Preferential Independence

The preference independence principle is an important principle from MCDA for choosing criteria: they should be preferential independent.

Suppose we have 4 alternatives $a,b,c,d$ and a subset of criteria $J \subset G$ such that

  • $g_i(a) = g_i(b), \forall i \not \in J$
  • $g_i(c) = g_i(d), \forall i \not \in J$
  • $g_i(a) = g_i(a), \forall i \in J$
  • $g_i(b) = g_i(d), \forall i \in J$

preferential-independence.png

  • for criteria that are in $J$, $a$ is the same as $c$ and $b$ is the same as $d$
  • for criteria not in $J$, $a$ is the same as $b$, $c$ is the same as $d$

Preferential Independence

  • $J \subset A$ is preferentially independent within $G$ when
  • if $\forall a,b,c,d \in A$ these conditions hold
  • then $a \ P \ b \iff c \ P \ d$


if a decision maker says $a \ P \ b$ we know that he bases his opinion on the set of $J$, because in $\overline{J}$ $a \ I \ b$ - they are the same


Examples

Example 1

$g_1$ $g_2$ $g_3$
$a$ 45 70 100
$b$ 50 70 80
$c$ 45 90 100
$d$ 50 90 80


criteria $\{g_1, g_3\}$ are preferentially independent

  • i.e. $a \ P \ b \iff c \ P \ d$


Example 2

In this example the Preferential Independence principle is not satisfied

We're in a restaurant and there are 2 dishes and 2 drinks

  • dishes: fish, meat
  • drinks: red wine, white wine
  • so we have 4 combinations:
drinks $\downarrow$
meal $\to$ $(a)$ fish + white $(c)$ fish + red
$(b)$ meal + white $(d)$ meal + red

So we have two criteria:

  • $g_1$ - meal
  • $g_2$ - drink


For meal:

  • $g_1(a) = g_1(c)$
  • $g_1(b) = g_1(d)$

For drink:

  • $g_2(a) = g_2(b)$
  • $g_2(c) = g_2(d)$

Not satisfied:

  • If, when asked "what would you prefer - meat or fish", the decision maker asks "with what drink"
  • then the preferential independence is not satisfied: these two criteria are dependent
  • usually the case in real life

Satisfied

  • if a DM can say
  • "I always prefer meat to fish" ($b \ P \ a \land d \ P \ c$) and
  • "I always prefer red wine to white wine" ($c \ P \ a \land d \ P \ b$)
  • then if he says "I prefer meat with red wine to meat with white line" then he will say "I prefer fish with red wine to fish with white line" ($d \ P \ c \Rightarrow c \ P \ a$)
  • usually not the case


Example 3

Taken from [1]

Suppose we are choosing a car and there are two criteria

  • style: sport, SUV
  • color: red, black

Color is preferentially independent from style when

  • if the DM prefers:
    • (red, sport) to (black, sport)
    • (red, SUV) to (black, SUV)
  • then the color is preferentially independent from style
    • red is preferred to black regardless of style
  • preferential-independence-ex0.png


However style is not necessarily independent from color

  • if DM prefers
    • (red, sport) to (red, SUV), but
    • (black, SUV) to (black, sport)
  • then the style is not preferentially independent from color
    • because the color influences what decision maker prefers


With graphical depiction it's clear:

  • preferential-independence-ex1.png
  • red is always preferred to black (all edges come from red to black)
  • but when it comes to style, it's not the case: one edge comes from sport to SUV, another from SUV to sport


Example 4

Consider a case when an employer wants to hire a new worker on the basis of their age, degree and professional experience.

Worker $g_A$: Age $g_D$: Degree $g_E$: Experience
$a$ 25 Master No Experience
$b$ 25 No Degree 3 Years
$c$ 35 Master No Experience
$d$ 35 No Degree 3 Years

We see that given $J = \{g_A\}$ and $\overline{J} = \{ g_D, g_e \}$:

  • $g_i(a) = g_i(b), \forall i \not \in J$
  • $g_i(c) = g_i(d), \forall i \not \in J$
  • $g_i(a) = g_i(a), \forall i \in J$
  • $g_i(b) = g_i(d), \forall i \in J$

However an employee would prefer:

  • $a \ P \ b$ but $d \ P \ c$


Preferential Independence in Methods


Sources