  We study the problem of making calibrated probabilistic forecasts for a
binary sequence generated by an adversarial nature. Following the seminal paper
of Foster and Vohra (1998), nature is often modeled as an adaptive adversary
who sees all activity of the forecaster except the randomization that the
forecaster may deploy. A number of papers have proposed randomized forecasting
strategies that achieve an $\epsilon$-calibration error rate of
$O(1/\sqrt{T})$, which we prove is tight in general. On the other hand, it is
well known that it is not possible to be calibrated without randomization, or
if nature also sees the forecaster's randomization; in both cases the
calibration error could be $\Omega(1)$. Inspired by the equally seminal works
on the "power of two choices" and imprecise probability theory, we study a
small variant of the standard online calibration problem. The adversary gives
the forecaster the option of making two nearby probabilistic forecasts, or
equivalently an interval forecast of small width, and the endpoint closest to
the revealed outcome is used to judge calibration. This power of two choices,
or imprecise forecast, accords the forecaster with significant power -- we show
that a faster $\epsilon$-calibration rate of $O(1/T)$ can be achieved even
without deploying any randomization.

Machine Learning;Data Structures and Algorithms;Computer Science and Game Theory;Methodology;Machine Learning