| |
Learning Paths / Theoretical Foundations / Strategic equivalence
4.3 Strategic equivalence
|
Definition 2.3.1 (Keeney anf Raiffa 1976)
|
 |
The value functions v1 and v2 are strategically equivalent, written v1 ~ v2, if v1 and v2 have the same indifference curves and induced preferential ordering
|
|
|
|
-
For example, value functions
 | ( 2.3.1) |
are strategically equivalent, since they imply the same preference structure.
-
The same argument can be used to show that positive affine transformations of a value function are strategically equivalent. That, is
 | ( 2.3.2) |
where
and
are constants and
> 0.
-
Consequently, there is no absolute value scale. For example, value "50" has no interpretation without the context and all value functions can be scaled to give outcomes within the desired range. Most commonly values are scaled either between 0 and 1, or between 0 and 100.
-
To conclude, a value function uniquely specifies the entirely preference structure, but a preference structure does not uniquely specify a value function.
 |
| Figure 2.3.1 Value function and preferences |
|
|
|
Axiomatic foundations 