Decision Under Uncertainty
This is a tool to model Decision Analysis problems. Unlike Decision Under Risk, here we cannot obtain the probability distribution of possible consequences, can only list the scenarios.
; Decision Under Uncertainty
- it is impossible to determine with certainty the consequences of implementing an alternative - there are no probabilities
- '’Nature’’ decides what will happen
- we make a decision and then the Nature decides what is going to happen based on the decision
- this is like a two-player game, where consequences of my actions are based on what the Nature decides to do
The Model
Based on that we define the following model:
- $A$ is a set of alternatives
- $a \in A$ is a decision that you may take
- $E$ is a set of states of nature
- $e \in E$ is a decision that the Nature can take and influence the consequences of at least one alternative $a \in A$
- $X$ is a set of consequences
- $c: A \times E \mapsto X$
- $c(a, e)$ is the consequence of implementing $a$ when the Nature decides to do $e$
Decision Table
A finite case is usually modeled with a decision table:
- $A$ and $E$ are both finite
- $c(a_i, e_j$ - the consequence of implementing $a_i$ when $e_j$ happens
| $c$ | $e_1$ | $e_2 $ | $…$ | $e_n$ | $a_1$ | $c(a_1, e_1)$ | $c(a_1, e_2)$ | $…$ | $c(a_1, e_n)$ || $a_2$ | $c(a_2, e_1)$ | $c(a_2, e_2)$ | $…$ | $c(a_2, e_n)$ || $…$ | $…$ | $…$ | … | $…$ || $a_n$ | $c(a_n, e_1)$ | $c(a_n, e_2)$ | $…$ | $c(a_n, e_n)$ | Filling this table in is already very difficult (the same is in MCDA problems)
Dominance
Using the dominance principle we can remove a set of actions from $A$
- we can always remove the actions that are dominated
- the definition is exactly the same as for Normal Form Games
- $a \ D \ b$ - $a$ dominates $b$
; $a \in A$ is ‘‘efficient’’
- if $a$ is not dominated by any other alternative
$A^*$ is a set of efficient solutions
- when $A$ and $E$ are finite, can define them $A^*$ as
-
$A^* = { a \in A \ \ \forall b: \overline{b \ D \ a} }$ - i.e. $A^*$ is a set of alternatives that are not dominated by any other - $A^*$ is always not empty
Problems with Using the Dominance Principle
- dominated solutions are sometimes also good
Examples
Example 1: The Omelet Problem
The Game:
- you want to cook an omelet and you have only 6 eggs
- but you know that the last egg may be bad
- so there are 2 states of nature:
- the egg is good ($e_g$) or
- the egg is bad ($e_b$)
- and we can implement the following actions:
- put the egg into the omelet without checking it ($a_b$)
- throw the egg away without checking it ($a_t$)
- check the egg: use an additional bowl for it ($a_c$)
| $c$ | $e_g$ | $e_b$ | $a_b$ | O with 6 eggs | No O || $a_t$ | O with 5 eggs | O with 5 eggs || $a_c$ | O with 6 eggs
+ a bowl to wash | O with 5 eggs
+ a bowl to wash |
Notes:
- We don’t know anything about the probability - just scenarios
- no way to add additional information
Example 2
In this case the consequences are real numbers:
- $X \equiv \mathbb{R}$
- this is usually the case in such models
| $c$ | $e_1$ | $e_2$ | $e_3$ | $a_1$ | 40 | 70 | -20 | $a_2$ | -10 | 40 | 100 | $a_3$ | 20 | 40 | -5 |
Methods
How to make a choice?
- Max Min Strategy - extreme pessimism
- Max Max Strategy - extreme optimism
- Hurwitz’s Index - between the extreme pessimism and the extreme optimism
- Min Max Regret Strategy - when we want to minimize the regret of a missed opportunity
- Laplace Rule - the principle of insufficient reasoning
Note that these methods, like in Voting Theory (see Voting Theory Examples)
| $c$ | $e_1$ | $e_2$ | $e_3$ | $e_4$ | $a$ | 2 | 2 | 0 | 1 || $b$ | 1 | 1 | 1 | 1 || $c$ | 0 | 4 | 0 | 0 || $d$ | 1 | 3 | 0 | 0 | Note that in this case all the methods will give different results
- MinMax - $b$
- MaxMax - $c$
- Laplace - $a$
- Savage - $d$