How to Fairly Reallocate Conference Seats After Student Transfers?

Problem

Three departments originally have 200 students (Dept A 100, Dept B 60, Dept C 40) and 20 conference seats are allocated proportionally as 10, 6, 4. After student transfers the numbers become 103, 63, 34. How should the 20 seats be re‑allocated, and what changes if the total seats increase to 21?

Model 1: Proportion + Convention

Using the proportional‑plus‑convention method, the seat distribution is calculated as follows:

Dept A: 103 students → 51.5 % → 10.3 seats → rounded to 10 (or 11 if 21 seats). Dept B: 63 students → 31.5 % → 6.3 seats → rounded to 6 (or 7 if 21 seats). Dept C: 34 students → 17.0 % → 3.4 seats → rounded to 4 (or 3 if 21 seats). Total remains 20 (or 21) seats.

However, this raises fairness concerns for Dept C.

Model 2: Quantitative Fairness Indicator

Assume two parties with given numbers of students and seats. When the proportion of seats to students for one party exceeds that of the other, the allocation is considered fair; otherwise it is unfair. The absolute unfairness degree can be defined, but it has limitations, such as cases where the same absolute unfairness yields very different perceived inequities.

Model 3: From Absolute to Relative Unfairness

Define a relative unfairness measure by normalizing the absolute unfairness with the total seats, yielding a value that should be minimized for a fair allocation. This converts a one‑time seat distribution problem into a dynamic one, where each additional seat is assigned to the party with the larger relative unfairness (Q‑value).

Problem Solving

Applying the Q‑value method to the three departments for a total of 21 seats, the integer parts of the proportional allocation fill 19 seats. The 20th seat is assigned to Dept A (largest Q‑value). The 21st seat is then assigned to Dept C, resulting in a final distribution of 11 seats for Dept A, 6 for Dept B, and 4 for Dept C.

Source

Mathematical Models (5th Edition) – Jiang Qiyuan, Xie Jinxing, Ye Jun