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

For more reference, click here (Stanford Encyclopedia of Philosophy).

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.