| Summary: | The quantitative study of choice under conditions of uncertainty dates back to the earliest applications of probability to games of chance. Over time, theories of choice have transitioned away from the `oughts' of rational econometrics toward more face-valid descriptions of observed behaviors. Throughout this period, the problem of subjective probability has posed a consistent difficulty for theories of choice. The most successful approach for modeling these distortions is use of `log-odds,' which provides a powerful description of two-alternative choice as a power law function of relative outcome probability. The log-odds approach can be generalized using the framework of `compositional analysis.' The core statistical methodology of this framework is introduced and described, with an eye towards developing models of choice across any number of alternatives. The viability of these models is demonstrated on several previously published datasets.
A series of experiments with rats explored the effect of changing the number of alternatives. Power-law models continued to provide an effective description of behavior, but subjective probabilities were also found to be less distorted when subjects made choices among a larger number of alternatives (eight at once) than among smaller numbers (four or six). This effect was robust against controls for age, order of experience, chamber configuration, and schedule richness. A working hypothesis is put forward based on an analysis of responses as a dynamical process: Subjects succeed at complex tasks by limiting their transitions between response alternatives to a highly stereotyped `default transition matrix,' making only slight deviations in order to adapt to changing task demands. This strategy is computationally efficient. However, severe mismatches between the schedule and a subject's default transition matrix are much more likely to occur when fewer alternatives are available, and behavior under such conditions is necessarily insensitive. Implications for other choice models are considered.
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