our betting strategy. We also see that our strategy works well against probability of precipitation announced by the Japan Meteorological Agency.

Organization of this paper is as follows. In Section \[sec:formulation\] we formulate the probability forecasting game with side information and derive some basic properties of betting strategies. It also serves as a brief introduction to game-theoretic probability theory. In Section \[sec:logistic-betting\] we introduce our betting strategy based on logistic regression model. In Section \[sec:forcing\] we prove some properties of our logistic betting strategy in the framework of game-theoretic probability. In Section \[sec:experiments\] we give numerical studies of our strategy. In particular we apply our strategy to the data on probability of precipitation announced by the Japan Meteorological Agency. We end the paper with some discussions in Section \[sec:summary\].

Formulation of the probability forecasting game and summary of preliminary results {#sec:formulation}
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In this section we formulate the probability forecasting game and extend it to include side information. We mostly follow the results in [@ktt-2008saa].

At the beginning of day $n$ (or at the end of day $n-1$) an agency (we call it “Forecaster”) announces a probability $p_n$ of certain event in day $n$, such as precipitation in day $n$. 