Introduction can decipher one such dilemma with assurance

Introduction               Something aboutthe concept of applied contradiction has always fascinated me.

A vague field offactors that exist, for the most part, purely in the human conscious andintegrated rarely into the real world. But what if our own perception of thesecontradictory values could affect us? How we interact with others or defysocial norm? How do the possible outcomes of a partnership alter how we takerisks before a single action is taken? Ultimately; how can human cognitionalter probability before it’s reflected in the real world?1               It’s aquestion rooted deep in human psychology—a field which has pondered the way wethink for centuries. Psychology, as a study, has presented to the world some ofmankind’s most puzzling dilemmas and paradoxes. Within these, a countless rangeof arguments could be made over the diversity of social and emotional factorsthat 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 brainthat enables us to, in part, explain the dynamics of this field through a moreconcrete medium. In this way, I can decipher one such dilemma with assuranceunachievable through any other medium.In short: the Prisoner’sDilemma, explained by mathematical probability. Restraintof CooperationThe essence of thePrisoner’s Dilemma can be explained in two ways: in its most classic,applied form, in contrast to the story stripped down, reduced to itsmathematical and structural core.

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Both of these hold equal value inconsideration—to gain the full picture of thePrisoner’s Dilemma, an understanding of the relationship between these twoforms is important.As a story:Two individuals have been placed under arrest pretext of havingcommitted a major crime—robbing a bank, for example—but have been capturedunder evidence only for a much less serious crime—possibly petty theft.This takes place under two conditions: for our dilemma tofunction, we, the observer, may take the assumption that both, neither, or oneof the individuals actually committed the major crime—it has no significant effecton the essence of the problem. Secondly, In line with the mindset of the commonhuman being (as proven by countless empirical studies—see Miller, et al.(1999)), both care far more about their personal outcome—their freedom—thanabout the welfare of the other individual.

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

From theperspective of one of the individuals, you and the other criminal can take oneof two choices, leading to one of four scenarios. The scenarios are:1.      Youadmit your partner committed the crime.

You go free (serving 0 years), beingpardoned for the minor crime, but your partner will have spends 3 years inprison.2.     Theopposite occurs: you stay silent (complicit) and your partner accuses you ofcommitting the major crime.

You spend 3 years in prison. They go free.3.     Neitherof you accuse the other.

The police don’t have evidence on either of you forthe major crime, so you both serve a 1 year sentence for the minor crime.4.     Bothof you accuse one another. Both of you are absolved of your minor charges for betrayingthe other, but you each serve two years for the major crime.

Now, note thetotal amount of years served between you and your partner in each of thescenarios. If you both accuse one another, four years are served (2+2=4). Ifone of you indicts the other, three years are served total (3+0=3). If neitherof you accuse the other, two years are served (1+1=2).Given thisinformation, it would be logical to determine that the best course of actionfor the group would be for both individuals to remain silent (complicit). Thetotal number of years served in this instance is lowest. However, psychologyand mathematics tell us that this will not occur.

Notice that, regardless ofthe action of your partner, it never has a detrimental effect on your ownsentence to accuse your partner of the crime. If your partner is planning onaccusing you, your sentence goes down from 3 years to 2 years if you accusethem back. If your partner is not planning on accusing you, your sentence goesdown from 1 year to being set free (0 years). As a result, at least oneindividual, theoretically, will accuse the other.They shouldhave both cooperated, but from an individual stand point they noticed theycould always gain by defecting. If they have no control over what the otherperson is going to do. So they’ll both defect to try to better their ownsituation. 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 improvetheir situation. Individually, they’re worse off than if they both cooperated. FreeCooperation (Market Application) Avariation on the Prisoner’s Dilemma,often reflected in the real world, can be found in marketing expenditure andcompetition between businesses. Two companies, for example, Company A andCompany B, might be in the process of deciding how much money they should spendon advertising. In our example, there exists a sample population of 100 consumers,all completely dependent on an essential $2.00 product.

The product is offeredwith equal access by both Company A and Company B. Given that they produceidentical, equally valuable products, advertising would become the primaryinfluencer on sales. In this situation, the companies would have two options:to expend significantly on advertisement, or not to expend on advertisement atall. If neither side advertises, the consumers would naturally be drawn equallyto each company—50 to Company A, and 50 to Company B. With consumersdistributed evenly, each company would make $100.If onecompany chooses to advertise, however, we might project that 80 people could beinfluenced to purchase their product. This leaves Company B with just 20 consumers.

If the advertiser makes $160 in sales, subtracted by, say, $30 for advertisingexpenditure, they would turn a profit of $130.4Thenon-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 bothspent $30 on advertising, they only come away with only $70 each. Both peoplecooperating and not advertising is the most preferable situation, but bothcompanies can see that advertising will always make them more money. But unlikethe prisoner’s in jail, these companies can interact and have the opportunityto try to influence each other. From here Company B would be better off ifCompany A didn’t expend on ads at all.

Company A wouldn’t go for that becausethat would be worse for them. Company B could try to convince Company A thatthey would both not advertise, the only other situation where they’re bothbetter off. But without any real obligation to each other, there’s nothingthat’s stopping them from trying to advertise to gain more of the marketanyway, and to take over the industry. If you think a competing company is notgoing to advertise, you’re better off advertising.

In general, in a marketeconomy, businesses dominating an industry can communicate to maximize salesfor both parties.5 MathematicalRepresentationHow exactly, then,does the introduction of communication change the dynamic of the Prisoner’s Dilemma? This can be mostclearly explained through mathematics. Before doing this, however, it isimportant to establish a basic mathematical understanding of the dilemma. Manyof these representations might seem overly straightforward, but they areimportant for reference in a deeper analysis of the problem. To do this, wewill first set actions and their implications to the following variables:Let x = the probability of PrisonerA accusingLet y = the probability of PrisonerB accusingLet z = the expected value ofPrisoner B’s sentenceIn order to assess the motivationof each participant, we can set the following constants from Prisoner B’sperspective: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 variablesand constants defined:                                (Eq.

2)Which, foiled anddistributed, presents us with a basic representation of any non-communicatory sentencethat Prisoner B might receive:                                            (Eq. 3)Having situationallydescribed the sentence of Prisoner B, we can make adjustments to the equationto 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 equationdemonstrates the desirability of accusation to Prisoner B. No matter the inputfor x , Prisoner B cannot serve a longersentence than if they had complied. When Prisoner A accuses , we find that .

When Prisoner A complies , we simply find that .The same basic processcould be applied to determine the conditions under which Prisoner B would bemotivated 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 Acomplies , we find that . Demonstrative Factor: Sentencing LengthWhich primary factors,then, can we alter in these equations to change how the prisoners respond totheir dilemma? The impact of communication, of course, remains our ultimatequery, but it is important to first consider other aspects of the dilemma. Thefirst and most obvious of these is sentencing length, or drawback (C and D in our scenario).

Logically, for PrisonerB to consider cooperation with Prisoner A, his expected value for complicitymust exceed his expected value for accusation. We can represent this with thefollowing inequality:                                                                                               (Eq. 7)                                                                                               (Eq. 8)In order to find , we can rearrange Eq.

8 to the followingfraction:                                                                                                                               (Eq. 9)With Eq. 9 we arrive atperhaps the most significant mathematical revelation of this process.Remembering that —a fraction—we can determine that theright side of Eq.

9 must, too, fall within a fractional range. For this to bethe case, the numerator, , must be greater in value than thedenominator, . Therefore:                                                                                                          (Eq. 10)                                                                                                                                           (Eq. 11)In the terms of theproblem, Eq. 11 states that, for Prisoner B to consider cooperation, thedrawback for complicity must exceed the drawback for accusation—a logicalconclusion to come to at first glance at the dilemma. Applying this back to ouroriginal equation, we can answer our question: how does sentencing lengthimpact this decision-making?                                                                                           (Eq.

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

13)Prisoner B will always accuse. Demonstrative Factor: CommunicationHow, then,does communication, as described in the MarketApplication section, impact how decisions are made between the two parties?Unfortunately, unlike the sentence length—a quantifiable, clear demonstrativefactor—the impact of communication is too vague to be illustrated throughequations of constants and variables. Personality and circumstance have too greatof an influence. There are, however, several clear principles of communicatorypractice in the Prisoner’s Dilemma whichwe can discuss.Based upon our mathematical foundation, we candetermine a number of real-world conditions to be most important in facilitatingcooperation between two parties (particularly in a free market). As an initialrule, the advertising party in the dilemma must find an immediate payoff. Groupsuccess and its implications are always secondary, as the negative aspects of drawbackare not immediate.

Should both sides find their future payoffs significantly discounted,the threat of drawback could be sufficient to deter expenditure onadvertisement.Secondly, the likelihood of advertisement isheightened by retaliatory motivation. Robert Axelrod of the University ofMichigan explained in 1984 that, in a communicatory atmosphere, many wouldchoose to advertise if and only if they believed their rival to have advertisedin a previous instance.

4 Without communication, the rival’s ofteninnocent actions will almost always be preemptively assumed to be accusatory innature—communication grants a greater chance to both participants to avoidthis.Aditionally, in order to reach an agreement,both parties must show commitment to recurring cooperation. A single or fixednumber of repetitions in a dilemma will prevent any agreement from beingreached—if both sides know that a certain trial will be their final interactionwith 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, alsoapply to the second-last play, the third-last, and so on.Finally, cooperation can also arise if thereis a third party present with both great interest in the mutual success of allgroups and a great authority over the operation as a whole. This party would befirst to exercise a degree of restraint, despite the likelihood of the othertwo advertising. Conclusion(Conceptual Interpretation)How has this mathematical model enabled us to analyze the Prisoner’s Dilemma—its principlesand some of its more demonstrative factors? By practical means, it gave us theclearest possible method to take apart every single comparative factor withinthe problem as a whole. It showed the dilemma to us at its most basic forms.

And, perhaps ironically, cutting away much of the fluff aroundthe Prisoner’s Dilemma may haveallowed us to understand its meaning in a psychological and emotive sense moreclearly.Many claim that the puzzle illustrates a conflict betweenindividual and group rationality. A group whose members pursue rationalself-interest may all end up worse off than a group whose members act contraryto rational self-interest. More generally, if the payoffs are not assumed torepresent self-interest, a group whose members rationally pursue any goals mayall meet less success than if they had not rationally pursued their goalsindividually.Theprisoner’s dilemma is, ultimately, a paradox in decision analysis. Individualsacting in this scenario act purely in their own self-interest (a pathdemonstrated by probability to be most profitable), yet still pursue a courseof action that does not result in the ideal outcome.

The typical prisoner’sdilemma is set up in such a way that both parties choose to protect themselvesat the expense of the other participant. As a result of following a logical andmathematical thought process, both participants finish the scenario with aworse outcome than if they had cooperated with each other in thedecision-making process.


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