4 Certainty
When a decision must be made under certainty, the state of nature is fully known, and the optimal choice is to select the alternative with the highest payoff. However, determining this payoff can be complex, as it may be the result of a sophisticated function involving multiple variables.
For example, imagine you need to choose between four different restaurants (\(a_1\), \(a_2\), \(a_3\), \(a_4\)). Each restaurant offers a unique combination of characteristics, such as the quality of the food (\(k_1\)), the quality of the music played (\(k_2\)), the price (\(k_3\)), the quality of the service (\(k_4\)), and the overall environment (\(k_5\)). The corresponding payoff Table 4.1 assigns a numerical value to each characteristic, with higher numbers indicating better quality.
In this scenario, \(a_i\) represents the different restaurant options, \(k_i\) refers to specific characteristics of each restaurant, and the numbers in the table indicate the payoffs associated with each characteristic.
Please note that the characteristics \(k_j\) of the scheme in Table 4.1 do not represent different states of nature but represent characteristics and its corresponding utility (whatever that number may mean in particular) of one particular characteristics if we choose a respective alternative.
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | |
|---|---|---|---|---|---|
| \(a_1\) | 3 | 0 | 7 | 1 | 4 |
| \(a_2\) | 4 | 1 | 4 | 2 | 1 |
| \(a_3\) | 4 | 0 | 3 | 2 | 1 |
| \(a_4\) | 5 | 1 | 2 | 3 | 1 |
Domination
To arrive at an overall outcome for each alternative and make an informed decision, the first step is to determine whether any alternatives are dominated by others. An alternative is considered dominated if it is not superior in any characteristic compared to at least one other alternative.
Dominated alternatives can be excluded from consideration. For example, in Table 4.2, we can see that alternative 2 outperforms alternative 3. This makes it unnecessary to consider alternative 3 in the decision-making process.
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | |
|---|---|---|---|---|---|
| \(a_1\) | 3 | 0 | 7 | 1 | 4 |
| \(a_2\) | 4 | 1 | 4 | 2 | 1 |
| \(a_3\) | |||||
| \(a_4\) | 5 | 1 | 2 | 3 | 1 |
Weighting
No preferences
Still, we have three alternative left. How to decide? Well, we need to become clear what characteristics matter (most). Suppose you don’t have any preferences than you would go for restaurant \(a_1\) because it offers the best average value, see Table 4.3.
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | Overall | |
|---|---|---|---|---|---|---|
| \(a_1\) | 3 | 0 | 7 | 1 | 4 | 14/5 |
| \(a_2\) | 4 | 1 | 4 | 2 | 1 | 12/5 |
| \(a_4\) | 5 | 1 | 2 | 3 | 1 | 12/5 |
Clear preferences
Suppose you have a preference for the first three characteristics, that are quality of the food (\(k_1\)), the quality of the music played (\(k_2\)), and the price (\(k_3\)). Specifically, suppose that your preference scheme is as follows:
\[ g_1 : g_2 : g_3 : g_4 : g_5 = 3 : 4 : 3 : 1 : 1 \]
This means, for example, that you value music (\(k_2\)) four times more than the quality of the service (\(k_4\)) and the overall environment (\(k_5\)). The weights assigned to each characteristic are:
\[ w_1=3/12; w_2=4/12; w_3=3/12; w_4 = w_5 =1/12. \]
To determine the best decision, you can calculate the aggregated expected utility for each alternative as follows:
\[ \Phi(a_i)=\sum_{c}w_p\cdot u_{ic} \rightarrow max, \]
where \(u_{ic}\) represents the utility (or value) of alternative \(i\) for a given characteristic \(c\). The results of this calculation are shown in Table 4.4.
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | \(\Phi(a_i)\) | |
|---|---|---|---|---|---|---|
| \(a_1\) | 3 | 0 | 7 | 1 | 4 | 35/12 |
| \(a_2\) | 4 | 1 | 4 | 2 | 1 | 31/12 |
| \(a_4\) | 5 | 1 | 2 | 3 | 1 | 29/12 |
Thus, alternative \(a_1\) offers the best value given the preference scheme outlined above. In summary, we express the choice as follows: \[a_1\succ a_2 \succ a_4 \succ a_3,\] where \(\succ\) represents the preference relation (that is, “is preferred to”). If two alternatives offer the same value and we are indifferent between them, we can use the symbol \(\sim\) to represent this indifference.
Maximax (go for cup)
If you like to go for cup, that is, you search for a great experience in at least one characteristic, then, you can choose the alternative that gives the maximum possible output in any characteristic. The choice would in our example be (see Table 4.5): \[a_1\succ a_4 \succ a_2 \sim a_3,\]
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | Overall | |
|---|---|---|---|---|---|---|
| \(a_1\) | 3 | 0 | 7 | 1 | 4 | 7 |
| \(a_2\) | 4 | 1 | 4 | 2 | 1 | 4 |
| \(a_3\) | 4 | 0 | 3 | 2 | 1 | 4 |
| \(a_4\) | 5 | 1 | 2 | 3 | 1 | 5 |
Minimax (best of the worst)
The Minimax (or maximin) criterion is a conservative criterion because it is based on making the best out of the worst possible conditions. The choice would in our example be (see Table 4.6): \[a_2\sim a_4 \succ a_1 \sim a_3,\]
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | Overall | |
|---|---|---|---|---|---|---|
| \(a_1\) | 3 | 0 | 7 | 1 | 4 | 0 |
| \(a_2\) | 4 | 1 | 4 | 2 | 1 | 1 |
| \(a_3\) | 4 | 0 | 3 | 2 | 1 | 0 |
| \(a_4\) | 5 | 1 | 2 | 3 | 1 | 1 |
Körth’s Maximin-Rule
According to this rule, we compare alternatives by the worst possible outcome under each alternative, and we should choose the one which maximizes the utility of the worst outcome. More concrete, the procedure consists of 4 steps:
- Calculate the utility maximum for each column \(c\) of the payoff matrix (see Table 4.7): \[\overline{O}_c=\max_{i=1,\dots,m}{O_{ic}}\qquad \forall c.\]
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | |
|---|---|---|---|---|---|
| \(a_1\) | 3 | 0 | 7 | 1 | 4 |
| \(a_2\) | 4 | 1 | 4 | 2 | 1 |
| \(a_3\) | 4 | 0 | 3 | 2 | 1 |
| \(a_4\) | 5 | 1 | 2 | 3 | 1 |
| \(\overline{O}_c\) | 5 | 1 | 7 | 3 | 4 |
- Calculate for each cell the relative utility (see Table 4.8), \[\frac{O_{ij}}{\overline{O}_j}.\]
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | |
|---|---|---|---|---|---|
| \(a_1\) | 3/5 | 0/1 | 7/7 | 1/3 | 4/4 |
| \(a_2\) | 4/5 | 1/1 | 4/7 | 2/3 | 1/4 |
| \(a_3\) | 4/5 | 0/1 | 3/7 | 2/3 | 1/4 |
| \(a_4\) | 5/5 | 1/1 | 2/7 | 3/3 | 1/4 |
- Calculate for each row \(i\) the minimum (see Table 4.9): \[\Phi(a_i)=\min_{j=1,\dots,p}\left(\frac{O_{ij}}{\overline{O}_j}\right) \qquad \forall i.\]
| \(k_1\) | \(k_2\) | \(k_3\) | \(k_4\) | \(k_5\) | \(\Phi(a_i)\) | |
|---|---|---|---|---|---|---|
| \(a_1\) | 3/5 | 0/1 | 7/7 | 1/3 | 4/4 | 0 |
| \(a_2\) | 4/5 | 1/1 | 4/7 | 2/3 | 1/4 | 1/4 |
| \(a_3\) | 4/5 | 0/1 | 3/7 | 2/3 | 1/4 | 0 |
| \(a_4\) | 5/5 | 1/1 | 2/7 | 3/3 | 1/4 | 1/4 |
- Set preferences by maximizing \(\Phi(a_i)\): \[a_2\sim a_4 \succ a_1 \sim a_3,\]