Coherent Predictions are Strategic

Abstract

Two random quantities $x$ and $y$, taking values in sets $X$ and $Y$, are to be observed sequentially. A predicter (bookie) posts odds on $(x, y)$ and on $y$ given $x$ according to functions $P$ and $q(x)$, respectively. The predicter is coherent (the bookie can avoid a sure loss) if and only if $P$ is a finitely additive probability distribution on $X \times Y$ and $q$ satisfies a general law of total probability: $P(A) = \int q(x)(Ax)P_0(dx)$ for $A \subset X \times Y, Ax = \{y: (x, y) \in A\}, P_0 = \text{marginal of} P \text{on} X.$