{{\cal V}}_n> 1$, then $\limsup_n \log {{\cal K}}_n^\pi = \infty$.

Monotonicity with respect to the forecast probability {#subsec:monotonicity}
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Here we consider the case that $\log (p_n/(1-p_n))$ itself is an element of the vector of the side information $c_n$ and hence is multiplied by a coefficient in . For notational convenience we here eliminate $\log (p_n/(1-p_n))$ from $c_n$ and write as $$\label{eq:logistic-monotone}
\log \frac{\hat p_n}{1-\hat p_n}= \beta \log\frac{p_n}{1-p_n} + \tau_n,$$ where $\tau_n$ denotes the effect of side information other than $\log (p_n/(1-p_n))$. Intuitively $\beta$ represents how much trust Skeptic puts in Forecaster. If $\beta=0$ then Skeptic entirely ignores Forecaster’s $p_n$ and if $\beta=1$ then Skeptic takes $p_n$ for granted. The value of $\beta\in (0,1)$ corresponds to partial trust in $p_n$. It is somewhat surprising to see that $\beta > 1$ in the case of probability of precipitation announced by the Japan Meteorological Agency in Section \[subsec:JMA\].

We now investigate how $\nu_n$ in behaves with respect to $p_n$ for given $p_1, c_1, x_1, \dots, p_{n-1},c_{n-1},\allowbreak x_{n-1}$. This is an important question from the viewpoint of defensive forecasting ([@defensive-forecasting], [@df-linear]), because in defensive forecasting we want to obtain $p_n$ for which $\nu_n=0$. For notational simplicity we now omit the subscript $n$ and 