Pareto improvement as partial order
We use the concept of Pareto improvement to motivate and illustrate partial and total orders.
We covered the notion of Pareto improvement in the preceding post. I briefly alluded to the fact that it’s a strict partial order. Let’s explore that a bit more.
Partial order
A strict partial order is a binary relation that is irreflexive (no element precedes itself), and transitive. Pareto improvement satisfies both of these criteria:
- Irreflexive: \(\neg (A \prec A)\)
- No scenario is a Pareto improvement over itself because no one strictly prefers it (person 1 doesn’t prefer scenario A to scenario A), trivially.
- Transitive: \(A \prec B, B \prec C \Rightarrow A \prec C\)
- If arbitrary scenario B is a Pareto improvement over arbitrary scenario A and arbitrary scenario C is a Pareto improvement over scenario B, then scenario C is a Pareto improvement over scenario A. In other words, if no one is worse off in scenario B than scenario A and no one is worse off in scenario C than scenario B, then, clearly, no one is worse off in scenario three than scenario A. And if at least one person is better off in scenario B than scenario A and at least one person is better off in scenario C than scenario B, then, clearly, at least one person is better off in scenario C than scenario A.