the chance of winning $1000$ dollars. Hence Skeptic may want to buy $1000$ tickets. Thus it is reasonable that $\nu$ and $p_n$ are inversely proportional when $p_n$ is small.

Experiments {#sec:experiments}
===========

In this section we give some numerical studies of our strategy. In Section \[subsec:simulation\] we present some simulation results and in Section \[subsec:JMA\] we apply our strategy against probability forecasting by the Japan Meteorological Agency.

Some simulation studies {#subsec:simulation}
-----------------------

We consider three cases and apply three strategies to these examples. In our simulation studies Reality chooses her moves probabilistically, either by Bernoulli trials or by a Markov chain model.

-   Case 1:$x_n$ is a Bernoulli variable with the success probability 0.7 and $p_n$ alternates between $0.4$ and $0.6$ (i.e. $0.4=p_1= p_3 = \cdots$ and $0.6=p_2 = p_4 = \cdots$).

-   Case 2:$x_n$ is a Bernoulli variable with the success probability 0.5 and $p_n$ alternates between $0.4$ and $0.6$.

-   Case 3:$p_n=0.5$ and $x_n$ is generated by a Markov chain model with transition probabilities shown in Figure \[fig:2\].

![Transition probabilities for $x_n$[]{data-label="fig:2"}](markov.eps){width="4.5cm"}

-   Strategy 1:$\theta$ is a scalar and $c_n=1$ in (\[eq:logistic-simple\]). Assume that the prior density for $\theta$ is given as uniform distribution for \[0,1\]. The capital 