Randomized Online Apportionment

• Randomized online apportionment methods are a set of algorithms that use randomization to assign indivisible resources in real time while balancing proportionality and fairness.
• They leverage techniques such as dependent rounding, probabilistic sampling, and network flow decomposition to meet both local and cumulative quota constraints.
• These methods offer robust solutions for applications like legislative seat allocation, dynamic committee formation, and resource distribution in adversarial settings.

Randomized online apportionment methods assign indivisible resources—typically legislative seats or committee positions—to entities in a sequential or dynamic setting, using randomization to reconcile proportionality, quota, and monotonicity requirements that are provably incompatible in deterministic frameworks. These methods operate in contexts where allocations must be made •in real time* and often without knowledge of future demands or arrivals. The design of such algorithms draws on dependent rounding, probabilistic sampling schemes, network flow characterizations, and relaxations of classical apportionment axioms. Their analysis leverages mathematical programming, combinatorial optimization, and concentration inequalities. The field addresses both theoretical limitations and practical needs for fairness, stability, and strategic robustness in apportionment.

1. Classical Impossibility and the Need for Randomized Online Methods

Deterministic apportionment methods such as Hamilton’s largest remainder or divisor methods (Jefferson/D’Hondt, Adams, Webster/Sainte-Laguë) traditionally assign seats so that each entity's allocation is either the floor or ceiling of its ideal fractional share (the "quota" axiom) (Cembrano et al., 31 Oct 2024). However, Balinski and Young's impossibility theorem establishes that no deterministic scheme can satisfy quota and population monotonicity simultaneously—where monotonicity dictates that increasing a party's votes never decreases its seat count (Correa et al., 6 May 2024). Additional constraints in online settings, such as irrevocable assignment and temporal quota compliance (cumulative allocations remain within the floor or ceiling of the running total share) further exacerbate these tensions (Cembrano et al., 16 Oct 2025).

Randomized procedures circumvent these limitations by ensuring quota compliance and monotonicity of expected allocations, rather than ex-post assignments. In online scenarios, this randomization can also mitigate adversarial input effects and smooth allocations over time, offering proportionality in expectation even when per-step quotas may be violated.

2. Dependent Rounding and Randomized Quota Allocation

A central primitive is dependent randomized rounding, where fractional quotas $q_i$ for $n$ entities are split into integral allocations $a_i \in \{\lfloor q_i\rfloor, \lceil q_i\rceil\}$ such that $\sum_i a_i = H$ (house size) (Correa et al., 6 May 2024, Gölz et al., 2022). Grimmett's random shift method arranges the fractional residues on the unit interval and selects a subset corresponding to a random shift, so marginally each entity receives its expected share and the set of entities awarded additional seats is uniformly distributed over all admissible selections (Correa et al., 6 May 2024). Sampford’s sampling refines this by yielding higher-order monotonicity: for any coalition, increasing all members' quotas strictly increases the probability of their joint selection (Correa et al., 6 May 2024). These properties are critical for avoiding paradoxes in coalition formation and strategic manipulation.

For clustered or partitioned apportionment, as in peer selection or resource allocation, randomized rounding algorithms such as AllocationFromShares produce integral allocations summing to $k$ such that each cluster’s expected allocation matches its fractional quota, with support over only a linear number of outcomes (Aziz et al., 2016).

3. Online Sequential Apportionment: Network Flows and Recursive Construction

The online apportionment setting requires that allocations be made as candidate sets, votes, or quotas arrive over time, often with irrevocability. Deterministic methods suffer a worst-case deviation in cumulative seat count that is linear in $n$; exact compliance with global quota is only attainable when $n\leq 3$ (Cembrano et al., 16 Oct 2025). For $n=2$ or $3$, randomized online methods are constructed using recursive network flows. At each timestep, given the history of cumulative allocations and quotas, a flow network is built whose structure encodes feasible assignments under local and global quota constraints. Fractional flows are decomposed via the hypersimplex (Birkhoff–von Neumann decomposition), producing a lottery over integral assignments that maintain both per-step and cumulative proportionality in expectation (Cembrano et al., 16 Oct 2025).

Formally, for each upper-quota set $u$ in the flow network FN$(V^t, A^t, v^{t+1}, \pi)$ (see the paper for edge-capacity definitions), the vector $z(u)$ is decomposed into integral assignment vectors $\{z_j\}$ with weights $\lambda_j$. The randomized mechanism chooses an assignment per round according to these weights, ensuring

$\mathbb{E}[a_i^{t+1}] = v_i^{t+1}$

and simultaneously

$$A_i^{t+1} \in \{\lfloor V_i^{t+1}\rfloor, \lceil V_i^{t+1}\rceil\} \quad \forall i, t$$

where $A_i^{t+1}$ is cumulative allocation.

4. Relaxations and Approximate Proportionality in Online Allocation

Exact quota or envy-free allocations are generally unattainable in adversarial online settings, even with randomization. The PROP1 property ("proportionality up to one good") relaxes proportionality: each agent’s bundle can meet their fair share upon addition of one extra item (Choo et al., 5 Aug 2025). Uniform randomized allocation achieves a non-trivial approximation of PROP1 against non-adaptive adversaries with high probability, as shown by Bernstein's concentration inequality analysis. Learning-augmented algorithms that exploit predictions of maximum item value (MIV) further improve robustness against adversarial input sequences. However, stronger notions (EF1, MMS, PROPX) remain non-approximable—highlighting PROP1 as the practical baseline for randomized online apportionment (Choo et al., 5 Aug 2025).

5. Monotonicity and Polyhedral Characterizations

Monotonicity in apportionment—ensuring that increasing votes never harms an entity’s allocation—is often violated by deterministic or naive randomized methods (e.g., the "Alabama paradox"). Dependent rounding schemes (Sampford, cumulative rounding) restore monotonicity both in expectation and for joint probabilities of coalitions (Gölz et al., 2022, Correa et al., 6 May 2024). Polyhedral network flow characterizations identify all house-monotone, quota-compliant allocations as extreme points of linear programs with flow constraints on per-step and cumulative assignments (Cembrano et al., 31 Oct 2024). Any convex combination of these points whose expected allocation matches quotas describes a valid randomized apportionment method.

Randomized divisor methods (parameterized by $\delta$) yield ex-ante proportional allocations via randomization over $\delta$, but worst-case deviation from quota can be linear in house size, motivating relaxed versions that allow small deviations from fixed house size in exchange for improved fairness (Cembrano et al., 31 Oct 2024).

6. Applications and Practical Implications

Randomized online apportionment algorithms are relevant for legislative seat allocation under dynamic population data, peer review or selection processes, dynamic committee formation (with diversity constraints), and resource allocation in time-varying environments. They have been used in simulations where rapid, repeated proportional allocations are needed, outperforming classical priority-queue-based approaches for very large house sizes (Cheng et al., 2014). The practical adoption of randomized apportionment can mitigate manipulation and paradoxes, increase transparency, and provide robust statistical guarantees for proportional fairness in expectation.

Cumulative rounding further applies to temporally repeated assignments in education, labor shifts, or sortition, ensuring fair temporal smoothing for agents or states (Gölz et al., 2022).

7. Open Challenges and Future Research

Current limitations include scaling randomized online apportionment for $n>3$ parties under strict global quota constraints; for larger $n$, relaxations and probabilistic guarantees (e.g., concentration bounds, approximate proportionality) supplant strict quota compliance (Cembrano et al., 16 Oct 2025). Efficient implementation of sequential dependent rounding and network flow decomposition, especially under adversarial or streaming constraints, remains a subject of ongoing research. Integration with learning-augmented algorithms—leveraging predictive side-information—suggests further potential for robust online fairness (Choo et al., 5 Aug 2025).

Recent connections to discrete geometry (k-level line arrangements) and combinatorial optimization foster new perspectives on the combinatorial complexity of monotone apportionment mappings and may inform algorithmic improvements and hardness results (Cembrano et al., 31 Oct 2024). Broader applications in mechanism design and real-time fair division are anticipated as randomized online apportionment methods mature.