Justified representation (JR) is a criterion of fairness in multiwinner approval voting. It can be seen as an adaptation of the proportional representation criterion to approval voting.

Proportional representation (PR) is an important consideration in designing electoral systems. It means that the various groups and sectors in the population should be represented in the parliament in proportion to their size. The most common system for ensuring proportional representation is the party-list system. In this system, the candidates are partitioned into parties, and each citizen votes for a single party. Each party receives a number of seats proportional to the number of citizens who voted for it. For example, for a parliament with 10 seats, if exactly 50% of the citizens vote for party A, exactly 30% vote for party B, and exactly 20% vote for party C, then proportional representation requires that the parliament contains exactly 5 candidates from party A, exactly 3 candidates from party B, and exactly 2 candidates from party C. In reality, the fractions are usually not exact, so some rounding method should be used, and this can be done by various apportionment methods.

In recent years, there is a growing dissatisfaction with the party system. A viable alternative to party-list systems is letting citizens vote directly for candidates, using approval ballots. This raises a new challenge: how can we define proportional representation, when there are no pre-specified groups (parties) that can deserve proportional representation? For example, suppose one voter approves candidate 1,2,3; another voter approves candidates 2,4,5; a third voter approves candidates 1,4. What is a reasonable definition of "proportional representation" in this case? Several answers have been suggested; they are collectively known as justified representation.

Below, we denote the number of seats by k and the number of voters by n. The Hare quota is n/k - the minimum number of supporters that justifies a single seat. In PR party-list systems, each voter-group of at least L quotas, who vote for the same party, is entitled to L representatives of that party.

A natural generalization of this idea is an L-cohesive group, defined as a group of voters with at least L quotas, who approve at least L candidates in common.

Ideally, we would like to require that, for every L-cohesive group, every member must have at least L representatives. This condition, called Strong Justified Representation (SJR), might be unattainable, as shown by the following example.

Example 1. There k=3 seats and 4 candidates {a,b,c,d}. There are n=12 voters with approval sets: ab, b, b, bc, c, c, cd, d, d, da, a, a. Note that the Hare quota is 4. The group {ab,b,b,bc} is 1-cohesive, as it contains 1 quota and all members approve candidate b. Strong-JR implies that candidate b must be elected. Similarly, The group {bc,c c,cd} is 1-cohesive, which requires to elect candidate c. Similarly, the group {cd,d,d,da} requires to elect d, and the group {da,a,a,ab} requires to elect a. So we need to elect 4 candidates, but the committee size is only 3. Therefore, no committee satisfies strong JR.

There are several ways to relax the notion of strong-JR.