Decision Maker's Best Choice: An Investigation into the Efficiency of Search Strategies with Partial Information

Decision makers (DMs) must often choose among alternative without knowing
the full choice set. Whenever alternatives are presented equentially - one at a
time - they must choose without perfect information about the actual and/ or
the eventually later upcoming alternatives. Decisions of this type are quite
frequent. Whether a DM has to decid when to buy an asset on the open market like an apartment in the housing market, a car in the used car market
or choose a job in the job market. Sequential decision problems of this type have been pecified formally in a number of ways (Bearden et al. [20061):
1. Full information problems: Problems where the DM is assumed to have
complete information about the distribution from which the observations are ampled.
2. Partial information problems: Problems where it is assumed that the DM knows either the distribution of one or more correlated Attributes
of the quality of the alternatives which can be observed with some cost
c or he knows only certain properties of the distribution of the quality of the alternatives from which the observations are drawn (eg. that it is
Gaussian or multinomial) but not the parameter of the distribution.
3. No-information problems: Problems where the DM has no information
of the distribution of the values describing the quality of the alternatives
but rather decides on the basis of ranks in terms of their quality.
In the following paper we consider the partial information problem only. First
we focus on the situations where DMs do not know exactly the underlying
distribution. Using the multinomial distribution to model the underlying dis
tribution approximately and "learning" the distribution by revising the prior
of its parameters while observing the alternatives quality sequentially give us
a practical procedure to make good and increasingly better decisions in such
cases. We also consider cases where the underlying distribution of the value of the alternatives is not permanent and changes over time. Adaptation of the coefficient of confidence a and the introduction of forgetting older alternatives
by using time stamps is our proposal for such situations. In the cond part
of this paper the other interpretation of the concept of decisions with partial
information is investigated. Instead of observing the single value X describing
the quality of the alternatives our alternative are described by a triple
of real-valued random variables X ( = (X1, X2, X3)) with known joint normal
distribution. The DM first pay c1 called search cost to observe an alternative
with the value X1. Because of the joint distribution x1 gives him only some
information about the true value X3 of the alternative. He can now stop with
this alternative or either continue and observe a new value X1 with cost c1 or
he can try to get more information about the true value X3 of the alternative
by taking an action called test and thus observing a test value X2 with cost c2.
In this case he might again either stop or continue in observing the value X1
of another new alternative again with cost c1. Whereas it is assumed that we
know the 3-dimensional distribution of X ( = (X1, X2, X3)) the DM can only
experience X3 but not observe it. Situations like this occur quite frequently
whenever for example a specialist can be consulted for a fee c2 when buying
a house or a used car to get a more reliable judgment on the quality of an
observed alternative. Similarly, we meet such situations in finding mates where
the test might be a costly date or in hiring where the test is a costly job inte
rview after an application. The case of more than one test is not considered
here.