Laplace Rule
How to choose an alternative in Decision Under Uncertainty? The Laplace Rule, also called the Principle of Insufficient Reasoning, helps to do that.
Main idea
- (using notation from Decision Under Uncertainty)
- since there is no way to assess probabilities in Decision Under Uncertainty models - assume the uniform distribution
-
so each state of nature $e \in E$ is expected to happen with probability $1 / E $ - and we compute the expected values of the consequences based on these probabilities
So,
- choose $a \in A$ that solves the following:
- $\max_{a \in A} \sum_{e \in E} \cfrac{1}
Example
Consider this matrix:
| $c$ | $e_1$ | $e_2$ | $e_4$ | $a_1$ | 40 | 70 | -20 | 90/3 | $a_2$ | -10 | 40 | 100 | 130/3 | $a_3$ | 20 | 40 | -5 | 55/3 |
So we choose $a_2$ because it gives the best result on average
Remarks
- it must be meaningful to make a linear combination of the consequences
- i.e. the scale should be numerical
- the principle should be used with caution
- you either become the king of Belgium or not - are these events equally likely?
Manipulation
can manipulate the results by adding new alternatives
- (Independence to Third Alternatives is not satisfied)
Suppose we have two scenarios:
- $E$ and $\overline{E}$
- by the principle we have $P(E) = P(\overline{E}) = 0.5$
- suppose we modify an alternative by adding something that is always true
- $\overline{E} \equiv \overline{E} \land (R \lor \overline{R})$
- then we can distribute the OR and have $[\overline{E} \land R] \lor [\overline{E} \land \overline{R}]$
- so now we have 3 alternatives:
- $E$, $\overline{E} \land R$ and $\overline{E} \land \overline{R}$
- and now $P(E) = P(\overline{E} \land R) = P(\overline{E} \land \overline{R}) = 1/3$
- i.e. we’ve manipulated the results