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Assume that a decision-maker in different contexts applies different preference structures and corresponding choice heuristics to solve problems. From the viewpoint of the analyst, there will always be some randomness, which can mathematically be included into the overall threshold, so that it gets , where is a probability density function. Because is a discrete set, between consecutive pairs of (in ascending order), there is a range of, satisfying . It represents the range of an invariant preference structure. The probability of this preference structure being applied, , equals the probability of being in this range, given f is a continuous distribution:


It may equivalently be viewed as the probability of applying decision heuristics implied in the preference structure.

Thus any single decision can be modeled as a two-layer process, choosing an appropriate preference structure and applying this structure to the choice task, forming preferences among alternatives and making the choice. Because the preference structure actually applied by the decision-maker is usually unknown by the researcher, the final probability of an alternative being chosen can be modeled as the expected result of choice aggregated from all possible choice outcomes under these latent preference structures,


where is the probability that object i is accepted when is applied. Because within an invariant range the value of does not affect the choice outcome, does not need to be identified. Instead,is enough as the critical values for the overall thresholds.

Although the process of selecting a preference structure itself may be susceptible to bounded rationality and simplifying processes, here only the outcome of this process is modeled. Assume the decision-maker selects the preference structure probabilistically based on its expected value, the probability of a preference structure being applied can be modeled as:

whereis the value that the decision-maker expects from applying preference structure k. In essence, it is a heuristic implied in the preference structure that is selected. Since within a certain preference structure different heuristics do not affect the choice outcome, Eq. (5.15) can also be formulated as the aggregated probability that all these heuristics are chosen,

where is the value of heuristic h implied in preference structure k. Assume that the value of a heuristic is influenced by mental effort, risk perception and expected outcome.

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