Personalized Bivalued Valuations

• Personalized bivalued valuations are additive models where each agent assigns one of two values per item, enabling fair allocation mechanisms.
• They support constructive algorithms like Match-and-Freeze that guarantee fairness concepts such as EFX and PMMS under specific conditions.
• Extensions to weighted settings and chores illustrate the model’s broad applicability, while also revealing complexity challenges and open research questions.

Personalized bivalued valuations are additive preference models for indivisible-item allocation in which each agent evaluates every item using one of two agent-specific levels. In the goods setting, the standard formulation assumes that for each agent $i$ there exist scalars $a_i>b_i\ge 0$ such that $v_i(\{g\})\in\{a_i,b_i\}$ for every good $g$, with bundle values extended additively as $v_i(S)=\sum_{g\in S}v_i(\{g\})$. Closely related formulations appear under the names “personalized bi-valued utilities” and, for chores, personalized bivalued cost functions with two disutility levels per agent. This two-level, agent-specific structure has become a focal restriction in discrete fair division because it supports constructive existence theorems for EFX, PMMS in a factored subcase, weighted relaxations such as WEFX and WEQX with fractional Pareto-optimality, and several online and chore-allocation guarantees, while still exhibiting nontrivial complexity and algorithmic pathologies (Byrka et al., 20 Jul 2025, Jin et al., 24 Jul 2025, Liu et al., 9 Apr 2026, Garg et al., 2021).

1. Formal model and terminology

The core model consists of a set of agents $N=\{1,\dots,n\}$ and a set of indivisible goods $M$. In the formulation used for personalized bivalued valuations, each agent $i$ has two values $a_i>b_i\ge 0$, and every good is either “high” or “low” for that agent in the sense that

$$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$

The literature also uses the labels “big” and “small” goods for agent $a_i>b_i\ge 0$0, depending on whether the singleton value is $a_i>b_i\ge 0$1 or $a_i>b_i\ge 0$2. A common normalization scales utilities so that $a_i>b_i\ge 0$3 and $a_i>b_i\ge 0$4, where $a_i>b_i\ge 0$5 is agent $a_i>b_i\ge 0$6’s value ratio (Byrka et al., 20 Jul 2025, Jin et al., 24 Jul 2025).

A notable subcase is the factored case: a personalized bivalued valuation is called factored if $a_i>b_i\ge 0$7 is an integer multiple of $a_i>b_i\ge 0$8, or $a_i>b_i\ge 0$9. This divisibility condition is central for stronger fairness guarantees, because the same constructive allocation that establishes EFX can be upgraded to PMMS when all valuations are factored (Byrka et al., 20 Jul 2025).

Weighted variants add a positive weight $v_i(\{g\})\in\{a_i,b_i\}$0 for each agent, with $v_i(\{g\})\in\{a_i,b_i\}$1. In that setting the allocation objective is no longer purely unweighted envy reduction, but weighted fairness notions normalized by $v_i(\{g\})\in\{a_i,b_i\}$2, together with fractional Pareto-optimality via Fisher-market equilibria. The same personalized two-value pattern is retained: for each good $v_i(\{g\})\in\{a_i,b_i\}$3, $v_i(\{g\})\in\{a_i,b_i\}$4, possibly after rescaling to $v_i(\{g\})\in\{a_i,b_i\}$5 and integer $v_i(\{g\})\in\{a_i,b_i\}$6 (Liu et al., 9 Apr 2026).

The chores analogue reverses the interpretation from utility to cost. There, each agent $v_i(\{g\})\in\{a_i,b_i\}$7 has an additive cost function with two cost levels, typically written as $v_i(\{g\})\in\{a_i,b_i\}$8 and $v_i(\{g\})\in\{a_i,b_i\}$9 for each chore $g$0. Many algorithmic treatments then specialize further to the common-ratio form $g$1 (Garg et al., 2021, Lin et al., 8 Jan 2025).

2. Fairness and efficiency notions

The most prominent fairness notion in this line of work is envy-freeness up to any good. For personalized bi-valued utilities, an allocation $g$2 is EFX if for every pair of agents $g$3 and every good $g$4,

$g$5

One paper frames PMMS as “a strictly stronger variant of EFX,” and uses personalized bivalued valuations as a regime in which the relationship between the two notions can be analyzed constructively (Byrka et al., 20 Jul 2025, Jin et al., 24 Jul 2025).

For chores, the corresponding notions are stated in terms of costs. EF1 requires that for all $g$6, there exists some chore $g$7 such that

$g$8

while exact EFX for chores requires the inequality after removal of any chore from the envied bundle. Approximate versions also appear: in $g$9-instances, an allocation is $v_i(S)=\sum_{g\in S}v_i(\{g\})$0-EFX if

$v_i(S)=\sum_{g\in S}v_i(\{g\})$1

for all $v_i(S)=\sum_{g\in S}v_i(\{g\})$2 and all $v_i(S)=\sum_{g\in S}v_i(\{g\})$3 (Lin et al., 8 Jan 2025, Garg et al., 2021).

Weighted generalizations replace absolute comparisons by normalized ones. An allocation $v_i(S)=\sum_{g\in S}v_i(\{g\})$4 is WEFX if for every pair $v_i(S)=\sum_{g\in S}v_i(\{g\})$5 and every good $v_i(S)=\sum_{g\in S}v_i(\{g\})$6,

$v_i(S)=\sum_{g\in S}v_i(\{g\})$7

and it is WEQX if

$v_i(S)=\sum_{g\in S}v_i(\{g\})$8

In the same weighted framework, fractional Pareto-optimality is defined by the absence of any fractional reallocation that weakly increases every agent’s utility and strictly increases at least one; equivalently, an integral allocation is fPO whenever it arises in some Fisher-market equilibrium. The reported relationships include: any fPO allocation is PO, an integral allocation coming from a market equilibrium is fPO, and WEQX implies WEFX (Liu et al., 9 Apr 2026).

Pareto-optimality itself is treated in both goods and chores formulations. For personalized bi-valued utilities, an allocation $v_i(S)=\sum_{g\in S}v_i(\{g\})$9 is PO if there is no other allocation $N=\{1,\dots,n\}$0 such that all agents weakly improve and at least one strictly improves. In chores models, the inequalities are reversed in the standard way because lower cost is better (Jin et al., 24 Jul 2025, Garg et al., 2021).

3. Constructive EFX and PMMS existence for goods

A central existence theorem states: for any instance with personalized bivalued valuations there exists an allocation that satisfies EFX, and if all valuations are factored then the same allocation is guaranteed to satisfy PMMS. The proof is constructive and yields a polynomial-time algorithm, “Match-and-Freeze,” built around ratio-weighted matching and controlled temporary exclusion of agents from future matching rounds (Byrka et al., 20 Jul 2025).

The algorithm maintains the set $N=\{1,\dots,n\}$1 of unallocated items and, in each round $N=\{1,\dots,n\}$2, a set $N=\{1,\dots,n\}$3 of active agents. It builds a bipartite graph on $N=\{1,\dots,n\}$4, where there is an edge $N=\{1,\dots,n\}$5 exactly when $N=\{1,\dots,n\}$6, and assigns that edge weight $N=\{1,\dots,n\}$7. It then computes a maximal matching of maximum total weight. Each matched agent receives one high-valued good. In any connected component that contains unmatched agents, all matched agents in that component are frozen for the next $N=\{1,\dots,n\}$8 rounds, where $N=\{1,\dots,n\}$9 over the unmatched agents $M$0 in the component. After the matching step, any remaining unmatched active agent picks an arbitrary remaining good in priority order (Byrka et al., 20 Jul 2025).

Two invariants drive the correctness proof. The first is the componentwise ratio inequality

$M$1

where $M$2 and $M$3 denote the unmatched and matched agents in a connected component of the round graph. The proof sketch is an alternating-path exchange argument: if an unmatched agent had larger ratio than a matched one in the same component, the total matching weight could be increased, contradicting maximality of the chosen matching. The second invariant tracks the last round $M$4 in which some agent obtains a good that agent $M$5 values at $M$6. Before round $M$7, whenever $M$8 receives an item it is of value $M$9; if $i$0 is ever frozen, that happens only after $i$1, and the total number of frozen rounds is at most $i$2 (Byrka et al., 20 Jul 2025).

The final case analysis establishes

$i$3

for every pair of agents $i$4 in the final allocation. Because every good in $i$5 has value at least $i$6 to $i$7, this implies

$i$8

which is precisely the EFX condition. The runtime is polynomial in $i$9 and $a_i>b_i\ge 0$0: each round allocates at least one item, so there are at most $a_i>b_i\ge 0$1 rounds, and the required maximum-weight maximal matching can be computed in polynomial time (Byrka et al., 20 Jul 2025).

A related paper gives a second constructive EFX proof for personalized bi-valued utilities via a “Match–Modify–Freeze” procedure. It maintains a bipartite graph between unfrozen agents and remaining goods, uses a maximum matching on large-good edges, and if the matching is not perfect applies a Modify step that swaps along alternating paths so that agents of higher ratio get matched whenever possible. Agents who receive a large good are then frozen for a number of future rounds proportional to the smallest ratio on the relevant alternating path. The stated invariant is that after each round the partial allocation remains EFX, and the resulting algorithm is polynomial-time (Jin et al., 24 Jul 2025).

This class is also used to separate fairness notions. One result constructs a three-agent instance with two monotone valuations and one additive valuation in which no PMMS allocation exists, even though EFX allocations are known to exist under those assumptions. That formal separation positions personalized bivalued valuations as a regime in which EFX is robust, while PMMS requires the additional factored assumption (Byrka et al., 20 Jul 2025).

4. Pareto-optimality, weighted fairness, and market structure

For personalized bi-valued utilities, Pareto-optimality admits a structural characterization when every ratio $a_i>b_i\ge 0$2 is an integer. The analysis introduces the item-exchange graph $a_i>b_i\ge 0$3, a directed multigraph whose edges record items moved between agents when passing from an allocation $a_i>b_i\ge 0$4 to a Pareto-improving allocation $a_i>b_i\ge 0$5, with labels $a_i>b_i\ge 0$6, $a_i>b_i\ge 0$7, $a_i>b_i\ge 0$8, or $a_i>b_i\ge 0$9 according to the source’s and target’s valuations. If $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$0 is Pareto-dominated, one chooses a Pareto improvement $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$1 minimizing the number of exchanged items and studies the corresponding minimum Pareto-improvement graph $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$2. The theorem states that, in the integer-ratio case, $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$3 must be exactly one of two directed cycles: a Type I “small-large exchange” cycle, or a Type II “one-many exchange” cycle. Consequently, if every $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$4, one can decide in polynomial time whether $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$5 is PO, and if $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$6 is not PO one can find a dominating PO allocation in polytime (Jin et al., 24 Jul 2025).

The complexity changes sharply for fractional ratios. When $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$7 may be fractional, even when all ratios belong to a two-point set $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$8, testing Pareto-optimality is coNP-complete. This establishes that personalization by itself does not make PO verification tractable; tractability in the reported characterization hinges on the integrality of the value ratios (Jin et al., 24 Jul 2025).

Weighted and market-based formulations extend the same two-level structure. In the setting with agent weights $$v_i(\{g\})\in\{a_i,b_i\}\quad \forall g\in M, \qquad v_i(S)=\sum_{g\in S}v_i(\{g\}).$$9, one paper gives a polynomial-time algorithm for WEFX and fPO and shows that the algorithm can be adapted to compute WEQX and fPO. The procedure has two phases. First, it computes a welfare-maximizing allocation $a_i>b_i\ge 0$00, sets each price $a_i>b_i\ge 0$01 to the assigned agent’s value, constructs the MBB graph $a_i>b_i\ge 0$02, pushes goods backward along certain paths, and then partitions agents into “reachability” groups $a_i>b_i\ge 0$03 defined from successive least spenders. Second, it repeatedly raises the prices of all goods in a currently processed group by factor $a_i>b_i\ge 0$04 and transfers goods from a big spender to the least spender until the required pWEFX condition holds. The invariants explicitly tracked include equilibrium preservation, internal group pWEFX, one-time raising of lower groups, monotonicity of the big-spender benchmark $a_i>b_i\ge 0$05, and growth of the set $a_i>b_i\ge 0$06 of agents already pWEFX toward the current big spender (Liu et al., 9 Apr 2026).

The weighted algorithm runs in $a_i>b_i\ge 0$07 time. Its efficiency guarantee is fPO because the final allocation remains a market equilibrium. A further implementation remark states that personalization with different $a_i>b_i\ge 0$08 values can be handled by raising each good in a group by factor $a_i>b_i\ge 0$09, or by repeating a “unit-raise” until pWEFX holds, with the same invariants extending to that fully personalized setting (Liu et al., 9 Apr 2026).

These market-based results clarify the distinction between unweighted EFX existence and weighted equitable variants. EFX is guaranteed for personalized bivalued goods allocations, but the weighted literature targets WEFX or WEQX together with fPO rather than exact unweighted EFX plus PO. This suggests that market equilibria are especially natural for weighted normalization and fractional efficiency, but not automatically sufficient for stronger simultaneous guarantees.

5. Chores, approximate fairness, and online extensions

In the chores domain, bivalued costs support several strong algorithmic guarantees. For indivisible chores with costs in $a_i>b_i\ge 0$10, there is a strongly polynomial-time algorithm returning an integral allocation that is EF1 and fPO. The method starts from a Fisher-market equilibrium in which each chore is priced at its assigned cost and given to a cost-minimizer, partitions agents into partial components that are pEF1 internally, and then iteratively eliminates inter-group pEF1-envy by transferring chores along mBB edges or raising the prices of all chores held by the current big spender’s group by a factor of $a_i>b_i\ge 0$11. The two-value structure is used critically: after a price raise, every unraised agent finds all those chores minimum bang-per-buck, creating the edges needed for subsequent envy-reduction transfers (Garg et al., 2021).

The same paper also proves that, in the divisible bivalued-chores setting, one can compute an envy-free and fPO fractional allocation in strongly polynomial time. It begins with a balanced-flow fractional Fisher equilibrium, partitions agents into pEF components with nonincreasing per-agent spending, then repeatedly raises prices for an initial segment of groups and performs “uniform draining” of infinitesimal mass from the biggest-spending pool to the least-spending pool along mBB edges until the spending gap closes. The stated outcome is full EF together with fPO (Garg et al., 2021).

Approximate EFX for chores has been studied more finely in $a_i>b_i\ge 0$12-instances. One result gives a polynomial-time algorithm that returns an allocation that is $a_i>b_i\ge 0$13-EFX and Pareto optimal for every bivalued instance with $a_i>b_i\ge 0$14. The algorithm starts from an integral pEF1 payment equilibrium $a_i>b_i\ge 0$15, partitions items into payment-1 and payment-$a_i>b_i\ge 0$16 sets, and then repeatedly identifies a “strong-envy” pair $a_i>b_i\ge 0$17 with $a_i>b_i\ge 0$18 and $a_i>b_i\ge 0$19, swaps one high-payment item $a_i>b_i\ge 0$20 against the entire bundle $a_i>b_i\ge 0$21, and updates the high/low groups. Each round preserves the equilibrium invariant and groupwise payment fairness, and since each round moves one agent from $a_i>b_i\ge 0$22 to $a_i>b_i\ge 0$23, the process ends after at most $a_i>b_i\ge 0$24 rounds (Lin et al., 8 Jan 2025).

For the special case $a_i>b_i\ge 0$25, the same paper strengthens the guarantee to exact EFX and PO in polynomial time. The proof exploits the fact that the initial pEF1 equilibrium has every agent’s payment in $a_i>b_i\ge 0$26 for some integer $a_i>b_i\ge 0$27, then performs either a high-for-low swap or a one-way move of a high chore, depending on the structure of the violating pair. The process terminates in at most $a_i>b_i\ge 0$28 steps while preserving MPB feasibility and hence Pareto optimality (Lin et al., 8 Jan 2025).

Online allocation results show that the two-value pattern remains useful under irrevocable arrivals. In the two-agent goods setting with additive bivalued valuations, there is a deterministic online algorithm, “Adapted Envy-Graph,” that always produces a non-wasteful allocation satisfying $a_i>b_i\ge 0$29-EF1 and $a_i>b_i\ge 0$30-MMS, and these guarantees are tight in the sense that no deterministic algorithm can achieve $a_i>b_i\ge 0$31-EF1 or $a_i>b_i\ge 0$32-MMS for any $a_i>b_i\ge 0$33. In the two-agent chores setting, an analogous deterministic algorithm always produces a complete allocation satisfying $a_i>b_i\ge 0$34-EF1 and $a_i>b_i\ge 0$35-MMS, and no deterministic algorithm can guarantee $a_i>b_i\ge 0$36-EF1 or $a_i>b_i\ge 0$37-MMS. The same work also states that if $a_i>b_i\ge 0$38 agents are binary and one agent is bivalued, a simple round-robin rule yields exact EF1 and exact MMS (Wang et al., 30 May 2025).

6. Boundary cases, counterexamples, and open problems

Several results delineate what personalized bivalued structure does not imply. First, Pareto-optimality is not uniformly easy to check: it is in $a_i>b_i\ge 0$39 when each value ratio $a_i>b_i\ge 0$40 is an integer, but coNP-complete in the general fractional-ratio case. This rules out a simplistic reading of “two values per agent” as a blanket tractability condition (Jin et al., 24 Jul 2025).

Second, EFX and Pareto optimality do not currently admit a general simultaneous existence theorem in the personalized bi-valued goods setting. One paper explicitly poses the open problem of whether there always exists an allocation that is simultaneously EFX and Pareto-optimal in that setting, and whether one can be found in polynomial time. The same source notes that this is known for uniform bi-valued utilities, but that known extensions via max-Nash-welfare or market algorithms fail in the personalized case (Jin et al., 24 Jul 2025).

The max-Nash-welfare failure is illustrated by a two-agent four-item example in which the MNW solution gives $a_i>b_i\ge 0$41 to agent 1 and $a_i>b_i\ge 0$42 to agent 2, yet agent 1 strongly envies agent 2 after removal of any single item. A separate counterexample shows that even when an allocation is forced by EFX considerations, it may fail to be fractionally Pareto-optimal, so a Fisher-market approach that inherently targets fPO cannot by itself enforce EFX (Jin et al., 24 Jul 2025).

Third, an earlier Fisher-market-based algorithmic claim for bivalued goods was shown to be incorrect. Garg and Murhekar (2021) had proposed a polynomial-time algorithm that purported to find an EFX and fPO allocation, but a later paper gives a counterexample in which the algorithm may fail to terminate: the least spender alternates, all prices in the relevant component are repeatedly multiplied by $a_i>b_i\ge 0$43, prices blow up, and the allocation never changes. The corrective result is a new polynomial-time algorithm computing WEFX and fPO, and an adaptation for WEQX and fPO (Liu et al., 9 Apr 2026).

Finally, the personalized bivalued EFX proofs highlight why the setting is both special and limited. One paper states that the personalized bivalued structure gives a natural ordering of agents by ratio $a_i>b_i\ge 0$44, and that the matching-plus-freezing argument hinges on all high-value edges sharing a common two-value scale per agent. In general additive valuations, singleton values can take arbitrarily many levels, so the ratio-based maximality argument and uniform freeze do not extend straightforwardly. On that basis, the existence of EFX for arbitrary additive valuations remains the central open problem (Byrka et al., 20 Jul 2025).

Personalized bivalued valuations therefore occupy a distinctive intermediate position in fair division. They are expressive enough to encode heterogeneous thresholds, weighted comparisons, nontrivial exchange cycles, and online impossibility phenomena, yet structured enough to admit constructive EFX algorithms, PMMS under a divisibility condition, and several market-based efficiency guarantees. The current frontier lies in understanding how far these techniques can be pushed beyond two-value structure without losing either tractability or exact fairness.