Eliciting a Utility Function

Recall, however, that there's another loss function included in this figure, one with a dollar sign at the top. Different management options will require different amounts of money. Because resources, including money, people, time, and energy, are limited, we need to consider the possibility that choosing the option with the lowest probability of extinction may cause us to lose opportunities to do other things. Simply fencing reserves, for example, would provide a 45% expected probability of survival. Is it really worth spending six times as much money for a captive breeding program that would increase the expected probability of survival to 81%?

One way of answering that question is simply to think about the decision as if we were gambling.⁹ To make it simple, let's suppose that our two choices are:

1. Spend $0.6M to fence reserve and have a 45% chance of survival.

2. Spend $3.7M on a captive breeding program and have an 81% chance of survival.

Let me change the wording of those options a little:

1. Buy a lottery ticket for $6 that gives you a 45% chance of winning a million bucks.

2. Buy a lottery ticket for $37 that gives you an 81% chance of winning a million bucks.

Which one of those would you choose? Well, the expected return from the first option is

$$0.45*1000000 - 6 \approx 450000$$

While the expected return from the second option is

$$0.81*1000000 - 37 \approx 810000 \quad .$$

Not too hard to pick, is it? The second option is clearly better. What if instead of that lottery ticket winning you a million bucks, though, it won you only fifty. Then the return from the first option is

$$0.45*50 - 6 = 16.5$$

and the return from the second option is

$$0.81*50 - 37 = 3.5 \quad .$$

Again, not too hard to pick, but now we'd pick the first option instead of the second one.

What does this have to do with rhinos? Just an illustration of the simple point that it's not only how much it costs to save rhinos that matters, but also how much it benefits us to save them. The challenge that we'll talk more about after Thanksgiving break is that we can measure costs fairly easily in dollars and cents (or Euros or pounds or Rand or ...). It's typically a lot harder to measure the benefits.

One approach is to use what economists call a utility function. The utility function is intended to describe how much a unit of something is ``worth.''¹⁰ I've already illustrated the idea with the two extremes in terms of lottery outcomes, but let's see how we might go about finding one for the rhino example.

• A zero probability of extinction (the desired outcome) is, by definition, associated with a utility value of 1.

• Certain extinction is associated with a utility value of 0.

• All intermediate probabilities of extinction are associated with a utility value between 0 and 1, inclusive, and the utility function should be non-increasing.

• How much is a 50% probability of extinction worth on this scale?

• How much is a 25% probability of extinction worth on this scale?

• How much is a 75% probability of extinction worth on this scale?

• Interpolate a curve. (1999 results $U(p) = 0.99 + 0.64p - 3.77p^2 + 2.13p^3$).

• Calculate utility of
  • status quo
    $$\begin{eqnarray*} (0.1)(U(0.95)) + (0.9)(U(0.85)) &=& (0.1)(0.022) + (0.9)(0.12) \\ &=& 0.11 \end{eqnarray*}$$
    
    

  • new reserve
    $$\begin{eqnarray*} (0.6)(U(0.9)) + (0.4)(U(0.4)) &=& (0.6)(0.065) + (0.4)(0.78) \\ &=& 0.35 \end{eqnarray*}$$
    
    

  • captive breeding
    $$\begin{eqnarray*} (0.8)(U(0)) + (0.2)(U(0.95)) &=& (0.8)(1) + (0.2)*(0.022) \\ &=& 0.80 \end{eqnarray*}$$
    
    

  • expand reserve
    $$\begin{eqnarray*} (0.1)(U(0.9)) + (0.2)(U(0.9)) + (0.7)(U(0.37)) &=& (0.1)(0.065) + (0.2)(0.065) + \\ & & (0.7)(0.82) \\ &=& 0.59 \end{eqnarray*}$$