  Sensitivity analysis for the unconfoundedness assumption is crucial in
observational studies. For this purpose, the marginal sensitivity model (MSM)
gained popularity recently due to its good interpretability and mathematical
properties. However, as a quantification of confounding strength, the
$L^{\infty}$-bound it puts on the logit difference between the observed and
full data propensity scores may render the analysis conservative. In this
article, we propose a new sensitivity model that restricts the $L^2$-norm of
the propensity score ratio, requiring only the average strength of unmeasured
confounding to be bounded. By characterizing sensitivity analysis as an
optimization problem, we derive closed-form sharp bounds of the average
potential outcomes under our model. We propose efficient one-step estimators
for these bounds based on the corresponding efficient influence functions.
Additionally, we apply multiplier bootstrap to construct simultaneous
confidence bands to cover the sensitivity curve that consists of bounds at
different sensitivity parameters. Through a real-data study, we illustrate how
the new $L^2$-sensitivity analysis can improve calibration using observed
confounders and provide tighter bounds when the unmeasured confounder is
additionally assumed to be independent of the measured confounders and only
have an additive effect on the potential outcomes.
