BRIMBERG & HURLEY
A Baseball Decision Problem
A Baseball Decision Problem
Jack Brimberg
Bill Hurley
Department of Business Administration
Royal Military College of Canada
Kingston, Ontario, Canada
[email protected]
[email protected]
Abstract
This paper presents a Kahneman-Tversky type probability example in a baseball context with a counter-intuitive
best strategy. The problem is ideally suited for classroom discussion in an introductory course on probability
or managerial decision-making.
Editor's note: This is a pdf copy of an html document which resides at http://ite.pubs.informs.org/Vo5No1/
BrimbergHurley/
an elementary probability calculation one can show
that this probability approaches 90%.
1. Introduction
In the application of probability and statistics to business problems, it is useful to teach students that intuition can be quite poor in the assessment of chance
events. There are a number of ways to make this point.
Our preference is to give students a series of problems
with multiple choice answers and ask them for their
judgements as to which answer is correct. One of the
examples we use is the classic Birthday Problem:
"Suppose there are 40 people at a cocktail party. We
are interested in the chance that at least two people in
the group have coincident calendar birthdates. If two
people have a birthdate of, say, June 12, but in different
years, then we would say these two have coincident
birthdates. Would you say the chance that at least two
people have the same birthday is
A: at least 80%
B: between 50% and 80%
C: between 20% and 50%
D: less than 20%."
Most of our students choose alternative D. They are
surprised to learn that the correct answer is A. Using
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The contribution of this paper is to offer a new problem
in an interesting context. It concerns a baseball manager trying to determine a pitching rotation.
2. A Baseball Problem
Here is the statement of the problem as we use it:
"Suppose a manager of one of the teams in the World
Series is faced with the following problem. He is down
three games to two so he must win the final two games
to win the Series. His top two pitchers (his Ace and
Number 2) are ready to go, however his Ace would be
pitching on three days rest if he were to pitch Game
6. The opposing team has its number 2 and 3 pitchers
ready to go; the manager assesses that both are of
about the same ability and sufficiently rested. The
manager feels he must win Game 6, and that his Ace
on three days rest is better than his Number 2 fully
rested. He therefore decides to go with his Ace in
Game 6. As it turns out, the team wins Game 6 and
the manager's choice of pitcher is applauded by the
national TV and print media "experts". What would
you have done?
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A Baseball Decision Problem
A: Pitch the Ace in Game 6 and Number 2 in Game 7.
B: Pitch the Number 2 in Game 6 and the Ace in Game 7"(1).
The correct answer is B since it gives the team the best
chance to win the World Series. Most students choose
A.
On occasion we have given this problem to some of
our colleagues. Surprisingly, each has argued that the
correct answer is A, pitch the Ace in Game 6. The
conversations have gone something like this:
Question: "So what would you do?"
Answer: "I would pitch the Ace in Game 6."
Question: "Why?"
Answer: "Because if you don't win Game 6, there's not going to
be a Game 7."
Question: "But if you pitch your Number 2 in Game 6, you maximize your chances of winning the Series."
Answer (slightly hostile): "I don't care what your probabilities
say, there is not going to be a Game 7 if you don't win Game 6!"
"I know that the [conjunction] is least probable, yet a
little homunculus in my head continues to jump up
and down, shouting at me - "but she can't just be a
bank teller; read the description!!!!"
Gould's reaction is remarkably similar to the attitude
we encountered. Most of our colleagues still do not
believe that it is best to save the Ace for Game 7, even
though a simple application of the laws of probability
suggests otherwise. The prospect of elimination if the
next game is lost appears to override the longer term
rational objective of winning the Series.
To see this, let a6 be the probability that the Ace wins
Game 6. Then the Ace's probability of winning Game
7 is
This problem has the flavour of those used by Kahneman and Tversky (1973) to demonstrate that human
intuition is notoriously bad at processing even the
simplest probability problems. Here is one of their
examples:
"Linda is 31 years old, single, outspoken and very
bright. She majored in philosophy. As a student, she
was deeply concerned with issues of discrimination
and social justice, and also participated in antinuclear
demonstrations. Which of the following statements is
more probable?
A: Linda is a bank teller.
B: Linda is a bank teller and active in the feminist movement.
C: Linda is a bank teller and takes yoga classes."
Most respondents choose B or C. Few choose A. Yet,
by the laws of probability:
Pr(A) ≥ Pr(A ∩ B).
That is, the simple event, "Linda is a bank teller" is at
least as probable as either of the compound events.
Regarding this result, Stephen Jay Gould has written:
a7 = a6 + ε
where ε > 0. ε is positive since he has an additional
day of rest and the opposing pitchers are judged to be
the same. Let the probability that the Number 2 wins
Game 6 be
b6 = a6 - δ
(3)
where δ > 0. Note that this definition operationalizes
the manager's belief that his Ace is better than his
Number 2 even on three days rest. The Number 2's
probability of winning Game 7 is b7 = b6.
Then the probability that the Series is won with the
Ace pitching Game 6 is w6 = a6b7, and with him pitching
Game 7, it is w7 = b6a7. Begin with w7:
w7 = b6a7 = b7(a6 + ε) = b7a6 + εb7.
(1)
(2)
(4)
Therefore
w7 = w6 + εb7
(1)
(5)
For those unfamiliar with baseball, we offer the following explanations for some of the terms we have used. For professional baseball
in North America, two teams play each year for the championship in a final tournament called the World Series. The first team to
win four games wins the World Series. Pitchers are those players who throw the ball to the batter. They are a very important part of
any baseball team. Starting pitchers normally require at least four days rest between starts. Anything less than four days usually
leads to a deterioration in performance. The coach of a baseball team, the man responsible for deciding which players play and how
they play, is termed a manager.
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A Baseball Decision Problem
or
w7 > w6
(6)
since εb7 > 0. Hence the probability of winning the Series is greater if the manager decides to save his Ace
for Game 7. Note that this is true irrespective of the
relative strength of the Ace and Number 2 (i.e., the
value of δ). By way of numerical example, suppose a6
= 0.50, a7 = 0.75, and b6 = b7 = 0.40. Then
w7 = 0.40 x 0.75 = 0.30
and
w6 = 0.50 x 0.40 = 0.20,
or, there is a 50% increase in the probability of winning
the Series by saving the Ace for Game 7.
An interesting side issue relates to the pitching rotation
favoured by the owner of the team whose objective
may be to maximize revenues. Clearly the probability
of two more games is highest when the Ace pitches
Game 6 (0.50 versus 0.40 for the numerical example).
Hence an owner with a sufficiently high preference
for dollars over winning the Series would prefer having the Ace pitch Game 6.
3. Classroom Use
In class we usually give students five short multiple
choice problems similar to those above and ask them
to take 10 minutes to finish. This short time frame
forces students to use their intuition to arrive at their
choices. The explicit instructions are as follows:
"Instructions: The following experiments present hypothetical situations with multiple choice answers. Please
answer these questions as best you can. If you want
to make calculations, do so. You should be aware that
some of the questions have counterintuitive answers,
but some do not. You have 10 minutes to answer all
the questions. Mark your answers on the question
sheet by circling your preferred answer. Do not sign
your name. At the end of this exercise, we will discuss
the answers."
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We usually take the problems up immediately upon
completion of the exercise by the students. We begin
by giving the correct answer and then explaining why
it is correct. Of the 6 to 8 problems we regularly use,
the Birthday Problem and the Baseball Problem generate the most discussion. The reaction of students to
the Baseball Problem is particularly interesting. Even
after you demonstrate the correct answer, there are a
significant number who still question that it is correct
to save the Ace for Game 7.
An interesting follow-up exercise is to ask students
where their intuitions went wrong. Usually we have
this discussion in a subsequent class. This delay gives
students time to reflect on where they made their
mistake. For the Baseball Problem, we believe the
source of the difficulty is choice of the wrong objective.
The objective of getting to Game 7 is different than the
objective of winning Game 6 and Game 7. And as
demonstrated above, each leads to a different rotation.
We have done this exercise in a variety of settings including undergraduate courses, graduate courses
(MBA and Masters in Defence Engineering and Management) at the Royal Military College of Canada
(RMC), and professional development seminars. Most
of the participants in these last two settings are military
officers ranging in rank from Captain to Colonel. It is
in these sessions that we have had the liveliest and
most interesting class discussions.
And finally a word of caution on problem selection.
One of the problems we have used is the infamous
Monte Hall Problem:
"A player faces three closed containers, one that holds
a prize and two that are empty. After selecting one
(without opening it), the player is shown that one of
the other two containers is empty. The player is now
given the option of switching from his or her original
choice to the other closed container. Which of the following statements do you agree with the most?
A: It makes no difference whether the player switches.
B: The player has a slightly better chance of winning if he or she
switches.
C: The player doubles his or her chance of winning if he or she
switches."
The correct answer to this question is C. Most students
choose A. One year we gave this to junior officers in
a professional development course at RMC. A large
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A Baseball Decision Problem
number of these officers felt that answer C was wrong
even after they were shown a Monte Carlo procedure
which demonstrated that it was correct. These students
spent a significant amount of time on the problem after
the class was over. For the most part, they would send
incorrect conditional probability arguments "demonstrating" that the correct answer was A. The students
spent so much time talking about and analyzing the
problem subsequent to the class that, in the end, we
were asked by the Directing Staff of the program (senior military officers) not to use it again!
References
Kahneman, D. and A. Tversky, (1973), "On the Psychology of Prediction," Psychological Review, Vol.
80, pp. 237-251.
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