Textbook
The vectors $a,b\in\mathbb R^d$ contain criteria, all to be maximized: larger means better. The componentwise comparison $a\succeq b$ is defined by $a_i\ge b_i$ for every $i\in\{1,\ldots,d\}$.
Dominance $a\succ b$ requires $a\succeq b$ and at least one index $j$ with $a_j>b_j$. Separately, strict improvement in every criterion is written $a\succ\succ b$: every $a_i>b_i$. Identical vectors do not dominate each other under the first definition.
The vectors $(5,3)$ and $(3,4)$ are incomparable: each wins one coordinate and loses the other. An average or weighted sum introduces an additional preference rule absent from dominance itself.
For minimization, the inequalities reverse. Symbols vary between authors: a single dominance sign may require strict improvement everywhere, and matrix comparisons may use it differently. Table I in Zitzler and colleagues uses $\succ\succ$ for strict dominance in a minimization setting; these examples explicitly maximize.
Dominance is transitive: if $a$ dominates $b$ and $b$ dominates $c$, then $a$ is no worse than $c$ in every coordinate. A strict improvement from the first comparison remains strict relative to $c$. Incomparability, however, is not equality and does not follow the same transitive rule. The relation need not order every pair of alternatives.
Definitions and notation: Performance Assessment of Multiobjective Optimizers: An Analysis and Review.
Related concepts: Coordinate Vectors: Order and Dimension.
In plain language
Laptops can be compared by battery life and memory, with larger values preferred for both. If one is no worse on either and better on at least one, its rival has no compensating advantage within those criteria. More memory at the cost of shorter battery life leaves the alternatives incomparable; Pareto does not secretly recruit a jury from two numbers. Each criterion needs its own direction because lower weight is usually preferable. A heavy battery earns no prize merely for an impressive kilogram figure.
Example
In this constructed scene, both coordinates are abstract criteria to maximize. Each alternative is compared with the baseline $(3,4)$.
Pareto dominance: comparing several criteria
Baseline
Equal coordinates mean no worse, but there is no improvement.
Step 1 of 4: Baseline
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Baseline
Equal coordinates mean no worse, but there is no improvement.
- Baseline
- 34
- Candidate
- 34
One criterion improves
The first coordinate improves and the second ties: dominance holds.
- Baseline
- 34
- Candidate
- 5included4
Both improve
Every coordinate improves strictly.
- Baseline
- 34
- Candidate
- 5included7included
A trade-off
One criterion wins and the other loses. Neither vector dominates.
- Baseline
- 34
- Candidate
- 5included3excluded
Against baseline $(3,4)$, alternative $(5,4)$ passes two “no worse” checks and one “strictly better” check. Alternative $(5,7)$ improves both coordinates, a stronger condition than ordinary dominance needs. For $(5,3)$, the second coordinate alone refutes dominance over the baseline; the first coordinate refutes dominance in the reverse direction. Two explicit obstructions establish incomparability rather than a reluctance to choose a winner.
The direction of each criterion and the definition of the symbols determine the comparison. Without them, “better” is still a blank field.