The pitcher-batter problem

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The game of baseball is a zero-sum game in that when one side wins, the other side loses. The goal is to win. One aspect of the game is the pitcher-batter problem.

Pitcher-batter problem: (zero-sum game)

Pitcher tries to throw a pitch the batter cannot hit.
Batter tries to hit the pitch thrown by the pitcher.

This ignores the DH (Designated Hitter), AL (American League) in 1973, NL (National League) in 2022, who does not want to be "outstanding in the field". That is what "stands out to me" from the spectator "stands", if you can "stand" the remarks that "stand" the test of time. Am I "grandstanding"?

Decision analysis techniques and sensitivity analysis can be used to gain further insight.

The baseball. The bat.
Country song: Sometimes I'm the baseball. Sometimes I'm the bat.

The pitcher tries to throw the batter a pitch that the batter cannot hit.
The batter tries to hit the pitch the pitcher throws.

To simplify the analysis, the following assumptions are made.

The only possible pitches are the fast ball or the curve ball.
Both the batter and pitcher know this.
Both know that the other knows.
Both know that the other knows that the other knows.
...and so on...

The speed of the pitch is so fast and the time for decisions so short that the batter must decide what pitch to look for before the pitch is thrown.

If the batter guesses correctly, the chance of a hit is increased.

When the pitcher throws a fast ball:

If the batter expects fast ball, then the hit probability is 0.3.

If the batter expects curve ball, then the hit probability is 0.1.

When the pitcher throws a curve ball:

If the batter expects fast ball, then the hit probability is 0.2.

If the batter expects curve ball, then the hit probability is 0.6.

A decision node is drawn as a rectangular box and indicates a decision to be made by a decision maker.

A state of nature node is drawn as an oval (i.e., rounded rectangle) and indicates an event that will happen and whose control is not under the decision maker's power.

A decision tree can be made from the batter's point of view.
Decision tree for pitcher

A decision tree can be made from the pitcher's point of view.

Hit average	batter expects	batter expectes
Pitcher throws	fast ball	curve ball
fast ball	0.3	0.1
curve ball	0.2	0.6

Question: What if the pitcher throws only fast balls? The batter will only look for fast balls hit with average 0.3.

Question: What if the pitcher throws only curve balls? The batter will only look for curve balls hit with average 0.6.

Question: What if the batter always looks for a fast ball? The pitcher will throw only curve balls and the hit average will be 0.2.

Question: What if the batter always looks for a curve ball?

The pitcher will throw only fast balls and the hit average will be 0.1.

Hit average	batter expects	batter expects
Pitcher throws	fast ball	curve ball
fast ball	0.3	0.1
curve ball	0.2	0.6

In terms of the payoff table:

Question: How can the pitcher minimize his/her risk in terms of the payoffs? by minimizing his/her maximum payoff.

Question: What is the goal of the pitcher?

minimize the maximum row payoffs

Hit average when	batter expects	batter expects
Pitcher throws	fast ball	curve ball
fast ball	0.3	0.1
curve ball	0.2	0.6

In terms of the payoff table:

Question: How can the batter minimize his/her risk in terms of the payoffs? by maximizing his/her minimum payoff.

Question: What is the goal of the batter?

maximize the minimum row payoffs

Question: Is there a saddle-point? No, since the maximum of the minimum row payoffs is not equal to the minimum of the maximum column payoffs.

Question: What strategy should be used? Since there is no saddle-point, a mixed strategy should be used.

Question: What does this mean?

Using a mixed strategy means that you cannot let your opponent know your next move.

The batter's expected value of getting a hit when looking for a fast ball, in terms of the probability p of the pitcher throwing a curve ball is

EV = 0.3 (1.0 - p) + 0.2 p = -0.1 p + 0.3 where 0.0 ≤ p ≤ 1.0

The batter's expected value of getting a hit when looking for a curve ball, in terms of the probability p of the pitcher throwing a curve ball is

EV = 0.1 (1.0 - p) + 0.6 p = 0.5 p + 0.1 where 0.0 ≤ p ≤ 1.0.

The break-even point is where the expected value when looking for a fast ball is equal to the expected value when looking for a curve ball.

- 0.1 p + 0.3 = 0.5 p + 0.1

which is when 0.2 = 0.6 p

which means that p = 1/3 ≈ 0.333.

The batters expected value (i.e., batting average) is

EV = (0.3)(1/3) + (0.2)(2/3) ≈ 0.1 + 0.167 = 0.267

Pitcher-batter problem: (zero-sum game)

Pitcher tries to throw a pitch the batter cannot hit.
Batter tries to hit the pitch thrown by the pitcher.

A sensitivity analysis reveals more insight.

Each player must randomly vary their viewpoint (with the proper mix) without letting the other player know what to expect. This is one reason managers, through coaches, signal players what to do.

What happens if the batter gets better at hitting a curve ball when expecting a fast ball?

The batter's hit average will increase, and so will the probability that the pitcher throws a curve ball. If the pitcher does not adjust, then the batter's hit average would increase even more.

The same analysis and results can be done with two people playing tennis and considering forehand and backhand volleys.

This area of analysis is part of executive decision making and strategic management.

It is also part of military and security analysis.

In all cases, one assumes that one is playing the game against an intelligent adversary.

In a data science application, the data does not change but some of the techniques are quite similar.