{"id":3713,"date":"2023-08-19T14:20:47","date_gmt":"2023-08-19T19:20:47","guid":{"rendered":"https:\/\/laurentlessard.com\/bookproofs\/?p=3713"},"modified":"2023-08-21T15:55:27","modified_gmt":"2023-08-21T20:55:27","slug":"optimal-baseball-lineup","status":"publish","type":"post","link":"https:\/\/laurentlessard.com\/bookproofs\/optimal-baseball-lineup\/","title":{"rendered":"Optimal baseball lineup"},"content":{"rendered":"

This week’s Fiddler<\/a> is a problem about how to set the optimal baseball lineup.<\/p>\n

\nEight of your nine batters are “pure contact” hitters. One-third of the time, each of them gets a single, advancing any runners already on base by exactly one base. (The only way to score is with a single with a runner on 3rd). The other two-thirds of the time, they record an out, and no runners advance to the next base. Your ninth batter is the slugger. One-tenth of the time, he hits a home run. But the remaining nine-tenths of the time, he strikes out. Your goal is to score as many runs as possible, on average, in the first inning. Where in your lineup (first, second, third, etc.) should you place your home run slugger?<\/p>\n

Extra Credit<\/em>: 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\u2019s 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?\n<\/p><\/blockquote>\n

My solution:
\n[Show Solution]<\/a><\/p>\n

\n

We will attack this problem using dynamic programming<\/a>. Rather than solving for the average number of runs for a particular configuration (of outs, runners on base, current batter, etc.), we will solve for all possible configurations<\/em>! To this effect define the following:
\n\\[
\nV(o,r,b,i) = \\left\\{
\n\\begin{array}{l}
\n\\text{Expected number of runs at end of}\\\\
\n\\text{game given that we are currently in}\\\\
\n\\text{the $i^\\text{th}$ inning with $o$ outs, $r$ runners}\\\\
\n\\text{on base, and the $b^\\text{th}$ batter is up.}
\n\\end{array}\\right\\}
\n\\]Each index can only take on certain values. Define the sets:
\n\\begin{align}
\nO &= \\{0,1,2,3\\} && \\text{(number of outs)} \\\\
\nR &= \\{0,1,2,3\\} && \\text{(runners on base)} \\\\
\nB &= \\{1,2,\\dots,9\\} && \\text{(batter currently up)} \\\\
\nI &= \\{1,2,\\dots,9\\} && \\text{(current inning)}
\n\\end{align}Therefore there are $4\\times 4\\times 9\\times 9 = 1296$ possible configurations we must solve for. Finally, some subset of the batters are the sluggers<\/em>. Let this subset be $S \\subseteq B$. The remaining batters are the pure contact hitters, which we denote $\\bar S$. Therefore $S\\cap \\bar S = \\emptyset$ and $S \\cup \\bar S = B$.<\/p>\n

Finally, define $p$ to be the on-base average for pure contact hitters, and $q$ to be the slugging percentage for the sluggers. In the problem statement, we have $p=\\frac{1}{3}$ and $q=\\frac{1}{10}$. <\/p>\n

We can now write equations that determine how the various configuration are related to each other:<\/p>\n