  In this paper, the author defines Generalized Unique Game Problem (GUGP),
where weights of the edges are allowed to be negative. Two special types of
GUGP are illuminated, GUGP-NWA, where the weights of all edges are negative,
and GUGP-PWT($\rho$), where the total weight of all edges are positive and the
negative-positive ratio is at most $\rho$. The author investigates the
counterpart of the Unique Game Conjecture on GUGP-PWT($\rho$). The author shows
that Unique Game Conjecture on GUGP-PWT(1) holds true, and Unique Game
Conjecture on GUGP-PWT(1/2) holds true, if the 2-to-1 Conjecture holds true.
The author poses an open problem whether Unique Game Conjecture holds true on
GUGP-PWT($\rho$) with $0<\rho<1$.
