Off-Chain Influence Proof (OffCIP)

• OffCIP is a property of transaction fee mechanisms that ensures miners cannot increase revenue via off-chain auctions beyond on-chain rules.
• It employs a burn identity and virtual-value optimization to tightly couple allocation and fee burning, anchoring its design in Bayesian mechanism principles.
• OffCIP informs robust TFM design by highlighting the limits of deterministic mechanisms and advocating for cryptographic or randomized approaches to deter miner profit extraction.

Off-Chain Influence Proof (OffCIP) is a property of transaction fee mechanisms (TFMs) that guarantees the miner cannot improve revenue by running an off-chain auction or otherwise extracting additional payments from users outside the prescribed on-chain protocol. OffCIP directly addresses the vulnerability—unaddressed by previous incentive-compatibility desiderata—that a miner, acting as a monopolist, may conduct off-chain fee extraction schemes and obtain revenue beyond what is possible via the canonical protocol. This property critically constrains the design space of “good” TFMs and reveals deep connections to classic results in Bayesian mechanism design and virtual-value optimization (Ganesh et al., 2024, Ganesh et al., 2 Dec 2025).

1. Formal Definitions and Economic Interpretation

Off-Chain Influence Proofness is defined relative to a block-building process $B$ which takes as input miner advice, user bids, and returns (randomized) allocations, payments, and burn amounts. Given a value distribution $T$ for user private values $v_i$, and a protocol-induced on-chain equilibrium $\sigma^C$, OffCIP requires that, for any off-chain mechanism $M$ and any corresponding Bayes–Nash equilibrium $\sigma^D$ in the extended off-chain/on-chain game, the miner's total (on-chain plus off-chain) expected revenue is maximized by adhering to the on-chain rule $B$ with “no-op” off-chain activity.

Expressed precisely: let $M^0$ denote the trivial off-chain mechanism that simply enforces on-chain strategies and charges no off-chain payments. $\sigma^C$ is OffCIP if: $$\mathbb{E}_{v\sim T^n}\Bigl[\mathrm{Rev}^D\bigl(M^0; \sigma^{(D,M^0)}(v)\bigr)\Bigr] \geq \mathbb{E}_{v\sim T^n}\Bigl[\mathrm{Rev}^D\bigl(M; \sigma^{(D,M)}(v)\bigr)\Bigr]$$ for all off-chain $M$ and all user BNE $\sigma^{(D,M)}$. Economically, this states that the on-chain protocol “disciplines” the miner such that no additional off-chain screening or blackmailing mechanism can improve her revenue (Ganesh et al., 2024, Ganesh et al., 2 Dec 2025).

2. The Burn Identity and Mechanism Characterization

The central technical insight is the “burn identity.” Any OffCIP TFM must tightly couple its allocation and burn rules via a convex, monotone function $U$ of the users' virtual values. For regular priors and i.i.d. bidders, letting $\varphi$ denote the Myerson virtual value map, there exists $U:\mathbb{R}^n \to \mathbb{R}$ with

$$X(\vec v) = \nabla U(\varphi(\vec v)), \quad \mathrm{Burn}(\vec v) = \sum_i \varphi(v_i)\frac{\partial U}{\partial \beta_i}(\varphi(\vec v)) - U(\varphi(\vec v))$$

This formalizes the economic intuition: OffCIP mechanisms behave as if the miner is a single additive buyer purchasing items (“inclusions”) at their virtual values, and the burn acts as her payment in a buyers' auction (Ganesh et al., 2 Dec 2025).

A TFM with direct-revelation rules $(\bar X, \bar B)$ is OffCIP if and only if, when interpreted as an allocation and pricing rule for a single virtual-value buyer, it is DSIC (dominant strategy incentive-compatible) in those virtual values.

3. Canonical Examples and Failures

A critical example is EIP-1559, the Ethereum base-fee mechanism. Under unlimited supply and posted price $p$, every bid above $p$ is included, the payment is burned, and the miner’s on-chain revenue is zero. Despite being DSIC and robust on-chain, EIP-1559 fails OffCIP: the miner can run an off-chain auction (e.g., requiring tips via side channels), threatening to censor non-tippers. This off-chain blackmail allows revenue strictly greater than the on-chain equilibrium, violating OffCIP (Ganesh et al., 2024).

By contrast, the Cryptographic Second-Price Auction (CSPA), wherein users submit encrypted bids and the miner can only choose the reserve, satisfies OffCIP (and on-chain user- and miner- simplicity). Because the allocation and pricing are determined via secure multiparty computation, the miner cannot conduct profitable off-chain deviation. Myerson theory ensures that choosing the monopoly reserve on-chain is revenue-optimal even in the broad off-chain game (Ganesh et al., 2024).

4. Structure and Uniqueness: Deterministic and Randomized OffCIP TFMs

For deterministic plaintext mechanisms with on-chain user- and miner-simplicity, OffCIP forces the TFM into a uniform posted-price mechanism with a precisely calibrated per-inclusion burn:

• All users with $v_i \geq p$ are included, pay $p$, and per-user fee $b(p)$ is burned.
• The price $p$ is the Myerson reserve and $b(p)=\varphi(p)$.
• This construction requires infinite supply and prior-dependence.

No such deterministic OffCIP TFM exists with finite supply, as no single threshold suffices for block size constraints while maintaining OffCIP (Ganesh et al., 2 Dec 2025).

For finite-capacity settings, randomized mechanisms can achieve OffCIP. For bounded priors, position auction mechanisms allocate with specified probabilities and burn amounts computed via virtual-value optimization, and these satisfy all requirements for OffCIP under DSIC (Ganesh et al., 2 Dec 2025).

Table: Summary of Main Mechanism Classes

Mechanism Type	Crypto Req?	Supply Type	OffCIP?	Notes
EIP-1559 Posted Price	No	Unbounded	No	Off-chain blackmail possible
Naïve Second-Price (Plaintext)	No	Finite/Infinite	No	Miner can insert phantom/fake bids
Cryptographic Second-Price	Yes	Any	Yes	Heavy cryptography for bidder privacy
Posted-Price + Burn	No	Infinite	Yes	Only deterministic plaintext OffCIP TFM
Randomized TFM (Position)	No	Finite/Infinite	Yes	Requires prior, randomized allocation

5. Impossibility Theorems and Design Limits

An impossibility result establishes that no non-trivial TFM can simultaneously be:

• On-chain user-simple,
• On-chain miner-simple,
• OffCIP,
• Strongly collusion-proof (resistant to miner–user side contracts).

If all these are required, only the trivial “no allocation” mechanism survives (i.e., no user with $v_i$ below the supremum is ever included), precluding any positive-probability allocation. The root of this impossibility is that collusion- and off-chain-proofness together force miner revenue to be “flat” (independent of $n$ and $v_i$), which is incompatible with revenue increasing in $n$ (as more users should mean higher optimal revenue per Myerson) (Ganesh et al., 2024).

6. Open Questions, Implementation and Practical Implications

OffCIP introduces stringent design constraints: deterministic, finite-capacity, and plain-text OffCIP TFMs are impossible. Even in infinite-supply, OffCIP necessitates precise “virtual-value” burning, tightly linking the protocol to the prior on user values. This presents several implementation challenges:

• Estimating or updating priors for value distributions in practice.
• Adapting burn parameters dynamically as demand and user behavior shift.
• The complexity/overhead of full cryptographic solutions (CSPA) in production environments.

Open research directions include exploration of approximate OffCIP (permitting small $\epsilon$ off-chain revenue), adaptive dynamic TFMs with sequential posted-price burns, hybrid partial-crypto protocols (to balance OnCS, OffCIP, and efficiency), and extensions to multi-item and heterogeneous cost auction environments (Ganesh et al., 2 Dec 2025).

7. Connections to Broader Mechanism Design and Future Work

The OffCIP concept reveals the essential alignment between on-chain and off-chain mechanism revenue properties. The underlying mathematical techniques invoke Myerson’s virtual value theory, DSIC auctions for single additive buyers, and the Rochet-style convex “taxation principle.”

A plausible implication is that future blockchain protocol designs must explicitly account for miners’ ability to coordinate off-chain, and embed cryptographic and/or burn-based defenses accordingly. Extensions to non-i.i.d. or non-regular value distributions, richer block types, and dynamic demand environments are open areas for significant theoretical advancement.

OffCIP thus refines what it means for a TFM to be robust, universal, and miner-proof, suggesting that simple posted-price+burn structures—as well as richer randomized protocols—will play a central role in next-generation transaction fee mechanism design (Ganesh et al., 2024, Ganesh et al., 2 Dec 2025).