We investigate the recent fee mechanism EIP1559 of the Ethereum network. Whereas previous studies have focused on myopic miners, we here focus on strategic miners in the sense of miners being able to reason about the future blocks. We derive expressions for optimal miner behavior (in terms of setting block sizes) in the case of two-block foresight and varying degrees of hashing power. Results indicate that a sufficiently large mining pool will have enough hashing power to gain by strategic foresight. We further use a simulation study to examine the impact of both two-block and three-block foresight. In particular, the simulation study indicates that for realistic levels of hashing power, mining pools do not gain from being able to reason more than two blocks ahead. Moreover, even though the presence of strategic miners increases the variation in block sizes and potentially empty blocks, overall system throughput tends to increase slightly compared with myopic mining. We further analyze the effect of varying the base fee updating rule.