Satisficing decisions are common in daily life, for instance, deciding to buy something or not. The decision outcome is simply accepting or rejecting an object of interest. Formulating this type of decisions starts from defining preference structures.
Based on the concept of bounded rationality, any decision-making process can be understood as a problem-solving process in which a decision-maker processes information to arrive at a decision that achieves a particular goal within some margin of accuracy that is satisfactory to him/her. Assume this cognitive process consists of at least three sequential procedures: filtering of information, attribute representation and judging the resulting states.
Let represent the set of attributes that are relevant to the decision-maker as the input for the decision process. The principle of bounded rationality suggests that people only discriminate between attribute levels with some margin of accuracy. Let be a set of successively increasing activation thresholds for , corresponding to stricter judgment standards (note that N can be attribute-dependent, so it should be , but for representation simplicity, the subscript is ignored). An attribute may then meet one or more of these increasingly stricter activation thresholds and hence the stimulation becomes stronger, formally
where represents the resulting mental state of the decision-maker on attribute against threshold . Thus, filtering and attribute representation transforms external attributes into a set of activated and non-activated (ignored) internal (mental) attribute states.
Assume that the decision-maker will arrive at preferences by attaching values, integrating these values for corresponding states to derive an overall value, and then judging the overall value against some overall threshold. The value of an attribute state is represented by . The resultant value of state n of attribute is
All attribute states that are relevant to the decision problem are combined according to some integration rule to arrive at an overall value for each choice alternative. Multiple integration rules are operationally appropriate as later it can be seen that only the relative relationship between attribute states matters. For operational convenience, if an additive integration rule is assumed, the overall value of choice alternative i equals:
In the final step, assume that the overall value is also discretized by checking them against a set of successively increasing overall thresholds , resulting in the overall states,. This can be expressed as:
In case this discretization only involves two preference orders (e.g. reject or accept as in satisficing decisions), only oneis needed and defines rejecting the alternative, whereas implies accepting it.
Define a state value set for each attribute, which includes all possible values related to the attribute,
Letrepresent a combination of state values across attributes, one from each attribute value set,
Ordering all the ascending forms an overall value set,
Checking this overall value set against the overall thresholdresults in a unique pattern of relationships withsome overall values above the threshold, and some below the threshold. Thus, can be divided into a subset of rejected overall values and a set of accepted ones. This pattern can be viewed as a preference structure (PS), Ф, that is used to classify overall values into an ordered set of preferences (in this case rejection or acceptation). Mathematically,