Decision Making Under Uncertainty

Joseph Software, Inc. (JSI) is exploring the development of a grammar-and-style checker for microcomputers capable of improving writing quality.
Estimated total cost for prototype development: $200,000.

Outcomes Based on Prototype Performance:
- Moderate Success:
- Expected Sale Price: $600,000
- Probability: P(moderate success) = 0.20
- Major Success:
- Expected Sale Price: $1,200,000
- Probability: P(major success) = 0.10
- Failure (prototype does not exceed performance of existing software):
- Loss of development costs: $200,000
- Probability: P(failure) = 0.70

**Decision Tree for JSI: **
- Start Node: Develop Prototype
- Branches:
  - Failure: No Revenue, Lose $200,000
  - Moderate Success: Gain $400,000 (Sale Price - Development Cost)
  - Major Success: Gain $1,000,000 (Sale Price - Development Cost)

Expected Monetary Value (EMV):
- EMV is calculated as follows:
  - Failure:
    ext{EMV}_{ ext{failure}} = P(failure) imes ext{Loss} = 0.70 imes (-200,000) = -140,000
  - Moderate Success:
    ext{EMV}_{ ext{moderate}} = P(moderate success) imes ext{Profit} = 0.20 imes 400,000 = 80,000
  - Major Success:
    ext{EMV}_{ ext{major}} = P(major success) imes ext{Profit} = 0.10 imes 1,000,000 = 100,000
- Total EMV:
  ext{Total EMV} = ext{EMV}{ ext{failure}} + ext{EMV}{ ext{moderate}} + ext{EMV}_{ ext{major}}
  = -140,000 + 80,000 + 100,000 = 40,000

Expected Profit with Perfect Prediction (EPPP):
- Calculated by assuming the best outcome can be selected based on the perfect information:
- If JSI could achieve the best performance outcome:
  - Major Success Gain: $1,200,000
  - Costs: $200,000
  - Expected Profit:
    ext{Expected Profit} = 1,200,000 - 200,000 = 1,000,000

EVPI Calculation:
- EVPI reflects the value of having perfect information before making a decision if allowed to act:
- Calculating the value of the informed decision:
  - Expected Profit under perfect prediction: 1,000,000
  - Expected Profit of optimal decision without perfect information: 40,000
- Thus, EVPI:
  ext{EVPI} = ext{EPPP} - ext{EMV} = 1,000,000 - 40,000 = 960,000

JSI can hire a consultant at a fee of $5000 who provides advice on whether to develop the prototype.
Based on historical data, the conditional probabilities for consultant recommendations are as follows:
- P(Recommendation to develop a prototype | failure) = 0.20
- P(Recommendation to develop a prototype | moderate success) = 0.60
- P(Recommendation to develop a prototype | major success) = 0.90

Updated Decision Tree Including Consultant:
- Adds a layer illustrating the possible recommendations by the consultant prior to developing the prototype.
- Branches include:
  - If consultant recommends to develop:
  - If consultant recommends not to develop:
  - Evaluate expected profits for both situations considering the fees.

Calculation of EVSI:
- Determine EMV if JSI follows the consultant's advice:
  - If the consultant recommends developing:
    ext{EMV}_{ ext{hire}} = P(failure) imes (0.20 imes -200,000) + P(moderate) imes (0.60 imes 400,000) + P(major) imes (0.90 imes 1,000,000)
  - If the consultant recommends not developing:
- Compare the expected profit of hiring the consultant against the expected profit from not hiring consulting services:
  - Calculate the difference to assess the expected value of hiring the consultant and the net benefit.
- Expected Profit from not hiring the consultant: 40,000
- Expected Profit from hiring consultant, considering recommendations:
  - If consultant recommends to develop: 316,000
  - If the consultant recommends not to develop: 0
- Final comparison for hiring:
  - Expected Profit of Hiring Consultant: 110,600
  - Since the fee of hiring is $5,000, it is less than value from hire, therefore JSI should opt to hire the consultant.
- Conclude Yes, JSI should hire the consultant since the EMV with consulting advice exceeds hiring costs and increases their expected profit significantly.