Price of MEV: Quantifying Extractable Value

• Price of MEV (PoMEV) is defined as the measurable cost and inefficiency induced by adversarial transaction manipulation in CFMM protocols.
• It employs game-theoretic models to analyze routing and reordering attacks, demonstrating a constant price of anarchy and an O(log n) reordering penalty.
• The framework guides MEV searchers and protocol designers in tuning liquidity and slippage parameters to mitigate user losses and enhance overall market efficiency.

The price of Maximal Extractable Value (PoMEV) formalizes the quantifiable cost, inefficiency, and price impact induced when miners/validators, MEV searchers, or adversarial agents extract extra value—beyond transaction fees—by optimizing transaction ordering or insertion in decentralized exchange protocols. In modern CFMM environments (such as Uniswap and similar AMMs), these activities lead to user losses, liquidity provider (LP) value erosion, and network-level inefficiency, and have become a principal research axis in the analysis of decentralized finance protocol security, market efficiency, and mechanism design.

1. Formal Definition and Core Models

PoMEV refers to the measurable increase in transaction costs or reduction in welfare due to extractable value captured via adversarial transaction manipulation. In the context of Constant Function Market Makers (CFMMs), MEV is typically realized through transaction reordering or the insertion of additional trades (e.g., sandwich attacks) that worsen the price received by the original trader, enabling extractors to realize profit over and above equilibrium exchange rates. This can be mathematically characterized as follows:

For a forward exchange function $G(\Delta)$ describing the output of a CFMM for an input $\Delta$, sandwich attacks are captured via the increment equation:

$$G(\Delta + \Delta^{sand}) - G(\Delta^{sand}) = (1 - \eta) G(\Delta)$$

where $\Delta^{sand}$ is the inserted attacker's trade and $\eta$ the user’s slippage limit. The MEV is then the additional value realized by the adversary, e.g., $PNL = TradeOut' - TradeIn$ on the CFMM's output and input.

The game is triadic: miners (controlling order), MEV searchers (designing sandwich or arbitrage bundles), and users (posting trades subject to slippage).

2. Game-Theoretic Analysis: Routing and Reordering MEV

Two principal facets of PoMEV are analyzed within a formal game-theoretic framework:

• Routing MEV: Users may optimally route trades across a network of CFMMs, which, under adversarial manipulation, becomes a nonatomic routing game where each path’s forward exchange function is affected by inserted MEV trades. The distinction between optimal (centralized) routing and equilibrium (selfish) routing yields the price of anarchy (PoA) for MEV:

$$PoA(\Delta_{AB}, \eta_{AB}) = \frac{W^{(sand)}(\alpha^*)}{W^{(sand)}(\bar{\alpha})}$$

where $W^{(sand)}(\cdot)$ is the sandwiched social welfare, $\alpha^*$ is the centralized routing outcome, and $\bar{\alpha}$ is the equilibrium. Under suitable liquidity and slippage bounds (curvature $\mu, \kappa$, liquidity $\beta$), the price of anarchy is shown to be bounded by a constant, independent of network size. Thus, the relative inefficiency of selfish routing in the presence of MEV is limited to at most a fixed multiplicative factor.

• Reordering MEV: Given a sequence of $n$ user trades, adversarial reordering to extract maximal MEV can be evaluated by defining the “cost of feudalism” (CoF):

$$CoF(T_n) = \frac{\max_{\pi \in S_n} \max_i |PNL_{\pi(i)} - PNL_i|}{\text{average}_{\pi} \left(\frac{1}{n} \sum_i |PNL_{\pi(i)} - PNL_i|\right)}$$

Theoretical results show that, under strongly local sandwich attacks and with sufficient liquidity and smooth price impact (bi-Lipschitz bounds $G$ with curvature $\kappa, \mu$ and minimum liquidity $\beta$),

$CoF(T_n) = O(\log n)$

i.e., even in the worst-case ordering (maximizing aggregate MEV), the additional price impact is only logarithmically greater than average scenarios.

3. Quantitative Asymptotic and Explicit Bounds

PoMEV can be concretely bounded in CFMM architectures as follows:

• Maximal Sandwich Profit Bound: Provided $G$ is sufficiently smooth and exhibits robust liquidity, the maximal value that can be extracted in a single attack is explicitly computable with

$$G(\Delta + \Delta^{sand}) - G(\Delta^{sand}) = (1 - \eta) G(\Delta)$$

Solving for $\Delta^{sand}$ gives attackers, MEV searchers, and protocol designers a robust profit estimate per trade and parameter regime.

• Routing Price of Anarchy: When sandwich attack impacts are sufficiently localized, the PoA is provably constant, not increasing with market or network size:

$PoA(\Delta_{AB},\eta_{AB}) \leq c$

for some constant $c$ depending only on liquidity and curvature parameters, not the number of CFMM pools or network scale.

• Reordering Prophets Bound: For $n$ user trades, the maximal adverse reordering penalty grows at most as $O(\log n)$ above average loss (i.e., the worst-case “prophet” scenario is only logarithmically worse than the average), paralleling prophet inequality results.

4. Practical Protocol and Design Implications

These theoretical insights have direct implications for both MEV searchers and CFMM protocol designers:

• For MEV Searchers: The profit obtainable from sandwiching individual trades, and the inefficiency introduced by selfish routing, is bounded and predictable given market liquidity, curvature, and user slippage limits. Explicit formulas for sandwich sizing ($\Delta^{sand}$ via CFMM equations) enable accurate profit forecasting.
• For Protocol Designers: Liquidity and curvature parameters, as well as recommended default slippage limits $\eta$, can be selected or tuned to minimize extractable MEV. “Flatter” price response (lower curvature) and higher pool liquidity reduce both sandwich profit and routing inefficiency. The $O(\log n)$ bound gives confidence that even in blocks with many trades, the maximal adverse price impact remains moderate.

5. Impact on User Costs and Social Welfare

The PoMEV is ultimately a quantification of the “tax” or price impact paid by users (and, in some settings, LPs) due to adversarial exploitation of order flow. For social welfare:

• Under optimal, centralized routing with MEV agents, the aggregate user loss is at most a constant factor below the theoretical maximum output, due to the constant PoA.
• In adversarial reordering scenarios, even full control of transaction ordering by attackers yields only logarithmic growth in maximal user loss relative to the number of trades per block, not linear growth.

This quantification offers both provable security and efficiency guarantees to users and protocols in the CFMM context.

6. Methodological Innovations and Mathematical Techniques

The analysis leverages convex optimization for centralized routing, nonatomic equilibrium models for selfish routing, smooth game analysis (inspired by Roughgarden’s $(\lambda,\mu)$-smoothness), and prophet inequalities for sequence prophet bounds. Explicit sandwich attack sizing is derived analytically from CFMM exchange functions with bi-Lipschitz properties.

The underlying game theory framework models miners/validators, searchers, and users as rational agents, and separates the contribution of routing (edge-wise sandwich profit) and reordering (within-block permutations). Mathematical results rely on properties such as strong locality (no bundled attack improves over attacking trades individually) and liquidity lower bounds.

7. Synthesis and Broader Consequences

PoMEV, as defined and analyzed for CFMMs, connects theoretical market analysis directly with adversarial blockchain security modeling and decentralized market mechanism design. The explicit and asymptotic bounds—constant PoA, $O(\log n)$ reordering penalty—enable robust estimation and mitigation of extractable value for all stakeholders. MEV searchers have reliable profit estimation tools, while protocol designers can adjust system parameters to keep MEV-induced inefficiency under formal control.

These results generalize to other settings where transaction ordering can be manipulated for private gain and provide a concrete pathway to designing decentralized financial infrastructures with provable limits on unfair “taxation” by miners and MEV agents. The analytical separation of routing and reordering inefficiency quantifies, with precision, the price of MEV in modern decentralized exchange networks (Kulkarni et al., 2022).