Two-Sided Matching¶
Core Idea¶
Two-sided matching is the structural pattern of forming pairings or assignments between the members of two sets — agents with agents, or agents with objects — where each side carries preferences or compatibility constraints, no money-price clears the market, and the central question is the stability and efficiency of who-gets-paired-with-whom. The defining move is that allocation happens by mutual selection across a bipartite relation rather than by a single clearing price. [1] The pattern was first made precise by Gale and Shapley (1962) in their stable marriage problem, which proved that for any set of preference rankings on two sides a stable matching always exists and can be found constructively. [2] The body of theory that grew from that result — studied under the names matching theory and stable matching — earned Roth and Shapley the 2012 Nobel Memorial Prize in Economic Sciences, recognizing both the mathematics of stability and its translation into the design of real allocation institutions. [1]
What makes the pattern a genuine prime is that it names a recurring shape rather than a single mechanism: two populations, a relation of mutual acceptability or preference between them, and a solution concept (stability) that asks not "what is the equilibrium price?" but "is there any pair who would both prefer each other to their assigned partners?" Whenever an allocation cannot or should not be settled by price, and the parties on both sides have something to say about whom they are paired with, the situation is a two-sided matching problem.
How would you explain it like I'm…
Picking partners both ways
Matching both sides
Two-sided matching
Structural Signature¶
Two-sided matching encodes a structural pattern: two populations with side-specific preferences → a bipartite relation of acceptable pairings → a stability/efficiency criterion → a matching with no blocking pair. It separates the space of all conceivable assignments from the much smaller space of assignments no pair would defect from, and names the rule that selects within that smaller space. [2] The signature is indifferent to what the two sides are: doctors and hospitals, students and schools, donors and recipients, antibodies and antigens. What travels is the bipartite-plus-stability shape, not the economic content.
Recurring features:
- Pairing two populations by mutual selection rather than price
- Bipartite preference structure with no clearing price
- Stability as the absence of any blocking pair
- Deferred acceptance resolving preferences into a stable assignment
- Proposer-optimal versus receiver-optimal outcomes
- Strategy-proofness of truthful preference revelation
- Money-free allocation under two-sided compatibility constraints
The structural insight is robust: a hospital ranking residents, a school district ranking applicants, a kidney-exchange registry searching for compatible cycles, and a population of immune receptors binding complementary antigens all exhibit the same logic of two sets resolved into pairings that no member would unilaterally abandon. [1] The deferred-acceptance procedure that proves stability in the marriage problem is the same procedure that clears the National Resident Matching Program; the structure is what carries the algorithm from one setting to the next.
What It Is Not¶
Two-sided matching is not any allocation problem whatsoever. It specifically requires two sides that each have preferences or compatibility constraints over the other. A problem in which only one side has preferences (a queue, a lottery, a first-come-first-served line) is one-sided allocation, not two-sided matching; the bilateral structure — and the blocking-pair logic that depends on it — is absent.
Nor does the prime claim that matching markets lack prices in some absolute sense. Residents are paid salaries; schools have budgets; kidney exchanges occur within healthcare systems that cost money. The claim is narrower and structural: there is no single clearing price that adjusts to equate supply and demand and thereby determine who is paired with whom. Prices may exist in the background, but they are not the allocation mechanism. When a price does the clearing, the situation is a priced market, not a matching market.
Two-sided matching is also not a guarantee of a unique or fair outcome. Stability is a weak property: it rules out pairs who would both defect, but it typically admits many stable matchings, and which one an algorithm returns depends on its design (who proposes). One stable matching can be best-possible for every member of one side and worst-possible for every member of the other. The prime names the structure and its solution concept; it does not assert that the structure yields equity, uniqueness, or optimality in any welfare sense.
Finally, the prime is not a claim about intent or design. A two-sided matching can arise spontaneously (mating markets, molecular binding) with no designer at all, or be engineered deliberately (the residency match, school-choice mechanisms). Recognizing a situation as two-sided matching says something about its structure — bipartite, preference-driven, price-free, stability-seeking — not about whether anyone arranged it.
Broad Use¶
Economics & market design: Labor markets (the National Resident Matching Program assigning graduating doctors to residency positions), school choice (centralized assignment of students to public schools), and entry-level professional markets are designed as deferred-acceptance mechanisms that produce stable matchings. [3] Market design as a field grew substantially out of repairing markets that had unraveled for lack of a stable clearinghouse.
Computer science & operations research: The stable-marriage problem and bipartite assignment underlie online ad-slot allocation, distributed task scheduling, content-delivery routing, and ride-hailing dispatch, where riders and drivers must be paired in real time under mutual constraints rather than by a posted price. [4] The Gale–Shapley algorithm and its weighted/optimal-assignment cousins (the Hungarian algorithm for the assignment problem) are standard tools.
Organ transplantation: Kidney-exchange programs match incompatible donor-recipient pairs into compatible cycles and chains, precisely because a direct cash market for organs is prohibited; the matching structure substitutes for the forbidden price. [5]
Biology (non-obvious): Antibody-antigen and receptor-ligand binding is a matching of complementary molecular shapes across two populations under affinity constraints — a money-free, preference-analog (affinity) pairing of two sets with its own notion of "stable" binding. The structure recurs with no economic content whatsoever.
Social institutions: Marriage markets, mentor-mentee programs, roommate assignment, and adoption matching all form bilateral pairings under mutual preference where price is unavailable or socially proscribed.
Clarity¶
Two-sided matching names the class of allocation problems where price cannot or should not clear the market, and in doing so it sharpens the central design question from "what is the equilibrium price?" to "which pairing rule yields stable, strategy-proof, and efficient assignments?" [1] This reframing is the prime's first contribution to clarity: it lets a practitioner recognize that a problem is not an ordinary priced market and stop searching for a price that will never do the allocating.
The concept also clarifies what "good" means in such markets. In a priced market, the diagnostic is whether the market clears at an equilibrium price. In a matching market, the diagnostic is stability — whether any pair exists who would both prefer each other to their current partners, since such a pair would defect and unravel the assignment. Reframing the goal as "no blocking pair" rather than "market clears" tells the designer what failure looks like (unraveling, exploding offers, side deals) and what success looks like (an assignment everyone is willing to live with). This is why centralized clearinghouses replaced chaotic decentralized hiring in markets like medical residency: the decentralized version produced unstable, ever-earlier offers, and the matching frame named the cure.
Manages Complexity¶
By reducing allocation to a bipartite preference structure plus a stability criterion, two-sided matching collapses a combinatorial explosion of possible assignments — for n members on each side there are n! possible perfect matchings — into the much smaller set satisfying "no blocking pair," and supplies algorithms that find such an assignment in polynomial time. [2] The designer does not have to search the factorial space; the deferred-acceptance procedure constructs a stable matching directly.
This management of complexity is what makes the prime practically powerful. A school district facing tens of thousands of students and hundreds of schools cannot enumerate assignments, but it can run a single mechanism that, given everyone's preferences, returns a stable assignment with provable properties. The structure also makes the relevant trade-offs legible: the designer can ask which side the mechanism favors (proposer-optimality), whether participants are safe revealing true preferences (strategy-proofness), and whether the outcome is Pareto-efficient. Each of these is a property of the structure, computable and comparable across mechanism choices, rather than a matter of negotiation case by case.
Abstract Reasoning¶
Recognizing the matching structure licenses inferences that would otherwise require fresh analysis in each domain. Gale and Shapley's existence proof guarantees that a stable matching always exists for strict preferences, so a designer need never wonder whether the goal is attainable. [2] The lattice structure of the set of stable matchings tells us that there is a proposer-optimal and a receiver-optimal stable matching, and that the proposing side's best stable outcome is the receiving side's worst — a fact about who is advantaged by the algorithm's design that transfers to any instance of the structure.
The structure also frames questions of incentive. Roth's impossibility and strategy-proofness results establish when truthful preference revelation is a dominant strategy and when no stable mechanism can be strategy-proof for both sides at once. [6] A practitioner who recognizes a new problem as two-sided matching immediately inherits this whole apparatus: existence, the proposer/receiver-optimal poles, the strategy-proofness boundary, and the unraveling pathology that afflicts decentralized matching markets. The prime turns disparate pairing problems into one mathematical object about which a great deal is already known.
Knowledge Transfer¶
The deferred-acceptance insight from residency matching transfers directly to school-choice assignment and to kidney-exchange chains: in each, a money-free bipartite preference structure is resolved into a stable matching by the same algorithmic logic, and a designer fluent in one can redesign another. [1] When economists were asked to fix school-choice systems, they did not start from scratch; they recognized the residency-match structure and ported the mechanism, adjusting only the domain-specific constraints (sibling priorities, walk-zone preferences) while keeping the stability core intact.
The transfer reaches even into biology, where the molecular shape-complementarity of immunology is the same pairing structure under affinity rather than preference: two populations, a relation of acceptable pairing, and a notion of which bindings persist. [7] An immunologist who recognizes the bipartite-matching shape can borrow combinatorial reasoning about coverage and competition; a market designer who studies binding affinities can borrow intuition about competition for scarce complementary partners. The transfer is not metaphorical alone: it rests on the shared bipartite-plus-stability structure, which is why a single algorithm and a single existence theorem cover cases as distant as doctors-to-hospitals and antibodies-to-antigens. [2]
Examples¶
Formal/abstract¶
The stable marriage problem: Consider n men and n women, each with a strict ranking of all members of the opposite set. A matching pairs everyone; it is unstable if some man and woman each prefer the other to their assigned partner, because that pair would defect. Gale and Shapley's deferred-acceptance algorithm runs in rounds: each unmatched man proposes to his most-preferred woman who has not yet rejected him; each woman tentatively holds her best proposal so far and rejects the rest; rejected men propose down their lists. The procedure terminates in at most n² proposals with a stable matching, and it is proposer-optimal — every man gets the best partner he could have in any stable matching, while every woman gets her worst stable partner. Mapped back: This abstract object is the whole prime in miniature: two populations, side-specific preferences, no price, and a solution concept (no blocking pair) reached by a constructive procedure whose design silently decides which side is favored. Every applied instance below is a relabeling of these same elements, which is why a single proof and a single algorithm suffice for all of them.
The assignment problem (one-sided objects): A weighted bipartite graph pairs n workers with n tasks, each worker-task pair carrying a value, and the goal is the maximum-weight perfect matching. The Hungarian algorithm solves it in polynomial time. Here only one side has preferences (the values), so "stability" reduces to optimality of total weight rather than the absence of a blocking pair. Mapped back: This is the boundary of the prime: when only one side ranks, the two-sided blocking-pair logic collapses into ordinary optimization. The contrast makes the defining feature visible — it is the mutual preference of two sides that creates the stability question and makes the problem genuinely two-sided rather than a single-objective assignment.
Applied/industry¶
The National Resident Matching Program: Before 1952, graduating medical students and hospitals hired through a decentralized scramble that "unraveled" — offers crept ever earlier, exploding deadlines forced premature commitments, and the market repeatedly broke down. The NRMP replaced this with a centralized clearinghouse: students rank residency programs, programs rank students, and a deferred-acceptance algorithm produces a stable matching released on a single Match Day. Because the matching is stable, no student-program pair has an incentive to make a side deal outside the match, and the unraveling stops. The mechanism was later redesigned (Roth–Peranson) to handle couples seeking jobs in the same city while preserving stability as far as possible. Mapped back: The residency match is the stable-marriage problem wearing professional clothing — doctors and hospitals are the two populations, rank-order lists are the preferences, salaries exist but do not clear the market, and stability is exactly what prevents the decentralized chaos that price-clearing would not have fixed. Recognizing the structure is what let economists diagnose unraveling and prescribe the cure.
Kidney-exchange chains: Many patients with willing but biologically incompatible living donors can be helped by swapping: donor A gives to patient B while donor B gives to patient A, or longer cycles and chains seeded by an altruistic donor. Because buying and selling organs is illegal, no price can clear this market; instead a registry collects compatibility data and searches for cycles and chains that maximize the number of compatible transplants. The problem is a two-sided matching over incompatible pairs, solved by combinatorial optimization under medical-compatibility constraints. Mapped back: Kidney exchange is two-sided matching forced into existence precisely because the price mechanism is forbidden — the structural prime is not a stylistic choice but the only available allocation logic when money is ruled out and both sides (here, biological compatibility on each side of every potential transplant) constrain who may be paired with whom. The blocking-pair intuition becomes "is there a feasible swap that would help patients currently left unmatched?"
Structural Tensions¶
T1: Stability and efficiency can pull in different directions. A stable matching rules out defecting pairs, but it need not maximize total welfare; in school choice, the stable (no-justified-envy) assignment can leave students worse off than an efficient (Pareto-optimal) but unstable one would. Designers must choose which property to guarantee, because in general no mechanism delivers both stability and full Pareto-efficiency together. The very stability that prevents unraveling can lock in assignments that everyone could improve upon by trading, and the trade that would improve them is exactly the kind of defection stability forbids.
T2: Every stable matching favors one side at the expense of the other. The set of stable matchings forms a lattice with a proposer-optimal pole and a receiver-optimal pole, and the proposing side's best stable outcome is the receiving side's worst. There is no neutral stable matching that is simultaneously best for both. The choice of who proposes — students or schools, doctors or hospitals — is therefore a distributive decision disguised as a technical one, and it is often invisible to the participants whose welfare it silently allocates.
T3: Strategy-proofness cannot be guaranteed for both sides at once. Deferred acceptance is strategy-proof for the proposing side — truth-telling is a dominant strategy — but the receiving side can sometimes gain by misrepresenting preferences, and no stable mechanism is strategy-proof for everyone. This means the structural promise "just tell us your true preferences" is honest for one side and a half-truth for the other, and sophisticated participants on the disadvantaged side may game the mechanism, eroding the data quality the mechanism depends on.
T4: Defining the preference order is itself a contested act. The mathematics assumes each side arrives with a clean strict ranking, but real preferences are incomplete, intransitive, constructed in the act of ranking, or shaped by what participants believe the mechanism will do with them. School-choice rankings are influenced by perceived odds; residency rankings by signaling norms. The structure treats preferences as exogenous inputs, yet the mechanism's own design reshapes the preferences it then processes, blurring the line between measuring choices and manufacturing them.
T5: Priorities and quotas smuggle policy into the stability concept. Real matching markets attach priorities (sibling status, walk-zones, affirmative-action reserves) and quotas that alter what counts as a "blocking pair." A matching can be stable with respect to one priority structure and unstable with respect to another, so the apparently neutral language of stability quietly encodes contested distributive choices. Whose priority counts, and how reserves interact, determines outcomes as much as the algorithm does, while presenting itself as mere constraint satisfaction.
T6: The price-free framing can obscure resources that still matter. Declaring a market "money-free" focuses attention on preferences and compatibility, but participants' ability to form good preferences, navigate the mechanism, or even enter it is often unequally distributed by wealth, information, and access. A school-choice system with no tuition can still advantage families who can decode the mechanism, tour schools, and game rankings. The structural prime is silent on these background inequalities, and treating allocation as purely a matching problem can launder them as neutral outcomes of a fair procedure.
Structural–Framed Character¶
Two-Sided Matching sits at the structural end of the structural–framed spectrum: it names the pattern of forming pairings between the members of two sets — agents with agents, or agents with objects — where each side carries preferences or compatibility constraints, no money-price clears the market, and the central question is the stability and efficiency of who-gets-paired-with-whom. Allocation happens by mutual selection across a bipartite relation rather than by a single clearing price.
The bipartite preference-and-stability structure, formalized by Gale and Shapley, is substrate-neutral and definable without reference to human practice; it carries no normative charge. Receptor-ligand binding instantiates a two-sided matching in which molecules pair according to compatibility constraints with no price involved. Only a mild matching-market vocabulary leans framed, while applying the prime recognizes a bipartite-matching structure already present rather than importing a stance. On every diagnostic, it reads structural.
Substrate Independence¶
Two-Sided Matching is a moderately substrate-independent prime — composite 3 / 5 on the substrate-independence scale. The pattern — pairing two sets by mutual preference with no clearing price, prioritizing stability and efficiency — is genuinely structural and reaches across residency and school-choice allocation, stable-marriage and ride-hailing dispatch algorithms, and even shape-complementary antibody-antigen binding. The deferred-acceptance transfer is real, but it stays inside an allocation-mechanism family and carries an economics-and-algorithmics flavor. Because it does not reach physical, formal, or cognitive substrates as such, the structure travels meaningfully but within a bounded neighborhood.
- Composite substrate independence — 3 / 5
- Domain breadth — 3 / 5
- Structural abstraction — 4 / 5
- Transfer evidence — 3 / 5
Relationships to Other Primes¶
Parents (2) — more general patterns this builds on
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Two-Sided Matching is a kind of Allocation
Two-sided matching is a specialization of allocation. The general allocation pattern assigns limited supply across competing claims under a feasibility constraint and a criterion. Two-sided matching specializes by structuring the assignment as pairings between members of two sets, where both sides carry preferences and no money-price clears the market. The same who-gets-how-much commitment of allocation applies, with mutual selection across a bipartite relation as the specific mechanism and stability and efficiency as the specific evaluative criteria.
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Two-Sided Matching presupposes Preference
Two-sided matching forms pairings across two sets where each side carries a ranking over potential partners, and central solution concepts like stability and efficiency are defined in terms of those rankings. Without Preference — an evaluator's ordering over a choice set — there is no stability to check (no blocking pair can be identified) and no efficiency criterion to apply. Matching presupposes preference as the primitive that supplies the rankings the allocation must respect.
Path to root: Two-Sided Matching → Preference
Neighborhood in Abstraction Space¶
Two-Sided Matching sits among the more crowded primes in the catalog (25th percentile for distinctiveness): several abstractions describe nearly the same structure, so a description that fits it will tend to fit its neighbors too — transporting it usually means disambiguating within this family rather than landing on it exactly.
Family — Preferences, Trade-offs & Commensuration (9 primes)
Nearest neighbors
- Competition — 0.83
- Exchange — 0.81
- Asymmetry — 0.81
- Conflict of Interest — 0.81
- Allocation — 0.81
Computed from structural-signature embeddings · 2026-05-29
Not to Be Confused With¶
Two-sided matching must be distinguished from the Correspondence Principle, which is the embedding-similarity artifact flagged as its nearest existing prime. The correspondence principle is a constraint on theory succession: it requires that a new, more general theory reproduce the predictions of the older theory it supersedes in the limiting regime where the old theory was known to work — quantum mechanics must reproduce classical mechanics for large quantum numbers, special relativity must reduce to Newtonian mechanics at low velocities. Its subject matter is the relationship between two descriptions of the same reality across a limit. Two-sided matching has nothing to do with theory succession or limiting regimes; its subject is the pairing of members of two distinct populations under mutual preference. The two share surface vocabulary — both involve a relation between two things — but the relata are utterly different: the correspondence principle relates an old theory to its successor, whereas matching relates a doctor to a hospital. The high embedding similarity is an artifact of both texts discussing "two-sided" relations and "correspondence," not of any shared structure. Where the correspondence principle is about epistemic continuity across theoretical revolutions, matching is about allocative stability across a bipartite market.
Two-sided matching is also not Auction Theory, and this is the most instructive contrast because the two are precisely complementary. An auction allocates goods via price and bids: participants compete by offering money, and the allocation is determined by who pays what, with the price doing the clearing. Two-sided matching is the priceless counterpart — it handles exactly those allocation problems where a price cannot or should not clear the market, either because money is prohibited (kidney exchange), because the goods are also choosers with their own preferences (hospitals choosing residents, schools ranking students), or because price-based allocation would be morally or socially unacceptable. In an auction, only the buyers have preferences and the seller wants the highest price; in a matching market, both sides rank each other and there may be no seller at all. The defining question differs accordingly: an auction asks "what allocation maximizes revenue or efficiency given bids?" while matching asks "what assignment has no blocking pair?" The two frameworks even share a common parent in mechanism design — both were studied by overlapping casts of economists — but they sit on opposite sides of the price/no-price divide, and conflating them leads designers to reach for a price where none can do the work, or to ignore the bilateral preferences that make a problem a matching problem in the first place.
Finally, two-sided matching is distinct from Duality, the structure-preserving correspondence between two equivalent formulations of a single object. Duality pairs descriptions: a linear program with its dual, a vector space with its dual space, a planar graph with its dual graph — in each case the two paired things are the same underlying object viewed from two sides, and a theorem in one translates into a theorem in the other. Two-sided matching pairs different individuals from two populations, not two descriptions of one thing. The doctor and the hospital are genuinely separate agents with conflicting interests, not dual descriptions of a single entity; their pairing is an allocation, not a translation. Duality is an equivalence (the two formulations carry the same information); matching is an assignment (it commits scarce partners to one another and forecloses other pairings). The confusion is tempting because both involve "two sides," but duality's two sides are two faces of one coin, whereas matching's two sides are two crowds of people who must be sorted into couples. One is about representational equivalence; the other is about who ends up with whom.
Solution Archetypes¶
No catalogued solution archetypes reference this prime yet.
Notes¶
The prime sits at composite substrate-independence 3: the bipartite-plus-stability shape is genuinely abstract, but the surrounding apparatus (preferences, strategy-proofness, market unraveling) carries an unmistakable economics-and-algorithmics flavor. The biological instance (antibody-antigen binding) shows the structure reaching a non-economic substrate, but it does so by reinterpreting "preference" as "affinity," which is exactly the kind of translation that signals a structurally framed rather than fully substrate-neutral prime. The pattern travels widely within the allocation-mechanism family and stretches into biology, but it does not reach physical, formal, or cognitive substrates in the way a pure structural prime such as activation energy does.
A recurring source of confusion is the assumption that "matching market" implies the absence of money. It does not. Residents are salaried; the point is only that no price clears the allocation. Keeping this distinction sharp is what prevents the prime from collapsing into "any market without explicit prices" on one side or "any allocation problem" on the other.
The stability solution concept is weaker than it first appears, and most of the prime's practical subtlety lives in that weakness. Stability admits many matchings, says nothing about fairness between the two sides, and can conflict with efficiency. Practitioners who treat "we ran a stable mechanism" as the end of the analysis routinely miss the distributive choices (who proposes, whose priorities count) that the mechanism makes on their behalf. The structural tensions above are not edge cases; they are the substance of real market design.
There is also a generative direction worth noting: many problems that present as one-sided allocation (lotteries, queues, posted-price sales) can be re-described as two-sided matching by making the second side's preferences explicit, and doing so often reveals blocking-pair instabilities that the one-sided framing concealed. The school-choice literature began precisely this way — by noticing that "assignment" systems were really suppressing schools' or students' preferences and thereby generating justified envy.
References¶
[1] Roth, A. E., & Sotomayor, M. A. O. (1990). Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis (Econometric Society Monographs No. 18). Cambridge University Press. Canonical monograph defining two-sided matching as price-free bilateral allocation by mutual selection; synthesizes the body of matching theory (recognized by the 2012 Nobel Memorial Prize to Roth and Shapley), the recurrence of the bipartite-plus-stability shape across markets, the reframing of the design question toward stability and strategy-proofness, and the transfer of deferred acceptance across residency, school choice, and kidney exchange. ↩
[2] Gale, D., & Shapley, L. S. (1962). College admissions and the stability of marriage. American Mathematical Monthly, 69(1), 9–15. Foundational stable-marriage paper: proves via the deferred-acceptance algorithm that a stable matching always exists for any preference profile on two sides and can be found constructively in polynomial time, covering instances from doctors-to-hospitals to abstract bipartite matching. ↩
[3] Roth, A. E. (1984). The evolution of the labor market for medical interns and residents: A case study in game theory. Journal of Political Economy, 92(6), 991–1016. Shows that the National Resident Matching Program implements a deferred-acceptance mechanism producing a stable matching, the model later generalized to centralized school-choice systems. ↩
[4] Kuhn, H. W. (1955). The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2(1–2), 83–97. Original polynomial-time algorithm for the maximum-weight bipartite assignment problem; the weighted-assignment cousin of stable matching underlying ad allocation, task scheduling, and real-time dispatch. ↩
[5] Roth, A. E., Sönmez, T., & Ünver, M. U. (2004). Kidney exchange. The Quarterly Journal of Economics, 119(2), 457–488. Formalizes kidney exchange as a matching over incompatible donor-recipient pairs solved by combinatorial optimization precisely because a cash market for organs is prohibited. ↩
[6] Roth, A. E. (1982). The economics of matching: Stability and incentives. Mathematics of Operations Research, 7(4), 617–628. Establishes the strategy-proofness boundary: no stable matching mechanism makes truthful preference revelation a dominant strategy for both sides at once, though deferred acceptance is strategy-proof for the proposing side. ↩
[7] Murphy, K., & Weaver, C. (2017). Janeway's Immunobiology (9th ed.). Garland Science. Standard immunology text: antibody-antigen recognition is a complementarity-of-shape binding governed by affinity, instantiating the bipartite-matching structure under affinity rather than preference. ↩