A Practical Decision Under Uncertainty: Heating or Buying a Pullover?

In this fourth edition, we analyze a relatable and simple decision problem using the framework of statistical decision theory. The goal is to determine the most cost-effective way for a student to stay warm during an unexpected cold spell in Germany.

Scenario

In April, an unexpected cold wave hits Germany. A student must decide whether to:

  1. Increase heating, which costs €7.50 per week, or
  2. Buy a pullover for €10 at a local discount store.

However, the pullover has an uncertain lifetime:

  • 50% chance it tears after 1 week
  • 40% chance it tears after 2 weeks
  • 10% chance it lasts until summer (i.e., 3 weeks)

Once a decision is made, it is final — if the pullover tears early, the student must buy another (no switching to heating midway).

Part a) What Should the Student Do Depending on the Duration of the Cold Spell?

We compare the expected costs of the two options (heating vs. pullovers) over 1, 2, and 3 weeks.

Heating Costs (Deterministic):

  • 1 week: €7.50
  • 2 weeks: €15.00
  • 3 weeks: €22.50

Pullover Costs (Stochastic):

1 Week Cold:

  • The pullover holds up for at least 1 week.
  • Expected cost: €10

Conclusion: Heating (€7.50) is cheaper → Choose heating

2 Weeks Cold:

  • 50% chance the pullover survives both weeks → Cost = €10
  • 50% chance it tears after week 1 → Need a second pullover → Cost = €20

Expected cost:
$0.5 \cdot 10 + 0.5 \cdot 20 = 15 \text{ EUR} $

Conclusion: Heating and pullover cost the same → Either is acceptable

3 Weeks Cold:

  • Probability tree:
    • 10% chance: Pullover lasts all 3 weeks → €10
    • 65% chance: Pullover tears in week 2 → need 2 pullovers → €20
    • 25% chance: Pullover tears in week 1 and again in week 2 → need 3 pullovers → €30

Expected cost:
$0.10 \cdot 10 + 0.65 \cdot 20 + 0.25 \cdot 30 = 21.50 \text{ EUR} $

Conclusion: Pullover (€21.50) is cheaper than heating (€22.50) → Choose pullover

Part b) What If the Duration of the Cold Spell Is Uncertain?

Now assume a weather report gives probabilities for the duration:

  • 75% chance: Cold spell lasts 1 week
  • 25% chance: Cold spell lasts 3 weeks
  • 0% chance: Cold spell lasts exactly 2 weeks

Expected Cost of Heating:

$0.75 \cdot 7.50 + 0.25 \cdot 22.50 = 5.625 + 5.625 = 11.25 \text{ EUR}$

Expected Cost of Pullovers:

$0.75 \cdot 10 + 0.25 \cdot 21.50 = 7.50 + 5.375 = 12.875 \text{ EUR}$

Conclusion: Heating is cheaper on average → Choose heating

Summary

  • Decision theory helps weigh fixed vs. uncertain costs.
  • In short or highly uncertain situations, fixed costs (like heating) may be safer.
  • When longer durations are more likely and predictable, taking risks (like buying a pullover) may be more economical.

This edition shows how everyday decisions, like how to stay warm,  can be modeled using tools from statistical decision theory.

Want to explore a similar real-life decision you’re facing? Ask the model!