Shared Order Books in Market Microstructure

• Shared order books are consolidated limit order systems where multiple trading platforms contribute and access a unified pool of liquidity for efficient trade execution.
• Mathematical frameworks such as fixed-point mappings, BSDEs, and dynamic contracting models ensure the existence of a unique Nash equilibrium in cross-venue order matching.
• The system generates competitive spillover benefits and free-rider risks, highlighting the need for coordinated liquidity incentives and regulatory interventions.

A shared order book is a market microstructure in which multiple trading platforms, exchanges, or market participants contribute orders to—and access liquidity from—a single consolidated limit order book. In this structure, a quote submitted on one exchange can be executed by incoming orders from any of the connected venues, leading to unified liquidity, more efficient price formation, and complex incentive structures. Shared order books play a critical role in high-frequency trading, fragmented cash and derivatives markets, intraday electricity markets, and decentralized exchanges.

1. Structural Features and Mathematical Formulation

A shared order book allows orders, quotes, and market information to propagate and be matched across multiple venues or market subsystems, subjects all participants to a common pool of liquidity, and enables cross-venue execution. The canonical mathematical representation treats the unified order book as the aggregation of all order submissions, cancellations, and trades from participating platforms.

Consider two exchanges, $E_0$ and $E_1$, sharing a single consolidated limit order book. Each exchange may have a market maker ($M_0$ for $E_0$, $M_1$ for $E_1$) posting quotes; any incoming market order, regardless of the arrival venue, can execute against available liquidity. This model is formalized in (Aïd et al., 12 Sep 2025) using a continuous-time Principal-Agent framework, treating exchanges as principals setting incentives for their market makers (the agents), both with complete access to the joint order book.

Key characteristics:

• Unified liquidity: The standing quotes (bid/ask) posted by market makers are visible and executable from all connected exchanges.
• Order-matching mechanism: Trades can be routed and executed regardless of where an order originated.
• Incentive interdependence: Liquidity provision incentives offered by one exchange to "its" market maker can benefit all market participants and rival exchanges.

2. Equilibrium Analysis and Principal–Agent Incentive Structures

The existence and uniqueness of equilibrium outcomes in shared order books are addressed through dynamic contracting models with risk-averse agents. Each exchange designs incentive contracts, subject to its own risk aversion and that of its market maker. The Nash equilibrium between two exchanges—each acting as a Principal to its own Agent—depends on the best response of each market maker, factoring in competition for order flow and inventory risk.

Mathematically, the equilibrium is specified by a fixed-point mapping $\Delta$, which for side $k \in \{\text{b},\text{a}\}$ and exchange $i \in \{0,1\}$, resolves the optimal quote as

$$\Delta^{(k,i)}(\boldsymbol{z}, \boldsymbol{q}) := \begin{cases} \Gamma^{(k,i)}(z^{(i,k,i)}, q^{i}) & \text{if } \Gamma^{(k,i)}(z^{(i,k,i)}, q^{i}) \leq \Gamma^{(k,1-i)}(z^{(1-i,k,1-i)}, q^{(1-i)}) \ \Gamma^{(k,i)}_\beta(\Gamma^{(k,1-i)}(z^{(1-i,k,1-i)}, q^{(1-i)}), z^{(i,k,i)}, q^{i}) & \text{otherwise} \end{cases}$$

where $\Gamma$ encodes the agent's optimal bid/ask quote absent competitive pressure, and $\Gamma_\beta$ accounts for matching the competitor's quote when undercut.

A critical result is that there exists a unique fixed point for this mapping, leading to a single Nash equilibrium that can be computed numerically by solving the associated backward stochastic differential equations (BSDEs) or the PDE system for the value function of each agent.

For instance, the risk-averse market maker’s certainty equivalent (in the absence of contracts) satisfies

$$0 = \partial_t \widehat{w}_m(t, q) - \frac{\gamma^m \sigma^2}{2} (q^m)^2 + \mathcal{H}^m(\boldsymbol{z}, q), \quad \widehat{w}_m(T, q) = 0,$$

where $\mathcal{H}^m$ is a Hamiltonian incorporating inventory costs and the effect of quoting.

3. Economic Implications: Competitiveness Spillover and the Free-Rider Problem

The structure of shared order books generates powerful economic externalities. When one exchange chooses to incentivize its market maker (e.g., through fee rebates or direct compensation for tighter quotes), all connected venues benefit: the improved liquidity provision (narrower spreads, deeper book) is accessible market-wide due to the shared order book architecture.

This creates a "public good" effect—liquidity incentivization by one platform increases the utility and execution quality for all, regardless of the source of incentives. Empirically, numerical solutions demonstrate:

• The first exchange to offer incentives secures a substantial increase in liquidity, as quantified by the exchange's value function and tighter quotes.
• The competing exchange, even without offering any incentives itself, experiences an "externality benefit" manifested in improved execution for its users.
• If both exchanges simultaneously incentivize, the incremental improvement over unilateral action is marginal.

This setting gives rise to a classic free-rider problem: since the benefits accrue to all, each exchange may rationally under-invest in costly liquidity incentives, hoping to benefit from the other's expenditures. In equilibrium, both may withhold incentives, resulting in a suboptimal outcome—thicker spreads and diminished competition—even though a single platform's unilateral incentives would have produced large welfare gains.

4. Numerical Characterization and Certainty Equivalent PDEs

Solutions to the equilibrium problem are obtained by explicit numerical integration of the derived PDE system using, for example, an explicit Euler scheme. Key variables are the market maker’s inventory and risk aversion, and the exchange’s contract design (parameters $Y_0$ and stochastic process $Z$ in the contract representation).

Representative numerical findings show:

• The exchange’s value function (certainty equivalent to expected profit minus incentive payments) increases sharply when switching from no-contract to an optimal incentive scheme.
• The competitor's value function increases due to the spillover, even when not offering incentives.
• Bid/ask quotes move less with inventory under incentive contracts, reflecting greater risk absorption by market makers.

Figures mapping the exchanges’ value functions versus market maker inventory and risk profiles illustrate quantitatively how much each platform gains or loses under different incentive regimes.

5. Policy and Regulatory Considerations for Shared Order Books

Because liquidity incentives operate as a public good and the equilibrium is characterized by under-provision, regulatory or coordinated interventions may be required to achieve the full potential benefits of shared order books. Policy implications for any market using shared order books include:

• Minimum incentive standards: Mandating a baseline level of incentive provision may avert the free-rider outcome.
• Subsidization: Compensating exchanges for offering incentives can correct the externality problem.
• Coordination: Facilitating cooperative agreements between connected exchanges (without violating competition law) can align incentives.

These interventions are especially pertinent in regulated sectors such as intraday electricity markets or in decentralized finance, where multiple platforms aggregate order flow for optimal market functioning.

6. Integration with Market Microstructure Theory

The shared order book framework unifies several microstructure mechanisms:

• Unified order exposure enables tighter best quotes and greater consolidated liquidity.
• Market makers internalize both the risks from their quote exposure and the incentives from multiple exchanges.
• Order arrival intensities and execution probabilities are directly influenced by the aggregate quoting behavior, as seen in arrival intensity-based models.
• Equilibrium spread, inventory risk, and execution metrics are driven endogenously by the interplay between competitive incentive schemes and the microstructure of shared execution.

The public good nature of liquidity incentives and the potential for free-riding distinguish the shared order book from fragmented or siloed liquidity pools, highlighting the importance of joint welfare analysis in market design.

7. Summary Table: Key Features of Shared Order Book Economics

Feature	Description	Consequence/Significance
Liquidity is cross-exchange	Quotes offered on one venue hit on another	Cross-platform spillover; unified liquidity
Incentives are non-excludable	All benefit when a single platform incentivizes	Free-rider problem; under-incentivization risk
Public good externality	Incentive provision improves execution market-wide	Policy intervention may be required
Nash equilibrium is unique	Fixed-point for bid/ask quotes via coupled PDEs/BSDEs	Tractable market design and welfare analysis

In summary, shared order books represent a microstructural architecture in which cross-platform liquidity and execution are both technically and economically intertwined. Principal–Agent models incorporating CARA risk aversion, incentive contract design, and dynamic programming analysis rigorously demonstrate how incentive provision becomes a public good with attendant free-rider risks. Numerical and theoretical results establish the uniqueness and robustness of equilibrium, inform practical contract design, and support policy recommendations to maximize market liquidity and competition in shared order book environments (Aïd et al., 12 Sep 2025).