide an algorithm for a minimum
spanning tree game that returns an allocation belonging to the $4$-approximate
core with a constant Lipschitz constant.
  The Shapley value is a popular allocation that satisfies several desirable
properties. Therefore, we investigate the robustness of the Shapley value. We
demonstrate that the Lipschitz constant of the Shapley value for the minimum
spanning tree is constant, whereas that for the matching game is $\Omega(\log
n)$, where $n$ denotes the number of vertices.
