Liquidity Games: Strategic Market Interactions

Updated 8 January 2026

Liquidity games are formal models where agents’ trading, quoting, or liquidity provision decisions dynamically determine market liquidity through endogenous interactions.
The models employ tools from Nash equilibria, FBSDEs, and mean field game theory to analyze how strategic behavior and incentive design drive liquidity levels and systemic stability.
These frameworks inform practical applications in decentralized finance and regulatory policy by linking contract design, trading costs, and market dynamics.

Liquidity games are formal interactions in financial markets where agents’ payoffs are directly determined by the aggregate liquidity generated through strategic actions such as asset trading, quoting, or liquidity provision in automated venues. Unlike classical optimal trading or microstructure models that take liquidity as external or fixed, liquidity games explicitly model how agents’ choices shape and are shaped by the endogenous liquidity environment, often yielding complex reflexive dynamics and nontrivial equilibrium behavior. The literature encompasses strategic run games, multi-agent stochastic games, mean field and swarm-based models, and decentralized finance mechanisms.

1. Foundational Models and Strategic Liquidity Provision

Liquidity games originate in formal models where the core resource—liquidity, i.e., the ability to transact rapidly at low cost—is endogenously determined by agents’ actions. Early continuous models appear in systemic banking risk, where depositors face strategic withdrawal decisions constrained by bank liquidity buffers: fire-sale potential ( $L(q)=q^\alpha$ ) and central bank collateral eligibility ( $M(c)=c^\beta$ ), both represented by power-law technologies over the unit-interval asset continuum. Depositor coordination incentives are captured as pure-strategy Nash games, with equilibrium existence hinging on whether the combined liquidity cushion $Y^* = \max_{z}\left[z^\alpha + (1-z)^\beta\right]$ suffices to cover aggregate withdrawal shocks. Policy levers such as the central bank's haircuts (tuning $\beta$ ) directly raise the systemic liquidity threshold, stabilizing the equilibrium and preventing runs—thereby embedding liquidity games within the architecture of financial stability and unconventional monetary policy (Bindseil et al., 2020).

Parallel formalizations appear in strategic trading environments. In multi-agent liquidation, open-loop Nash equilibria are constructed via coupled forward-backward systems (FBSDEs), incorporating both permanent price impact and slippage (quadratic trading cost). Existence and uniqueness results require weak interaction bounds among agents, and closed-form solutions emerge for symmetric and constant-coefficient cases. Mean field limits and propagation of chaos arguments show that aggregate liquidity flows stabilize in large populations, and constraints such as market drop-out (i.e., absorption at zero) enforce no-round-trip trading and change the shape of equilibrium liquidation rates, effectively encoding no-short-selling in sellers-only games (Fu et al., 2023, Drapeau et al., 2019, Fu et al., 2018, Fu et al., 2024).

2. Liquidity Event Games and Contractual Liquidity Design

In corporate finance and venture liquidity events, convertible instruments such as SAFE contracts are analyzed as finite deterministic games. Each investor optimizes strategic choices between cashing out and converting to shares, with payoffs depending not only on the exit value, but on competition over seniority and conversion ratios. The general framework admits cases where pure-strategy Nash equilibria fail to exist (especially in heterogeneous contract mixtures), but guarantees for existence and polynomial-time computation arise in uniform SAFE-type profiles. Distinct equilibrium structures are characterized: γ-threshold partitions, shifting of boundary players, and optima in minimal seniority settings. Algorithmic results show that uniformity in contract design is essential for ensuring strategic stability in liquidity events (Meyden, 2021).

3. Auction-Based and Decentralized Liquidity Games

Periodic auctions and decentralized market makers provide environments where liquidity games manifest through bid-ask quoting and liquidity mining. In double-auction models under imperfect information, strategic quoting collapses trade in the absence of incentives—exhibiting a pure Nash equilibrium with infinite spreads and zero liquidity. The introduction of quadratic incentive fees aligns private and social objectives, restoring unique interior Nash equilibria with quantifiable liquidity and providing explicit formulae for optimal fee schedules. This links incentive design directly to liquidity generation, showing how exchanges' fee structures can counteract the tendency toward illiquidity in adverse informational environments (Derchu et al., 2023).

Decentralized liquidity mining, exemplified by Uniswap v3, is modeled as a multi-player zero-sum allocation game over discrete bins (price intervals), with equilibrium strategies approximately proportional to the local reward distribution. The reward provider can thus optimize liquidity distribution to minimize slippage or maximize price stability by controlling binwise rewards. Empirical and theoretical results show the Nash equilibrium achieves allocation proportionality, but is sensitive to assumptions such as complete rationality and the importance of trading fees relative to mining rewards (Yin et al., 2021).

4. Mean Field and Swarm-Based Liquidity Games

Mean field game (MFG) approaches capture the collective strategic behavior of infinitely many agents, with each agent internalizing only the statistical law of the population. Coupled forward-backward systems or master PDEs determine equilibrium strategies, whose empirical distributions (e.g., in inventory holdings and liquidity provision rates) feed back into market prices and transaction costs. Applications include liquidation under trading constraints (no direction change), constant product AMM pool games where liquidity provider (LP) behavior influences price drift and slippage, and stability tests where approximate Nash equilibria guarantee near-optimality even in finite agent sets (Sequeira et al., 2024, Fu et al., 2024, Fu et al., 2018, Fu et al., 2023).

"Rational Swarm" models extend this paradigm to fully decentralized learning regimes, where independent agents use local difference rewards—representing their own marginal contribution to total liquidity—in multi-agent reinforcement learning frameworks. Difference reward shaping ensures individual learning naturally aligns with global liquidity optimization, enabling rapid convergence of the swarm to high-liquidity equilibria without coordination or collusion. Empirical evaluation demonstrates superior liquidity, efficiency, and clearing rates compared to global or local reward designs, substantiating the potential for liquidity games as a basis for scalable decentralized market design (Vidler et al., 1 Jan 2026).

5. Automated Market Makers and Competitive Liquidity Provision

Advanced game-theoretic analysis of concentrated liquidity market makers (CLMMs), such as Uniswap v3, reveals nontrivial Nash equilibria in competition among heterogeneous LPs. The original game’s quadratic size in price intervals is reduced by atomic decomposition to a linear-size game, admitting a unique waterfilling Nash equilibrium: low-budget LPs allocate their full budget evenly across price intervals, while rich LPs leave unused capacity. Empirical studies show real-world LPs, particularly in volatile pools, adopt strategies (few wide ranges, low-frequency updates) far from Nash equilibrium, but history-based inert equilibria closely match actual behavior and can substantially improve returns. The framework enables direct assessment of strategic inefficiency and guides future liquidity product design (Tang et al., 2024).

Principal-agent models between AMM venues and strategic LPs capture leader-follower Stackelberg dynamics, where the venue crafts fee contracts to maximize processed order flow while preserving LP profitability. Optimal contracts dynamically reflect pool reserves, external price, and liquidity-dependent order flow sensitivity. Equilibrium analysis shows LPs provide liquidity only when increased depth attracts more noise trading, and closed-form reward structures prescribe how fee-sharing and risk compensation schemes should be calibrated (Aqsha et al., 28 Mar 2025).

6. Reflexivity, Regulation, and Market Design Implications

Liquidity games constructed with endogenous utility—where agent payoffs depend directly and reflexively on aggregate liquidity—represent a conceptual shift in market modeling, with implications for regulatory design and agent incentives. Analytical and simulation studies suggest that local marginal-contribution incentives (difference rewards, fee rebates) efficiently increase overall system liquidity. Contractual and structural constraints (drop-out, no round-trip, collateral eligibility) produce distinct stability properties and shape equilibrium dynamics. Regulatory and design policies leveraging game-theoretic liquidity competition can mitigate coordination failures, promote financial stability, and optimize market efficiency (Bindseil et al., 2020, Vidler et al., 2024, Vidler et al., 1 Jan 2026).

7. Representative Mathematical Structures

Liquidity games formalize agent interaction through combinations of:

Pure and mixed strategy Nash equilibria: depositors in run games, converters in liquidity events, LP allocations.
FBSDE and Riccati-type systems: multi-agent liquidation, mean field games, sequential Stackelberg problems.
Reward-sharing mechanisms: atomic decomposition and waterfilling equilibria for CLMMs, proportional allocation, difference reward shaping.
Algorithmic and dynamic programming: polynomial-time computation (γ-thresholding in contract games), reinforcement-learning convergence.

These frameworks unify the analysis of liquidity as a strategic, endogenous resource in both centralized and decentralized markets.

Table: Liquidity Game Archetypes

Archetype	Principal Mechanism	Reference
Bank run and fire-sale games	Power-law liquidity buffers, depositor coordination, LOLR	(Bindseil et al., 2020)
Auction and quoting games	Spread competition, quadratic incentives, Nash equilibrium	(Derchu et al., 2023)
Contractual liquidity events	Convertible instrument seniority, Nash/payoff computation	(Meyden, 2021)
Multi-agent liquidation	Price impact (temporary/permanent), FBSDE frameworks	(Drapeau et al., 2019 Fu et al., 2023 Fu et al., 2018 Fu et al., 2024)
Liquidity mining & CLMM games	Atomic allocation, waterfilling, fee sharing	(Yin et al., 2021 Tang et al., 2024)
Decentralized MFG and swarms	Mean field equilibrium, difference rewards, Q-learning	(Sequeira et al., 2024 Vidler et al., 1 Jan 2026)
Principal-agent AMM contracts	Stackelberg optimal fee/reward design	(Aqsha et al., 28 Mar 2025)

References

Fire sales, LOLR, and pure-strategy run games (Bindseil et al., 2020)
Portfolio tracking and predatory trading (Voß, 2019)
Liquidity event games in convertible contracts (Meyden, 2021)
Equilibria and incentives in illiquid auctions (Derchu et al., 2023)
Liquidity mining and Nash equilibrium allocation (Yin et al., 2021)
FBSDE for strategic liquidation (Drapeau et al., 2019)
Market entry with trading constraints (Fu et al., 2024)
Closed-loop Nash competition (Micheli et al., 2021)
Market drop-out and absorption (Fu et al., 2023)
Mean field liquidation and equilibrium convergence (Fu et al., 2018)
State-constrained differential game (Schied et al., 2013)
Price formation in mean-field settings (Evangelista et al., 2022)
Self-exciting order flow liquidation games (Fu et al., 2020)
Crowd-based transient price impact (Neuman et al., 2021)
Non-cooperative liquidity games in bond trading (Vidler et al., 2024)
Game-theoretic CLMM provisioning (Tang et al., 2024)
AMM principal-agent reward contracts (Aqsha et al., 28 Mar 2025)
Broker-informed trader competition (Donnelly et al., 11 Mar 2025)
Mean field games for liquidity pools (Sequeira et al., 2024)
Liquidity games in Markov team/swarms (Vidler et al., 1 Jan 2026)

Liquidity games form a rigorous mathematical backbone for analyzing and designing market microstructures, liquidity incentives, fintech contract mechanisms, and decentralized financial engineering, elucidating both theoretical equilibria and practical design implications across a broad spectrum of financial systems.