Recursively Balanced Picking Sequences

• Recursively balanced picking sequences are allocation protocols that partition picks into rounds, ensuring every agent's turn differs by at most one pick for balanced participation.
• They guarantee fairness metrics such as egalitarian welfare and maximin share, with tight bounds derived via counting and combinatorial arguments.
• An optimal 'reverse-after-first' sequence outperforms naive round-robin by optimizing MMS guarantees, demonstrating practical improvements in fair allocation.

A recursively balanced picking sequence is a class of allocation protocols for indivisible goods, defined by the property that, at every prefix of the sequence, the difference in the number of choices granted to any two agents is at most one. Formally, let $N = \{1,\ldots,n\}$ denote a set of $n \geq 2$ agents and $M = \{g_1, \ldots, g_m\}$ a set of $m \geq n$ indivisible goods. A picking sequence is described by $\pi = (a_1,\ldots,a_m)$, where $a_j \in N$ indicates which agent picks at step $j$. For recursively balanced sequences, $\pi$ can be partitioned into rounds of length $n$ (except possibly the last one), with each agent appearing exactly once per full round and at most once in a partial round. When states of strict additive utilities and preferences are assumed, recursively balanced picking sequences produce allocations that are envy-free up to one good and admit rich structural and fairness guarantees (Celine et al., 19 Dec 2025).

1. Formal Structure and Recursive Balance

A recursively balanced picking sequence enforces the constraint that, for every prefix of the sequence and every pair of agents $i, j \in N$, the number of times $i$ appears minus the number of times $j$ appears is at most one in magnitude. This ensures uniform participation in each "round," defined as a contiguous block of $n$ picks, and implies that the sequence is decomposable into such rounds as:

$$(a_1,\ldots,a_n \mid a_{n+1},\ldots,a_{2n} \mid \ldots \mid a_{(R-1)n+1},\ldots,a_m),$$

with each agent present once per full round and at most once in a partial final round. The allocation $A^\pi = (A_1^\pi,\ldots,A_n^\pi)$ is generated by sequentially granting pick rights to $a_j$ at step $j$ to select her most-preferred remaining good. This structure precludes trivial imbalances and aligns with distributing opportunity equitably over time.

2. Measures of Fairness: Egalitarian Welfare and Maximin Share

Two primary fairness metrics are employed:

• Egalitarian Welfare (EW):

$\mathrm{EW}(A) = \min_{i \in N} u_i(A_i)$

where $u_i$ is the additive utility of agent $i$ for her allocated goods. The price of fairness under the egalitarian metric is measured by comparing the worst-case ratio of maximum possible EW achievable by any sequence (or within a given class) to the EW achieved by a specific sequence:

$$\mathrm{PoF}_{EW}(\pi; S') = \sup_{\text{instances } I} \frac{\max_{\pi' \in S'} \mathrm{EW}(A^{\pi'}, I)}{\mathrm{EW}(A^\pi, I)}$$

where $S'$ is a relevant class of sequences.

• Maximin Share (MMS):

$$\mathrm{MMS}_i = \max_{\{P_1,\ldots,P_n\}} \min_{j = 1,\ldots,n} u_i(P_j)$$

The MMS guarantee for a sequence $\pi$ is quantified by the greatest $\rho$ such that, for every agent $i$ and every instance, $u_i(A_i^\pi) \geq \rho \cdot \mathrm{MMS}_i$.

These metrics allow rigorous comparison of fairness properties across recursively balanced as well as more general picking sequences (Celine et al., 19 Dec 2025).

3. Egalitarian Price of Recursively Balanced Sequences

If all sequences are initiated with the same first-round prefix $(1,\ldots,n)$, trivial pathological instances are avoided. For recursively balanced sequences $S$ versus the class $S'$ of all sequences starting with $(1,\ldots,n)$, main results include:

• Theorem 4.1: For any $n \geq 2$, $m \geq n$, and $\pi \in S$,

$\mathrm{PoF}_{EW}(\pi; S') = \min\{m-n+1, n\}$

• Theorem 4.2: For the price of fairness relative to other recursively balanced sequences,

$$\mathrm{PoF}_{EW}(\pi; S) = \min\{\lceil m/n \rceil, \lfloor \log_2 n \rfloor + 1\}$$

Proof Methods: The upper bounds arise by bounding the delay between possible picks for any agent under recursive balance, ensuring that multiplicative loss in utility is not excessive. Lower bounds are realized by crafting instances where early valuable goods are denied to one participant, leveraging adversarial sequencing. For the logarithmic bound, a combinatorial argument constructs a directed dependency graph; supposing a higher ratio leads to a contradiction via the impossibility of embedding an overly large binary tree in $n$ nodes (Celine et al., 19 Dec 2025).

4. Maximin Share Guarantees and Sequence Classification

4.1 Agent-Specific MMS Bounds

• Lemma 5.1: For agent $i$'s pick-times $\pi_i = (t_1, \ldots, t_R)$ (with $t_{R+1} = m+1$),

$$u_i(A_i) \geq \min_{r=2..R+1} \frac{(r-1)\cdot(n+1-i)}{t_r-i}\;\cdot\;\mathrm{MMS}_i$$

• Corollaries:
  • $u_i(A_i) \geq \frac{n+1-i}{2n-i}\,\mathrm{MMS}_i$
  • $$u_i(A_i) \geq \frac{1}{\lfloor \frac{m+1-i}{n+1-i} \rfloor} \,\mathrm{MMS}_i$$

Tightness is established by matching instances (Lemma 5.2).

4.2 Regular and Irregular Sequence Types

A sequence $\pi$ is termed irregular if, in the partial second round, (a) the round contains an even number $k\geq 2$ of picks, (b) agent $n-1$ does not pick in it, and (c) agent $n$ picks in each of the first $k/2$ picks. Otherwise, $\pi$ is regular.

• Theorem 5.3 (Regular): Let $\pi$ be regular, with $t_1=n, t_{R+1}=m+1$ the pick times of agent $n$. Define

$\alpha = \min_{r=2..R+1} \frac{r-1}{t_r-n}.$

Then $u_i(A_i) \geq \alpha \cdot \mathrm{MMS}_i$ for all $i$, and this is tight for agent $n$.

• Theorem 5.4 (Irregular): Every agent receives at least $2/(m-n+2)~\mathrm{MMS}$, and this bound is tight for agent $n-1$.

4.3 Best and Worst Case Sequences

• Theorem 5.5 (Best Guarantee):

$$\alpha_{\max} = \min \left\{\frac{\lfloor m/n \rfloor}{\lfloor m/n \rfloor n - n + 1},\; \frac{\lceil m/n \rceil}{m-n+1}\right\}$$

There exists a sequence $\pi^*$ attaining $u_i(A_i^{\pi^*}) \geq \alpha_{\max} \mathrm{MMS}_i$ for all $i$, all instances. $\pi^*$ starts $(1,\ldots,n)$, followed by reversals $(n,n-1,\ldots,1)$ in all subsequent rounds.

• Theorem 5.6 (Worst Guarantee):

$\alpha_{\min} = \max\{1/n,\,1/(m-n+1)\}$

Round-robin is among the worst, i.e., there exists a $\pi$ for which some agent receives only $\alpha_{\min} \mathrm{MMS}$.

Classification is determined by the position and frequency of the worst-placed agent's picks.

Sequence Type	MMS Guarantee	Worst-Case Example
Regular, optimal	$\alpha_{\max}$	Reverse-after-first sequence
Irregular	$2/(m-n+2)$	Sequence skipping agent $n-1$ in 2nd round
Worst-case	$\alpha_{\min}$	Round-robin

5. Algorithmic Construction of Optimal Sequences

Although no explicit pseudocode is provided, the optimal recursively balanced picking sequence $\pi^*$ with maximal MMS guarantee $\alpha_{\max}$ is fully characterized:

• Initialize with the round $(1,2,\ldots,n)$.
• For rounds $r=2,\ldots,\lceil m/n\rceil$, set round $r$ to $(n,n-1,\ldots,1)$.
• If the last round is partial, truncate as needed.
• Concatenate the rounds to obtain $\pi^*$.

For example, with $n=3$, $m=7$: rounds are $(1,2,3)$, $(3,2,1)$, $(3,2,1)$, and $(3)$, concatenated.

This structure compensates the agent picking last in the first round by assigning her the first pick in each subsequent round, optimizing MMS guarantees (Celine et al., 19 Dec 2025).

6. Analytical Techniques and Characteristic Proof Methods

The analysis utilizes several core techniques:

• Counting Arguments: Upper bounds for price-of-fairness and MMS approximation ratios are derived by counting the number of goods accessible to each agent by their $r$-th pick, and quantifying the utility that can be obtained versus an unconstrained adversary.
• Instance Construction: Lower bounds (tightness) are demonstrated by concentrating an agent's entire value on a small prefix in the sequence, so that in the adverse sequence the agent misses high-value goods.
• Combinatorial Arguments: For logarithmic bounds, a directed dependency graph is constructed; assuming a higher bound leads to a contradiction via binary tree size limitations.
• Inductive and Telescoping Sums: The MMS lower-bound lemma aggregates value over successive picks and applies weighted-averaging, leveraging telescoping sum identities.

7. Comparative Significance and Implications

Recursively balanced picking sequences provide a structural basis for achieving allocations that are envy-free up to one good. While all such sequences guarantee the same worst-case performance regarding egalitarian welfare—captured by $\min\{m-n+1,n\}$ and $$\min\{\lceil m/n \rceil, \lfloor \log_2 n \rfloor + 1\}$$—they differ markedly in their approximate MMS guarantees. The so-called "reverse-after-first" sequence uniquely achieves the optimal lower bound $\alpha_{\max}$, uniformly across all agents and instances, while round-robin sequences enforce only the worst-case guarantee $\alpha_{\min}$. For applications where maximin-share fairness is paramount, the optimal regular "reverse-after-first" structure is preferred over naive round-robin (Celine et al., 19 Dec 2025).