BoBW Fair Allocation Algorithm

• The Best-of-Both-Worlds algorithm is a fair allocation method that guarantees each agent at least her ex-ante proportional share and at least half of her truncated proportional share ex-post.
• It employs linear programming and combinatorial matching phases to manage exceptional items and allocate remaining goods efficiently in settings without transfers.
• The algorithm overcomes limitations of prior up-to-one-item fairness approaches by achieving near-optimal, constant-factor MMS/TPS approximations in every allocation outcome.

A best-of-both-worlds (BoBW) algorithm in fair allocation is a deterministic polynomial-time mechanism that constructs a distribution over allocations of indivisible items among $n$ agents with additive valuations. It guarantees that each agent receives at least her proportional share in expectation (ex-ante proportionality), and in every realized outcome (ex-post), each agent attains at least half of her truncated proportional share (TPS), a quantity between her maximin share (MMS) and her proportional share (PS). This dual guarantee is almost information-theoretically best possible, and the algorithm is efficiently computable without transfers or monetary payments.

1. Problem Formulation and Fairness Notions

The allocation problem considers a set of indivisible goods $M$, $|M| = m$ allocated among $n$ agents. Each agent $i$ assigns an additive value $v_i(S) = \sum_{j\in S} v_i(j)$ to bundles $S\subseteq M$, with equal entitlement and no monetary transfers.

The algorithm seeks to guarantee both:

• Ex-ante proportionality: For each agent $i$, the expected value of her allocation is at least her proportional share $\mathrm{PS}_i = v_i(M)/n$.
• Ex-post strong value: In every realized allocation, every agent $i$ receives at least $1/2$ of her TPS, where TPS is defined per-agent as the largest $t$ satisfying

$\frac{1}{n} \sum_{j\in M} \min\{v_i(j),\ t\} = t.$

This always satisfies $$\mathrm{MMS}_i \leq \mathrm{TPS}_i \leq \mathrm{PS}_i$$. Notably, $\mathrm{TPS}_i$ smooths the impact of highly skewed item values and is easier to compute or approximate than MMS.

2. Algorithmic Structure and Implementation

The algorithm consists of three core phases, each linked to a critical linear program (LP) or combinatorial structure.

(a) Preprocessing and LP1 for Exceptional Items

The first step identifies "exceptional" items for each agent $i$, those satisfying $v_i(j)\geq \mathrm{TPS}_i$. The algorithm solves LP1, which assigns to each agent either an exceptional item or a specially constructed auxiliary item (representing a share equivalent to $\mathrm{TPS}_i$). The linear program, over variables $x_{ij}$ (assignment fractions) and $s_i$ (allocated value), is:

$$\begin{align*} \max \quad & \sum_i s_i \ \text{subject to:} \quad & \sum_{i} x_{ij} \leq 1 \quad \forall j \in M_1, \ & \sum_{j\in M_1} x_{ij} = 1 \quad \forall i, \ & s_i = \sum_{j\in M_1} u_i(j) x_{ij} \quad \forall i, \ & s_i \geq \mathrm{PS}_i \quad \forall i, \ & x_{ij}\geq 0, \end{align*}$$

where $u_i(j)=v_i(j)$ for exceptional items $j$ and $u_i(j)=0$ otherwise.

(b) Rounding and Completion (Matching Phase)

The (fractional) solution of LP1 is decomposed (by a Birkhoff–von Neumann–style decomposition) into a lottery over integral allocations. Agents receiving an exceptional item do so definitively. Those with auxiliary items are paired to remaining real items via maximum matching (with valuations truncated to TPS) so as to achieve the best possible ex-post value.

(c) Allocation of Remaining Items (LP3)

After the primary assignment, leftovers $M_3$ are distributed via a second linear program (LP3), which ensures for each agent that the leftovers raise her total value (from primary plus leftovers) up to at least $\mathrm{TPS}_i$. Constraints enforce (for those not holding an exceptional item):

$s_i \geq f_i \cdot v_i(M_3),$

where $f_i$ is the fraction needed to reach TPS. The LP is rounded (again via a faithful randomized procedure) with a final support reduction step to compress the size of the allocation support.

Throughout, the critical concept is faithful randomized rounding: regardless of fractional values, every agent's loss in value compared to the fractional benchmark is at most the value of one item.

3. Ex-Ante and Ex-Post Guarantee Analysis

The algorithm achieves:

• Ex-ante proportionality: By construction, every agent $i$ satisfies $$\mathbb{E}[v_i(\text{allocation})] \geq \mathrm{PS}_i$$. This is enforced in LP1 and maintained through rounding.
• Ex-post guarantee: Every realization supplied by the lottery (the algorithm outputs a lottery of at most $n$ integral allocations) gives every agent at least $1/2$ of her TPS, and thus at least $1/2$ of her MMS. This guarantee follows case-wise: either an agent receives an exceptional item of value at least $\mathrm{TPS}_i/2$, or the largest leftover item available is at most $\mathrm{TPS}_i/2$, so even after rounding, the sum of primary+leftover ensures the $1/2$ threshold.

No prior mechanism based on “up-to-one-item” fairness (such as EF1) attains this $1/2$-fraction ex-post guarantee; prior BoBW mechanisms could deliver only a $1/n$-fraction of MMS in some worst-case instances.

4. Theoretical Limits and Impossibility Boundaries

The provided guarantees are tight up to small constants:

• Ex-ante limit: One cannot surpass proportional share in expectation for every agent.
• Ex-post limit: No algorithm can guarantee strictly more than an $n/(2n-1)$-fraction of TPS ex-post for all agents. For large $n$ this is approximately $1/2$. This limit arises from explicit constructions (e.g., with $2n-1$ items) where some agent inevitably receives at most this share.

Moreover, if additional requirements such as ex-ante fractional Pareto optimality (fPO) are imposed, it is impossible to maintain high ex-post MMS approximations uniformly. The algorithm thus relaxes efficiency to deliver stronger ex-post value.

5. Comparison to Previous and Related Approaches

Classical BoBW mechanisms (e.g., mechanisms relying on EF1 or “up-to-one-item” benchmarks) could, in certain cases, yield allocations where an agent's ex-post value is a negligible fraction of her MMS or even TPS. The approach in (Babaioff et al., 2021) breaks through this “up-to-one-item” barrier:

• Guarantees a $1/2$-fraction of TPS ex-post rather than $1/n$ for MMS.
• Employs two LPs and a finer combinatorial allocation process (matching and fractional redistribution without economically efficient transfers).
• Demonstrates that “up-to-one-item” property is insufficient when agent value distributions are highly asymmetric.

By carefully analyzing the structural dependencies in the LP and matching steps, this method achieves a uniform, constant-factor MMS/TPS approximation in every realization.

6. Practical Applications and Impact

This BoBW approach is directly applicable in any setting where divisible allocation or monetary compensation is not an option and fairness is required both in expectation and in every realized outcome. Examples include:

• Assignment of indivisible tasks or roles in organizations where ensuring both perceived and realized fairness is critical.
• Dividing public resources (e.g., school placements, public housing) without side payments and with public trust contingent on both process and result.
• Allocation in platforms such as Spliddit, where users expect both fair ex-ante compensation and to avoid highly unfair ex-post outcomes even in randomization-laden environments.

Because the implementation requires only efficient deterministic polynomial-time LP solutions and matching (and randomized rounding with bounded spread), it is practical even for sizable problem instances.

7. Summary Table of Guarantees and Limits

Guarantee Type	Bound Achieved	Tight?
Ex-ante value	$\mathrm{PS}_i$ (exact)	Yes; cannot improve
Ex-post value	$1/2 \cdot \mathrm{TPS}_i$	Yes, up to $n/(2n-1)$
LPs required	2 (LP1, LP3), plus matchings	N/A
Complexity	Deterministic poly-time	Yes

This structure contrasts with prior mechanisms, for which the ex-post approximation could degrade to $1/n$ in adversarial instances.

• "Best-of-Both-Worlds Fair-Share Allocations" (Babaioff et al., 2021)

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The deterministic polynomial-time algorithm from (Babaioff et al., 2021) achieves fair-share allocations that optimally balance ex-ante proportionality and strong, instance-independent ex-post guarantees, setting a new state-of-the-art for resource allocation without transfers among agents with additive utilities.