We axiomatize subjective probabilities on finite domains without requiring richness in the outcome space or restrictions on risk preference using Event Exchangeability (Chew and Sagi, 2006), which has been implicit in the prior literature (Savage, 1954; Machina and Schmeidler, 1992; Grant, 1995). In three successively stronger theorems, we characterize a probability representation of the exchangeability relation, followed by characterizing a unique subjective probability, and finally endowing this subjective probability with the property of reduction consistency—acts inducing the same lottery are indifferent. This subjective probability can serve as foundation to derive expected utility and rank linear utility by imposing the sure-thing principle and its comonotonic variant. Moreover, our finite-domain setting reveals a novel possibility of state dependence arising from our subjective probability not being reduction consistent. Our axiomatic treatment can be further adapted to smaller collections of events and deliver small worlds probabilistic sophistication.
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