Boltzmann Fair Division¶

A proposed probabilistic allocation model that converts contribution, need, and preference into a distribution potential and assigns resources through a Boltzmann-like probability rule.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7587
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Probabilistic Allocation → Economics & Finance
Aliases: Boltzmann fair-division model

Core Idea¶

Boltzmann Fair Division is a proposed resource-allocation model that borrows the mathematical shape of a Boltzmann distribution.^[1] It assigns each possible recipient or allocation a “distribution potential” constructed from human factors such as contribution, need, and preference; a probability rule then makes allocations with potential-sensitive frequency rather than through direct bargaining or a fixed deterministic ranking.^[2]

The statistical-mechanics analogy does not make fairness a physical law.^[3] The model's normative content lies in how potential is defined, whose needs and contributions count, and which temperature-like or scaling parameters control dispersion.^[4] Probabilistic assignment can reduce strategic negotiation, but it does not remove value judgments from model design.^[5]

How would you explain it like I'm…

The Fair-Share Ticket Lottery

Imagine sharing cookies by lottery, where each person gets tickets. People who helped more, need more, or want it more get more tickets, so they win more often, but everyone has some chance. Boltzmann Fair Division is an idea for sharing like that. People still have to decide what counts as a good reason for more tickets.

Sharing by Weighted Chance

Boltzmann Fair Division is an idea for sharing things out using chance instead of arguing or a strict ranking. Each person, or each possible way of dividing things, gets a score called a 'distribution potential,' based on things like how much they contributed, how much they need, and what they prefer. Then a probability rule, borrowed from a formula in physics, decides who gets what, so options with better scores happen more often. Using math from physics doesn't make fairness a law of nature. People still have to choose what goes into the score and how spread out the chances should be.

Potential-Weighted Probabilistic Allocation

Boltzmann Fair Division is a proposed model for allocating resources that borrows the mathematical form of the Boltzmann distribution from statistical mechanics. Each possible recipient or allocation gets a 'distribution potential' built from human factors such as contribution, need, and preference. A probability rule then chooses allocations so that ones with more favorable potential happen more often, instead of settling allocations through bargaining or a fixed ranking. A temperature-like or scaling parameter controls how spread out the outcomes are. The physics analogy does not turn fairness into a law of nature; the real moral choices lie in how the potential is defined, whose contributions and needs count, and how the parameters are set. Randomized assignment can reduce strategic negotiation, but it does not eliminate value judgments from the design.

Boltzmann Fair Division is a proposed resource-allocation model that adopts the functional form of a Boltzmann distribution. Each candidate recipient or allocation is assigned a 'distribution potential' constructed from human factors such as contribution, need, and preference, and allocations are then drawn with probabilities that vary with that potential rather than determined by direct bargaining or a fixed deterministic ranking. Temperature-like or scaling parameters govern dispersion: how sharply the allocation concentrates on high-potential options versus spreading across alternatives. The statistical-mechanics analogy is formal, not substantive; it does not turn fairness into a physical law. The model's normative content resides in how potential is defined, whose needs and contributions count, and how the dispersion parameters are set. Probabilistic assignment may reduce incentives for strategic negotiation, but it relocates rather than eliminates value judgments, which move into model design.

Structural Signature¶

Sig role-phrases:

Allocable resource — a divisible pool or repeated sequence of resource units supplies what the rule distributes.
Eligible participants — the declared recipient set fixes whose claims enter the probability normalization.
Human-factor inputs — measured contribution, need, and preference provide the normative quantities from which participant potentials are built.^[6]
Distribution-potential function — a disclosed weighting rule compresses those heterogeneous inputs into one value x_j for each participant.
Dispersion parameter — β controls how strongly differences in encoded potential affect allocation odds.
Exponential normalization — the rule P_j = e^(βx_j) / Σ_k e^(βx_k) converts all potentials jointly into a probability vector.^[7]
Probabilistic assignment — a draw from that vector allocates a resource unit, while repeated draws yield expected shares rather than guaranteed individual outcomes.
Concentration regimes — β equal to zero yields equal probabilities, whereas increasing positive β concentrates probability on participants with larger potentials.
Institutional-design boundary — eligibility, feasibility, factor measurement, weights, and the fairness interpretation remain external choices that the probability formula does not settle.

What It Is Not¶

Not a physical law of social fairness. The exponential form is borrowed from statistical mechanics; human need, contribution, and preference are designed normative inputs rather than particle energies.^[8]
Not proof that one realized draw is morally fair. The rule specifies ex-ante probabilities and expected repeated shares, while a particular sampled outcome can remain unequal or surprising.
Not allocation without value judgments. Eligibility, factor definitions, scales, weights, feasibility, and the meaning of fairness are chosen before the probability formula operates.
Not arbitrary lottery. Randomness qualifies only when the outcome probabilities are derived from the declared distribution potentials and normalized across eligible participants.
Not equal division except at a parameter regime. Equal probabilities arise when the dispersion parameter removes potential differences from the odds; they are not the general finite-parameter result.
Not deterministic merit ranking. Concentration on the largest potential is a limiting regime, whereas ordinary finite settings retain probabilistic allocation to lower-potential participants.
Not negotiation or market pricing. Bargaining strategies and price exchange use different allocation mechanisms; calling this rule spontaneous does not make preferences truthful or eliminate strategic behavior around its inputs.

Scope of Application¶

Boltzmann Fair Division is a domain-bounded proposed allocation model for settings in which an eligible population, allocable resource, contribution–need–preference potential, and exponential probability rule can all be specified; its preserved source base supports model analysis and proposals in the habitats below, not a claim of universal adoption or validated fairness in every setting.

Abstract fair-division models. The rule can compare probabilistic allocation driven by a combined distribution potential with equal, merit-based, or need-based schemes.
Distributive-justice analysis. Competing normative considerations can be encoded in the potential and exposed as explicit weights, while the formula itself does not decide which encoding is just.
Repeated resource allocation. Successive resource units can be drawn from the probability vector so that expected long-run shares can be distinguished from any single realization.
One-unit lotteries. A single indivisible unit can be allocated probabilistically, provided ex-ante fairness is not confused with equality of the realized outcome.
Divisible-resource modeling. Expected shares or repeated assignments can represent a divisible pool only after feasibility, minimum shares, and joint constraints have been supplied outside the simple participant-level formula.
Income-distribution studies. The model has been used to analyze income redistribution and feasible equality across national cases, with income, contribution, need, and preference measures requiring local definition.^[9]
Emissions-allocation simulations. Country-level emissions-trading scenarios can compare Boltzmann allocations with free allocation or auction baselines under declared efficiency and fairness criteria.^[10]
Public-goods allocation proposals. Public resources may be assigned through the rule when eligibility, the public objective, and the meaning of each human-factor input are institutionally specified.
Vaccine-distribution proposals. Need, contribution, and preference can be modeled as allocation inputs, but medical priorities, minimum guarantees, feasibility, and ethical constraints remain external requirements.
Government-budget planning proposals. Competing recipients or programs can receive potential-sensitive allocation probabilities only after budget indivisibilities and policy constraints are represented.
Social-welfare optimization. The probability rule can be coupled to a welfare function, while the welfare objective and admissible allocations remain separate from the Boltzmann normalization.
Entropy-based allocation research. Entropy maximization and the dispersion parameter can be studied as formal mechanisms for balancing concentration against spread within the proposed fair-division framework.
Sensitivity and scenario analysis. Analysts can vary factor weights or β to examine how normative inputs and concentration regimes alter expected allocations.
Comparisons with strategic allocation. The model can serve as a nonnegotiated benchmark against bargaining, auctions, or game-theoretic procedures, without implying that its reported inputs are strategy-proof.
Pedagogical thought experiments. Small-group examples such as allocating a scarce shared good can make the potential-to-probability operation legible without counting as evidence of field deployment.
Emerging empirical evaluation. Applications remain within scope only as explicitly proposed, simulated, or studied instances until primary-source verification and independent evidence establish operational uptake and performance.

Clarity¶

A clear account writes the potential and probability equations, defines factor direction and scale, and explains what parameter changes make allocation more concentrated or diffuse. “Spontaneous” means generated by the rule without negotiation; it does not mean uncaused or norm-free.

Fairness claims should name the criterion being advanced and compare results with relevant alternatives.

Manages Complexity¶

Boltzmann Fair Division compresses a multidimensional allocation dispute into a distribution potential \(x_j\) for each eligible participant and one dispersion parameter \(\beta\). Contribution, need, and preference enter through the declared construction of \(x_j\); the exponential normalization \(P_j=e^{\beta x_j}/\sum_k e^{\beta x_k}\) then turns all participants' potentials into a probability vector. The analyst can read expected shares and repeated-allocation frequencies from that vector without resolving every claim through a separate negotiation.

The parameter exposes the model's principal branches: \(\beta=0\) gives equal probabilities, increasing positive \(\beta\) concentrates allocation on larger potentials, and the large-\(\beta\) limit approaches selection of the highest-potential claims while lower-ranked participants can retain nonzero probability at finite values. The compression stops before normative and institutional design. It does not decide how incomparable contributions, needs, and preferences should be measured or weighted, which allocations are feasible, whether reports are manipulable, or whether ex-ante probabilities make a realized unequal draw fair. Different resource divisibility, eligibility, welfare, and minimum-guarantee rules must be restored before using the formula in a policy setting.

Abstract Reasoning¶

The characteristic diagnostic move runs from the declared probability vector back to the modeled distribution potentials. For two participants under the same rule, the probability ratio satisfies \(P_j/P_k=e^{\beta(x_j-x_k)}\); with a known positive \(\beta\), higher allocation odds therefore diagnose a higher encoded potential, not greater need, contribution, or preference separately. To identify which consideration created that ordering, the analyst must reopen the construction of \(x_j\), its factor weights, and its scales. A result attributed to “fairness” can thus be decomposed into the value judgments already embedded in the potential.

An interventionist move changes one modeling choice and predicts how the allocation distribution responds. Increasing a participant's potential, or increasing the weight on a factor on which that participant scores relatively highly, raises that participant's odds relative to lower-potential participants when \(\beta>0\). Holding potentials fixed while moving \(\beta\) from zero upward moves the rule from equal probabilities toward increasing concentration on the largest potentials; the large-positive-\(\beta\) limit approaches highest-potential selection. These are predictions about probabilities and expected repeated allocations, not guarantees about the recipient of one random draw.

A boundary move runs from the resource and institutional design to whether the simple probability rule is an adequate allocation model. The analyst must specify eligible participants, feasible outcomes, resource divisibility, whether draws are repeated, and whether minimum guarantees or linked constraints apply. Arbitrary randomization is outside the named construction if probabilities do not arise from the potential, while a deterministic highest-score rule is its limiting case rather than the ordinary finite-\(\beta\) regime. Sensitivity to factor definitions, weights, and \(\beta\) reveals whose claims control the modeled distribution; it does not establish that the inputs are truthful, immune to strategy, or normatively justified.

Knowledge Transfer¶

Within fair-division modeling, the construction transfers literally across divisible resources, repeated assignments, income models, emissions allocations, or public-resource proposals only when the eligible set, feasible outcomes, distribution-potential function, factor scales, and randomization rule are redefined transparently. What carries is the mapping from contribution, need, and preference inputs into x_j, followed by normalized probabilities P_j = e^(β x_j) / Σ_k e^(β x_k). The vocabulary of potential, factor weight, dispersion parameter, feasible allocation, ex-ante probability, realized draw, and expected repeated share supports diagnostics for hidden normative weights, incomparable input scales, manipulation, and claims that a surprising draw disproves the rule. Interventions include varying one factor weight or β, checking the equal-probability and concentrated limits, and comparing sensitivity under alternative fairness criteria; empirical or moral conclusions do not transfer between applications merely because the formula does.

Beyond this proposed fair-division model, the honest reach is B — shared abstract mechanism through Allocation plus A — analogy to statistical mechanics. Other allocation rules can share the general mechanism of assigning scarce resources among eligible claims, but they do not inherit the Boltzmann potential or its fairness interpretation. The exponential mathematical form is borrowed from a physical distribution; particles, energy states, and thermodynamic equilibrium do not literally supply the social values of need, contribution, or preference. Those human factors, feasibility rules, and institutional judgments remain home-bound to the allocation setting. Transfer stops before mathematical resemblance is treated as normative justification, before “spontaneous” is taken to eliminate model design or strategic reporting, or before a finite-β probability rule is confused with equal or deterministic division.

Examples¶

Canonical¶

Suppose one resource unit is to be assigned among three eligible participants whose disclosed contribution–need–preference rule yields potentials x = (0,1,2). Choose β = ln 2. The exponential weights are then (1,2,4), so normalization gives probabilities (1/7,2/7,4/7). A single draw can still award the unit to the lowest-potential participant; over repeated independent allocations, the model concerns expected frequencies, not a guarantee about each realization. At β = 0, the same participants would instead receive equal probabilities of 1/3.

Mapped back: The one unit is the Allocable resource, and the three recipients are the Eligible participants. Their underlying contribution, need, and preference scores are the Human-factor inputs, which the declared Distribution-potential function compresses into (0,1,2). The Dispersion parameter is ln 2; Exponential normalization produces the checked probability vector, and Probabilistic assignment produces the draw. Comparing β = ln 2 with β = 0 exposes the Concentration regimes without settling the Institutional-design boundary.

Applied / In Practice¶

The preserved source describes an eight-country emissions-trading simulation in which country-level human-factor inputs are encoded as distribution potentials and the Boltzmann rule allocates emissions units probabilistically. Researchers can compare the resulting expected allocations with free-allocation and auction baselines under declared efficiency and fairness criteria, then vary β or the factor weights to see which normative choices drive the comparison. The simulation demonstrates use of the proposed rule; it does not establish that the factors are uniquely measurable, that strategic reporting disappears, or that the same encoding should govern an operational policy.

Mapped back: Emissions units instantiate the Allocable resource, and the eight countries the Eligible participants. Country measures enter as Human-factor inputs and become values through the Distribution-potential function. Exponential normalization and Probabilistic assignment generate the modeled allocations, while sensitivity to β traverses the Concentration regimes. Measurement, weights, feasibility, and the meaning of fairness remain within the Institutional-design boundary.

Structural Tensions¶

T1: Normative plurality versus scalar distribution potential. Contribution, need, and preference are different kinds of claim, and compressing them into one x_j makes a common probability calculation possible. The compression also hides compensations: a large score on one factor can offset another under weights and scales chosen by the designer. Refusing aggregation leaves the rule unable to rank claims; accepting the scalar as self-justifying disguises contestable moral choices as measurement. Diagnostic: How are the factors defined, scaled, weighted, and combined, and would a plausible alternative encoding materially reorder the allocation probabilities?

T2: Ex-ante fairness versus unequal realized outcomes. A probability vector can give each participant a defensible chance or expected repeated share while a single draw awards the whole indivisible unit to one person. Judging only the realization can condemn every lottery after the fact; judging only the rule can ignore repeated streaks, vulnerable recipients, or settings in which minimum outcomes matter. The relevant fairness object must therefore be declared. Diagnostic: Is the criterion evaluating opportunity before the draw, expected shares over repetition, or the realized distribution, and does the resource setting make that object normatively adequate?

T3: Dispersion control versus designer discretion. The parameter β exposes a useful continuum from equal probabilities toward concentration on higher encoded potentials. Its apparent technical neutrality can conceal a major distributive choice: the same claims yield radically different odds under different values. Fixing β by convention improves comparability but may not fit the stakes; tuning it to obtain a favored outcome reverses the intended transparency. Diagnostic: What substantive criterion selects the dispersion parameter, and was it chosen independently of the allocation pattern the designer hoped to produce?

T4: Nonnegotiated assignment versus strategic input formation. Sampling from a formula can avoid direct bargaining over each resource unit, but participants or institutions may still influence reported need, credited contribution, stated preference, eligibility, or the weights applied. Calling the output “spontaneous” can therefore move strategic behavior upstream rather than remove it. Rejecting the rule because inputs can be contested would also apply too broadly to allocation systems. Diagnostic: Which inputs can interested parties shape, and what makes the encoded potential sufficiently reliable for the claimed nonnegotiated advantage?

T5: Participant-level probabilities versus jointly feasible allocations. Exponential normalization produces valid probabilities over eligible recipients for one draw, but divisible pools, bundles, quotas, complementarities, and minimum guarantees can constrain combinations of outcomes. Applying the simple vector repeatedly is transparent yet may create an infeasible or normatively prohibited aggregate. Adding every institutional constraint can obscure the model's core comparative mechanism. Diagnostic: Does independent sampling respect the resource's joint feasibility and guarantees, or must the potential rule be embedded in a larger constrained allocation design?

T6: Mathematical analogy versus normative justification. The Boltzmann form supplies a tractable way to transform potentials into probabilities and gives entropy and concentration tools for analysis. Its origin in statistical mechanics does not show that social claims behave like energies or that the resulting distribution is just. Dismissing the form because the analogy is nonliteral ignores its mathematical utility; treating physical pedigree as moral authority naturalizes a designed policy. Diagnostic: What fairness argument supports the human-factor encoding and allocation criterion independently of the borrowed exponential form?

T7: Formal sensitivity versus empirical legitimacy. The model makes consequences of weights and β calculable, which supports scenario analysis. That internal clarity does not establish that contribution, need, and preference are measured validly, that participants accept the rule, or that a proposed application improves on alternatives. Requiring mature deployment evidence before theoretical study would block evaluation of an emerging construct; presenting simulations as validated governance would outrun the narrow evidence base. Diagnostic: Is the claim about the formula's formal behavior, a simulated comparison, or demonstrated institutional performance, and does the evidence match that level?

T8: Boltzmann Fair Division autonomy versus reduction to Allocation (Allocation). The parent Prime carries the portable assignment of scarce resources among eligible claims under a rule. Every Boltzmann Fair Division is a strict kind of Allocation because it produces resource-assignment probabilities, but the child specifically builds a contribution–need–preference potential and applies a β-controlled exponential normalization. Reduction loses that named transformation and its disclosed human factors; total autonomy hides the general allocation structure. Diagnostic: Does the case retain the human-factor potential and Boltzmann-like probability rule as differentia of this Allocation?

Structural–Framed Character¶

Boltzmann Fair Division is framed-leaning. Its evaluative_weight is strong because the model presents a way to combine contribution, need, and preference under a fairness interpretation. Its human_practice_bound character is strong: eligibility, resource claims, factor definitions, and realized assignment all belong to allocation practice. Its institutional_origin is substantial because an institution must specify the claimant set, feasible outcomes, weights, and acceptable fairness criterion. Its vocab_travels result is mixed: probability and allocation language generalize, while the named distribution potential retains its proposed fair-division meaning. Its import_vs_recognize result favors import, because the formula does not reveal a pre-existing social distribution without a designed encoding of claims and parameter choices.

The smallest portable skeleton is Allocation: limited supply is assigned among competing claimants under a feasibility constraint and selection criterion. Boltzmann Fair Division supplies the contribution–need–preference potential, exponential normalization, dispersion parameter, and proposed fairness interpretation. Portable and cross-domain reach belongs to that Prime; probability is a constitutive formal mechanism, while fairness evaluates the rule rather than owning its operative structure.

Its character: a framed-leaning allocation model whose formal probability rule is real but whose identity and interpretation remain inseparable from human decisions about claims, values, and feasible division.

Structural Core vs. Domain Accent¶

Boltzmann Fair Division is domain-specific rather than a prime because it specializes Allocation (Allocation) with a proposed human-factor potential and exponential random-assignment rule whose fairness interpretation remains contestable.

What is skeletal (could lift toward a cross-domain prime). A limited supply is contested by plural eligible claimants, a feasibility envelope bounds possible shares, an assignment maps claimants to realized or expected portions, and a selection criterion chooses among feasible mappings. Holding supply and claimants fixed while changing the criterion can change the distribution, which makes the criterion's work inspectable. Recognition requires all five roles—bounded supply, competing claims, feasibility, assignment, and criterion; scarcity alone, delivery with no contention, or probability with no resource-to-claimant mapping fails the Allocation skeleton.

What is domain-bound. The accent defines each claimant's distribution potential from declared contribution, need, and preference inputs, then applies P_j = e^(βx_j) / Σ_k e^(βx_k) across the eligible set. The dispersion parameter controls the equal-probability and increasingly concentrated regimes, and random draws produce realized assignments or expected repeated shares. Eligibility, factor definitions and scales, weights, feasibility constraints, resource divisibility, guarantees, and the adopted fairness criterion are institutional choices that the exponential normalization does not settle.

Why this does not clear the prime bar. Allocation recurs literally in economic budgets, operating-system scheduling, and organismal energy partition, but the complete Boltzmann Fair Division signature does not recur literally across at least three unrelated domains: the contribution–need–preference potential, β-controlled normalization, and proposed distributive interpretation are its specific accent. Knowledge Transfer carries the Allocation roles to other division problems and treats the statistical-mechanics source as analogy rather than normative authority. Remove the Boltzmann potential and its fairness framing and a criterion-governed resource assignment can remain, which is Allocation but not this model. Remove bounded supply, plural claimants, feasibility, or claimant-to-share assignment while retaining the exponential formula and one has a probability distribution, not the domain-specific fair-division abstraction.

This entry is a kind of Allocation.

Instantiates — Allocation (Allocation). The allocable resource is the limited supply, eligible participants are the competing claimants, externally declared feasibility rules bound what can be assigned, and the normalized Boltzmann probabilities map those claimants to realized or expected shares. The distribution-potential function and dispersion parameter jointly supply the selection criterion: holding the pool and claimant set fixed while changing those inputs changes the assignment. Remove the bounded pool, plural claimants, feasibility envelope, or claimant-to-share mapping and the construction may remain an exponential probability model, but it is no longer Boltzmann Fair Division. Its domain-specific residual is the proposed contribution–need–preference potential and its interpretation as a fair-division rule.

Related to — Probability (Probability). Exponential normalization produces a probability measure over eligible recipients, so probabilistic reasoning governs draws and expected repeated shares. Probability is a constitutive formal mechanism here, but it does not supply the scarce-resource, claimant, feasibility, or normative-criterion roles that make Allocation the broader abstraction.

Decline — Fairness (Fairness) as the broader abstraction. The model proposes one contestable way to encode contribution, need, and preference; its formula does not establish that the inputs, weights, or realized outcomes satisfy a defensible fairness standard. Fairness therefore evaluates the allocation rather than subsuming its complete operative structure.

Relationships to Other Abstractions¶

Current abstraction Boltzmann Fair Division Domain-specific

Parents (1) — more general patterns this builds on

Boltzmann Fair Division is a kind of Allocation Prime

The allocable resource is the limited supply, eligible participants are the competing claimants, externally declared feasibility rules bound what can be assigned, and the normalized Boltzmann probabilities map those claimants to realized or expected shares.

Hierarchy path (1) — routes to 1 parentless root

Boltzmann Fair Division → Allocation → Scarcity → Constraint

Neighborhood in Abstraction Space¶

Boltzmann Fair Division sits in a sparse region of the domain-specific corpus (71^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Strategic Decision Biases & Mechanisms (29 abstractions)

Nearest neighbors

Individual-Pieces Set — 0.86
Buzen's Algorithm — 0.84
Participation constraint (mechanism design) — 0.83
Principle of Maximum Entropy — 0.83
Belief Aggregation — 0.83

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Boltzmann Distribution. The physical Boltzmann distribution assigns probabilities to energy states under a thermodynamic model, whereas Boltzmann Fair Division applies an analogous exponential form to designed human-factor potentials. Tell: particle energies and temperature identify the physical distribution; contribution, need, preference, and an allocation-dispersion parameter identify the fair-division model.
Random Allocation. Random allocation is the broad class of lotteries or randomized rules, whereas Boltzmann Fair Division derives each participant's odds from a declared distribution-potential function and exponential normalization. Tell: arbitrary or uniform draw probabilities give a random allocation; potential-sensitive normalized probabilities give the Boltzmann model.
Equal Division. Equal division gives participants equal shares or equal chances by rule, whereas Boltzmann Fair Division generally weights probabilities by differences in encoded potential. Tell: equality independent of contribution, need, and preference is equal division; equality appearing only at the zero-dispersion regime is a special Boltzmann case.
Market Allocation. Market allocation coordinates distribution through prices, exchange, and purchasing decisions, whereas Boltzmann Fair Division samples from institutionally constructed probabilities without a price-clearing mechanism. Tell: bids, prices, or trades allocate in the market case; the potential-to-probability rule allocates in the Boltzmann case.
Fair Division. Fair division is the broader field of criteria and procedures for allocating resources, whereas Boltzmann Fair Division is one proposed probabilistic model with a specified potential and exponential rule. Tell: an envy-free, need-based, merit-based, or negotiated procedure can be fair division; it is Boltzmann Fair Division only when the named probability mechanism is preserved.

References¶

[1] The Boltzmann Fair Division for Distributive Justice registry ↩ Show verification details

[2] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[3] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[4] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[5] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[6] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[7] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[8] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[9] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩

[10] Unverified encyclopedia synthesis; claim-specific authoritative support was not established in this verification pass. ↩