Editing Satisficing (section)

==== Applied to the utility framework ====
In [[economics]], satisficing is a [[behavior]] which attempts to achieve at least some [[minimum]] level of a particular [[Variable (mathematics)|variable]], but which does not necessarily maximize its value.<ref>{{cite book |chapter-url=http://huwdixon.org/SurfingEconomics/chapter7.pdf |chapter=Artificial Intelligence and Economic Theory |title=Surfing Economics: Essays for the Inquiring Economist |author-link=Huw Dixon |first=Huw |last=Dixon |year=2001 |location=New York |publisher=Palgrave |isbn=978-0-333-76061-1 }}</ref> The most common application of the concept in economics is in the behavioral [[theory of the firm]], which, unlike traditional accounts, postulates that producers treat [[Profit (economics)|profit]] not as a goal to be maximized, but as a constraint. Under these theories, a critical level of profit must be achieved by firms; thereafter, priority is attached to the attainment of other goals.

More formally, as before if {{math|<var>X</var>}} denotes the set of all options {{math|<var>s</var>}}, and we have the payoff function '''{{math|<var>U</var>(<var>s</var>)}}''' which gives the payoff enjoyed by the agent for each option. Suppose we define the optimum payoff {{math|<var>U</var><sup>*</sup>}} the solution to

:<math>\max_{s\in X} U(s)</math>

with the optimum actions being the set '''{{math|<var>O</var>}}''' of options such that {{math|<var>U</var>(<var>s<sup>*</sup></var>) {{=}} ''U''<sup>*</sup>}} (i.e. it is the set of all options that yield the maximum payoff). Assume that the set '''{{math|<var>O</var>}}''' has at least one element.

The idea of the {{em|aspiration level}} was introduced by [[Herbert A. Simon]] and developed in economics by Richard Cyert and James March in their 1963 book ''[[A Behavioral Theory of the Firm]]''.<ref>{{cite book |last1=Cyert |first1=Richard |last2=March |first2=James G. |year=1992 |title=A Behavioral Theory of the Firm |edition=2nd |publisher=Wiley-Blackwell |isbn=978-0-631-17451-6 }}</ref> The aspiration level is the payoff that the agent aspires to: if the agent achieves at least this level it is satisfied, and if it does not achieve it, the agent is not satisfied. Let us define the aspiration level '''{{math|''A''}}''' and assume that {{math|''A'' ≤ ''U''<sup>*</sup>}}. Clearly, whilst it is possible that someone can aspire to something that is better than the optimum, it is in a sense irrational to do so. So, we require the aspiration level to be at or below the optimum payoff.

We can then define the set of satisficing options '''{{math|<var>S</var>}}''' as all those options that yield at least '''{{math|<var>A</var>}}''': {{math|<var>s</var> &isin; <var>S</var>}} {{em|if and only if}} {{math|<var>A</var> ≤ <var>U</var>(<var>s</var>)}}. Clearly since {{math|<var>A</var> ≤ <var>U</var><sup>*</sup>}}, it follows that {{math|<var>O</var> &sube; S}}. That is, the set of optimum actions is a subset of the set of satisficing options. So, when an agent satisfices, then she will choose from a larger set of actions than the agent who optimizes. One way of looking at this is that the satisficing agent is not putting in the effort to get to the precise optimum or is unable to exclude actions that are below the optimum but still above aspiration.

An equivalent way of looking at satisficing is {{em|epsilon-optimization}} (that means you choose your actions so that the payoff is within epsilon of the optimum). If we define the "gap" between the optimum and the aspiration as '''{{math|&epsilon;}}''' where {{math|&epsilon; {{=}} ''U''<sup>*</sup> − ''A''}}. Then the set of satisficing options '''{{math|<var>S</var>(&epsilon;)}}''' can be defined as all those options '''{{math|<var>s</var>}}''' such that {{math|<var>U</var>(<var>s</var>) ≥ <var>U</var><sup>*</sup> − &epsilon;}}.