Existence of the Core in Approval-Based Committee Elections

Abstract: We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

• The paper demonstrates that every instance of an approval-based committee election has a core committee, a stronger fractional stability condition called core+, and presents a polynomial-time algorithm for computing these committees.
• The authors introduce 'core+' a fractional relaxation of the blocking condition for core stability, defined through the construct of 'harmonic entropy' allowing a continuous certificate that can be optimized.
• The existence of a core+ committee directly implies the existence of an ordinary core committee.
• Core+ committees can be found in polynomial time even under the strict Droop quota.
• This is assured by a continuous potential function that organizes suboptimal solutions through maximizing the harmonic entropy of reserve payments, based on the sum of maximum values of income.
• A polynomial time approximation method for determining committees is validated through the empirical study of local search and replacing within a defined interval.
• For indexed committees of size k, every solution is bounded by the linear combination of the payment to the reserve.
• And based on water filling algorithms the solution translates from the potential gain to real gain.
• The formulation is extendable to maximizing full resolutions post relaxing variables to manage linear feasibility problems.
• For a given target proposal the given validity should be verified.
• The highest efficiency approaches should adhere to committee monotony,

Problem setting and principal contribution

"Existence of the Core in Approval-Based Committee Elections" (2609.11912) resolves the central non-emptiness question for the core in approval-based committee elections (ABCEs). Given $n$ voters, candidates $C$, committee size $k$, and approval sets $A_i\subseteq C$, voter $i$ receives utility $u_i(W)=|A_i\cap W|$ from committee $W$. A coalition $S$ blocks $W$ if it can afford a proposal $T$ under quota $C$0,

$C$1

and every member of the coalition strictly prefers $C$2 to $C$3. Under the standard Hare quota, $C$4.

The paper proves three results. First, every ABCE instance has a core committee. Second, the result is strengthened: a committee exists that satisfies core+, a fractional relaxation of the blocking condition that is strictly stronger than integral core stability. Third, a core+ committee, and therefore a Hare-core committee, can be computed in polynomial time. The existence theorem applies to every quota in the interval

$C$5

and the paper also establishes existence under the strict Droop quota $C$6. The formal existence proof has additionally been verified in Lean, as reported by the authors (2609.11912).

The result is stronger than establishing a particular conventional proportionality axiom such as EJR or FJR. Core stability allows a deviating coalition to combine candidates in an arbitrary way; its members need not share a common candidate or a common approved set. Consequently, satisfying the core requires ruling out a much broader class of coalitional deviations.

Core+ and its payment characterization

The paper introduces core+ by relaxing the binary variables that represent coalition membership, candidate selection, and voter-to-candidate assignments to continuous variables in $C$7. A violation of core+ is therefore a fractional objection: voters may participate fractionally in the blocking coalition, candidates may be selected fractionally, and approval assignments may also be fractional. Since every integral objection is a fractional objection, core+ implies the ordinary core.

The importance of core+ is not only conceptual. Its fractional definition admits a dual certificate in terms of voter payments. For a committee $C$8, a payment system assigns each voter payments $C$9 to approved winning candidates and a reserve $k$0, with

$k$1

while each winning candidate receives total payment at most $k$2. The paper proves the following equivalence:

A committee satisfies core+ if and only if it has a payment system such that payments to approved winners satisfy $k$3, and every losing candidate $k$4 has reserve load

$k$5

This is obtained by applying Farkas' lemma to the linear feasibility system describing fractional objections. The result gives core+ a positive certificate: reserves represent unspent voter purchasing power, and a losing candidate is harmless precisely when the total reserve of its approvers remains below the quota. The strict inequality for losing candidates is essential. If a losing candidate accumulated reserve load at least $k$6, its supporters could finance a fractional deviation.

This characterization connects core+ to priceability notions, including frugal Lindahl priceability. It also explains why the proof uses payment systems rather than directly manipulating coalitions. The coalitional definition is combinatorial and negative—there must be no blocking coalition—whereas the payment characterization supplies a continuous certificate that can be optimized.

Harmonic entropy as the central potential

The paper's main technical construction is a new potential function, harmonic entropy. For a probability vector $k$7, the authors define water-filling levels

$k$8

where $k$9 is the sum of the $A_i\subseteq C$0 largest coordinates. Equivalently, for $A_i\subseteq C$1, $A_i\subseteq C$2 is the unique water level at which the excess mass above that level equals the mass required to create $A_i\subseteq C$3 new coordinates at the same level.

Harmonic entropy is then defined by

$A_i\subseteq C$4

The function is continuous, concave, symmetric, and invariant under appending a zero coordinate. It is maximized by the uniform distribution. For a uniform vector with $A_i\subseteq C$5 coordinates, its value is the harmonic number

$A_i\subseteq C$6

This differs structurally from Shannon entropy. Harmonic entropy is tailored to the discrete operation of adding or deleting one candidate coordinate, rather than to an abstract information-theoretic interpretation. Its decisive property is induced by the shift operator

$A_i\subseteq C$7

which truncates large coordinates and creates one new coordinate. The water-filling levels satisfy

$A_i\subseteq C$8

Consequently, the series defining $A_i\subseteq C$9 telescopes:

$i$0

This identity is the mechanism that makes candidate exchanges analyzable. When a new candidate is added, the reserve of each approving voter becomes the newly created coordinate, and the potential increase equals the voter's relevant reserve. When a candidate is deleted, the reverse operation yields a precisely controlled potential loss.

The voting rule and exchange inequalities

For each committee $i$1 and payment system $i$2, voter $i$3 has payment vector

$i$4

consisting of payments to approved winners and the reserve. The paper defines the committee value

$i$5

The proposed voting rule selects committees maximizing $i$6. The optimization jointly chooses the committee and a feasible payment system, rather than optimizing over committees alone.

A secondary convex objective is used to select an optimal payment system with useful regularity properties. In particular, one can choose an optimum satisfying:

• every payment to an approved winner is at most the voter's reserve;
• if a winning candidate receives less than quota, every approving voter pays that candidate exactly her reserve;
• every reserve is at least the uniform-share lower bound $i$7.

These properties support the two exchange lemmas on which the existence proof rests.

Adding a candidate

Let $i$8 and define its reserve load as

$i$9

If $u_i(W)=|A_i\cap W|$0, the paper proves

$u_i(W)=|A_i\cap W|$1

The construction applies the water-filling operator to every voter approving $u_i(W)=|A_i\cap W|$2. The resulting new payments to $u_i(W)=|A_i\cap W|$3 sum to $u_i(W)=|A_i\cap W|$4. If this exceeds the candidate's capacity $u_i(W)=|A_i\cap W|$5, the authors convexly interpolate with the original payment system so that the new candidate receives exactly quota. Concavity of harmonic entropy preserves the corresponding objective gain.

Thus, a losing candidate with reserve load at least $u_i(W)=|A_i\cap W|$6 produces an objective increase of at least one quota.

Deleting a candidate

Conversely, if a committee $u_i(W)=|A_i\cap W|$7 has size $u_i(W)=|A_i\cap W|$8 satisfying $u_i(W)=|A_i\cap W|$9, the paper proves that some candidate $W$0 can be removed at objective loss at most

$W$1

The proof is substantially more involved. It identifies underfilled winners, shows that their remaining capacities can absorb reserves, and uses a max-flow/min-cut argument to redistribute those reserves among approved candidates. A candidate with average payment load at most $W$2 is then selected. A quadratic balancing argument ensures that this candidate is a largest payment coordinate for every voter who approves it, allowing the telescoping identity for harmonic entropy to bound the loss from deletion.

For a $W$3-candidate set, the bound becomes

$W$4

The numerical gap between the addition and deletion bounds,

$W$5

is the quantitative margin that converts local optimality into core+.

Existence of core+ committees

The principal theorem states that for every quota satisfying

$W$6

there exists a size-$W$7 committee satisfying core+. More specifically, every global maximizer of $W$8 satisfies core+.

The argument is a contradiction based on single-candidate replacements. Suppose an optimal committee $W$9 had a losing candidate $S$0 with reserve load at least $S$1. Adding $S$2 would increase $S$3 by at least $S$4. The resulting committee has size $S$5, so some candidate could be deleted at cost at most $S$6. Therefore, some replacement of a member of $S$7 by $S$8 would improve the objective by at least

$S$9

This contradicts the optimality of $W$0. Hence every losing candidate has reserve load strictly below $W$1, and the payment characterization establishes core+.

The argument also proves a stronger local statement. If a size-$W$2 committee satisfies

$W$3

for every $W$4 and $W$5, then $W$6 satisfies core+. Therefore, core+ is guaranteed not only by global maximization but by a sufficiently strong local optimum under single-candidate swaps.

The extension to the strict Droop quota is obtained by taking a sequence of quotas converging from above to $W$7. Because the set of committees is finite, one committee occurs infinitely often among the corresponding core+ committees. If that committee admitted a strict Droop fractional objection, the same objection would violate core+ for all sufficiently close quotas above $W$8, yielding a contradiction. The proof is nonconstructive at the endpoint, but it establishes non-emptiness under the strict Droop condition.

Polynomial-time computation

Although $W$9 is defined through an infinite series and a continuous maximization over payment systems, the authors derive a polynomial-time implementation through truncation and linear programming.

For a payment vector with at most $T$0 coordinates, truncating harmonic entropy after $T$1 terms gives the uniform error bound

$T$2

Consequently,

$T$3

Each truncated water-filling level can be represented by a variable $T$4 satisfying $T$5. The positive-part constraints defining the water level are linearized with auxiliary variables. The resulting linear program has $T$6 variables and constraints, with polynomially bounded rational coefficients.

The local-search algorithm chooses

$T$7

and sets

$T$8

At each iteration it evaluates all single-candidate replacements using $T$9. A replacement is made whenever the truncated objective increases by at least $C$00. Since every payment vector has truncated harmonic entropy at most $C$01, the objective lies between $C$02 and $C$03. Each replacement increases the objective by at least $C$04, so the number of replacements is bounded by

$C$05

At termination, approximation error guarantees that no true replacement improves the objective by $C$06 or more. The local criterion therefore implies core+. Since the number of neighboring committees examined per iteration is at most $C$07 and each objective evaluation is a polynomial-size linear program, the entire procedure runs in polynomial time.

The algorithm computes a committee that is core+ for an interior quota below the Hare quota. Because a smaller quota makes blocking more difficult for a committee to avoid, core+ at this quota implies ordinary core stability at the Hare quota. This supplies the claimed polynomial-time construction of a Hare-core committee.

Limitations and open questions

The paper establishes existence and computation but does not characterize the broader behavioral properties of the harmonic-entropy rule. In particular, it does not provide empirical comparisons with PAV, MES, or other proportional rules, nor does it establish standard axioms such as committee monotonicity, candidate monotonicity, participation, or strategyproofness.

The computational result is based on a specific interior quota and on truncated approximations to the harmonic-entropy objective. Although this suffices to compute a Hare-core committee, the paper does not provide an exact polynomial-time algorithm for globally maximizing the untruncated objective. The local-search method instead exploits a quantitative gap and approximate LP evaluations.

The relationship between core+ and existing priceability concepts is also only partially developed. The payment characterization identifies core+ with frugal Lindahl priceability, but the paper does not determine whether core+ is strictly stronger than core on natural classes of approval profiles, or whether the harmonic-entropy rule satisfies additional structural properties beyond those needed for the proof. Finally, the paper leaves open how the rule behaves under alternative quotas, tie-breaking, and restrictions on approval profiles, despite proving existence throughout the interval above the strict Droop threshold.

Conclusion

The paper resolves the non-emptiness problem for the approval-based committee core by constructing a continuous potential over committees and voter payment systems. Harmonic entropy is designed so that water-filling operations exactly track the potential changes caused by candidate additions and deletions. The resulting exchange inequalities imply that every global or sufficiently strong local optimum has a payment certificate for core+, yielding core existence as a corollary. The paper also converts the infinite-dimensional-looking optimization into polynomial-size linear programs after truncation and obtains an explicit polynomial bound of $C$08 on the number of local-search replacements. The central contribution is therefore both existential and algorithmic: core-stable committees always exist, a stronger fractional stability property is attained, and such committees can be computed in polynomial time (2609.11912).