Decision making under uncerainty

Decision Making under Uncertainty

When a decision-maker doesn’t know the probabilities of various outcomes (states of nature), they use decision criteria under uncertainty. Here are five key methods:

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i) Maximax (Gain) or Minimin (Loss) Criterion

• Maximax: Optimistic approach — chooses the alternative with the maximum possible gain.

• Minimin: If dealing with costs/losses, chooses the alternative with the minimum possible loss.

Example:

Alternatives	Outcome 1	Outcome 2	Outcome 3
A1	40	10	20
A2	30	25	15
A3	20	50	10

• Maximax:
  A1 → Max = 40
  A2 → Max = 30
  A3 → Max = 50 → Choose A3

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ii) Maximin Criterion

• Pessimistic approach — choose the alternative whose minimum payoff is the highest.

Example:

From the same table above:

• A1 → Min = 10

• A2 → Min = 15

• A3 → Min = 10
  → Choose A2

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iii) Hurwicz Alpha Criterion

• Weighted average of best and worst outcomes.

• Formula: Hurwicz value = α × (best payoff) + (1 – α) × (worst payoff)

• α is the coefficient of optimism (0 ≤ α ≤ 1)

Example:

Let α = 0.6

• A1 → 0.6×40 + 0.4×10 = 24 + 4 = 28

• A2 → 0.6×30 + 0.4×15 = 18 + 6 = 24

• A3 → 0.6×50 + 0.4×10 = 30 + 4 = 34 → Choose A3

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iv) Laplace Criterion

• Assumes all outcomes are equally likely.

• Take the average payoff for each alternative.

Example:

• A1: (40+10+20)/3 = 23.33

• A2: (30+25+15)/3 = 23.33

• A3: (20+50+10)/3 = 26.67 → Choose A3

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v) Minimax Regret Criterion

• Builds a regret table:

  1. For each state, find the maximum payoff.

  2. Subtract actual payoff from the maximum to get regret.

  3. Pick the alternative with the minimum of the maximum regrets.

Regret Table:

Alternatives	Regret 1	Regret 2	Regret 3	Max Regret
A1	0	40	0	40
A2	10	25	5	25
A3	20	0	10	20 → Choose A3

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🌳 Simple Decision Tree Example

Scenario:

You can invest in either Project A or Project B.

Project	Success (₹)	Failure (₹)
A	100,000	20,000
B	80,000	50,000

But you don’t know the probabilities, so use a decision tree under uncertainty.

                [Decision Node]
                    /       \
               Project A   Project B
                /    \       /    \
          Success  Failure  Success  Failure
         (100k)    (20k)    (80k)    (50k)

• Use any of the criteria (maximax, maximin, etc.) to decide:

  • Maximax: Choose A (100k vs 80k)

  • Maximin: Choose B (50k vs 20k)