Enhanced Proof-of-Work Incentive Mechanisms: Protocol Analysis Based on Directed Acyclic Graph (DAG)

1. Introduction

This work, originating from ETH Zurich, aims to address a fundamental flaw in Satoshi Nakamoto's original incentive argument for Bitcoin. The paper points out that rational economic behavior does not necessarily equate to protocol compliance, as demonstrated by selfish mining strategies. The core issue is that in traditional, tree-structured Proof-of-Work blockchains, miners with advantageous network positions or significant computational power can profit by deviating from the protocol (e.g., withholding blocks), thereby jeopardizing system stability.

1.1. Blockchain Game

Standard blockchains like Bitcoin form a tree-like structure. Forks occur naturally or maliciously, leading to chain reorganizations that cause some blocks to become orphaned, with their creators losing rewards. This structure creates perverse incentives; for instance, factors like network latency can affect miner profitability, thereby encouraging non-cooperative behavior.

1.2. Our Contribution

The authors propose a novel blockchain design whose data structure is composed of blocks forming aDirected Acyclic Graph, rather than a tree structure. The accompanying incentive scheme is rigorously designed, ensuring thatAdhering to the protocol constitutes a strict, strong Nash equilibrium.. Any deviation (such as creating unnecessary forks) will strictly reduce the deviator's reward. This guarantees protocol adherence through purely self-interested motives.

1.3. Intuitive Overview

The protocol ensures that miners are incentivized to reference all known unreferenced blocks when creating new blocks. This results in a dense DAG with no blocks being discarded. Consensus on transaction ordering is achieved by selecting a "main chain" from this DAG, similar to other protocols, but the reward mechanism is the key to enforcing honest behavior.

2. Protocol Terminology and Definitions

The framework defines key concepts:Blockis a vertex in the DAG, containing transactions and references (edges) to previous blocks.Tip blockis a block that has not yet been referenced by any other block.Main chainis a specific path selected from the DAG according to a deterministic rule (e.g., based on cumulative proof-of-work). The block $B$'sReward Function$R(B)$ is defined based on its position and reference relationships within the DAG structure.

3. Protocol Design and DAG Interpretation

When creating a new block, a miner must reference all the tip blocks in its local DAG view.Alltip blocks. This rule is not enforced by a protocol mandate but is guaranteed through reward design: omitting references reduces the potential reward of the new block itself. The resulting structure is a growing DAG where blocks have multiple parent blocks.

3.1. Main Chain and Total Order

To achieve consensus on transaction order (e.g., to prevent double-spending), a single chain must be extracted from the DAG. The paper suggests using established methods, such as the GHOST rule or the heaviest chain rule applied to the DAG. All blocks not on the main chain are still included and receive rewards, but their transaction order will be determined relative to the timeline of the main chain, as discussed in works like "Secure High-Rate Transaction Processing in Bitcoin" by Sompolinsky and Zohar.

4. Reward Scheme Construction

This is the core of the proposal. The reward for block $B_i$ is not a fixed coinbase reward. It is calculated as a function reflecting its contribution to DAG stability and connectivity. One possible formulation (inspired by the original text) could be: $R(B_i) = \alpha \cdot \text{base reward} + \beta \cdot \sum_{B_j \in \text{references}(B_i)} f(\text{depth}(B_j))$, where $\text{references}(B_i)$ are the blocks referenced by $B_i$, and $f$ is a decay function. This makes referencing older, unreferenced blocks profitable.

4.1. Details of Incentive Mechanism

This scheme aims to satisfy two key properties: 1) Citation Incentive:For any new block, adding a citation to a known terminal block never decreases, and usually increases, its expected reward. 2) Forking Penalty:If miners attempt to create parallel chains (forks) by not referencing the latest blocks, the reward mechanism ensures that the cumulative rewards for blocks in the fork are strictly less than what they could have earned by building honestly on the main DAG. This makes forking economically irrational.

5. Core Insights and Analytical Perspectives

6. Technical Details and Mathematical Framework

Incentive compatibility is proven using game theory. Consider a miner $m$ with computing power $\alpha$. Let $\mathbf{s}$ be the strategy profile of all miners. Let $U_m(\mathbf{s})$ be the utility (expected reward) of miner $m$. The protocol strategy $\mathbf{s}^*$ (always referencing all tips) is a Nash equilibrium if, for every miner $m$ and every alternative strategy $\mathbf{s}'_m$, it satisfies:

$$U_m(\mathbf{s}^*_m, \mathbf{s}^*_{-m}) \geq U_m(\mathbf{s}'_m, \mathbf{s}^*_{-m})$$

论文构建了一个奖励函数$R$，使得对于任何涉及扣留引用或创建不必要分叉的偏离$\mathbf{s}'_m$，该不等式是严格的（$ > $）。该函数可能包含：

• Time-based decay:The reward for referencing a block decreases as the block ages, encouraging timely inclusion.
• Connectivity reward:The reward a block receives is proportional to the number of previous blocks it directly or indirectly helps confirm.

A simplified model for the reward of block $B$ might look like this:

$$R(B) = \frac{C}{\sqrt{k(B) + 1}} + \sum_{P \in \text{parent blocks}(B)} \gamma^{\text{distance}(P)} \cdot R_{\text{base}}(P)$$

where $k(B)$ is the number of blocks published concurrently that $B$未引用的同时发布的区块数量（衡量分叉创建），$\gamma < 1$是衰减因子，$R_{基础}(P)$是父区块$P$的基础奖励。

7. Experimental Results and Performance

Although the provided PDF excerpt does not contain explicit experimental results, the paper's claims imply significant performance improvements relative to tree-based blockchains:

Chart description (conceptual): The simulation chart will display two lines over time: 1) In the proposed DAG protocolThe cumulative reward of honest miners, and 2) The cumulative reward of deviating miners attempting a withholding attack. The line of honest miners will consistently remain above that of the deviators, intuitively demonstrating a strict Nash equilibrium. The second chart will compareTraditional blockchain(flat or slow growth) withDAG-based chain(shows steeper, more efficient growth)Transaction throughput (TPS)。

8. Analytical Framework: A Game Theory Case

Scenario: Two rational miners, Alice (30% hashrate) and Bob (20% hashrate), compared in a traditional PoW chain versus the proposed DAG chain.

Traditional chain (tree-like): Alice发现一个区块。她可以立即广播（诚实），或者扣留它并开始挖掘一条秘密链（自私）。如果她扣留并在网络找到一个区块之前找到第二个区块，她可以同时发布两个区块，导致重组，使Bob可能找到的区块成为孤块，从而将其在该时期的奖励份额从30%可能提高到100%。Eyal和Sirer的模型表明，这对于$\alpha > 25\%$的矿工可能是有利可图的。

Proposed DAG chain: Alice found a block $A_1$. The reward function $R(A_1)$ is maximized only when referencing all known tip blocks (including Bob's latest block if he found one). If she withholds $A_1$ to secretly mine $A_2$, she will forfeit the reference reward for not linking to Bob's public block. When she eventually reveals her chain, calculations show that:

$$R(A_1) + R(A_2)_{\text{秘密}} < R(A_1)_{\text{诚实}} + R(A_2)_{\text{诚实}}$$

Even if she creates a small fork, the protocol's reward mechanism ensures her cumulative reward is less. The rational choice is to immediately publish $A_1$ and include all references. Bob faces the same calculation. Therefore, the only stable strategy for both parties is to follow the protocol.

Wannan shari'ar ba ta amfani da lambar code ba, amma ta bayyana yadda sabon tsarin ƙarfafawa ya canza matrix na yanke shawara na dabarun.

9. Application Prospects and Future Directions

Direct Application:

• Next-Generation L1 Public Chain: New proof-of-work blockchains can adopt this design from the genesis block to ensure stronger security against mining pools.
• Hybrid Consensus: The DAG incentive model can be adapted to proof-of-stake or delegated proof-of-stake systems to mitigate stake grinding or similar attacks.
• Layer 2 and Sidechains: This principle can be applied to protect sidechains or Rollup sequencers with faster finality, where incentive misalignment is also a concern.

Future Research Directions:

• Dynamic Fee Market: Integrate robust transaction fee auctions (such as EIP-1559) into the DAG reward model without compromising incentive compatibility.
• Quantum-Resistant Readiness: Exploring how larger post-quantum cryptographic signatures affect the scalability and incentive models of DAG.
• Formal Verification: Using model checkers such as the Coq proof assistant or TLA+ to formally verify the game-theoretic properties of the implemented protocol.
• Cross-Chain Incentives: Apply similar incentive alignment principles to protocols governing blockchain interoperability (cross-chain bridges) to prevent cross-chain MEV exploitation.

10. References

1. Nakamoto, S. (2008). Bitcoin: A Peer-to-Peer Electronic Cash System.
2. Eyal, I., & Sirer, E. G. (2014). Majority is not Enough: Bitcoin Mining is Vulnerable. In Financial Cryptography.
3. Sompolinsky, Y., & Zohar, A. (2015). Secure High-Rate Transaction Processing in Bitcoin. In Financial Cryptography.
4. Nisan, N., Roughgarden, T., Tardos, É., & Vazirani, V. V. (2007). Algorithmic Game Theory. Cambridge University Press.
5. Lewenberg, Y., Sompolinsky, Y., & Zohar, A. (2015). Inclusive Block Chain Protocols. In Financial Cryptography.
6. Buterin, V. (2014). Slasher: A Punitive Proof-of-Stake Algorithm. Ethereum Blog.
7. Daian, P., et al. (2019). Flash Boys 2.0: Frontrunning, Transaction Reordering, and Consensus Instability in Decentralized Exchanges. IEEE Symposium on Security and Privacy.
8. Sliwinski, J., & Wattenhofer, R. (2022). Better Incentives for Proof-of-Work. arXiv:2206.10050.