Satisficing is to accept the first-seen option that met certain thresholds, known as aspirations. Assume options are presented as vectors \(\vec{x}=(x^1,x^2,\cdots,x^k)\), and the aspiration level as \(\vec{\theta}=(\theta^1,\theta^2,\cdots,\theta^k)\). There is an indicator function for accept,

\[\sigma=\mathbb{1}\{x^i \ge \theta^1 \ \forall i\},\]

and a payoff function of the option, \(\phi(\vec{x})\in\mathbb{R}\). The option is assumed to be stochastic with density function \(f(\vec{x})\).

The probability of an option is acceptable is

\[\Pr[\sigma=1 \mid \theta] = \int_\theta^\infty f(\vec{x})d\vec{x}\]

the expected payoff of any option is

\[E[\phi(\vec{x})] = \int_\Omega f(\vec{x})\phi(\vec{x})d\vec{x}\]

and the expected payoff of an acceptable option is

\[E[\phi(\vec{x}) \mid \sigma=1] = \frac{\int_\theta^\infty f(\vec{x})\phi(\vec{x})d\vec{x}}{\int_\theta^\infty f(\vec{x})d\vec{x}}.\]

Consider \(N\) options presented in tandem, the decision maker must take one of them. Thus the last one must be accepted if presented. The policy would be on setting the aspiration \(\vec{\theta}\) to maximize the expected payoff. Let the value of a policy at state \(n\) to be \(V_n(\vec{\theta})\). Then we have

\[V_N(\vec{\theta}) = E[\phi(\vec{x})]\]


\[V_n(\vec{\theta}) = E[\phi(\vec{x}) \mid \sigma=1]\Pr[\sigma=1 \mid \vec{\theta}] + V_{n+1}(\vec{\theta})\Pr[\sigma=0 \mid \vec{\theta}]\]

for \(n=1,\cdots,N-1\).

The goal of optimal satisficing is to find \(\arg\max_{\theta\in\Omega} V_n(\theta)\). As \(E[\phi(\vec{x})] < E[\phi(\vec{x}) \mid \sigma=1]\) for positive function \(\phi(\vec{x})\), the optimal \(\theta\) usually decreases as \(n\) increases.

Heuristic satisficing is to use a fixed \(\theta\) for all \(n\). That is, with

\[V_H(\theta) = \sum_{n=1}^{N-1}\Big((1-P_\sigma)^{n-1}P_\sigma E[\phi(\vec{x}) \mid \sigma=1]\Big) + (1-P_\sigma)^{N-1}E[\phi(\vec{x})]\]

to find \(\arg\max_{\theta\in\Omega} V_H(\theta)\). It is found that, the value of heuristic satisficing \(V_H\) has only a slight decrease from the optimal satisficing \(V_1\).

Satisficing can be converted into infinite horizon: Introduce a time cost of \(c\) per unit time in heuristic satisficing. The value is then

\[V_\infty = \int_0^\infty (1-P_\sigma)^t \Big(P_\sigma E[\phi(\vec{x}) \mid \sigma=1]-c\Big) dt\]

It is found that, as \(c\) increases, \(\theta\) decreases. This reflects that the higher the time cost, the more relaxed the aspiration have to be.

Bibliographic data

   title = "On Optimal Satisficing: How Simple Policies can Achieve Excellent Results",
   author = "J. N. Bearden and T. Connolly",
   booktitle = "Decision Modelling in Uncertain and Complex Environments",
   editor = "T. Kugler and J. C. Smith and T. Connolly and Y. J. Son",
   address = "New York",
   publisher = "Springer",
   year = "2008",