Let us check the Polity to begin with:
1. N is the set of all individuals in society.
And that which their politics concerns – the state of society.
2. S is the set of all possible information contained within society, so that a set s ∈ 2S (2S being the set of all possible subsets of S) contains all extant information about a particular iteration of society and will be called the state of society. S is an arbitrary topological space.
And the means by which individuals make judgements about that which their politics concerns. Their preferences over the information contained within the state of society.
3. Each individual i ∈ N has a complete and transitive preference relation ≽i defined over a set of preference-information Si ⊂ S such that si ≽ s′i can be read “individual i prefers preference information si at least as much as preference-information s′i”.
Any particular set of preference-information si ⊂ Si can be thought of as the state of society as viewed by individual i. The set of preference-information for individual i is a subset of the information contained within a particular iteration of society, so si ⊂ s ⊂ S.
A particular state of society s is a Pareto efficient if there is no other state of society s′ for which one individual strictly prefers their preference-information s′i ⊂ s′ to that particular state si ⊂ s, and the preference-information s′j ⊂ s′ in the other state s′ is at least as preferred by every other individual j ≠ i.
4. A state s ∈ S is said to be Pareto efficient iff ∄ s′ ∈ 2S & i ∈ N : s′i ≻ si & s′j ≽ sj ∀ j ≠ i ∈ N.
To put it crudely, a particular state of society is Pareto efficient if no individual can be made “better off” without making another individual “worse off”. A dynamic concept which mirrors this is the concept of a Pareto improvement – whereby a change in the state of society leaves everyone at least indifferent, and at least one individual in a preferable situation.
5. A movement between two states of society, s → s′ is called a Pareto improvement iff ∃ i ∈ N : s′i ≻ si & s′j ≽ sj ∀ j ≠ i ∈ N .
Note that this does not imply that s′ is a Pareto efficient state, because the same could potentially be said of a movement s′ → s′′. The state s′ is only a Pareto efficient state if we cannot find yet another state for which the movement to that state is a Pareto improvement. The following Theorem, demonstrates this distinction and gives an alternative definition of Pareto efficiency.
Theorem: A state s ∈ 2S is Pareto efficient iff there is no other state s′ for which the movement s → s′ is a Pareto improvement.
If one adheres to a consequentialist political doctrine (such as classical utilitarianism) rather than a deontological doctrine (such as liberalism) in which action is guided by some categorical imperative other than consequentialism, the guide offered by Pareto improvement is the least controversial, and least politically committal criterion to decision-making one can find. Indeed if we restrict political statements to those which concern the assignation of losses, it is a-political. It makes a value judgement only about who ought gain (whosoever stands to).
Unless one holds a strict deontological doctrine in the style, say, of Robert Nozick’s Anarchy state and Utopia (in which the maintenance of individual freedom is the categorical imperative), or John Rawls’ A Theory of Justice (in which again individual freedom is the primary categorical imperative and the betterment of the “poorest” the second categorical imperative), it is more difficult to argue against implementing some decision which will cause a change of society which all individuals in society will be at worst indifferent to. Than arguing for some decision rule which will induce a change of society which some individual will find less preferable. To the rationalisitic economist it seems almost petty, certainly irrational to argue against this criterion, like those individuals who demand “fairness” in the famous “dictator” experiment rather than accept someone else becoming “better off”, and themselves no “worse off”.