The first time I ever flew to Canada, I was flying to Toronto and I forgot to bring my passport.  This was pre 9/11 and so the immigration authorities still had a sense of humor.  Upon landing they brought me to the basement to interrogate me.  Only one question was required and it was ingenious.  “What is the infield fly rule?”  Only an American would know the answer to this question.  (The immigration authorities knew the right trade-off between Type I and Type II errors.  A quick survey of my dinner companions tonight revealed that indeed only Americans knew the answer, but not all Americans.)

Suppose there is a runner on first base and the ball is hit in the air where it is catchable by an infielder.  As the ball hangs in the air, it sets off a tiny zero sum game-within-the-game to be played by the players on the ground watching it fall to Earth.  For if the ball is caught by the infielder, then the runner must be standing on first base, else he will be out after a quick throw to the first baseman, a double play.  But, if he does stay standing on first base then the infielder can allow the ball to fall to the ground forcing the runner to advance.  Then a quick throw to second base will get the runner out.  And again a double play unless the batter has already made it to first.

Apparently this goes against some moral code deeply ingrained in American culture.  Is it that the optimal strategy is random?  Is it that we don’t want our heroes stranded between bases waiting to see which way they will meet their end?  Is it that we don’t want to see the defense gaining advantage by purposefully dropping the ball?  Whatever it is, we have ruled it out.

The infield fly rule states that in the above situation, the batter is immediately called out and the runner must stay on first base.  No uncertainty, no inscrutability, no randomization. No intrigue.  No fun.

But we are game theorists and we can still contemplate what would happen without the infield fly rule.  Its actually not so bad for the runner.  The runner should stand on first base.  Usually by the time the ball descends, the batter will have made it to first base and a double play will be avoided as the infielder can either catch the ball and get the batter out or drop the ball and get the runner out.  But not both.

In fact, if the ball is hit very high in the infield (usually the case since an infield fly almost always occurs because of a pop-up) the batter should advance to second base and even to third if he can.  That is, run past the base runner, a strategy that otherwise would never be advisable. This forces the infielder to catch the ball as otherwise the best he can do is force out the runner on first, leaving a runner in scoring position.

So in these cases an infield fly does not really introduce any subtle strategy to the game.  When the team at-bat plays an optimal strategy, the outcome entails no randomness.  The final result is that there will be one out and a runner will remain on first base.  And the fielder can always catch the ball to get the out.

However, this is not the only situation where the infield fly rule is in effect.  It also applies when there are runners on first and second and also bases loaded.  In those situations, if we did away with the infield fly rule the strategy would be a bit more subtle.  And interesting!  Lets try to figure out what would happen.  Post your analysis in the comments.

(conversation with Roberto, Massimo, Itai, Alesandro, Wojciech, Alp, Stefan and Takashi acknowledged)

Next time:  eliminating stalemate (another mysterious artificial rule) in Chess.

Update: Ah, a reader points out that the infield fly rule is waived when there is only a runner on first.  Good thing my friendly immigration officer didnt know that!  (even if he did, he would know that only an American would get the question wrong in that particular way).

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