  A set of probabilistic forecasts is calibrated if each prediction of the
forecaster closely approximates the empirical distribution of outcomes on the
subset of timesteps where that prediction was made. We study the fundamental
problem of online calibrated forecasting of binary sequences, which was
initially studied by Foster & Vohra (1998). They derived an algorithm with
$O(T^{2/3})$ calibration error after $T$ time steps, and showed a lower bound
of $\Omega(T^{1/2})$. These bounds remained stagnant for two decades, until
Qiao & Valiant (2021) improved the lower bound to $\Omega(T^{0.528})$ by
introducing a combinatorial game called sign preservation and showing that
lower bounds for this game imply lower bounds for calibration.
  We introduce a strengthening of Qiao & Valiant's game that we call sign
preservation with reuse (SPR). We prove that the relationship between SPR and
calibrated forecasting is bidirectional: not only do lower bounds for SPR
translate into lower bounds for calibration, but algorithms for SPR also
translate into new algorithms for calibrated forecasting. In particular, any
strategy that improves the trivial upper bound for the value of the SPR game
would imply a forecasting algorithm with calibration error exponent less than
2/3, improving Foster & Vohra's upper bound for the first time. Using similar
ideas, we then prove a slightly stronger lower bound than that of Qiao &
Valiant, namely $\Omega(T^{0.54389})$. Our lower bound is obtained by an
oblivious adversary, marking the first $\omega(T^{1/2})$ calibration lower
bound for oblivious adversaries.
