Matching Games: Theory, Models, and Applications

• Matching games are a heterogeneous class of models where pairings, b-matchings, or game moves define coalition values in diverse settings.
• They examine cooperative, capacitated, and partitioned structures with precise results on core stability, allocation rules, and associated computational thresholds.
• Advanced research spans dynamic and decentralized matching, algorithmic stabilization techniques, and recreational games that reveal deep combinatorial and algebraic insights.

Matching games are a heterogeneous class of mathematical, economic, and computational models in which the central object is a matching: a set of pairwise disjoint edges, a capacity-respecting $b$-matching, a many-to-one assignment, or a game move that creates a prescribed pattern. In current research usage, the term covers cooperative games whose coalition values are induced by optimal matchings in graphs, strategic matching markets in which matched agents generate payoffs endogenously by playing a game, dynamic and decentralized matching processes, and recreational or algorithmic games based on forming matches (Benedek et al., 2022, Garrido-Lucero et al., 2020, Gualà et al., 2014). Across these lines of work, the recurring questions are how matchings encode surplus or utility, which stability notions are appropriate, and which variants remain computationally tractable.

1. Cooperative graph-based matching games

In the classical cooperative formulation, a matching game is defined on an edge-weighted graph $G=(V,E;w)$ whose vertices are players and whose coalition value is the maximum weight of a matching in the induced subgraph. For a coalition $S\subseteq V$, this value is

$$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$

If $G$ is bipartite, the model is an assignment game; thus assignment games are exactly the bipartite special case of matching games (Benedek et al., 2022). This formulation makes the cooperative worth of a coalition entirely combinatorial: a coalition can secede and realize the best internal pairing available to it.

Several extensions replace ordinary matchings by richer packing structures. In multiple partners matching games, each player $i$ has a capacity $b_i\in \mathbb{Z}_{\ge 1}$, and a coalition’s value is the weight of a maximum $b$-matching in the induced subgraph; when every $b_i=1$, the model collapses to the classical matching game (Xiao et al., 2021). A closely related capacitated language is used for edge-weighted, vertex-capacitated graphs $G=(V,E)$ with capacities $G=(V,E;w)$0, where a $G=(V,E;w)$1-matching allows each vertex $G=(V,E;w)$2 to be incident to at most $G=(V,E;w)$3 selected edges (Gerstbrein et al., 2022). These generalizations are structurally significant because they preserve the pairing interpretation while introducing congestion, repeated partnering, or quota effects.

Partitioned matching games shift the player set from vertices to ownership classes. Here $G=(V,E;w)$4 is partitioned as $G=(V,E;w)$5, player $G=(V,E;w)$6 owns $G=(V,E;w)$7, and the coalition value $G=(V,E;w)$8 is the maximum weight of a matching in the subgraph induced by $G=(V,E;w)$9. The parameter

$S\subseteq V$0

is the width of the game, and $S\subseteq V$1 recovers the classical matching game (Benedek et al., 2023). This model was introduced for international kidney exchange, where each country controls a pool of patient–donor pairs and fairness is defined over countries rather than individual vertices.

2. Core, allocation rules, and complexity

The core is the standard cooperative stability concept. An allocation $S\subseteq V$2 is in the core if $S\subseteq V$3 and $S\subseteq V$4 for every coalition $S\subseteq V$5. For ordinary matching games, there is a particularly sharp characterization: an allocation $S\subseteq V$6 is in the core if and only if

$S\subseteq V$7

The survey literature emphasizes that assignment games always have non-empty core, whereas matching games on general graphs may have empty core; the smallest example is a triangle with unit weights (Benedek et al., 2022). When the core is empty, the least core, nucleolus, and Shapley value become natural substitutes or refinements. The same survey records a major algorithmic asymmetry: the nucleolus of a matching game is polynomial-time computable, while computing the Shapley value is $S\subseteq V$8-complete for uniform matching games (Benedek et al., 2022).

Capacities induce sharp complexity thresholds. For core-related tasks $S\subseteq V$9 (core membership), $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$0 (core non-emptiness), and $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$1 (finding a core allocation), the survey reports that ordinary matching games are polynomial-time solvable, $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$2-matching games are polynomial-time solvable when $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$3 but already hard when $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$4, and partitioned matching games show the analogous threshold at width $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$5 versus $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$6 (Benedek et al., 2022). The kidney-exchange formulation refines this picture: for partitioned matching games, $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$7, $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$8, and $$v(S)=\max\{w(M)\mid M \text{ is a matching of } G[S]\}.$$9 are polynomial-time solvable if $G$0, while for $G$1, $G$2 is co-complete and $G$3 and $G$4 are co-hard even for uniform games (Benedek et al., 2023).

Exact stability is often unavailable or hard to verify, which motivates approximate notions. For multiple partners matching games, the $G$5-approximate core requires

$G$6

An LP-based construction yields allocations in the $G$7-approximate core, and therefore guarantees that the $G$8-approximate core is always non-empty. The same work shows that $G$9 is optimal relative to the natural LP relaxation because the integrality gap is already $i$0 on disjoint triangles with unit capacities and unit weights (Xiao et al., 2021).

Population monotonicity imposes a different cooperative discipline: every player should weakly benefit when the coalition containing that player expands. For matching games, this property is exceptionally restrictive. A game admits a population monotonic allocation scheme if and only if every connected component of the underlying graph is a double-star whose two centers form a dominant pair. The proof proceeds through local obstructions such as induced $i$1, $i$2, paw, and diamond configurations, and the resulting recognition problem is polynomial-time solvable (Xiao et al., 2021).

The relation between core allocations and LP dual optima is exact in some models and only partial in others. For the assignment game, core imputations are precisely the optimal solutions to the dual of the matching LP. For concurrent games, meaning general graph matching games with non-empty core, analogous characterizations survive but are weaker because the relevant polyhedral structure is only half-integral. For unconstrained and constrained bipartite $i$3-matching games, every optimal dual solution induces a core imputation, yet not every core imputation arises from a dual optimum (Vazirani, 2022). This separates “core = dual optima” from mere non-emptiness of the core.

3. Capacitated stability and stabilization

A central capacitated notion is graph stability defined by equality of integral and fractional matching values. For an edge-weighted, vertex-capacitated graph $i$4 with weights $i$5 and capacities $i$6, the integral optimum $i$7 is the value of a maximum-weight $i$8-matching, and the fractional relaxation has optimum $i$9. The graph is stable exactly when

$b_i\in \mathbb{Z}_{\ge 1}$0

In the unit-capacity case $b_i\in \mathbb{Z}_{\ge 1}$1 for all $b_i\in \mathbb{Z}_{\ge 1}$2, this reduces to the familiar matching-LP notion of graph stability (Gerstbrein et al., 2022).

Stable graphs are important because, in the unit-capacity setting, they characterize the existence of stable outcomes in two canonical game models: network bargaining games and cooperative matching games. In a cooperative matching game, a payoff vector $b_i\in \mathbb{Z}_{\ge 1}$3 is stable when it lies in the core, meaning

$b_i\in \mathbb{Z}_{\ge 1}$4

for every coalition $b_i\in \mathbb{Z}_{\ge 1}$5; the capacitated variant replaces $b_i\in \mathbb{Z}_{\ge 1}$6 by $b_i\in \mathbb{Z}_{\ge 1}$7. The classical unit-capacity equivalence is

$b_i\in \mathbb{Z}_{\ge 1}$8

The capacitated case breaks this harmony. The implication $b_i\in \mathbb{Z}_{\ge 1}$9 still holds for capacitated network bargaining games, and $b$0 still holds for capacitated cooperative matching games, but $b$1 is false. An explicit counterexample satisfies

$b$2

yet admits the core allocation $b$3 (Gerstbrein et al., 2022).

The stabilization problem asks how to repair instability by deleting vertices. A set $b$4 is a vertex-stabilizer if $b$5 is stable. The more structured $b$6-vertex-stabilizer problem is given a maximum-weight $b$7-matching $b$8 and asks for a minimum-cardinality set $b$9 such that $b_i=1$0 is stable and $b_i=1$1 is preserved as a $b_i=1$2-matching. In the capacitated setting, the restricted problem remains tractable: the paper proves that the capacitated $b_i=1$3-vertex-stabilizer problem is solvable in polynomial time, and a related variant where $b_i=1$4 need not be maximum admits a polynomial-time $b_i=1$5-approximation (Gerstbrein et al., 2022).

The unrestricted problem is substantially harder. For capacitated graphs, even with all edge weights equal to $b_i=1$6, the vertex-stabilizer problem becomes NP-complete, and unless $b_i=1$7, there is no efficient $b_i=1$8-approximation for any $b_i=1$9. The proof is by approximation-preserving reduction from Minimum Independent Dominating Set. Algorithmically, the tractable side relies on expanding each capacitated vertex into multiple unit-capacity copies, detecting feasible augmenting walks, and decomposing them into augmenting paths, cycles, flowers, and bi-cycles (Gerstbrein et al., 2022). This marks a precise separation between matching-preserving stabilization and unrestricted stabilization.

4. Endogenous utilities, externalities, and generalized transfer

A different research line treats matching as the outer layer of a strategic game. In stable matching games, each potential pair $G=(V,E)$0 is endowed with a two-player game

$G=(V,E)$1

and an allocation $G=(V,E)$2 specifies both a matching $G=(V,E)$3 and the strategies played inside each matched pair. Pairwise stability is defined relative to a family of admissible action sets $G=(V,E)$4: no pair $G=(V,E)$5 should be able to deviate to some $G=(V,E)$6 and make both strictly better off. Without commitment, one sets $G=(V,E)$7, yielding pairwise-Nash stability. With commitment, the admissible set expands to feasible contracts or Pareto-optimal profiles. The deferred-acceptance-with-competitions algorithm produces $G=(V,E)$8-pairwise stable allocations, and renegotiation proofness is characterized by constrained Nash equilibrium: a pairwise stable allocation is renegotiation proof if and only if every matched couple plays a constrained Nash equilibrium relative to its reservation payoffs. Feasible game classes include constant-sum games with a value, strictly competitive games with an equilibrium, potential games, and infinitely repeated games (Garrido-Lucero et al., 2020).

These ideas extend to one-to-many and more general market structures. In one-to-many matching games, a hospital may be matched to multiple doctors, and its utility is the sum of the outcomes of the games it plays with its matched doctors; the deferred-acceptance-with-competitions algorithm and the renegotiation process remain polynomial whenever couples play bi-matrix games in mixed strategies (Garrido-Lucero et al., 2021). The broader framework of general matching games allows coalition-dependent payoffs, encompasses one-to-many markets and roommates models, and isolates two tractable submodels. In additive separable one-to-many matching games, any pairwise stable allocation is core stable; in the roommates submodel, pairwise stability and core stability coincide. The same framework gives polynomial-time procedures for $G=(V,E)$9-pairwise stable and $G=(V,E;w)$00-renegotiation-proof outcomes in zero-sum, strictly competitive, potential, and repeated-game settings (Garrido-Lucero et al., 21 Jul 2025).

Externalities further enlarge the state space. In matching games with additive externalities, agent $G=(V,E;w)$01’s utility under matching $G=(V,E;w)$02 is

$G=(V,E;w)$03

so each match may affect not only its endpoints but all agents. Stability depends on the deviators’ assumptions about outsider reactions: neutral, optimistic, or pessimistic. The complexity landscape is mixed. In many-to-many settings, neutral and pessimistic stable-set membership are coNP-complete and non-emptiness is NP-hard, while optimistic membership is in $G=(V,E;w)$04 and optimistic non-emptiness is NP-complete; in the one-to-one setting, pairwise stability becomes polynomial-time computable under nonnegative externalities in the neutral and pessimistic cases (Brânzei et al., 2012).

A related but distinct generalization modifies transferability rather than preferences. In linearly transferable utility matching, the standard TU equation $G=(V,E;w)$05 is replaced by

$G=(V,E;w)$06

where $G=(V,E;w)$07 varies by pair. Stable outcomes of such LTU problems are in one-to-one correspondence with Nash equilibria of a generalized hide-and-seek bimatrix game. The paper also proves that the linear-programming structure of TU survives only in the TU case itself: a LTU matching problem is a linear program if and only if it has the TU property (Galichon et al., 2024). This recasts non-TU matching stability as nonzero-sum equilibrium computation rather than LP duality.

5. Dynamic, decentralized, and online matching

Dynamic matching games replace one-shot stability by equilibrium paths in repeated interaction. In a dynamic two-sided matching game with firms $G=(V,E;w)$08, workers $G=(V,E;w)$09, utilities $G=(V,E;w)$10 and $G=(V,E;w)$11, and discounted payoff

$G=(V,E;w)$12

the market evolves through repeated offer-and-accept stages. The model distinguishes three commitment regimes: no commitment, firms’ commitment, and workers’ commitment. Without commitment, any stationary equilibrium outcome is the same in every period and is stable, and conversely every stable matching can be supported as a stationary equilibrium. Under firms’ commitment and workers’ commitment, every stable matching can still be supported, but more flexible equilibrium constructions depend on discount-factor thresholds such as

$G=(V,E;w)$13

The common conclusion is that stable matchings remain implementable, while impatience determines whether profitable switching is deterred (Guiñazú et al., 2024).

In engineering applications, stability may be subordinate to throughput or delay. For distributed user association in B5G/mmWave heterogeneous networks, the problem is modeled as a two-sided matching game between user equipments and base stations with quotas $G=(V,E;w)$14. The classical benchmark is deferred acceptance, which is stability-optimal but maintains BS waiting lists and delays final decisions. The early acceptance family of matching games accepts a user as soon as that user is in the BS’s preference list with available quota. The paper proves finite convergence for its variants, gives worst-case complexities $G=(V,E;w)$15 for DA and $G=(V,E;w)$16 for EA, and argues that stability does not imply optimality for performance metrics such as throughput. Empirically, the EA games achieve higher network throughput while exhibiting a significantly faster association process (Alizadeh et al., 2020).

Online matching games in bipartite expanders study a different form of dynamics. A requester switches left nodes on and off, at most $G=(V,E;w)$17 left nodes may be on simultaneously, and each newly switched-on node must be irrevocably matched to a neighbor. A bipartite graph has online matching up to $G=(V,E;w)$18 with load $G=(V,E;w)$19 if the matcher has a winning strategy while no right node receives more than $G=(V,E;w)$20 active matches. If the graph has $G=(V,E;w)$21-expansion up to $G=(V,E;w)$22 and all left nodes have degree $G=(V,E;w)$23, then it admits polynomial-time online matching up to $G=(V,E;w)$24, and also an $G=(V,E;w)$25-time strategy with load $G=(V,E;w)$26, where $G=(V,E;w)$27 is the number of left nodes (Bauwens et al., 2022). These results are then used to build dynamic bitprobe storage schemes, explicit static and dynamic dictionaries with non-adaptive access patterns, and constant-depth non-blocking connectors.

6. Recreational, combinatorial, and arithmetic matching games

Outside market design, the term also appears in games whose moves explicitly create or discover matches. In the solitaire memory game with perfect recall, a deck contains $G=(V,E;w)$28 identical pairs, a move flips two cards, and the optimal strategy is to remove known pairs immediately, otherwise reveal an unknown card and, if its mate is unknown, reveal another unknown card. The paper derives the asymptotic formulas

$G=(V,E;w)$29

so the expected number of moves is asymptotically $G=(V,E;w)$30; the expected number of unwitting matches tends to $G=(V,E;w)$31; and the expected number of flips until two matching cards have been seen is asymptotically $G=(V,E;w)$32 (Velleman et al., 2012). The analysis proceeds through block decompositions of standard deals and recurrences for the first repeated card.

Match-three video games provide another technically distinct usage. In generalized Bejeweled-like games, the board size is the parameter while swap rules, gravity, refill mechanics, and the number of gem types remain fixed. The central theorem shows that deciding whether a player can pop a specific gem is NP-hard, by reduction from 1-in-3 Positive 3SAT; corollaries establish NP-hardness for reaching a target score, achieving a target score within a move limit, causing at least $G=(V,E;w)$33 gems to pop, and surviving at least $G=(V,E;w)$34 turns (Gualà et al., 2014). Complementing worst-case hardness, automated playtesting work treats Match-3 as a benchmark for procedural personas realized by evolving the utility function of a Monte Carlo Tree Search agent. On a custom $G=(V,E;w)$35 board, the study defines four personas—MaxS, MinS, MaxM, and MinM—and shows that evolved utility functions can bias MCTS toward distinct playstyles and help estimate the difficulty envelope of a level (Mugrai et al., 2019).

Graph-theoretic matcher games replace the formation of edges by the formation of larger rooted subgraphs. In the generalized matcher game with $G=(V,E;w)$36, rooting at the center yields the star-game and rooting at an endpoint yields the stripe-game. The central parameter is the $G=(V,E;w)$37-packing number $G=(V,E;w)$38, the maximum number of vertex-disjoint $G=(V,E;w)$39s. The paper proves several exact values and lower bounds, including that in the unrooted-$G=(V,E;w)$40 game with Maximizer responding, the value is exactly $G=(V,E;w)$41; it also characterizes perfect trees for the rooted variants through recursively defined families $G=(V,E;w)$42, $G=(V,E;w)$43, and $G=(V,E;w)$44 (Bachstein et al., 2019).

There is also a purely arithmetic usage. In $G=(V,E;w)$45-matching games, a player draws two balls from a bag containing $G=(V,E;w)$46 colors and wins if both balls have the same color. Fairness is equivalent to the Diophantine equation

$G=(V,E;w)$47

For $G=(V,E;w)$48, the fair games are pairs of consecutive triangular numbers. For $G=(V,E;w)$49, the nontrivial fair games form an infinite full binary tree rooted at $G=(V,E;w)$50, and the coordinate set admits a number-theoretic characterization in terms of the arithmetic of $G=(V,E;w)$51 (Hui et al., 2011). This suggests that “matching game” can denote not only a market or algorithmic interaction, but also a family of combinatorial objects with unexpectedly deep algebraic structure.