This third edition presents a real-world-inspired decision-theoretic scenario rooted in the classical formulation of I. J. Good (1952), where human incentives, ethical dilemmas, and corporate strategy intersect.

A new invention is presented to a company’s scientific advisor, who must decide whether to recommend its implementation. This decision involves uncertainty and differing incentives between the advisor and the company.

Scenario Setup

The advisor makes the following assessments:

  • \(p\): Probability that the invention works
  • \(V\): Gain for the company if the invention is applied and works
  • \(-L\): Loss for the company if the invention is applied and fails
  • \(v\): Personal gain for the advisor if he recommends and it works
  • \(-l\): Personal loss for the advisor if he recommends and it fails

Assume there is no gain or loss for either party if the invention is not implemented.

We define:

  • Parameter space: \(\Theta = \{ \theta_0: \text{Works}, \theta_1: \text{Fails} \}\)
  • Decision space: \(D = \{ d_0: \text{Apply invention}, d_1: \text{Do not apply} \}\)

Loss Functions

Company’s Loss Function:

\begin{align*} L_{\text{Company}}(\theta, d_0) &= \begin{cases} -V & \text{if } \theta = \theta_0 \\ -L & \text{if } \theta = \theta_1 \end{cases} \\ L_{\text{Company}}(\theta, d_1) &= 0 \end{align*}

Advisor’s Loss Function:

\begin{align*} L_{\text{Advisor}}(\theta, d_0) &= \begin{cases} -v & \text{if } \theta = \theta_0 \\ -l & \text{if } \theta = \theta_1 \end{cases} \\ L_{\text{Advisor}}(\theta, d_1) &= 0 \end{align*}

Expected Losses

  • Company:

\begin{align*} E[L_{\text{Company}}(d_0)] &= -pV – (1 – p)L \\ E[L_{\text{Company}}(d_1)] &= 0 \end{align*}

  • Advisor:

\begin{align*} E[L_{\text{Advisor}}(d_0)] &= -pv – (1 – p)l \\ E[L_{\text{Advisor}}(d_1)] &= 0 \end{align*}

The preferred decision minimizes expected loss (i.e., maximizes expected utility).

Decision Conditions

Company prefers applying the invention if:

\begin{align*} -pV – (1 – p)L &< 0 \\ p(V – L) &> -L \\ p &> \frac{-L}{V – L} \end{align*}

Advisor prefers applying the invention if:

\begin{align*} p &> \frac{-l}{v – l} \end{align*}

Ethical Dilemmas and Conflict Scenarios

Range of \(p\)Company PositionAdvisor PositionOutcome
\(p > \frac{-l}{v – l}\) and \(p > \frac{-L}{V – L}\)Apply inventionApply inventionBoth benefit → Apply invention
\(p < \frac{-l}{v – l}\) and \(p < \frac{-L}{V – L}\)Do not applyDo not applyBoth avoid losses → Do not apply
\(\frac{-L}{V – L} < p < \frac{-l}{v – l}\)Apply inventionDo not applyEthical dilemma for advisor (firm wins, advisor loses)
\(\frac{-l}{v – l} < p < \frac{-L}{V – L}\)Do not applyApply inventionEthical dilemma for advisor (advisor wins, firm loses)

Possible Remedies

  1. Fixed Salary: Advisor receives a fixed compensation independent of outcome.
  2. Aligned Incentives: Tie advisor’s gain/loss to a fraction of the company’s:

$[ v = aV, \quad l = aL, \quad a \in (0,1) ]$

This alignment ensures the advisor’s decisions are consistent with the company’s interest.

I. J. Good’s Recommendation (1952)

“In my opinion the firm should ask the adviser for the estimates of \(p\), \(V\) and \(L\), and should take the onus of the actual decision on its own shoulders. In other words, leaders of industry should become more probability-conscious.”

This perspective highlights the need for organizations to internalize and own strategic decisions under uncertainty rather than outsourcing both assessment and accountability.

Conclusion:

This case study illustrates how decision theory goes beyond mathematics into real-life implications involving ethics, communication, and incentive design. The alignment of individual and organizational interests is crucial when decisions under uncertainty are made.

Stay tuned for the next part!

We gratefully acknowledge Dr. Dany Djeudeu for preparing this course.