This is an leadin example (a.k.a. de Finetti’s betting model) of STAT 775 Bayesian Analysis course.

Suppose we have two actor: Client and Broker, and a series of events $$E_1,E_2,\cdots$$ which are uncertain for both of them. These two actors will do transactions based on the events. The rules are as follows:
The broker specifies a collection of ‘previsions’: $$q_1,q_2,\cdots$$, and then the client (knowing $$q_i$$) enter the transactions with a set of stakes $$s_1,s_2,\cdots$$. For each transaction, the client gives $$q_is_i$$ to the broker and the broker gives $$s_i\mathbb{I}(E_i)$$ to the client. Our goal is to prevent the “dutch book” meaning that win or lose is not certain.
First, we consider the range of $$q_i$$. For any $$q_i<0$$ or $$q_i>1$$, it will lead to Dutch Book in this transaction. Thus, we first have $$0\leqslant q_i\leqslant 1$$.
For each transaction, the client will gain $s_i\mathbb{I}(E_i)-q_is_i=s_i(\mathbb{I}(E_i)-q_i).$
For two disjoint events $$E_1,E_2$$, we assume $$E_3=E_1\cup E_2$$. Then, after the first three transaction, we know the clinet will gain \begin{equation} s_1\mathbb{I}(E_1)+s_2\mathbb{I}(E_2)+s_3\mathbb{I}(E_3)-(q_1s_1+q_2s_2+q_3s_3)\\ = \left\{ \begin{aligned} &s_1+s_3-(q_1s_1+q_2s_2+q_3s_3)=:g_1, \mathrm{if\ } E_1\\ &s_2+s_3-(q_1s_1+q_2s_2+q_3s_3)=:g_2, \mathrm{if\ } E_2\\ &s_1+s_2-(q_1s_1+q_2s_2+q_3s_3)=:g_3, \mathrm{if\ } (E_1\cup E_2)^c \end{aligned} \right. \end{equation}
Thus, we have $\left( \begin{matrix} 1-q_1&-q_2&1-q_3\\ -q_1&1-q_2&1-q_3\\ 1-q_1&1-q_2&-q_3 \end{matrix} \right) \left( \begin{matrix} s_1\\s_2\\s_3 \end{matrix} \right) = \left( \begin{matrix} g_1\\g_2\\g_3 \end{matrix} \right).$ We name it $$RS=g$$. If $$R$$ is invertable, we have $$S=R^{-1}g$$, which will lead to Dutch Book. So, we need $$R$$ singular. We compute the determinant of $$R$$, and that is $$\det (R)=q_1+q_2-q_3=0$$. It meas that if we need the “coherence” we need $$q_3=q_1+q_2$$, which is the axiom of probability.