In eikeluedeling/decisionSupport: Quantitative Support of Decision Making under Uncertainty

Decision making processes can be categorized in two levels of decision making.

1. The actual problem of interest of a policy maker which we call the underlying decision on how to influence an ecological-economic system based on a particular information on the system available to the decision maker and
2. the meta decision on how to allocate resources to reduce uncertainty, i.e to increase the current information to improve the underlying decision making process.

Value of Information Analysis deals with the meta decision problem 2.

The underlying decision problem and its notational framework

We are considering a decision maker who can influence an ecological-economic system having two alternative decisions $d_1$ and $d_2$ at hand. We assume, that the system can be characterized by the $n-$dimensional vector $X$. The characteristics $X$, are not necessarily known exactly to the decision maker. However, we assume furthermore that she is able to quantify this uncertainty which we call an \emph{estimate} of the characteristics. Mathematically, an estimate is a random variable with probability density $\rho_X$.\par

Furthermore, the characteristics $X$ determine the welfare $W_d$ according to the welfare function $w_d$: [ W_d = w_d (X) ] Thus, the welfare of decision $d$ is also a random variable which probability distribution we call $\rho_{W_d}$. The welfare function $w_d$ values the decision $d$ given a certain state $X$ of the system. In other words, decision $d_2$ is preferred over decision $d_1$, if and only if, the expected welfare of decision $d_2$ is greater than the expected welfare^[ For a comprehensive discussion of the concept of social preference ordering and its representation by a welfare function cf. \citet{GravelleRees2004}. ] of decsion $d_1$, formally [ d_1 \prec d_2 \Leftrightarrow E[W_{d_1}] < E[W_{d_2}]. ] This means the best decision $d^$ is the one which maximizes welfare: [ d^ := \arg \max_{d=d_1,d_2} E[W_d] ] This maximization principle has a dual minimization principle. We define the net benefit $NB_{d_1} := W_{d_1} - W_{d_2}$ as the difference between the welfare of the two decision alternatives. A loss $L_d$ is characterized if a decision $d$ produces a negative net benefit. No loss occurs if the decision produces a positive net benefit. This is reflected in the formal definition [ L_d := \begin{cases} - NB_d &, \textrm{ if } NB_d < 0\ 0 &, \textrm{ otherwise}. \end{cases} ] Using this notion it can be shown that the maximization of expected welfare is equivalent to the minimization of the expected loss $EL_d := E[L_d]$.

eikeluedeling/decisionSupport documentation built on April 12, 2024, 7:25 a.m.