Gavin what the other suspect does, each can

Gavin Dagenhardt

Ms. Egloff

IB Math Seniors

October 17, 2017

The Prisoner’s Dilemma

Introduction   

The Prisoner’s Dilemma is an example
of a game that involves two people who would benefit from their combined
cooperation, but may not cooperate.  One
version of the game says that:

the police have arrested two
suspects and are interrogating them in separate rooms. Each can either confess,
thereby implicating the other, or keep silent. No matter what the other suspect
does, each can improve his own position by confessing. If the other confesses,
then one had better do the same to avoid the especially harsh sentence that
awaits a recalcitrant holdout. If the other keeps silent, then one can obtain
the favorable treatment accorded a state’s witness by confessing. Thus,
confession is the dominant strategy for each. But when both confess, the
outcome is worse for both than when both keep silent (Dixit and Nalebuff 2008).

The participants A & B are then separately asked decide
whether or not they would like to remain silent.  There are a total of 4 outcomes that can
occur due to the decisions of the participants.  They are:

?       Dual Cooperative Response (A):  Both participants chose to remain silent and
therefore only receive one year of jail time.

?       Singular non-cooperative Response
(B): Participant A turns in Participant B remains silent.  B receives 3 years and A is sent free.

?       Singular non-cooperative Response
(C): Participant B turns in Participant A remains silent.  A receives 3 years and B is sent free.

?       Dual non-cooperative Response (D):
Both participants turn in the other and both receive 2 years.

The main goal of the game is to achieve the lowest jail time
between your counterpart and yourself.

Experiment

            For this
paper, I will be investigating the probability of the answers of a class of IB
math students.  In order to examine the
probability of their choices, the class of students will participate in the
dilemma.  They will be given two options
to choose from.  They must decide if they
will remain silent or agree to testify against their counterpart.  They will be given an online form to fill out
which explains the situation and their options. 
The form also states that a student agrees to the use of their data in
the experiment when they submit the form.  
The students will begin with one round of questions in which they are
not told who the counterpart is.  They will
first be asked if they will stay silent or testify against this mystery.  I, then, decided that the students will be
asked if they would like to change their answer and, if so, what they would
like to change it to.  After this
round,  the students will be paired up
and told that this is their counterpart in the next round.  Students will then be asked to answer the
first set of questions once again and then given the option to change it.  After this the information will be compiled
and the probabilities will be calculated for several different options.  I hypothesize that we will see a link between
between personal association and an increase in the likelihood of the
participant agreeing to testify.

Round One Results and Analysis

            After
completing the first round of questions, the answers were compiled and
collected in a table (Table 1).  Next, I
separated the two groups of those who chose to remain silent and those who
chose to testify.  I, then, calculated
the percentage that each group made up of the whole.

I decided to represent this data in a pie chart so that the
information could be better visualized.

 

 

 

 

 

 

 

 

During this round, only two students chose to change their
response when asked if they would like to do so.  I believe that very few students changed
their answer because they did not have know who they were against and,
therefore, felt like this was not necessary. 
Additionally, both students who changed their answer shifted from
silence to testify.  After their change,
the percentages are:

           

Using the data above, I found that if another student were
asked to answer the same questions in the same ways, there is 68.2% chance that
they would initially choose to remain silent with only a 32.8% chance that they
would agree to testify.  The data also
suggests that, statistically, there is a 7.7% chance that the same student would
change their mind.  Interestingly, the
two students who chose to change their mind initially chose silence and then
decided to switch to testifying.  The
next round will test the idea on whether or not personal association can
attribute to decision making and whether there is a link between initial
decision and probability of changing.

 

Round Two Results and Analysis

            For this round, the students were
then randomly paired with another student and asked the same questions.  The partners know who the other student was,
but could not communicate.  After
answering, their responses were recorded and place in a table (Table 2).  The percentages were then calculated like the
round before.

I am using the same type of pie chart to represent the
percentages for this round.

During this second round, three students chose to change
their answer.  Once again, all of the
students went from silence to testifying. 
However, in this round, the number of After the change, the percentages
were:

This is the first time in the experiment where more
participants chose to testify than remain silent.  At the beginning of the round, 57.7% of
students chose to remain silent and this number dropped to 46.2% after the
students were given the chance to change. 
While not conclusive, there seems to be a link present between personal
association and testifying.  There was a
7.7% increase in initially testifying rates between the first and second
rounds.  After the chance to change their
answer, this same increase rose to 11.5%. 

 

Limitations

One limitation of the experiment
could have been the limited sample size. 
The use of a single class contributed to a low number of participants
which means there is a larger margin of inaccuracy in the predictions.  Another limitation could be the lack of
tangible rewards for those who were able to receive a shorter sentence.  This may have contributed to the higher
numbers of those who chose to remain silent in the first round.

 

 

 

 

 

 

 

 

 

 

 

 

Works Cited

Dixit, Avinash K, and Barry J Nalebuff.
Thinking Strategically: The Competitive Edge in Business, Politics, and
Everyday Life. W. W. Norton & Company,
terpconnect.umd.edu/~pswistak/GVPT%20100/Dixit%20and%20Nalebuff.pdf.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tables

Table 1: The Responses of the 26 Students from
Round One

Student #

What will you choose to do?

Do you wish to change your answer?

If you answered yes above, what do
you wish to change your answer to?

1

Remain Silent

No

 

2

Remain Silent

No

 

3

Agree to Testify

No

 

4

Remain Silent

No

 

5

Agree to Testify

No

 

6

Remain Silent

No

 

7

Agree to Testify

No

 

8

Remain Silent

No

 

9

Remain Silent

Yes

Agree to Testify

10

Remain Silent

Yes

Agree to Testify

11

Remain Silent

No

 

12

Agree to Testify

No

 

13

Agree to Testify

No

 

14

Remain Silent

No

 

15

Agree to Testify

No

 

16

Remain Silent

No

 

17

Remain Silent

No

 

18

Remain Silent

No

 

19

Remain Silent

No

 

20

Remain Silent

No

 

21

Agree to Testify

No

 

22

Agree to Testify

No

 

23

Agree to Testify

No

 

24

Remain Silent

No

 

25

Remain Silent

No

 

26

Remain Silent

No

 

 

 

 

Table 2: The Responses of the 22 Students from
Round Two

Student #

What do you choose to do?

Do you want to change your answer?

If you answered yes, what would you
like to change it to?

1

Agree to Testify

No

 

2

Remain Silent

No

 

3

Agree to Testify

No

 

4

Agree to Testify

No

 

5

Remain Silent

No

 

6

Agree to Testify

No

 

7

Remain Silent

No

 

8

Remain Silent

No

 

9

Agree to Testify

No

 

10

Agree to Testify

No

 

11

Remain Silent

No

 

12

Agree to Testify

No

 

13

Remain Silent

Yes

Agree to Testify

14

Agree to Testify

No

 

15

Remain Silent

Yes

Agree to Testify

16

Remain Silent

No

 

17

Remain Silent

No

 

18

Remain Silent

No

 

19

Agree to Testify

No

 

20

Remain Silent

No

 

21

Remain Silent

Yes

Agree to Testify

22

Remain Silent

No

 

23

Agree to Testify

No

 

24

Remain Silent

No

 

25

Agree to Testify

No

 

26

Remain Silent

No

 

 

 

Written by