To address the risks of validator centralization, Proposer-Builder Separation (PBS) was introduced in Ethereum to divide the roles of block building and block proposing, fostering a more equitable and decentralized block production environment. PBS creates a two-sided market in which searchers submit valuable bundles to builders for inclusion in blocks, while builders compete in auctions for block proposals. In this paper, we formulate and analyze a role-selection game that models how profit-seeking participants in PBS strategically choose between acting as searchers or builders, using a co-evolutionary framework to capture the complex interactions and payoff dynamics in this market. Through agent-based simulations, we demonstrate that agents' optimal role-acting as searcher or builder-responds dynamically to the probability of conflict between bundles. Our empirical game-theoretic analysis quantifies the equilibrium frequencies of role selection under different market conditions, revealing that low conflict probabilities lead to equilibria dominated by searchers, while higher probabilities shift equilibrium toward builders. Additionally, bundle conflicts have non-monotonic effects on agent payoffs and strategy evolution. Our results advance the understanding of decentralized block building and provide guidance for designing fairer and more robust block production mechanisms in blockchain systems.