Proof of Work With External Utilities
Yogev Bar-On, Ilan Komargodski, Omri Weinstein
TL;DR
The paper tackles the problem of Proof-of-Work blockchains under heterogeneous costs and the PoUW paradigm, where external rewards subsidize mining via coupons. It develops a quadratic-cost framework with $c_i(h)=\frac{\alpha_i}{2}h^2-\alpha_i\beta_i h$ and a reward function $R(H)$, proving a unique pure Nash equilibrium for hash rates and deriving a closed-form expression in the linear-reward case $R(H)=\rho H$ where $h_i=\beta_i+\frac{\rho}{\alpha_i}$. A key finding is that miners with external incentives tend to concentrate useful work in a single block, while decentralization depends on how coupons are distributed, analyzed via Shannon entropy. The results show PoUW can preserve PoW-like security with lower costs and environmental impact if coupon accessibility remains sufficiently decentralized, with empirical plausibility for AI workloads as external tasks. Overall, the work provides a rigorous game-theoretic account of PoUW economics and a practical lens on how external rewards shape participation and security.
Abstract
Proof-of-Work (PoW) consensus is traditionally analyzed under the assumption that all miners incur similar costs per unit of computational effort. In reality, costs vary due to factors such as regional electricity cost differences and access to specialized hardware. These variations in mining costs become even more pronounced in the emerging paradigm of \emph{Proof-of-Useful-Work} (PoUW), where miners can earn additional \emph{external} rewards by performing beneficial computations, such as Artificial Intelligence (AI) training and inference workloads. Continuing the work of Fiat et al., who investigate equilibrium dynamics of PoW consensus under heterogeneous cost structures due to varying energy costs, we expand their model to also consider external rewards. We develop a theoretical framework to model miner behavior in such conditions and analyze the resulting equilibrium. Our findings suggest that in some cases, miners with access to external incentives will optimize profitability by concentrating their useful tasks in a single block. We also explore the implications of external rewards for decentralization, modeling it as the Shannon entropy of computational effort distribution among participants. Empirical evidence supports many of our assumptions, indicating that AI training and inference workloads, when reused for consensus, can retain security comparable to Bitcoin while dramatically reducing computational costs and environmental waste.