A theory of case-based decision making
Chapter 2. Decision rules
Elementary formula and interpretations
While evidence has been accumulating that the expected utility theory is too restrictive (at
least from a descriptive viewpoint), its various generalizations only attest to the strength and
appeal of the expected utility paradigm. With few exceptions, all suggested alternatives retain
the framework of the model. Yet it seems that in many situations of choice under uncertainty,
the very language of expected utility models is inappropriate. For instance, states of the
world are neither naturally given, nor can they be simply formulated. Furthermore, sometimes
even a comprehensive list of all possible outcomes is not readily available or easily
imagined.
2.1 Motivating examples
When the decision maker attempt to think in the language of EUT, he/she would have to
imagine all possible outcomes and all relevant states of the world. Many times, the number of
states is huge and the states themselves would not be defined in an intuitive way. Moreover,
even if the decision maker managed to image all outcomes and states, the task would by no
means be done. Next she would have to assess the utility of each outcome, and to form a
prior over the state space. It is not clear how the utility and the prior are to be defined,
especially since past experience appears to be of limited help in these examples.
Correspondingly, it is doubtful that EUT is the most useful tool for predicting behavior in
decision problems of this nature. A theory that will provide a more faithful description of how
people think would have a better chance of predicting what they will do. How, then, do
people think about such decision problems? We suggest that people chose acts based on
their performance in similar problems in the past. Case-based decision theory (CBDT)
attempts to formally capture this mode of reasoning as it applies to decision making.
2.2 Model
Generally, a decision maker would remember some of the problems that she and other
decision makers encountered in the past. When faced with a new problem, the similarity of
the situation brings this memory to mind, and with it the recollection of the choice made and
the outcome that resulted. We refer to the combination of these three, the problem, the act,
and the result, as a “case”. Thus similar cases are recalled, and based on them each
possible decision is evaluated. The specific model we propose here evaluates each act by
the sum, over all cases in which it was chosen, of the product of the similarity of the problem
to the one at hand and the resulting utility.
Formally, we start with three sets: let P be a set of decision problems, A – a set of acts that
may be chosen at the current problem, and R – a set of possible outcomes. A case is a triple
(q, a, r) where q is a problem, a is an act and r – an outcome. Thus, the set of conceivable
cases is the set of all such triples:
C=PxAxR
The next two components of the formal model are similarity and utility functions. The
similarity function (P x P). The similarity function is assumed to provide a quantification of
similarity judgments between decision problems. The term “similarity” should not be taken
too literally. Past decision problems affect the decision maker’s choice only if they are
recalled. Thus, while we use the suggestive term “similarity”, we think of this function as
Chapter 2. Decision rules
Elementary formula and interpretations
While evidence has been accumulating that the expected utility theory is too restrictive (at
least from a descriptive viewpoint), its various generalizations only attest to the strength and
appeal of the expected utility paradigm. With few exceptions, all suggested alternatives retain
the framework of the model. Yet it seems that in many situations of choice under uncertainty,
the very language of expected utility models is inappropriate. For instance, states of the
world are neither naturally given, nor can they be simply formulated. Furthermore, sometimes
even a comprehensive list of all possible outcomes is not readily available or easily
imagined.
2.1 Motivating examples
When the decision maker attempt to think in the language of EUT, he/she would have to
imagine all possible outcomes and all relevant states of the world. Many times, the number of
states is huge and the states themselves would not be defined in an intuitive way. Moreover,
even if the decision maker managed to image all outcomes and states, the task would by no
means be done. Next she would have to assess the utility of each outcome, and to form a
prior over the state space. It is not clear how the utility and the prior are to be defined,
especially since past experience appears to be of limited help in these examples.
Correspondingly, it is doubtful that EUT is the most useful tool for predicting behavior in
decision problems of this nature. A theory that will provide a more faithful description of how
people think would have a better chance of predicting what they will do. How, then, do
people think about such decision problems? We suggest that people chose acts based on
their performance in similar problems in the past. Case-based decision theory (CBDT)
attempts to formally capture this mode of reasoning as it applies to decision making.
2.2 Model
Generally, a decision maker would remember some of the problems that she and other
decision makers encountered in the past. When faced with a new problem, the similarity of
the situation brings this memory to mind, and with it the recollection of the choice made and
the outcome that resulted. We refer to the combination of these three, the problem, the act,
and the result, as a “case”. Thus similar cases are recalled, and based on them each
possible decision is evaluated. The specific model we propose here evaluates each act by
the sum, over all cases in which it was chosen, of the product of the similarity of the problem
to the one at hand and the resulting utility.
Formally, we start with three sets: let P be a set of decision problems, A – a set of acts that
may be chosen at the current problem, and R – a set of possible outcomes. A case is a triple
(q, a, r) where q is a problem, a is an act and r – an outcome. Thus, the set of conceivable
cases is the set of all such triples:
C=PxAxR
The next two components of the formal model are similarity and utility functions. The
similarity function (P x P). The similarity function is assumed to provide a quantification of
similarity judgments between decision problems. The term “similarity” should not be taken
too literally. Past decision problems affect the decision maker’s choice only if they are
recalled. Thus, while we use the suggestive term “similarity”, we think of this function as
