In chapter 4 of Darrell P. Rowbottom's Probability, the author explains the Dutch Book:
Imagine we're going to have a bet together, you and I, about whether something will happen. It could be about whether your favourite sportsperson or team will win their next match, or something more trivial like whether it will rain tomorrow anywhere in your country. We then go on to agree on a stake, $S$, which is the maximum amount of money that could change hands.
I will choose whether to bet for or against the event happening. (Note that my talk of ‘events happening’ is just for convenience, and can easily be translated into talk of ‘statements being true’. So we can understand the bets to concern propositions or statements. For example, betting on Manchester United winning their next match is equivalent to betting that ‘Manchester United will win their next match’ is true.) But before I make that choice, you're going to pick a number, a betting quotient $b$, on the understanding that the bet will be conducted as follows:
- If I bet against the event occurring, you will pay me $bS$. If the event occurs, I will give you $S$.
- If I bet for the event occurring, you will pay me $(1 − b)S$. If the event doesn't occur, I will give you $S$.
To choose $b$ is to determine ‘the odds’ of the bet, which is usually expressed as a ratio, namely $\frac{b}{1 − b}$. To select a value of one half for $b$, for example, would be to give ‘even odds’ on the event; you'll double your initial payment if you win the bet, otherwise you'll lose it. (You may notice that if you select a value of one for $b$, there is no value for the odds when they're defined in the way above. This is an intended feature of the setup, as we'll see.)
I think there is a mistake in points 1 and 2. It can be corrected in two ways:
- You pay me $(1-b)S$ or I pay you $bS$.
- You pay me $bS$ or I pay you $(1-b)S$.
Or using odds:
- You pay me $\frac{1-b}{b} S$ or I pay you $S$
- You pay me $\frac{b}{1-b} S$ or I pay you $S$
Is there indeed a mistake and are my proposed solutions correct?