5  Uncertainty

When a decision must be made under uncertainty, the state of nature is fully unknown. That is, different possible states of nature exist but no information on their probability of occurrences are given. The optimal rational choice can’t be made without a criterion that reflect preferences such as risk aversion. In the following, I discuss some popular criteria.

Laplace criterion

The Laplace criterion assigns equal probabilities to all possible payoffs for each alternative, then selects the alternative with the highest expected payoff. An example can be found in Finne (1998). In Table 5.1 are the data for another example. According to the expected average utility, the decision should be \[a_2\succ a_1 \succ a_3.\]

Table 5.1: Example data for uncertainty
Alternatives \(N_1\) \(N_2\) \(N_3\) Laplace Maximax Minimax
\(a_1\) 30 40 50 120/3 50 30
\(a_2\) 25 70 30 125/3 70 25
\(a_3\) 10 20 80 110/3 80 10

Maximax criterion (go for cup)

If you’re aiming for the best possible outcome without regard for the potential worst-case scenario, you would choose the alternative with the highest possible payoff. This “go for cup” approach focuses on maximizing the best-case outcome.

In the example of Table 5.1, the decision applying the Maximax strategy is \[a_3\succ a_2 \succ a_1.\]

Minimax criterion (best of the worst)

The Minimax (or Maximin) criterion is a conservative approach, aimed at securing the best outcome under the worst possible conditions. This approach is often used by risk-averse decision-makers. For examples on how to apply this criterion, see Finne (1998).

In the example of Table 5.1, the decision applying the Maximax strategy is \[a_1\succ a_2 \succ a_3.\]

Savage Minimax criterion

The Savage Minimax criterion minimizes the worst-case regret by selecting the option that performs as closely as possible to the optimal decision. Unlike the traditional minimax, this approach applies the minimax principle to the regret (that is, the difference or ratio of payoffs), making it less pessimistic. For more details, see Finne (1998).

Using the example data of Table 5.1, the regret table is constructed by subtracting the maximum payoff in each state from the payoffs in that state, see Table 5.2.

Table 5.2: Savage regret table
Alternatives Regret for \(N_1\) Regret for \(N_2\) Regret for \(N_3\) Maximum Regret
\(a_1\) \(30-30 = 0\) \(70-40 = 30\) \(80-50 = 30\) 30
\(a_2\) \(30-25 = 5\) \(70-70 = 0\) \(80-30 = 50\) 50
\(a_3\) \(30-10 = 20\) \(70-20 = 50\) \(80-80 = 0\) 50

Based on the Savage Minimax criterion, the alternative with the smallest maximum regret should be chosen and the decision is \[a_1\succ a_2 \sim a_3.\]

Hurwicz criterion

The Hurwicz criterion allows the decision-maker to calculate a weighted average between the best and worst possible payoff for each alternative. The alternative with the highest weighted average is then chosen.

For each decision alternative, the weight \(\alpha\) is used to compute Hurwicz the value: \[ H_i=\alpha \cdot \overline{O}_i + (1-\alpha)\cdot \underline{O}_i \] where \[\overline{O}_i=\max_{j=1,\dots,p}{O_{ij}}\quad \forall i\] and
\[\underline{O}_i=\min_{j=1,\dots,p}{O_{ij}}\qquad \forall i,\] that is, the respective maximum and minimum output for each alternative, \(i\).

This formula allows for flexibility in decision-making by adjusting the value of \(\alpha\), which reflects the decision-maker’s optimism (with \(\alpha=1\) representing complete optimism and \(\alpha=0\) representing complete pessimism).

The Hurwicz criterion calculates a weighted average between the best and worst payoffs for each alternative. Using the data of Table 5.1 once again, we need to assume an optimism index. Let’s say we are slightly optimistic and willing to take some risks by setting \(\alpha = 0.6\).

In \(a_1\), the maximum payoff is 50 and the minimum payoff is 30. \[ H_1 = 0.6 \cdot 50 + (1 - 0.6) \cdot 30 = 30 + 12 = 42 \]

In \(a_2\), the maximum payoff is 70 and the minimum payoff is 25. \[ H_2 = 0.6 \cdot 70 + (1 - 0.6) \cdot 25 = 42 + 10 = 52 \]

In \(a_3\), the maximum payoff is 80 and the minimum payoff is 10. \[ H_3 = 0.6 \cdot 80 + (1 - 0.6) \cdot 10 = 48 + 4 = 52 \]

Thus, the decision is \[a_2\sim a_3 \succ a_1.\]

Exercise 5.1 Use the data of Table 5.1 to make a decision for a complete pessimist (\(\alpha = 0\)) and an optimist (\(\alpha = 1\))

When \(\alpha = 0\):

  • \(a_1\): ( H_1 = 30 )
  • \(a_2\): ( H_2 = 25 )
  • \(a_3\): ( H_3 = 10 )

Thus, the decision is \[a_1\succ a_2 \succ a_3.\]

When \(\alpha = 1\):

  • \(a_1\): ( H_1 = 50 )
  • \(a_2\): ( H_2 = 70 )
  • \(a_3\): ( H_3 = 80 )

Thus, the decision is \[a_3\succ a_2 \succ a_1.\]

Error in Finne (1998)

The example that is shown in Figure 7 of Finne (1998, p. 401) contains some errors. Here is the correct table including the Hurwicz-values (we assume a \(\alpha=.5\)):

\(O_{ij}\) \(min(\theta_1)\) \(max(\theta_2)\) \(H_i\)
\(a_1\) 36 110 73
\(a_2\) 40 100 70
\(a_3\) 58 74 66
\(a_4\) 61 66 63.5

Thus, the order of preference is \(a_4\succ a_3 \succ a_1 \succ a_2\).

Exercise 5.2 Three categories

Read Finne (1998) and answer the following questions:

  1. Explain the three categories of decision making.
  2. Give examples of the three categories of decision making.
  3. Explain the four criteria for decision making under uncertainty.
Finne, T. (1998). The three categories of decision-making and information security. Computers & Security, 17(5), 397–405.

Exercise 5.3  

Table 5.3: Payoff table
z1 z2 z3
a1 0 9 8
a2 8 5 9
a3 1 10 1
a4 4 5 2
a5 9 10 10

In Table 5.3 you see a payoff table where high numbers indicate high disutility, ranging from \(0\) (no disutility) to \(10\) (high disutility).

  1. Using this payoff table, indicate which alternatives can be excluded because they are dominated by other alternatives. (Hint: Exclude dominated alternatives in the calculations of the following questions).
  2. Using the , calculate the complete order of preferred alternative(s), \(a_i\).
  3. Using the , calculate the complete order of preferred alternative(s), \(a_i\).
  4. Using the , calculate the complete order of preferred alternative(s), \(a_i\).