  It is well-known that in certain scenarios weakly entangled states can
generate stronger nonlocal effects than their maximally entangled counterparts.
In this paper, we consider violations of the CHSH Inequality when one party has
inefficient detectors, a scenario known as an asymmetric Bell experiment. %We
show that violations can occur if and only if the detection efficiency is above
$50\%$. For any fixed detection efficiency, we derive a simple upper bound on
the entanglement needed to violate the inequality by more than some specified
amount $\kappa\geq 0$. When $\kappa=0$, the amount of entanglement in all
states violating the inequality goes to zero as the detection efficiency
approaches $50\%$ from above. %This provides another scenario in which weakly
entangled states are advantageous for violating the CHSH Inequality in the
presence of detection inefficiency. We finally consider the scenario in which
detection inefficiency arises for only one choice of local measurement. In this
case, it is shown that the CHSH Inequality can always be violated for any
nonzero detection efficiency and any choice of non-commuting measurements.
