The Procedure

1. Define a gain function or a loss function. This is the thing that measures the value or cost of a particular outcome. The value function may be qualitative--allows persistence, doesn't allow persistence--or it may be quantitative--the probability of extinction in 100 years is $p$ or the present value of the population expected to exist 100 years from now is $x$.⁸

2. Associate utility values with each possible outcome in your decision tree. Utilities are a measure of how valuable (or how costly) a particular outcome is. Costs are often measured in dollars (or other currencies), but properly speaking the money is used merely as a metric to make different choices comparable. The dollar value per se isn't important. It's what the dollar value represents. To make this concrete, let's take a look at the decision tree Maguire and her co-authors [2] present for management of the Sumatran rhino (Figure 2). In this case, we have two possible loss functions: (1) an estimate of the probability of extinction associated with the particular management alternative and (2) an estimate of how much it would cost to implement each of the management alternatives.

   Figure 2: Decision tree for analysis of rhino conservation.

   

3. Having associated values with each possible outcome, you associate probabilities with each branch on the tree that's outside your control. These probabilities should reflect the probability of a particular outcome, given the events that have preceded it in the tree. Thus, the probabilities of particular events may depend on their context, as they should if we really think our management decision is going to have an effect.

4. Now that you have values associated with each possible outcome and probabilities associated with each branch of the tree, it's possible to calculate the expected values associated with each possible decision. The expected value of a decision is the weighted average of all outcomes associated with that decision, where the weights are the probabilities associated with each step in the decision tree.

   So for example, if we were to intervene by establishing a captive breeding program, the diagram tells us that

   • There is an 80% chance that the captive breeding program will succeed, in which case there's zero chance of extinction in 30 years.

   • There is a 20% chance that the captive breeding program will fail, in which case there's a 95% chance of extinction in 30 years.

   • The expected probability of extinction is
     
     $$(0.8)(0.0) + (0.2)(0.95) = 0.19$$
     
     

   This compares with a 0.53 expected probability of extinction if reserves are simply expanded.

   If we take the probability of extinction as our loss function, we're done. We simply choose the alternative that has the lowest probability of extinction, namely captive breeding, which has $pE = 0.19$.