Temporal Function Market Makers (TFMMs)

• TFMMs are time-dependent automated market makers that adjust asset weights to enable dynamic liquidity provision, progressive price formation, and adaptive market making.
• They incorporate mechanisms like Liquidity Bootstrap Pools where evolving weights and scheduled adjustments are used to manage arbitrage opportunities and mitigate MEV risks.
• Analytical models and simulations demonstrate that careful parameterization and guardrails can effectively balance risk management with functional adaptability in TFMMs.

Temporal Function Market Makers (TFMMs) are a class of automated market making algorithms in decentralized finance (DeFi) characterized by trading functions whose parameters, specifically asset weights, are explicit functions of time. Unlike constant function market makers (CFMMs)—such as the canonical constant product AMM—they introduce a temporal dimension into the market making invariant, leading to time-dependent liquidity provision, dynamic price formation, and new considerations for both risk management and adversarial exploitation across discrete time intervals (e.g., blockchain blocks). TFMMs are deployed in mechanisms including Liquidity Bootstrap Pools (LBPs) and Replicating Market Makers (RMMs), where asset weights evolve according to predetermined or externally-signaled schedules.

1. Mathematical Structure of TFMMs

TFMMs generalize CFMMs by encoding explicit time-dependent weights into the trading invariant. For an $N$-token pool, the invariant is given by:

$\prod_{i=1}^N R_i^{w_i(t)} = k(t)$

where $R_i$ are the reserves of token $i$, $w_i(t)$ is the time-dependent weight ($\sum_{i=1}^N w_i(t)=1$, $0 < w_i(t) < 1$), and $k(t)$ is a normalization function. The instantaneous quoted price for token $j$ (using token $i$ as numéraire) is

$$m^{\mathrm{TFMM}}_{i,j}(t) = \frac{1}{\gamma}\frac{R_i}{R_j}\frac{w_j(t)}{w_i(t)}$$

where $\gamma$ is the fee factor, typically $\gamma = 0.997$. Prices thus evolve between blocks as the weights $w_i(t)$ change, even if the reserves $R_i$ remain constant. This time-dependent architecture enables the introduction of mechanisms such as liquidity bootstrapping, progressive price curves, and adaptive market making (Willetts et al., 23 Apr 2024).

2. Temporal Dynamics and Price Formation

Within a block, the trading function remains fixed, but between blocks, the weights $w_i(t)$ may update according to prescribed schedules or as a function of exogenous signals (e.g., governance, oracle events, or external time series). This leads to discrete jumps in spot prices at the boundary of each time interval. The quoted trade amount (for trading $\Delta_i$ of token $i$ for token $j$) is determined by:

$$\Lambda_j = R_j \left(1 - \frac{1}{\left(1 + \gamma \frac{\Delta_i}{R_i}\right)^{w_i(t)/w_j(t)}}\right)$$

At equilibrium, the pool price $m_u$ aligns with the external market price $m_p$ up to the fee adjustment:

$\gamma m_p < m_u < \gamma^{-1} m_p$

This formulation ensures arbitrage bounds while enabling flexible temporal liquidity curves.

3. MEV Vulnerabilities and Inter-Block Attack Vectors

TFMMs’ discrete weight transitions introduce unique attack vectors for Maximal Extractable Value (MEV), particularly multi-block manipulation. Attacks proceed by pre-conditioning the reserves in anticipation of a favorable weight update. In the first block, an attacker executes a deliberate trade to move the quoted price away from $m_p$. After weights update in the subsequent block, the changed price results in an amplified arbitrage discrepancy, which can be exploited for profit.

The net extractable value from the attack is

$$Z(\epsilon) = X(\epsilon) - C(\epsilon) - X(\epsilon_0)$$

where $C(\epsilon)$ is the attacker’s trade cost, $X(\epsilon)$ is the arbitrage profit post-update, and $X(\epsilon_0)$ is the vanilla arbitrage baseline absent manipulation. The key insight is that dynamic weights, by causing endogenous pool price adjustments, enable attackers to “boost” arbitrage returns outside of the classical single-block sandwich or front-run attacks (Willetts et al., 23 Apr 2024).

Table: TFMM MEV Attack Process

Step	Description	Mathematical Formulation
Pre-manipulation trade	Attacker skews quoted price	$$m_{\mathrm{AMM},2}^{\mathrm{manip}} = (1+\epsilon) m_p$$
Weight update	Pool price jumps due to new $w_i(t)$	$m^{\mathrm{TFMM}}_{i,j}(t+1)$
Arbitrage exploit	Attacker or arbitrageur profits from amplified deviation	$Z(\epsilon)$

4. Analytical Models and Guardrails

The fee-inclusive trading condition for TFMMs is expressed as:

$$(R_1 + \gamma \Delta_1)^{w_1} (R_2 - \Delta_2)^{w_2} = R_1^{w_1} R_2^{w_2}$$

Post-manipulation, the relationship between reserve fractions and fee factor ties directly to the attack’s efficiency:

$$\left(1+\frac{\Delta_1}{R_1}\right)\left(1+\gamma\frac{\Delta_1}{R_1}\right)^{\frac{w_1}{w_2}} = \gamma^2 (1+\epsilon)$$

To defend against these multi-block attacks, the following guardrails are proposed:

• Maximum Allowed Trade Size: Prohibit trades altering reserves by more than a preset fraction (e.g., 20% of pool total).
• Maximum Allowed Weight Change: Impose cap on permitted shift in $w_i(t)$ per block.
• Minimum Allowed Weight Value: Ensure all weights remain above a threshold avoiding extreme price sensitivity.

These can be formalized via sufficiency conditions for preventing profitable manipulation, such as requiring the gradients of the net return with respect to manipulation to be nonpositive:

$$\frac{\partial}{\partial \epsilon}\Bigl(\Delta_1' - \Delta_1\Bigr) \le 0$$

The inequalities specify “safe regions” of parameter space wherein attack returns do not exceed baseline arbitrage opportunities.

5. Simulation Evidence for Robustness

Extensive numerical simulation—over 450 million attack scenarios—was performed to empirically verify attack efficacy and defense effectiveness. Scenarios varied reserves, weights, weight transitions, and trade sizes. Results show:

• With tight guardrail parameters (small trade sizes, minimal weight changes, higher minimum weights), no profitable attacks above baseline arbitrage were observed.
• When guardrails are relaxed, attack vectors successfully generate MEV, demonstrating the necessity of disciplined parameterization.
• Results were visualized with attack return heatmaps, where white regions denote absence of profitable attacks and colored regions indicate MEV extraction.

This evidence supports the conclusion that TFMMs, when properly parameterized, can be safeguarded against the multi-block MEV attack enabled by dynamic temporal weights (Willetts et al., 23 Apr 2024).

6. Relationship to Prediction Market Microstructures

The temporal adaptivity seen in TFMMs is conceptually related to adaptive market makers in prediction markets, such as the Bayesian Market Maker (BMM) and Logarithmic Market Scoring Rule (LMSR). BMM maintains a belief distribution using a sliding-window consistency index for uncertainty adaptation:

$$C(\mathrm{history}) = L(\mu_t, 2\sigma_t) - L(\mu_t, \sigma_t)$$

This mechanism enables equilibrium convergence while remaining responsive to shocks, analogous to the temporal smoothness and adaptation achieved by TFMM weight functions. LMSR uses bounded-loss inventory mechanisms with a single parameter, offering loss bounds but less adaptable pricing (Brahma et al., 2010).

A plausible implication is that time-dependent variants of Bayesian updating, or inventory-based risk smoothing, may inform future TFMM algorithm design. Both market microstructures seek to balance liquidity provision, price stability, and adaptability in the face of evolving market states.

7. Implications and Design Considerations for TFMMs

Design of TFMMs requires deliberate balancing of liquidity, adaptability, and security. Key points include:

• Dynamic Liquidity Provision: TFMMs enable scheduled or responsive liquidity, facilitating applications like progressive bootstrapping or moving-weight auctions.
• Adaptation to Market Information: Temporal structure accommodates information flows, oracle updates, or exogenous risk events.
• Risk Management: Guardrail parameterization is necessary to prevent MEV extraction that targets inter-block weight updates.
• Equilibrium and Convergence: Bayesian updating and smoothing mechanisms may improve steady-state performance, with tradeoffs in adaptivity and risk exposure.

In summary, TFMMs extend the mathematical and operational landscape of AMMs by introducing temporality into pool invariants. Their dynamic weights create both new opportunities for adaptive market making and new vulnerabilities, particularly to multi-block adversarial strategies. Analytical modeling and empirical evidence demonstrate that with appropriate guardrails, TFMMs can deliver robust, time-sensitive liquidity while limiting attack surfaces and preserving market integrity (Willetts et al., 23 Apr 2024, Brahma et al., 2010).