It’s the bottom of the 9th. With a man on first and no outs the hometown team is down a run. While the pitcher and catcher conference at the mound the TV play-by-play commentator hypothesizes, “They’re thinking about how to handle Speedy Joe on first. If he steals he’ll be in scoring position.”
The color commentator opines, “They’re over thinking this. The statistics show that attempting to steal in this case is no better than not. The chance of reaching second is the same if Speedy Joe attempts to steal as it is if he does not. Lightning Lefty should just pitch and not worry about Speedy Joe.”
It is likely the color commentator is right that the probability of reaching second is the same whether a runner attempts to steal or not. But the color commentator cannot be right about his advice for Lightning Lefty. In fact, one can draw these conclusions knowing almost nothing about baseball and a little bit about game theory (or by having an very logical mind, which is almost the same thing).
This situation in baseball is a well-known game within the game. With speed on first, it is a duel between pitcher and runner. The runner presses for advantage with a big lead. The pitcher changes his delivery to the stretch and demonstrates his best pick-off moves forcing the runner to dive back to first. TV networks know about this game, which is why they often split the screen and show both runner and pitcher, or they show the action from the third base camera so both can be seen in the same shot.
Simplifying the situation a bit, the runner has two choices: to run or not. The pitcher has two choices: to throw to first (in attempt to pick the runner off) or not. Of course neither does exclusively one or the other all the time. They randomize their play to keep the opponent guessing. In fact, the color commentator told us that pitchers and runners have randomized their play such that the probability of the runner reaching second is exactly balanced in the case the runner attempts to steal and he does not. (I’m ignoring the role of the other players involved in the game for simplicity.)
In fact, it must be so, or else the pitcher or the runner is not playing optimally. Imagine if the probability of reaching second were higher if Speedy Joe didn’t attempt to steal. Well, in that case his best play is to not attempt to steal because it maximizes his chance of reaching second. Knowing that, Lightning Lefty’s best play is not to try to pick him off (which runs the risk of throwing the ball away, letting Speedy reach second with ease). But if Lightning isn’t going to attempt a pick-off, Speedy should attempt to steal. But if Lightning knows Speedy will attempt to steal, he’ll try to pick him off, and so on, back and forth. You can see this is not an equilibrium. One can make a similar argument by imagining the other case, that the probability of reaching second is higher if the runner attempts to steal.
In game theory lingo, there is no pure strategy solution (Nash equilibrium) for either player. Neither Speedy nor Lightning can do just one thing all the time and neither can assume the other will do just one thing all the time. Both must play a mixed strategy of randomizing between the two choices each has. As argued above, it is illogical for the optimal random mixes of the two players over their respective strategies to give rise to a probability of success for one choice higher than the probability of success for another. Once you make such an assumption, you get caught in an infinite loop of indecision, as above.
This is a fundamental truth in game theory. The probable payoff for each of the pure strategies (attempt to steal, not attempt to steal; pick-off throw, no pick-off throw) that are mixed together by a player must be equal. The probability Speedy will reach second if he attempts to steal must equal the probability he will reach second if he does not.
However, that does not mean Lightening Lefty should ignore Speedy Joe as the color commentator suggested. That would be akin to Lightening playing the pure strategy of not making any pick-off throws to first. We know from above that a pure strategy cannot be optimal. The only way optimal (Nash equilibrium) play is achieved is if Lightening plays the mixed strategy that all pitchers play: throw to first with some probability strictly greater than zero and strictly less than one.
The color commentator got it wrong. Lightening Lefty is correct to worry about Speedy Joe and to talk things over with his catcher. Were I in the booth I’d have said, “The catcher is reminding Lightning to play a mixed strategy Nash equilibrium.” Maybe there’s a reason I’m not on TV.