Instead of scoring as many runs as possible in the first inning, you now want to score as many runs as possible over the course of nine innings. What’s more, instead of just having one home run slugger, you now have two sluggers in your lineup. The other seven batters remain pure contact hitters.
Where in the lineup should you place your two sluggers to maximize the average number of runs scored over nine innings?
I ran code that, for each of the 36 unique lineups, simulated cycling through the lineup, the hitters' probabilities, scoring runs, three outs per inning, and terminating at the end of the ninth inning.
For each slugger position, I averaged the total number of home runs over 10,000,000 trials.
I found different answers emerged from different simulations. The greatest consistent home run values occured when the first slugger was the third or fourth position, and the second slugger was the seventh or eighth position.