---
title: Binary MMS-Feasible Valuations
url: https://www.emergentmind.com/topics/binary-valued-mms-feasible-valuations
type: topic
---

# Binary MMS-Feasible Valuations

Searching arXiv for the specified paper and closely related work on binary valuations, MMS/WMMS, APS, and matroid-rank settings.
arXiv search query: 2601.09299 Binary XOS WMMS asymmetric agents
arXiv search query: 2312.08509 APS MMS binary marginals XOS matroid rank
arXiv search query: 2204.03803 weighted Nash welfare binary valuations
Binary-valued MMS-feasible valuations form a research umbrella for valuation classes over indivisible goods in which maximin-share-type guarantees become exact, approximately exact, or provably unattainable under binary structural restrictions. In the recent literature, “binary” has two distinct formal meanings: additive singleton values in \(\{0,1\}\), and more general binary marginals \(v(S\cup\{g\})-v(S)\in\{0,1\}\). That distinction is decisive. Binary additive and binary submodular valuations exhibit strong exact-feasibility phenomena, while binary XOS valuations already lose exact WMMS/MMS feasibility and admit only worst-case approximation guarantees [2601.09299][2312.08509].

## 1. Formal scope and basic notions

The canonical setting is fair allocation of indivisible goods among \(n\) agents. For symmetric entitlements, agent \(i\)’s maximin share is
\[
MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).
\]
For asymmetric entitlements \(b_i\), the weighted maximin share is
\[
WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.
\]
When \(b_1=\cdots=b_n=\frac1n\), WMMS reduces exactly to MMS. The weighted formulation is therefore the natural extension of MMS to asymmetric agents [2601.09299].

The phrase “binary-valued” is not uniform across papers. In additive models, binary means \(v_i(g)\in\{0,1\}\) for each good \(g\), so \(v_i(S)\) is the number of liked goods in \(S\). In XOS and submodular work, binary instead refers to binary marginals:
\[
v(S\cup\{g\})-v(S)\in\{0,1\}\qquad \forall S,g.
\]
This does not mean \(v(S)\in\{0,1\}\) for every bundle. For XOS valuations, the representation is
\[
v(S)=\max_{t\in[L]} l_t(S),
\]
where each \(l_t\) is additive. Binary XOS is thus strictly broader than binary additive, and binary submodular valuations coincide with matroid rank functions [2601.09299][2312.08509].

A second benchmark that repeatedly interacts with MMS is Any Price Share (APS), defined through a price simplex and known to dominate MMS for monotone valuations:
\[
APS_i \ge MMS_i.
\]
The extent to which APS and MMS coincide, or diverge, is one of the main structural separators among binary classes [2312.08509].

## 2. Exact MMS-feasible binary classes

The cleanest exact-feasibility result is for binary additive valuations. In the asymmetric setting, if agent \(i\) has a demand set
\[
D^i=\{g\in M\mid v_i(g)=1\},\qquad v_i(S)=|S\cap D^i|,
\]
then exact WMMS allocations always exist and can be found in polynomial time. The same result immediately yields exact MMS in the symmetric special case. The allocation algorithm uses a balancing rule based on normalized load \(\frac{|A_i|}{b_i}\), and the stated runtime is \(O(m\log n)\) [2601.09299].

A second exact class is binary submodular valuations, equivalently matroid rank functions. In that class, APS and MMS collapse exactly:
\[
APS_i=MMS_i.
\]
Consequently, exact APS allocations exist and can be computed efficiently; since exact MMS allocations are already known for matroid rank valuations, the same allocation is also exact APS. The literature summarized in this line of work further states that such exact allocations can be computed while maximizing social welfare [2312.08509].

These two domains already show that binaryity can be highly favorable, but only under substantial structural restrictions. Binary additivity yields exact weighted and unweighted maximin guarantees; binary submodularity yields exact MMS and exact APS, but the positive theorem is fundamentally tied to matroid rank structure rather than to binary marginals alone [2601.09299][2312.08509].

| Valuation class | Exact MMS/WMMS status | Representative statement |
|---|---|---|
| Binary additive | Exact WMMS and exact MMS | Polynomial-time exact WMMS allocation [2601.09299] |
| Binary submodular / matroid rank | Exact MMS; \(APS=MMS\) | Efficient exact APS/MMS allocation [2312.08509] |
| Binary XOS | Not exact in worst case | Only \(\frac1n\)-WMMS guaranteed [2601.09299] |

## 3. Binary XOS and the breakdown of exact feasibility

The sharpest negative result concerns binary XOS valuations. For general XOS valuations, there always exists a \(\frac1n\)-WMMS allocation:
\[
v_i(A_i)\ge \frac1n\,WMMS_i \qquad \forall i.
\]
The paper then proves that this factor is tight even when all valuations have binary marginals: for every \(n\ge 2\), there exists an instance of \(n\) agents with binary XOS valuations where no allocation is better than \(\frac1n\)-WMMS [2601.09299].

The lower-bound instance uses \(m=2n-1\) goods and asymmetric entitlements
\[
b_1=\cdots=b_{n-1}=\frac{1}{2n-1},\qquad b_n=\frac{n}{2n-1}.
\]
Agents \(1,\dots,n-1\) value a bundle at \(1\) iff it contains at least one of \(g_1,\dots,g_n\); agent \(n\) either values the whole block \(\{g_1,\dots,g_n\}\) additively or values each remaining good \(g_{n+1},\dots,g_{2n-1}\) as a separate singleton clause of value \(1\). In this construction,
\[
WMMS_1=\cdots=WMMS_{n-1}=\frac1n,\qquad WMMS_n=n,
\]
yet any allocation that gives positive value to the first \(n-1\) agents forces
\[
v_n(A_n)\le 1,
\]
hence
\[
\frac{v_n(A_n)}{WMMS_n}\le \frac1n.
\]
The theorem is formally a weighted result, but it establishes the central separation: binary XOS does not exhibit the exact-feasibility behavior enjoyed by binary additive valuations [2601.09299].

The same line of work gives an APS-side approximation result for binary XOS in the asymmetric setting: a polynomial-time \(\frac12\)-APS allocation, which matches the known upper bound. In the symmetric case, APS dominates MMS, so this implies a polynomial-time \(\frac12\)-MMS allocation. This is an approximation theorem, not an exact-feasibility theorem [2601.09299].

## 4. APS, MMS, and class separations under binary structure

APS is especially informative because it dominates MMS for every monotone valuation but behaves very differently across binary classes. For matroid rank valuations, the strongest possible relation holds:
\[
APS_i=MMS_i.
\]
The proof uses capping at the APS value, matroid exchange, and union-matroid rank conditions to derive \(n\) disjoint bases, which then realize an MMS partition of value equal to APS. This makes binary submodularity a particularly robust exact share-based domain [2312.08509].

Binary XOS breaks this equivalence. In that class,
\[
MMS_i \le APS_i \le 2\cdot MMS_i + 1,
\]
and the upper bound is almost tight: there exists an instance with three agents and six goods, all with identical binary XOS valuations, for which
\[
APS \ge 2\cdot MMS.
\]
The construction uses two additive binary clauses, one supported on \(\{g_1,g_2,g_3\}\) and one on \(\{g_4,g_5,g_6\}\), so that APS can fractionally support many value-\(2\) bundles while MMS is bottlenecked by integral partitioning. From this bound, a known \(0.3666\)-MMS algorithm yields a polynomial-time \(0.1222\)-APS allocation, while the lower-bound instance implies that no better than \(0.5\)-APS allocation may exist even with identical valuations [2312.08509].

The asymmetric setting introduces a further separation. APS guarantees do not translate into WMMS guarantees when entitlements are unequal: there are additive instances where, for some agent,
\[
\frac{APS_i}{WMMS_i}<\delta
\]
for any \(\delta>0\). This shows that APS and WMMS can diverge dramatically outside the symmetric case, and explains why binary XOS requires a separate weighted analysis rather than a transfer from APS bounds [2601.09299].

## 5. Algorithmic, strategic, and communication perspectives

Existence, implementability, and strategic realizability separate sharply in binary domains. For XOS WMMS, the \(\frac1n\)-guarantee is given constructively, but the construction assumes access to WMMS values, WMMS partitions, and additive clauses maximizing selected bundles. The result is therefore best read as a constructive existence theorem rather than a full polynomial-time algorithm under standard oracle access. By contrast, the binary additive case comes with a genuine polynomial-time exact algorithm [2601.09299].

Mechanism design introduces another boundary. Under matroid-rank valuations, exact MMS allocations, and even MMS plus Pareto-efficient allocations, exist by prior work cited in the literature, but there is no mechanism that is simultaneously truthful, index-oblivious, Pareto efficient, and maximin fair. The impossibility persists even if Pareto efficiency is weakened to local efficiency. The same paper proves a characterization of truthful non-wasteful mechanisms—truthfulness iff gradualness—and notes that the proof extends from matroid-rank valuations to binary XOS functions. The overall message is that existential MMS-feasibility does not imply truthful exact implementation [2109.05810].

Communication complexity reveals a different sense in which binary additive valuations are special. For equally entitled agents with binary additive valuations, the expected average number of bits transmitted per agent for randomized protocols computing an MMS allocation is
\[
\Theta\!\left(\log\frac{m}{n}\right),
\]
with total randomized communication
\[
\Theta\!\left(n\log\frac{m}{n}\right).
\]
The lower bound holds even when all agents have the same binary valuation function. A nearby class already breaks this compressibility: for 2-valued additive valuations, the paper gives a lower bound of
\[
\Omega\!\left(\frac{m}{n}\right)
\]
on the expected average number of bits transmitted per agent [2407.07641].

Weighted Nash welfare adds useful indirect structure in the binary additive setting, but not direct MMS theorems. The lexicographically refined MWNW rule studied for binary additive valuations is resource-monotone, population-monotone, group-strategyproof, and computable in polynomial time, with runtime
\[
\mathcal O\big(mn(m+n^2)\big).
\]
However, that work does not define MMS or prove any MMS guarantee; its relevance to binary-valued MMS-feasibility is therefore structural rather than theorem-level [2204.03803].

## 6. Online and mixed-domain extensions

Online models expose how fragile binary MMS-feasibility becomes under irrevocability. In the model with sequentially arriving agents and additive valuations, no online algorithm can ensure any non-trivial MMS approximation without suitable information about future valuation functions, even with only two agents and binary valuations. Under the \(k\)-type model, adversarial arrivals admit a \(\frac1k\)-MMS algorithm but also a lower bound ruling out any \(\Omega(\frac1{\sqrt{k}})\)-MMS-competitive algorithm, even for binary valuations. Under stochastic arrivals, the picture improves substantially: an asymptotic, arbitrarily close-to-\(\frac12\)-MMS competitive guarantee is achievable under the paper’s stated distributional assumptions [2503.02088].

A different online model studies sequentially arriving items allocated to offline agents. There the exact-feasibility frontier is class-dependent. For goods, additive binary valuations admit deterministic online allocations that are exact MMS, EF1, and max-USW; submodular binary valuations admit \(\frac12\)-MMS, \(\frac12\)-EF1, and \(\frac12\)-max-USW, and no deterministic non-wasteful algorithm can do better than \(\frac12\)-MMS. For chores, additive binary costs admit exact MMS, EF1, and min-USC, whereas supermodular binary costs admit no deterministic complete algorithm achieving \(\alpha\)-MMS for any \(\alpha\ge 1\). The same paper also identifies mixed binary-plus-bivalued classes where exact MMS remains possible, and two-agent bivalued settings where only approximation guarantees survive [2505.24321].

Mixed divisible/indivisible environments do not provide direct MMS theorems, but they show that binary structure alone does not trivialize fair allocation once continuous and discrete resources are combined. Under binary additive valuations for mixed goods, the feasible utility region is generally neither convex nor \(M\)-convex; fair-allocation optimization is NP-hard even when all indivisible goods are identical; and a polynomial-time algorithm is obtained only when all divisible goods are identical. The main structural contribution is a proximity theorem stating that if \(z^*\) is a \(\Phi\)-fair integral utility vector and \(\bar z\) is a fair relaxed one, then
\[
\lfloor \bar z_i\rfloor \le z_i^* \le \lceil \bar z_i\rceil \qquad (\forall i\in N).
\]
This suggests that binary mixed domains retain strong additive structure, but not the universal exact-feasibility behavior seen in pure binary additive allocation [2306.05986].

Taken together, these results yield a precise landscape. Binary additive valuations are an exact MMS- and WMMS-feasible class, with efficient algorithms and unusually low communication. Binary submodular valuations remain exact in the symmetric offline sense through the identity \(APS=MMS\). Binary XOS valuations do not preserve this behavior: in weighted settings the best worst-case guarantee is exactly \(\frac1n\), and even APS separates from MMS by an almost factor-\(2\) gap. Online and mixed settings then show that binary structure is powerful but not sufficient by itself; feasibility depends critically on additivity, matroid structure, temporal model, and whether fairness is weighted, strategic, or communication-constrained [2601.09299][2312.08509].

Source: https://www.emergentmind.com/topics/binary-valued-mms-feasible-valuations