# Game Theory: The Branch Of Mathematics

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BACKGROUND INFORMATION

The game theory generally is referred to as the branch of mathematics concerned with the analysis of strategies for dealing with competitive situations where the outcome of a participant’s choice of action depends critically on the actions of other participants. The game theory was introduced by John von Neumann, when he concluded that theory began with an idea in regards to the existing mixed-strategy equilibria. The prisoner’s dilemma is something that is relevant in terms of the game theory. The dilemma itself is categorized into two parties where each of the parties have two options. These options will then lead to it being a possible 4 outcomes, 22=4. The first prisoners dilemma was introduced by employees at the RAND corporation. The employees described the dilemma as:Two individuals are arrested. They are both then separated and independently interrogated by the police for a crime.

They are both interrogated to determine whether or not they are going to testify against the other individual. However if one of the suspects confesses, while the other suspect denies or remains silent, the suspect who confess will walk free, while the other is remained captive in jail and must serve a 4-10 year sentence. Then if both suspects remain silent then each of the individuals will serve approximately 6 months. Although if both individuals testify then both will serve around 5 years. The Prisoner’s Dilemma in Strategic FormEquilibrium Equilibrium is defined as a state of balance between opposing forces or actions that is either static or dynamic. Equilibrium comes into play because each of the suspects in the dilemma contain the same strategy, meaning either remain silent or testify.

This situation can be analyzed in a mathematical point of view, but certain aspects must first be introduced. D = testify reward – the sentence that is presented with the suspect that testifies rather than the suspect who remains silent, is walking freeE = remain silent – the sentence that is presented if both suspects remain silent is total of 6 months in jail F = testify penalty – the sentence that is presented if each testify is a total of 5 years in jailG = remain silent penalty – the sentence that is presented with the suspect remaining silent rather than testifying is a total of 4-10 years in jailFrom looking at the options, it is more than likely that the suspect is going to remain silent. Based on the information, if both suspects remain silent, then they both will serve around 5 to 6 months, which is definitely less time than the other options. However if the suspect is looking to serving the less amount of time in the jail, the testifying would be the more reliable and realistic approach.

This is because if the other suspect remains silent, then the result would be the other suspect being released to be set free. In order to witness and clearly see the point, the penalties and rewards are now going to be displayed as points. Therefore the more amount of points that the prisoner contains, the more that suspect has of avoiding being sent to jail. D= 6 E= 4 F= 2 G= 0When looking at it through the points it seems that testifying will always resort to being the best option. If a person chooses to testify they will either come out with the letter ¨W¨ or the letter ¨Y¨. Choosing to remain silent would result in the letters ¨X¨ and the letter ¨Z¨ being selected. This game acknowledges a term known as ´Nash Equilibrium´. This mathematical term is introduced and named after a mathematician, known as John Forbes Nash, Jr. Nash states that equilibrium exists in a game where both individuals are aware of the penalty and reward structure, the competitors choices.

He also makes it known that when an individual makes a decision where the options are considered, there is always a decision that is more beneficial than the other but is typically not selected. In the dilemma where, D>E>F>G, both players are testifying is noted as a Nash Equilibrium. Non-equilibriumAssume that instead of the suspects that remains silent penalty isn’t as harsh and severe and the testifying penalty. Meaning, while making one suspect remain silent while the other suspect testifies, does not introduce a longer jail sentence, whereas it would rather each prisoner testifying against each other. D=6 E= 4 F= 0 G= 2This concludes that now a suspect who wants to testify must come with the realization that testifying would be far more worthless than remaining silent if the other suspect wishes to testify. If D>E>G>F, this would result into the prisoner’s dilemma being considered a non-equilibrium game.

The more valid option would be some form of probability that the subject would testify. For example, say that there are two best friends. They both come to an agreement that they are willing to play a game. If Carolyn is likely to testify, which decision should Barbara go with and how exactly should Barbara react to the situation? If it is evident that Barbara is a trusting person who is more than likely willing to remain silent, should Carolyn take it upon herself to actually testify? InvestigationLastly, I performed a hands on trial based on the actual prisoners dilemma. The prize during this trial was their favorite candy. During this trial I used my younger cousins, being that candy is appealing to every little child. My cousins were giving outcomes in relation to them either testifying against the other cousin or remaining silent against the other cousin, which would then enhance which decision they would actually make.

Mathematics Variables being used during the situation:X = the probability that Barbara will testifyY = the probability of Carolyn testifyingZ = the expected value of Carolyn Recall of the constants from aboveD = testify reward (subject that is testifying gains a score when other remains silent)E = remain silent reward (both subjects gain a point when both remain silent)F = testify penalty (both subjects receive score when both testify)G = remain silent penalty (subject that remains silent gains a point when other testifies) An equation can then be generated to determine the expected value of Carolynz=xy(F)+x(1-y)(G)+y(1-x)(D)+(1-x)(1-y)(E)(1)This equation can then be reduced to:z=Fxy+Gx-Gxy+Dy-Dxy+E-Ex-Ey+Exy(2)I am then going to first conclude that Carolyn is going to to testify which would then cause the y value to remain at 1z=Fx+Gx-Gx+D-Dx+E-Ex-E+Ex(3)Which would then simply to being:z=Fx+D-Dx(4)The above equation shows that if Barbara testifies then it would be reasonable for Carolyn to testify. It also shows that if Carolyn was to remain silent it would be beneficial for Barbara to testify.

Later if it is concluded that Carolyn chooses to testify, then the y value when then remain constant at 0, which would then reduce the second equation to:z=Gx+E-Ex(5)The argument that was discussed in the introduction suggests that a Nash Equilibrium exists when both subjects chooses to always testify when D>E>F>G. In order for Carolyn to determine whether or not to remain silent, the expected value for remaining silent must be greater than the expected value for testifying. Which is shown in the following equation:Gx+E-Ex>Fx+D-DxRemember from the variables that x is the probability that Barbara will testifyGx-Ex-Fx+Dx>D-ESimplifying that equation will then give you:x>D-EG-E-F+DTo reiterate what was already discussed, in order for Carolyn to consider remaining silent, her expected value for remaining silent must be significantly higher than the probability of her testifying. Based on mathematical notes, a probability will always equal either 0 or 1.

Another mathematical notation is that the denominator must be greater than the numerator, or else it would be considered an improper fraction. Which results into the next equation:G-E-F+D>D-EThis can then be simplified to:G-F>0 or G>FThe answer above brings back the same argument that was also discussed in the introduction, non-equilibrium. Basically stating that in order for it to be non-equilibrium, the circumstance for testifying must be greater than the subject remaining silent. The conclusion taken from this is that the only possible way to actually elude the Nash Equilibrium is to follow the following restriction based on the values:D>E>G>FResults From this point I have displayed mathematically that a subjects best bet in the prisoners experiment involving, D>E>F>G and the Nash Equilibrium decision is to agree to testify regardless of anything. By nature, I actually wonder what real human beings would actually, who was not as aware of the mathematical reasoning and results of the prisoner’s dilemma.

REAL-LIFE EXPERIMENT

The final section through this investigation was my actual hands on trial based on the prisoner’s dilemma plan, which consisted of the use of my cousins. The reward for this game was a dollar given. The observation went along like this:Two participants were selected (younger cousins) and they were informed that they would receive a dollar by just participating in the simple math experimentThey were told that they would be given the opportunity to either testify (snitch) or either remain silent and the distribution of the dollar would come in to play based on their response as well as the other cousins response. They both received 3 colored index cards, where the included the words ¨testify, join, and the possible outcomes.¨

The outcome of the cards is shown below:

Experiment focused on D>E>F>G youpartnerYou receiveShe receivestestifysilent5 dollars1 dollarsilentsilent3 dollars3 dollarstestifytestify2 dollars2 dollarssilenttestify1 dollar5 dollarsExperiment focused on D>E>G>F youpartnerYou receiveShe receivestestifyremain silent5 dollars3 dollarsremain silentremain silent4 dollars4 dollarstestifytestify1 dollar1 dollarremain silenttestify3 dollars5 dollarsOnce they made up their mind to select their decisions they had to tell their partner the reason for them selecting what they didRevealed the choices that they selectedMoney was givenResults of trialFor trials D>E>F>G50% of the subjects remained silent, where 50% of the subjects testified90% of the trials included testifying85% of both participants testified90% of trials included remaining silent35% of both participants remained silentThe average of money received to the testifier was \$9.00, whereas for the person who remained silent was \$3.25For trials D>E>G>F70% of the participants remained silent, where 30% of the participants testified60% of the trials included testifying86% of both participants testified87% of the trials included remaining silent39% of both participants remained silentConclusionThe prisoner’s dilemma served the hypothesis.

The reward trial concludes that the participant is focused more on what they actually gain rather than their partner. The high results of percentages suggests that the subjects displays a hesitation to accept the compromises. In relevance to the study, prisoners will end up going free while trusting their partners, while their partners actually end up serving the maximum sentence in jail.

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