Hurwitz’s Index
How to choose an alternative?
Idea: in-between the extreme pessimism and extreme optimism
- this is a “compromise” between Max Min Strategy and Max Max Strategy
Coefficient of Pessimism:
- define $\alpha \in [0, 1]$ as the coefficient of pessimism
- then choose the solution $a \in A$ that is the closest to
- $\max_{a \in A} \big[ \alpha \cdot \min_{e \in E} c(a, e) + (1 - \alpha) \cdot \max_{e \in E} c (a, e) \big]$
Example:
$\alpha = 0.5$
| $c$ | $e_1$ | $e_2$ | $e_3$ | $\alpha =0.5$ | $a_1$ | 40 | 70 | -20 | (-20 + 70) / 2 | 25 | $a_2$ | -10 | 40 | 100 | (-10 + 100) / 2 | 45 | $a_3$ | 20 | 40 | -5 | (-5 + 40) / 2 | 17.5 | max: | 45 | $\to a_2$ |
Downsides
- bad use of information
- we combine two approaches (Max Min Strategy and Max Max Strategy) that both suffer from bad use of information
- now the scale matters - since we multiply by $\alpha$
- how to determine and justify $\alpha$?