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.