Free Option Problem in Blockchain

• The free option problem is the emergence of post-commitment decision rights that let builders cancel or defer block payloads without penalty, impacting on-chain liveness and price accuracy.
• Mathematical models highlight that higher market volatility, longer decision windows, and increased CEX-DEX exposure boost both the expected profit (V*) and the exercise probability (P*), with quantifiable impacts such as a 77% reduction in option exercise when the window is minimized.
• Mitigation strategies like reducing the option window and imposing dynamic penalties effectively decrease the free option’s abuse, thereby enhancing system throughput and ensuring more robust, predictable protocol performance.

The free option problem refers to the emergence of decision rights without appropriate economic cost or penalties, allowing the holder to cancel, defer, or otherwise manipulate outcomes after initial commitment, with direct analogs in both financial engineering and blockchain consensus protocols. In its classical financial context, “free option” typically denotes the ability—arising from contractual asymmetries or poorly designed mechanisms—for an agent to act after learning new information, often with negative externalities or distorted market signals. Recently, the problem has become prominent in blockchain protocol design, especially in the context of Ethereum’s enshrined Proposer-Builder Separation (ePBS) introduced in EIP-7732, where builders can, at no additional penalty, exercise a “free option” to withhold block payloads post-commitment, degrading system liveness and distorting on-chain prices (Mazorra et al., 29 Sep 2025).

1. Mechanism Design and the Genesis of the Free Option

The canonical setting for the free option problem in Ethereum’s ePBS involves two key agents: proposers, who commit to block headers at the beginning of a slot, and builders, who assemble and commit execution payloads (including critical financial transactions such as DEX trades or CEX-DEX arbitrage). After the proposer’s commitment, the builder faces an “option window” (≈8s in current implementations) to decide whether to release or withhold the full payload. During this interval, the builder can observe new external information (notably off-chain market prices). If this information renders the committed trades less profitable or loss-making, the builder can opt to withhold, resulting in the finalized block being empty (no state transition).

This arrangement is “free” in that, conditional on winning the slot and submitting a valid bid, the builder incurs zero penalty for withholding or failing to deliver the finalized payload, leading to the right—but not the obligation—to enforce execution on favorable terms.

2. Mathematical Model and Option Valuation

This scenario is formalized as a real option problem where the builder’s per-block payoff, when including a risky DEX position correlated to external prices, is

$$\Pi_\tau(y) = \mu + (1 + r_\tau) y - P_{\mathrm{DEX}}(y / P_0)$$

where:

• $\mu$ is baseline block value,
• $y$ is the DEX trade size,
• $r_\tau = (P_\tau - P_0)/P_0$ is the spot price return on (for example) a CEX,
• $P_{\mathrm{DEX}}(\cdot)$ denotes execution cost on the DEX (including AMM price impact).

The builder’s optimization is

$$V^* = \max_{y} \mathbb{E}\left[\max\{0, \Pi_\tau(y)\}\right]$$

with the associated exercise probability

$P^* = \mathbb{P}\big[\Pi_\tau(y) < 0\big]$

The value $V^*$ represents the expected profit with the free option, and $P^*$ quantifies how often the builder will optimally exercise (i.e., withhold) in response to adverse outcomes.

Critical parameters influencing $V^*$ and $P^*$ are:

• Volatility $\sigma$: Higher volatility in $r_\tau$ (as in periods of large cross-domain price movements) increases both the option value and exercise frequency.
• Option window length $\tau$: A longer decision window increases the mean-preserving spread of returns, raising both $V^*$ and $P^*$.
• Exposure to external signals: The more block value relies on exogenous markets (e.g., CEX-DEX arbitrage), the larger the economic incentive to exercise the free option since the builder can condition on the late-breaking information.

Formally, for $r_\tau \sim \mathcal{N}(0, \sigma^2)$, the conditional expectation $E[r_\tau \mid \Pi_\tau(y) > 0]$ is

$\sigma \cdot \frac{\varphi(z^*)}{1 - \Phi(z^*)}$

where $z^* = \Pi_0(y^*)/(\sigma y^*)$ and $\varphi, \Phi$ are standard normal density and distribution functions. The optimal trade size scales with volatility and on-chain liquidity (e.g., $y^* \approx 0.61 \sigma L$ for an AMM with liquidity $L$); the post-trade DEX price overshoots CEX by $1.22\sigma$ in this regime.

3. Empirical Impact and Consequences for Ethereum Liveness

Empirical analysis provided in (Mazorra et al., 29 Sep 2025), using historical Ethereum blocks under simulated ePBS, demonstrates:

• With an 8s option window under normal volatility, only $0.82\%$ of blocks have a profitable exercise.
• During periods of high ETH price volatility, the exercise probability surges to $6\%$ and above—precisely when the need for timely state transitions is most acute.
• Builders whose revenue mostly depends on CEX-DEX arbitrage have dramatically higher exercise rates (7–23%) compared to those with more diversified, endogenous block value (<1%).

Network-level consequences include:

• Liveness degradation: Empty blocks arise when builders exercise, reducing throughput.
• On-chain price distortion: Systematic exercise skews DEX prices away from spot (overshoots proportional to volatility), penalizing liquidity providers (LPs) and degrading price discovery.
• Strategic builder behavior: The value of the free option incentivizes riskier, larger trades or more volatile strategies near expiry of the window.

4. Mitigation Strategies

To address the free option problem, two primary mitigation strategies are analyzed:

a. Window Reduction

Shortening the option window reduces both the variance of returns $r_\tau$ within the decision interval and the probability of adverse price shocks. Data from (Mazorra et al., 29 Sep 2025) supports:

• Reducing the window from 8s to 6s lowers $P^*$ by >33%.
• Further reduction to 2s suppresses exercise by up to 77%. Trade-off: Excessive reduction may compromise blob propagation and impose infeasibly tight deadlines for honest builders, counteracting protocol scalability objectives.

b. Explicit Penalties

Imposing a monetary penalty $p$ on exercised (withheld) options transforms the profit function to

$$V^*(p) = \max_{y} \mathbb{E}[ \max\{0, \mu + p + r_\tau y - P_{\mathrm{DEX}}(y/P_0) \} ] - p$$

Theoretical analysis confirms $\frac{\partial V^*}{\partial p} = -P^*$ so that higher penalties reduce both $V^*$ and $P^*$. Even modest static penalties (e.g., 0.075 ETH) can cut exercise probability by 75% on volatile days.

A dynamic penalty mechanism—based on online convex optimization (projected online gradient descent)—allows the protocol to adaptively adjust $p$ in real time, targeting a specified maximum exercise frequency $\alpha$ (e.g., ensuring $P^* \leq \alpha$), with provable regret guarantees.

5. Broader Implications and Protocol-Level Trade-offs

The introduction of a “free” builder option exacerbates liveness risks, especially in the presence of volatile exogenous signals, and causes LP losses due to systematic mispricing. While the option is infrequently exercised on average, the tail risk is severe in stressed markets—precisely when economic coordination and finality are most needed.

A summary of the relationships is given below:

Variable	Effect on $V^*$ / $P^*$	Protocol Consequence
Volatility ($\sigma$)	Increase	More empty blocks, DEX mispricing
Option window ($\tau$)	Increase	Higher risk of liveness failure
Share of block value from CEX-DEX arb	Increase	Targeted exercise in high-exposure blocks
Penalty ($p$)	Decrease	Reduces exercise, liveness improves

6. Model Generality and Theoretical Perspective

The free option problem as studied in (Mazorra et al., 29 Sep 2025) generalizes to any protocol or financial system where agents are granted post-commitment decision rights without commensurate penalties. The analytical structure—option value as a function of conditional expectations, exercise frequencies as tail probabilities conditioned on market volatility, and mitigation through constrained decision windows or financial penalties—is highly general and pertinent across decentralized system design, financial market architecture, and optimal contract theory.

7. Summary

The free option problem in ePBS arises from the interplay of delayed commitment, external price signals, and a lack of penalty for cancellation, producing an American-style option embedded in the block production process. The problem’s magnitude scales with volatility, window length, and exogenous value share, and manifests as worsened liveness and mispricing in stressed markets. Robust mitigation requires careful parameterization of window size and/or dynamic penalty mechanisms that protect liveness while sustaining protocol throughput and scalability (Mazorra et al., 29 Sep 2025).