This paper examines the centralization in block-building and the impact of proposer-builder separation on decentralization. It addresses two main questions: whether heterogeneity in skills and knowledge among block producers inevitably leads to centralization, and whether proposer-builder separation can reduce this heterogeneity and preserve decentralization. 1. **Centralization in Block-Building:** - The paper develops mathematical models to quantify the extent to which heterogeneous rewards lead to concentration in the equilibrium staking distribution. - In a game-theoretic model with endogenous staking, heterogeneous block producer rewards, and staking costs, the main result (Theorem 3.3) shows that the concentration of stake is influenced by the heterogeneity in reward multipliers. - In a stochastic model where block producers repeatedly reinvest rewards into staking, the main result (Theorem 4.2) quantifies the rate at which stake concentrates on the most sophisticated block producers. 2. **Proposer-Builder Separation:** - The paper studies a model with heterogeneous proposers and specialized builders to understand how proposer-builder separation reduces heterogeneity in rewards. - The main result (Theorem 5.1) shows that the competitiveness of the builder ecosystem significantly reduces the heterogeneity in rewards across different proposers. The paper concludes that while economic forces generally lead to oligopolistic outcomes, a small amount of heterogeneity can be managed through proposer-builder separation. However, the effectiveness of this separation depends on the competitiveness of the builder ecosystem. The results provide quantitative insights into the dynamics of centralization and decentralization in blockchain protocols.This paper examines the centralization in block-building and the impact of proposer-builder separation on decentralization. It addresses two main questions: whether heterogeneity in skills and knowledge among block producers inevitably leads to centralization, and whether proposer-builder separation can reduce this heterogeneity and preserve decentralization. 1. **Centralization in Block-Building:** - The paper develops mathematical models to quantify the extent to which heterogeneous rewards lead to concentration in the equilibrium staking distribution. - In a game-theoretic model with endogenous staking, heterogeneous block producer rewards, and staking costs, the main result (Theorem 3.3) shows that the concentration of stake is influenced by the heterogeneity in reward multipliers. - In a stochastic model where block producers repeatedly reinvest rewards into staking, the main result (Theorem 4.2) quantifies the rate at which stake concentrates on the most sophisticated block producers. 2. **Proposer-Builder Separation:** - The paper studies a model with heterogeneous proposers and specialized builders to understand how proposer-builder separation reduces heterogeneity in rewards. - The main result (Theorem 5.1) shows that the competitiveness of the builder ecosystem significantly reduces the heterogeneity in rewards across different proposers. The paper concludes that while economic forces generally lead to oligopolistic outcomes, a small amount of heterogeneity can be managed through proposer-builder separation. However, the effectiveness of this separation depends on the competitiveness of the builder ecosystem. The results provide quantitative insights into the dynamics of centralization and decentralization in blockchain protocols.