Expected utility analysis of infinite compound lotteries

Abstract

Lotteries can be used to model alternatives with uncertain outcomes. Decision theory uses compound ordinary lotteries to represent a structure of lotteries within lotteries, but can only rank the finite compound lottery structure. We expand upon this approach to introduce solutions for infinite compound ordinary lotteries (ICOL). We describe a novel procedure to simplify any ICOL as much as possible to a maximum reduced ICOL, which is not a unique representation. We limit our discussion to ICOLs of first order, which are defined as maximum reduced ICOLs with a single maximum reduced ICOL in their direct outcome. Two special cases of ICOLs of first order are discussed. These are recursive and semi-recursive ICOLs. We provide an analytical approach to find the expected utility of recursive ICOLs, and a numerical algorithm for semi-recursive ICOLs. We demonstrate our solution methods by evaluating example decision problems involving: a randomizing device with unsuccessful trials, the St. Petersburg paradox, and training with virtual reality.

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