  The phenomenon of monogamy of Bell inequality violations is interesting both
from the fundamental perspective as well as in cryptographic applications such
as the extraction of randomness and secret bits. In this article, we derive new
and stronger monogamy relations for violations of Bell inequalities in general
no-signaling theories. These relations are applicable to the class of binary
output correlation inequalities known as XOR games, and to a restricted set of
unique games. In many instances of interest, we show that the derived relation
provides a significant strengthening over previously known results. The result
involves a shift in paradigm towards the importance in monogamies of the number
of inputs of one party which lead to a contradiction from local realistic
predictions.
