Temporal Maximin Share (TMMS) in Fair Allocation

• TMMS is a fairness concept for online allocation that extends the classical maximin share by ensuring each agent’s cumulative value meets evolving benchmarks across rounds.
• The feasibility of TMMS depends on scheduling buffers and valuation restrictions, with impossibility in minimal scheduling setups and positive results under sufficient buffering conditions.
• Concrete counterexamples and polynomial-time algorithmic procedures illustrate TMMS limitations and constructive approaches in settings like two-agent identical-days scenarios.

Temporal Maximin Share (TMMS) is a fairness concept for online allocation scenarios in which indivisible goods arrive over multiple rounds, and agents must be kept above evolving maximin-share benchmarks throughout the process. TMMS extends the classical maximin-share guarantee to the temporal setting, formalizing cumulative fairness requirements in dynamic, multi-period environments. This framework is motivated by impossibility results in static and temporal fair division, and by practical applications where inter-temporal equity is essential.

1. Formal Definition and Framework

Let $N = [n]$ be a set of $n$ agents, $T$ the number of allocation rounds, and $O^t = \bigcup_{s=1}^t O_s$ the cumulative set of goods available up to round $t$. For any agent $i$ and set of goods $X$, the (static) maximin-share (MMS) value is defined as:

$$\mu_i(X) = \max_{(P_1,\dots,P_n)\, \text{partition of } X} \min_{j \in [n]} v_i(P_j)$$

where $v_i(\cdot)$ denotes agent $i$’s valuation function.

A temporal allocation consists of sequences $(A^t_1, ..., A^t_n)$ such that $A^t_i \subseteq O^t$ and $\bigcup_{i} A^t_i = O^t$. An allocation $A = (A^T_1, ..., A^T_n)$ achieves the Temporal Maximin Share (TMMS) guarantee if for every agent $i \in [n]$ and round $t \in [T]$,

$v_i(A^t_i) \geq \mu_i(O^t)$

meaning each agent’s cumulative value up to any round meets or exceeds the static MMS for the goods received so far.

2. Existence and Impossibility Results

The existence of TMMS allocations is constrained by both the structure of the temporal process and the nature of agents’ valuations. The following results delineate this landscape:

Theorem/Proposition	Model Assumptions	TMMS Existence
Theorem 2.1	$n=2$, identical additive valuations, $r=1$ (no scheduling), $T>1$	Not guaranteed
Theorem 2.2	$n=2$, identical valuations, $r \leq T-1$ (limits on scheduling), $T>1$	Not guaranteed
Proposition 2.3	Generalized binary valuations ($v_i(g) \in \{0, b\}$)	Polynomial-time algorithm exists
Theorem 2.4	Identical days, $n=2$, identical valuations, no scheduling, $T>1$	Not guaranteed
Theorem 2.5	Identical days, $n=2$, $r \geq \lfloor T/2 \rfloor$ (sufficient scheduling)	Polynomial-time algorithm exists

The impossibility results (Theorems 2.1, 2.2, and 2.4) show TMMS allocations may fail to exist even with only two agents and highly symmetric valuation structures, unless strong scheduling resources are provided or valuations are sufficiently restricted. In particular, the positive existence results for TMMS arise only in domains with generalized binary valuations, or under identical-days conditions with sufficient buffer size in scheduling.

3. Approximation Results and Limits

No nontrivial approximation guarantees for TMMS are established beyond the exact results in restricted domains. The absence of positive approximation results highlights a fundamental discontinuity relative to other temporal fairness notions, such as temporal envy-freeness up to one good (TEF1), for which certain constant-factor approximations are attainable. For TMMS, the only guarantees shown are for exact attainment within the specialized cases enumerated above; in general, TMMS may not exist and is not approximable by any meaningful factor under the studied constraints.

4. Influence of Scheduling and Buffer Size

Scheduling, and in particular the buffer size $r$, is critical for determining TMMS feasibility. Three key findings characterize this dependence:

• Without scheduling ($r=1$), TMMS is generally impossible even in the most basic two-agent, identical-valuation scenarios.
• Allowing scheduling with buffer $r \leq T-1$ is insufficient to overcome the impossibility barrier.
• TMMS is achievable only when $r \geq \lfloor T/2 \rfloor$ in the identical-days, two-agent model, enabling sufficient aggregation of goods across rounds for balanced allocations.

These results quantify the minimum structural requirements for ensuring cumulative fairness, with buffer size acting as a controlling parameter for temporal aggregation.

5. Key Proofs, Counterexamples, and Argument Structure

The principal impossibility proofs use explicit counterexamples with additive valuations and two agents. In one construction, three goods $\{g_1,g_2,g_3\}$ arrive over two rounds (first $\{g_1,g_2\}$ with $v(\cdot)=1$ each, then $\{g_3\}$ with $v(\cdot)=2$). Regardless of how the first round is split, the agent not receiving $g_3$ in the second round ends with value 1, whereas the maximin-share for all goods at round two is 2, violating the TMMS requirement. This construction is robust to scheduling when $r \leq T-1$, since it is not possible to sufficiently delay allocation to allow both agents the opportunity to achieve the evolving MMS threshold.

Under identical-days, TMMS impossibility is exemplified by daily arrival of bundles $\{1,3,10\}$. Each day must be split as $(1+3)$ vs. $10$, but in subsequent rounds, no reassignment can lift both agents above the round-specific MMS threshold, illustrating the temporal compounding of fairness deficits.

Positive results follow directly from existing algorithmic procedures established for related fairness concepts in restricted domains. For generalized binary valuations, Elkind et al. provide a polynomial-time TEF1 algorithm, and since EF1 implies MMS in this domain, a TMMS allocation is achieved. In the identical-days, sufficient-buffer case, agents are each given one copy of each good after delaying the required rounds, yielding identical value bundles and satisfying TMMS.

6. Illustrative Examples

Two canonical counterexamples substantiate the negative existence results:

• General two-agent case: First round delivers $\{g_1, g_2\}$ (values 1 each), second round delivers $\{g_3\}$ (value 2). Each agent gets one "1" in round one, but only one can receive "2" in round two, resulting in one agent not reaching the MMS threshold of 2 after both rounds.
• Identical-days case: Each day brings $\{1,3,10\}$. The splitting $(1+3)$ vs. $10$ on day one does not admit a continuation that keeps both agents above their time-evolving MMS thresholds in future rounds.

These examples demonstrate the inherent temporal fragility of the TMMS guarantee when scheduling resources or valuation restrictions are inadequate.

7. Algorithmic Procedure for Identical-Days, Two-Agent, Sufficient Buffer Case

A polynomial-time algorithm exists only for the setting with identical-days, two agents, and scheduling buffer $r \geq \lfloor T/2 \rfloor$. The process is as follows:

1. Delay allocation of all goods from days $1$ to $H-1$ into day $H$ (with $H = \lfloor T/2 \rfloor$).
2. In round $H$, allocate to each agent exactly one copy of each good present.
3. Delay goods from days $H+1$ to $T-1$ into day $T$.
4. In round $T$, again allocate to each agent one copy of each good.

After round $H$ both agents attain identical bundles on $O^H$, meeting TMMS for $O^H$; after round $T$ the same holds for $O^T$. This procedure leverages scheduling to synchronize allocations and ensure cumulative fairness.

8. Context and Significance

TMMS marks a critical boundary in temporal fair division, distinguishing settings where cumulative fairness is structurally feasible from those where it is provably unattainable. Its analysis elucidates the role of inter-temporal dependencies, valuation domain restrictions, and scheduling resources in governing fairness guarantees. These findings clarify the limitations of temporal fairness and provide informed guidance for mechanism design in dynamic resource allocation environments (Choi et al., 19 Jan 2026).