Water Drilling Decision Analysis

A step-by-step guide with decision trees and financial evaluations

water drilling landscape, borehole construction, rural farmland

Highlights

Decision Tree Evaluation: A structured approach comparing drilling versus not drilling.
Expected Monetary Value (EMV): Calculation shows drilling has a positive EMV despite low probability.
Diviner’s Maximum Fee: The fee must not exceed the added value that improved probability brings.

Overview of the Scenario

Mr Irri Gation is a farmer in need of water for irrigation, who must decide whether to invest in drilling a borehole on his land. The benefits of a successful borehole are estimated at R2,000,000, but the probability of striking water is only 10%. Drilling itself comes at an upfront cost of R100,000.

There are two clear alternatives: drilling or not drilling. If Mr Irri Gation drills and is successful, the net benefit is a high reward reduced by the drilling cost. In the event of failure, the drilling cost becomes a loss. The alternative—choosing not to drill—results in neither cost nor benefit.

Part (a): Decision Tree Analysis

Constructing the Decision Tree

The decision tree for this problem can be divided into two main parts: the decision node (to drill or not) and the outcome nodes once drilling is chosen. Below is a simplified description of the branches:

Option 1: Drill for Water

This option splits into two scenarios:

Strike Water: There is a 10% chance of success. Here, Mr Irri Gation realizes the full benefit of R2,000,000. After subtracting the cost of drilling (R100,000), the net profit is:
R2,000,000 - R100,000 = R1,900,000
No Water: There is a 90% chance of failure. In this case, Mr Irri Gation suffers a loss equal to the drilling cost:
0 - R100,000 = -R100,000

Option 2: Do Not Drill

This option entails no costs and no benefits, giving an outright value of R0.

Expected Monetary Value (EMV) Calculation

For the option of drilling for water, the Expected Monetary Value (EMV) can be calculated by weighing each outcome by its associated probability:

EMV = (Probability of Success × Net Benefit if Water is Struck) + (Probability of Failure × Net Benefit if No Water is Struck)

For Mr Irri Gation:

EMV = (0.10 × R1,900,000) + (0.90 × -R100,000)
= R190,000 - R90,000
= R100,000

This means that the positive expected monetary value from drilling is R100,000, which is higher than the alternative of not drilling (R0). In terms of decision-making, since the EMV is positive, it is theoretically advantageous for Mr Irri Gation to drill.

Decision Tree Summary Table

Decision Node	Probability	Outcome	Net Benefit
Drill for Water	10%	Strike Water	R1,900,000
Drill for Water	90%	No Water	-R100,000
Do Not Drill	100%	No Action	R0

Part (b): Valuing the Water Diviner's Services

Assessing the Diviner's Impact

There is an alternative option available to Mr Irri Gation. A nearby farmer claims to have supernatural powers to “divine” underground water and offers his services at a fee, which includes a travel cost. The key question here is: What is the maximum amount Mr Irri Gation should pay for these services?

The diviner's value lies in improving the probability of finding water. In simpler terms, paying for the diviner’s services should only be justified if the fee is less than the increase in expected monetary value brought by an improvement in the chance of success.

Calculating Maximum Diviner Fee

To find the maximum amount which Mr Irri Gation should pay, we compare the difference between the EMV when not using the diviner and the EMV when the diviner’s service is considered to boost the chance of success.

Let X represent the maximum fee Mr Irri Gation is willing to pay for the diviner’s services.

Without the diviner, the EMV of drilling is already calculated as R100,000, based on the initial 10% success rate.

If the diviner is used, the diviner might be able to identify a high-probability location. For instance, if the diviner’s involvement shifts the effective success probability to a significantly higher value such as 50%, the scenario changes as follows:

EMV with Diviner’s Services

Assume the diviner increases the probability of a successful strike from 10% to 50%. The new outcomes would be:

Strike Water: There is a 50% chance of success. The net benefit would be R1,900,000 (R2,000,000 - R100,000), but now this amount is reduced by the fee X paid for the diviner.
No Water: There is a 50% chance of failure. The net benefit remains -R100,000.

The new EMV for drilling with the diviner’s services would then be:

EMV (with diviner) = (0.50 × (R1,900,000 - X)) + (0.50 × -R100,000)

We determine the maximum acceptable fee by setting the enhanced EMV equal to the original EMV without the diviner (R100,000):

(0.50 × (R1,900,000 - X)) + (0.50 × -R100,000) = R100,000
0.50R1,900,000 - 0.50X - 50,000 = R100,000
R950,000 - 0.50X - R50,000 = R100,000
R900,000 - 0.50X = R100,000
- 0.50X = R100,000 - R900,000
- 0.50X = -R800,000
X = R800,000 ÷ 0.50
X = R1,600,000

However, this initial calculation leads to an unexpectedly large fee, which signals the need for a critical re-appraisal of how the diviner's value is assessed. In fact, the correct reasoning is comparing the incremental gain in EMV from using the diviner’s service over the base case without the service. Without the diviner the EMV is R100,000. With the diviner, if the probability is increased to 50%, the increment in EMV (the additional benefit) can be thought of as:

Incremental Benefit = EMV (with diviner without fee) - EMV (without diviner)
= [(0.50 × R1,900,000) + (0.50 × -R100,000)] - R100,000
= [R950,000 - R50,000] - R100,000
= R900,000 - R100,000
= R800,000

This shows that the additional value provided by the diviner is R800,000, meaning Mr Irri Gation should not pay more than R800,000 for the diviner’s ability to increase his drilling success rate. Paying more than this fee would erode the net benefit derived from drilling, leaving Mr Irri Gation worse off.

Interpreting the Results

The analysis reveals two major results:

Drilling Decision: With an expected monetary value (EMV) of R100,000, drilling is a favorable option despite the low probability of success. This is because the high reward of a successful borehole outweighs the risk of incurring the drilling cost when no water is found.
Maximum Fee for Diviner’s Services: The maximum amount Mr Irri Gation should be willing to pay for the diviner is equivalent to the incremental benefit provided by the diviner. As the diviner increases the EMV by R800,000 under our assumed improvement in probability, this amount sets the upper limit for what Mr Irri Gation would pay for these services. In other words, he should not pay more than R800,000 for the diviner’s services.

Supplementary Details and Visualization

Diagrammatic Representation

The decision tree diagram provides a visual aid to understand the different outcomes. Below is a textual representation:

                    [Decision: Drill or Not Drill]
                           /           \
                 [Drill for Water]   [Do Not Drill (R=0)]
                    /          \
              [Strike Water]   [No Water]
            (10% probability) (90% probability)
             Net Benefit:    Net Benefit:
             R1,900,000      -R100,000

This diagram emphasizes the binary structure of the decision-making process and demonstrates how the overall EMV is computed based on the associated probabilities of each branch.

Key Variables and Their Impact

Probability of Success

The core challenge is the small probability (10%) of success when drilling without any additional help. This, however, is counterbalanced by the massive benefit (R2,000,000), leading to a net gain in the successful scenario. Even small improvements in probability—for instance, through the water diviner's services—could have a profound impact on the expected value, underscoring the economic importance of risk modification in such projects.

Drilling Cost

The cost of drilling is fixed at R100,000 irrespective of the outcome, and is subtracted directly from the benefits. This cost anchors the downside risk in the decision tree, ensuring that failure always leads to an immediate financial loss.

Service Fee for the Diviner

The maximum fee the diviner can command is determined by the diviner’s ability to increase the probability of striking water. Given that the baseline EMV without the diviner is R100,000 and the incremental gain provided by the diviner is calculated to be R800,000, Mr Irri Gation should ideally not pay more than R800,000 for these potentially transformative insights.