In economics, the overtaking criterion is used to compare infinite streams of outcomes.
Mathematically, it is used to properly define a notion of optimality for a problem of optimal control on an unbounded time interval.
[1] Often, the decisions of a policy-maker may have influences that extend to the far future.
Economic decisions made today may influence the economic growth of a nation for an unknown number of years into the future.
In such cases, it is often convenient to model the future outcomes as an infinite stream.
Then, it may be required to compare two infinite streams and decide which one of them is better (for example, in order to decide on a policy).
The overtaking criterion is one option to do this comparison.
E.g., it may be the set of positive real numbers, representing the possible annual gross domestic product.
is the set of infinite sequences of possible outcomes.
Given two infinite sequences
is called the "overtaking criterion" if there is an infinite sequence of real-valued functions
such that:[2] An alternative condition is:[3][4] Examples: 1.
: This shows that a difference in a single time period may affect the entire sequence.
are incomparable: The partial sums of
are larger, then smaller, then equal to the partial sums of
, so none of these sequences "overtakes" the other.
This also shows that the overtaking criterion cannot be represented by a single cardinal utility function.
I.e, there is no real-valued function
: Hence, there is a set of disjoint nonempty segments in
In contrast, every set of disjoint nonempty segments in
is called the "overtaking criterion" if it satisfies the following axioms: 1.
is a continuous relation in the obvious topology on
is preferentially-independent (see Debreu theorems#Additivity of ordinal utility function for a definition).
are essential (have an effect on the preferences).
Every partial order that satisfies these axioms, also satisfies the first cardinal definition.
[2] As explained above, some sequences may be incomparable by the overtaking criterion.
This is why the overtaking criterion is defined as a partial ordering on
The overtaking criterion is used in economic growth theory.
[5] It is also used in repeated games theory, as an alternative to the limit-of-means criterion and the discounted-sum criterion.
See Folk theorem (game theory)#Overtaking.