Introduction a single action is taken? Ultimately; how

Introduction

               Something about
the concept of applied contradiction has always fascinated me. A vague field of
factors that exist, for the most part, purely in the human conscious and
integrated rarely into the real world. But what if our own perception of these
contradictory values could affect us? How we interact with others or defy
social norm? How do the possible outcomes of a partnership alter how we take
risks before a single action is taken? Ultimately; how can human cognition
alter probability before it’s reflected in the real world?1

               It’s a
question rooted deep in human psychology—a field which has pondered the way we
think for centuries. Psychology, as a study, has presented to the world some of
mankind’s most puzzling dilemmas and paradoxes. Within these, a countless range
of arguments could be made over the diversity of social and emotional factors
that drive our decision-making and determine our behavior. As it turns out,
however, it is the rationality—the ability to calculate—of our own human brain
that enables us to, in part, explain the dynamics of this field through a more
concrete medium. In this way, I can decipher one such dilemma with assurance
unachievable through any other medium.

In short: the Prisoner’s
Dilemma, explained by mathematical probability.

 

Restraint
of Cooperation

The essence of the
Prisoner’s Dilemma can be explained in two ways: in its most classic,
applied form, in contrast to the story stripped down, reduced to its
mathematical and structural core. Both of these hold equal value in
consideration—to gain the full picture of the
Prisoner’s Dilemma, an understanding of the relationship between these two
forms is important.

As a story:

Two individuals have been placed under arrest pretext of having
committed a major crime—robbing a bank, for example—but have been captured
under evidence only for a much less serious crime—possibly petty theft.

This takes place under two conditions: for our dilemma to
function, we, the observer, may take the assumption that both, neither, or one
of the individuals actually committed the major crime—it has no significant effect
on the essence of the problem. Secondly, In line with the mindset of the common
human being (as proven by countless empirical studies—see Miller, et al.
(1999)), both care far more about their personal outcome—their freedom—than
about the welfare of the other individual.

The police
under whom the individuals are under arrest need a confession to convict either
individual of the major crime. They take them and put them in separate rooms so
they can’t talk, and interview both of them in exactly the same way.2
Keep it in mind that, in this simplified scenario, the two prisoners have
absolutely no means of communication. In short, they lack the ability to
cooperate.

From the
perspective of one of the individuals, you and the other criminal can take one
of two choices, leading to one of four scenarios. The scenarios are:

1.     

You
admit your partner committed the crime. You go free (serving 0 years), being
pardoned for the minor crime, but your partner will have spends 3 years in
prison.

2.     
The
opposite occurs: you stay silent (complicit) and your partner accuses you of
committing the major crime. You spend 3 years in prison. They go free.

3.     
Neither
of you accuse the other. The police don’t have evidence on either of you for
the major crime, so you both serve a 1 year sentence for the minor crime.

4.     
Both
of you accuse one another. Both of you are absolved of your minor charges for betraying
the other, but you each serve two years for the major crime.

Now, note the
total amount of years served between you and your partner in each of the
scenarios. If you both accuse one another, four years are served (2+2=4). If
one of you indicts the other, three years are served total (3+0=3). If neither
of you accuse the other, two years are served (1+1=2).

Given this
information, it would be logical to determine that the best course of action
for the group would be for both individuals to remain silent (complicit). The
total number of years served in this instance is lowest. However, psychology
and mathematics tell us that this will not occur. Notice that, regardless of
the action of your partner, it never has a detrimental effect on your own
sentence to accuse your partner of the crime. If your partner is planning on
accusing you, your sentence goes down from 3 years to 2 years if you accuse
them back. If your partner is not planning on accusing you, your sentence goes
down from 1 year to being set free (0 years). As a result, at least one
individual, theoretically, will accuse the other.

They should
have both cooperated, but from an individual stand point they noticed they
could always gain by defecting. If they have no control over what the other
person is going to do. So they’ll both defect to try to better their own
situation. But they actually come away not only hurting the group, but also themselves.
Their individual outcome is hurt by an action that should mathematically always either not effect or improve
their situation. Individually, they’re worse off than if they both cooperated.

 

Free
Cooperation (Market Application)

A
variation on the Prisoner’s Dilemma,
often reflected in the real world, can be found in marketing expenditure and
competition between businesses. Two companies, for example, Company A and
Company B, might be in the process of deciding how much money they should spend
on advertising. In our example, there exists a sample population of 100 consumers,
all completely dependent on an essential $2.00 product. The product is offered
with equal access by both Company A and Company B. Given that they produce
identical, equally valuable products, advertising would become the primary
influencer on sales. In this situation, the companies would have two options:
to expend significantly on advertisement, or not to expend on advertisement at
all. If neither side advertises, the consumers would naturally be drawn equally
to each company—50 to Company A, and 50 to Company B. With consumers
distributed evenly, each company would make $100.

If one
company chooses to advertise, however, we might project that 80 people could be
influenced to purchase their product. This leaves Company B with just 20 consumers.
If the advertiser makes $160 in sales, subtracted by, say, $30 for advertising
expenditure, they would turn a profit of $130.4

The
non-advertiser didn’t spend money, but only made $40. If they both advertise,
again half will buy Company A, and half will buy Company B. Since they both
spent $30 on advertising, they only come away with only $70 each. Both people
cooperating and not advertising is the most preferable situation, but both
companies can see that advertising will always make them more money. But unlike
the prisoner’s in jail, these companies can interact and have the opportunity
to try to influence each other. From here Company B would be better off if
Company A didn’t expend on ads at all. Company A wouldn’t go for that because
that would be worse for them. Company B could try to convince Company A that
they would both not advertise, the only other situation where they’re both
better off. But without any real obligation to each other, there’s nothing
that’s stopping them from trying to advertise to gain more of the market
anyway, and to take over the industry. If you think a competing company is not
going to advertise, you’re better off advertising. In general, in a market
economy, businesses dominating an industry can communicate to maximize sales
for both parties.5

 

Mathematical
Representation

How exactly, then,
does the introduction of communication change the dynamic of the Prisoner’s Dilemma? This can be most
clearly explained through mathematics. Before doing this, however, it is
important to establish a basic mathematical understanding of the dilemma. Many
of these representations might seem overly straightforward, but they are
important for reference in a deeper analysis of the problem. To do this, we
will first set actions and their implications to the following variables:

Let x = the probability of Prisoner
A accusing

Let y = the probability of Prisoner
B accusing

Let z = the expected value of
Prisoner B’s sentence

In order to assess the motivation
of each participant, we can set the following constants from Prisoner B’s
perspective:

Let A = reward for accusation
(years subtracted from sentence)

Let B = reward for complicity
(years subtracted from sentence)

Let C = drawback for accusation
(years added to sentence)

Let D = drawback for complicity
(years added to sentence)

Using this mathematical scenario,
we can summarize Prisoner B’s persective with the following:

Or, in the terms of the variables
and constants defined:

                               (Eq. 2)

Which, foiled and
distributed, presents us with a basic representation of any non-communicatory sentence
that Prisoner B might receive:

                                           (Eq. 3)

Having situationally
described the sentence of Prisoner B, we can make adjustments to the equation
to determine the conditions under which Prisoner B would be motivated to accuse.
Set y, the probability of Prisoner B accusing, as certain

. We are then left with the following:

                             

                                                        (Eq. 4)

                                                                                                                      (Eq. 5)

This equation
demonstrates the desirability of accusation to Prisoner B. No matter the input
for x

, Prisoner B cannot serve a longer
sentence than if they had complied. When Prisoner A accuses

, we find that

. When Prisoner A complies

, we simply find that

.

The same basic process
could be applied to determine the conditions under which Prisoner B would be
motivated to comply. This time, set y to

. Eq. 3 then reduces to:

                                                                                                                     (Eq. 6)

Inversely to Eq. 5, Eq.
6 then reduces to

 when Prisoner A accuses. When Prisoner A
complies

, we find that

.

 

Demonstrative Factor: Sentencing Length

Which primary factors,
then, can we alter in these equations to change how the prisoners respond to
their dilemma? The impact of communication, of course, remains our ultimate
query, but it is important to first consider other aspects of the dilemma. The
first and most obvious of these is sentencing length, or drawback (C and D in our scenario).

Logically, for Prisoner
B to consider cooperation with Prisoner A, his expected value for complicity
must exceed his expected value for accusation. We can represent this with the
following inequality:

                                                                                              (Eq. 7)

                                                                                              (Eq. 8)

In order to find

, we can rearrange Eq. 8 to the following
fraction:

                                                                                                                              (Eq. 9)

With Eq. 9 we arrive at
perhaps the most significant mathematical revelation of this process.
Remembering that

—a fraction—we can determine that the
right side of Eq. 9 must, too, fall within a fractional range. For this to be
the case, the numerator,

, must be greater in value than the
denominator,

. Therefore:

                                                                                                         (Eq. 10)

                                                                                                                                          (Eq. 11)

In the terms of the
problem, Eq. 11 states that, for Prisoner B to consider cooperation, the
drawback for complicity must exceed the drawback for accusation—a logical
conclusion to come to at first glance at the dilemma. Applying this back to our
original equation, we can answer our question: how does sentencing length
impact this decision-making?

                                                                                          (Eq. 12)

If we look back at our
initial, simplified Prisoner’s Dilemma scenario,
this makes perfect sense. Having proven that Eq. 12 represents the only
scenario in which Prisoner B will consider cooperation, our initial scenario of

 cannot
function:

                                                                                            (Eq. 13)

Prisoner B will always accuse.

 

Demonstrative Factor: Communication

How, then,
does communication, as described in the Market
Application section, impact how decisions are made between the two parties?
Unfortunately, unlike the sentence length—a quantifiable, clear demonstrative
factor—the impact of communication is too vague to be illustrated through
equations of constants and variables. Personality and circumstance have too great
of an influence. There are, however, several clear principles of communicatory
practice in the Prisoner’s Dilemma which
we can discuss.

Based upon our mathematical foundation, we can
determine a number of real-world conditions to be most important in facilitating
cooperation between two parties (particularly in a free market). As an initial
rule, the advertising party in the dilemma must find an immediate payoff. Group
success and its implications are always secondary, as the negative aspects of drawback
are not immediate. Should both sides find their future payoffs significantly discounted,
the threat of drawback could be sufficient to deter expenditure on
advertisement.

Secondly, the likelihood of advertisement is
heightened by retaliatory motivation. Robert Axelrod of the University of
Michigan explained in 1984 that, in a communicatory atmosphere, many would
choose to advertise if and only if they believed their rival to have advertised
in a previous instance.4 Without communication, the rival’s often
innocent actions will almost always be preemptively assumed to be accusatory in
nature—communication grants a greater chance to both participants to avoid
this.

Aditionally, in order to reach an agreement,
both parties must show commitment to recurring cooperation. A single or fixed
number of repetitions in a dilemma will prevent any agreement from being
reached—if both sides know that a certain trial will be their final interaction
with the other, they have no deterrent to betraying the other, and will always advertise.
In a fixed number of repetitions, the same principle would, by extension, also
apply to the second-last play, the third-last, and so on.

Finally, cooperation can also arise if there
is a third party present with both great interest in the mutual success of all
groups and a great authority over the operation as a whole. This party would be
first to exercise a degree of restraint, despite the likelihood of the other
two advertising.

 

Conclusion
(Conceptual Interpretation)

How has this mathematical model enabled us to analyze the Prisoner’s Dilemma—its principles
and some of its more demonstrative factors? By practical means, it gave us the
clearest possible method to take apart every single comparative factor within
the problem as a whole. It showed the dilemma to us at its most basic forms. And, perhaps ironically, cutting away much of the fluff around
the Prisoner’s Dilemma may have
allowed us to understand its meaning in a psychological and emotive sense more
clearly.

Many claim that the puzzle illustrates a conflict between
individual and group rationality. A group whose members pursue rational
self-interest may all end up worse off than a group whose members act contrary
to rational self-interest. More generally, if the payoffs are not assumed to
represent self-interest, a group whose members rationally pursue any goals may
all meet less success than if they had not rationally pursued their goals
individually.

The
prisoner’s dilemma is, ultimately, a paradox in decision analysis. Individuals
acting in this scenario act purely in their own self-interest (a path
demonstrated by probability to be most profitable), yet still pursue a course
of action that does not result in the ideal outcome. The typical prisoner’s
dilemma is set up in such a way that both parties choose to protect themselves
at the expense of the other participant. As a result of following a logical and
mathematical thought process, both participants finish the scenario with a
worse outcome than if they had cooperated with each other in the
decision-making process.