Game Theory: An unorthodox take on Prisoner’s Dilemma

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 09 Issue: 11 | Nov 2022 www.irjet.net p-ISSN: 2395-0072

Game Theory: An unorthodox take on Prisoner’s Dilemma

Pankaj Pilaniwala1

1University of Arizona***

Abstract -

This paper discusses the basic Prisoner’s Dilemma and focusses on the analysis of each outcome. The paper also highlightsIterativePrisoner’sDilemmaandexplainsthesame to the readers. The main intention of the paper is to focus on a particular outcome and present theories andlogic tosubscribe to the premise. There are multiple arguments that can be made with regards to how the game is played and outcomes are perceived. The paper analyzes each of those arguments and compares them to come up with a point to prove the argument that was made. The paper does this by giving multiple examples.

The paper utilizes tools from Philosophical Decision Theory to present arguments for defaulting position and analyzes the arguments to show that the position of cooperation is weakly preferred. The paper uses Regret principle to show how the different arguments of both the positions stack up against each other and then comes to a conclusion of defaulting.

Key Words:

Game Theory, Prisoner’s Dilemma, Philosophy, Ethics, Tragedy of Commons, Management

1.INTRODUCTION

The Prisoner’s Dilemma is a concept used to model a strategic interaction in which actors choosing their behaviors rationally according to their own self-interest make everyone worse off than they could have been otherwise.Thisparticular“game”isusedbothtounderstand failures of cooperation such as arms races and ethnic warfare and to prescribe particular solutions designed to elicitcooperation.Thekeyfeatureofthegameisthat,when thegameisplayedonlyonce,nomatterwhatanotherplayer does (cooperating with me or trying to exploit me), I am betterofftryingtoexploittheotherplayer–sointheend, everyplayerexploitsratherthancooperates,andtheyareall worseoffthantheywouldhavebeencouldsomeonehave “forced”themtocooperate.ThisisthebeautyofPrisoner’s Dilemmawhereindefaultingispreferredovercooperation. Thepaperexaminesthisindetailoverthenextparagraphs.

Prisoner’s Dilemma is an example of a game analyzed in gametheoryshowinghowtworandomindividualsmightnot cooperate,evenifitappearsthatitisintheirbestinterests todoso.ItwasoriginallyframedbyMerrillFordandMelvin Dresher.Thegameissetas:twoprisonersarearrestedand

imprisoned.Theyarekeptinasolitaryconfinementwithno means of communicating with the other. The prosecutors don’thaveenoughevidencetoconvictthepair.Hence,they givethemasituationwhichwillhelptheprosecutorstoget enoughevidencetoconvictthepairoratleastoneofthem. Thesituationislikethis:

Confess–B Don’tConfess-B

Confess-A -10,-10 -1,-20

Don’tConfess-A -20,-1 -2,-2

Morepreciselyputthesituationislikethis,ifbothofthem confessestheyget10yearseach.Ifoneofthemconfesses, theothergets10yearsandtheconfessorgetsoneyear.But ifnooneconfessestheygettwoyearseachfortheirother offenses.

Now,let’slookatthe matrix fromA’s pointofview.Acan choose between confessing or denying the charges. There arefourdifferentoutcomesforAdependingonwhateither oftheprisonersdo.IfAknowsthatBwillconfessthenit’s betterforAtoconfessaswell,becauseifAdecides not to confess and B confesses then A gets 20 years of prison whereasifAconfessesgivenBisconfessing,Agets10years ofprison.Now,let’ssaythatAknowsBwillnotconfess,soA caneitherconfessorhecannot.Thechoiceisbetween1year in prison vs 2 years in prison. In this case also, rationally, confessing is better for A as compared to not confessing, givenBisNotConfessing.So,ineithercaseit’sbetterforAto confess.

Now,let’slookatthe matrix fromB’spointofview.Bcan choose between confessing or denying the charges. There arefourdifferentoutcomesforBdependingonwhateither of the prisoners does. The outcome for B is different dependingwhatAdecidestodo.InonecaseBhastomakea choicebetween10yearsinprisonor20yearsinprison.And theotherisbetween1yearinprisonor2.ThechoicesBgets issimilartowhatAgets.Henceineithersituationasseenin theA’scondition,confessingisbetterthanNotConfessing. So,Bwillalsoendupconfessingandbothwillget10yearsin prison,versusthepotentialopportunitytogetjust2yearsin prison.

Itisimpliedthattheprisonerswillhavenoopportunityto reward or punish their partner other than the prison sentencestheygetandthattheirdecisionwillnotaffecttheir reputationinthefuture.Becauseconfessingoffersagreater

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rewardthancooperatingwiththem,allpurelyrationalselfinterestedprisonerswillconfess,meaningtheonlypossible outcome for two purely rational prisoners is for them to confess.Theinterestingpartofthisresultisthatpursuing individual reward logically leads both of the prisoners to confess when they would get a better reward if they both keptsilent.Thepaperwillattempttoanalyzethiscaseand present arguments on both the positions and conclude pointstopreferonepositionovertheother.

Iftheoutcomeispurelytobekeptinmindandsomehowthe prisoners got into agreement that they won’t confess, it’d seems to be the preferable outcome as both get just two yearsinprison.Boththeprisonershavetotrusteachother,

andbothhavetoseethegreatergoodforthemselves.Bynot confessing, both of them, they definitely can reduce their sentences from 10 years to 2 years. Of course, the next immediatethoughtistobetraytheirpartner.Butonething tounderstandhereisthesameoptionisalsoavailabletothe other prisoner and he can too betray. Hence, rather than betrayingtheotherprisoner,oneshouldtrusttheagreement andnotconfess.

Theprisoner’sdilemmashowsthatwhatisoptimalforone might not be for the group. People often put their selfinterest before the group interest. Individual rationality sometimescomeintoconflictwithgrouprationality.Inthis game,rationalityforceseachplayertochooseanoutcome that is worse than the best outcome. Hence, if the players think more about the group outcome, each of them might endupwithbetterresults.Andthisistruenotjustforthis specific case but for all the cases in numerous real-life examples.

2. REAL WORLD FAILURES OF NON_COOPERATION

The notorious ‘free rider’ problem is an example of a Prisoner'sDilemmainvolvingmanyindividuals.Considerthe bussystemsinmanyEuropeancountries.Theyarepaidfor bywhatisalmostanhonorsystemofpassengerspayingtheir fares.Thebussescanonlybekeptrunningifenoughpeople paythefare,butnoteveryoneneedstopaythefaretokeep thebussesrunning.IfIthinkfromapassenger’sPOV–they shouldn’twanttopayandwouldhopethatorherspaysuch that the service continues. But if everyone adopts this strategy,thebuseswillnolongerrunandeveryonewillhave to take taxis, or buy a car which are worse situations for everyonethanifeveryonepaidthefare.

Asimilarexampleasaboveisencounteredinoneofthe courses at Eller. As part of the coursework, students are requiredtoprovidefeedbacktotheprofessorattheendof thecourses.Someprofessorswilloffersomebonuspointsto the entire class if 75% of the class strength provided the feedback. Now, 25% of the students can earn these bonus pointswithoutprovidinganyfeedback.So,ifallthestudents startthinkingthattheywillnotfillthesurveyasothersare

doingitandtheycanjustgetfreebonuspointsandifenough studentsthinkso,thatmightresultinnooneearningthese free bonus points. In this case as it turns out to be cooperatingisbetterthandefecting.

2.1 Symmetric 2x2 Prisoner’s Dilemma with Ordinal Payoffs

Aprisoner’sdilemmagamecanbedefinedinthefollowing matrix:

Where–Y>Z>W>X

There are two players, Row and Column. Each has two possiblemoves,“cooperate”-Cor“defect,”-Dcorresponding, respectively,totheoptionsofremainingsilentorconfessing intheillustrativeanecdoteabove.Foreachpossiblepairof moves, the payoffs to Row and Column (in that order)are listedintheappropriatecell.ZZisthe“reward”payoffthat each player receives if both cooperate. WW is the “punishment” that each receives if both defects. YY is the “temptation”thateachreceivesassoledefectorandXXisthe “sucker” payoff that each receives as sole cooperator. We assumeherethatthegameissymmetric,i.e.,thatthepayoffis the same for each player, and payoffs have only ordinal significance,i.e.,theyindicatewhetheronepayoffisbetter thananother,buttellusnothingabouthowmuchbetter.Itis noweasytoseethatwehavethestructureofadilemmalike theoneinthestory.SupposeColumncooperates.ThenRow getsZZforcooperatingandYYfordefecting,andsoisbetter offdefecting.SupposeColumndefects.ThenRowgetsXXfor cooperatingandWWfordefecting,andsoisagainbetteroff defecting.ThemoveDDforRowissaidtostrictlydominate the move CC: whatever Column does, Row is better off choosing DD than CC. By symmetry DD also strictly dominatesCCforColumn.Thustwo“rational”playerswill defect and receive a payoff of WW, while two “irrational” players can cooperate and receive greater payoff ZZ. In standardtreatments,gametheoryassumesrationalityand common knowledge. Each player is analytical, knows the otheristoo,knowsthattheotherknowsheisrational,etc. and the players are aware that all the players value the outcome.ButsinceDDstrictlydominatesCCforbothplayers, theargumentfordilemmahererequiresonlythateachplayer knows his own payoffs. It is also worth noting that the outcome(D,D)(D,D)ofbothplayersdefectingisthegame's onlystrongNashequilibrium,i.e.,itistheonlyoutcomefrom which each player could only do worse by unilaterally changing its move. Flood and Dresher's interest in their dilemma seems to have stemmed from their view that it provided a counterexample to the claim that the Nash equilibriaofagameconstituteitsnatural“solutions”.Hence,

C D C Z,Z X,Y D Y,X W,W

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inthissituationalso,wherethepayoffisordinal,prisoners arebetteroffwithdefaulting.

3. TRAGEDY OF COMMONS

“Eachmemberofagroupofneighboringfarmersprefers to allow his cow to graze on the commons, rather than keepingitonhisowninadequateland,butthecommonswill berenderedunsuitableforgrazingifitisusedbymorethan somethresholdnumberuseit.”Thetragedyofthecommons isa termusedinsocialsciencetodescribea situationina shared-resourcesystemwhereindividualsactaccordingto theirownself-interestbehaveanti-ethicallytothecommon goodofallotherpeoplebyplummetingthecommonresource whichisavailabletoallandcouldbethereforever,through theircollectiveactions.

Takethisexample-Everyfishermanknowsthatifthereis toomuchfishingtheneventuallyfishstockswillrunout.Ifall thefishermencouldagreetofishatsustainablelevels,then thefishstockscouldlastforever.However,ifonefisherman startstoactselfishlyandstartstooverfishtheneventually thefishstockswillrunout.Andwhenotherfishermensee this, they will also start to over fish. It just starts with someoneactingselfishlyandthentheotherfollowsuit.The first person might start because they think that just them overfishingwillnotmakeanydifferencebutoncetheystart theneveryoneelsesoonjoinsthemandthestocksrunout.

Thisissimilartotheprisoners’dilemma.However,inthe prisoners’dilemmaindividualscannotcommunicateandso, iftheyactlogically,thentheywon’tco-operatewiththeother playersandwillendupwithaworseoutcome.In

the tragedy of the commons where all the players can communicate to each other and make a common ethical choicetosaveandharvestthecommonresourcebutstillendupoverusingthecommonresource.TheconceptofTragedy of Commons can be paralleled with Prisoner’s Dilemma, where each prisoner acts on his best self-interest and ultimatelygetsonlysecondtobestoutcome.Iftheinvolved individuals start acting or behaving in a manner that is thoughtfuloftheotherplayersinvolvedeveryonecanbenefit morethantheywouldotherwise.

4. ITERATIVE PRISONER’S DILEMMA

Sofar,thepaperdiscussedwhathappenswhenthePDis played once. But what if it’s played multiple times by the sameplayers?Now,let’ssaytheplayersaremadeawareof the number of iterations the game will be played, mutual cooperationwillbeunstable.Butiftheplayersdon’tknow whichorwhenisthelastround,thingsbecomeinteresting, undersuchconditions,cooperativestrategiesworkwell.

Aparticularlyeffective(andsimple)strategyistit-for-tat, inwhichtheplayercooperatesinthefirstround,andfrom there on the player does what the other player did in the

previousround.Thispunishesdefectionwithimmediatebut short-lived consequences “you just defected, so now I’m denyingyouthebenefitsofmycooperation.ButI’llcooperate if youcooperate.”In The Evolution ofCooperation,Robert Axelrodsaysoftit-for-tat,“[w]hataccountsfor[its]robust successisitscombinationofbeingnice,retaliatory,forgiving, andclear.”[3]it’sagreatstrategyifzoomed-outandviewed–it starts with trust, avoiding trouble. As soon as it detects defection, the system repays in a similar fashion and eventually rewards as soon as the system detects cooperation.Andit’seasytorecognizeandunderstand.

The point to note here is tit-for-tat is one of the many availableandpossible“nicestrategy”.IniterativePrisoner’s Dilemma,strategiesthatareniceandcordialtendtodeliver reasonableoutcomesascomparedtostrategiesthatmight hurt the others. The lesson:be cooperative until the other playerdefects,andthenreactbutforgive.Thinklong-term and gain trust. An important feature accounting for tit for tat’seffectivenessisitsniceness,whereanindividualshould neverfirstdefectandcomeoutasnon-trustworthyplayer. Titfortatisalsoeffectivebecauseitisretaliatory:Theplayer whobetraysfirstandshowssignofdefectionisalsotreated with the same gesture. It gives the other player an opportunitytorepaybackthesameflavorofnon-cooperative behavior. Additionally, this strategy is also forgiving, such thatwhentheplayerthatbehavedinanon-cooperativeway getsthesamereciprocation,theygetachancetochangetheir behavior and regain the trust. As soon as the other player seesthatthefirstplayerhasstartedcooperatingitgivesthem somehopeandtheyreciprocateincooperatingmode.Finally, tit for tat is also a clear strategy, readily understood by others, and the final goal of it is to establish trust and cooperation.

Oneimportantconcepttodiscusshereisifthegameisa limitediterationgamethentheplayerswillrushtodefault. Let’ssayit’sa‘n’iterationPrisoner’sDilemmagame.PlayerA knowsthatplayerBwilldefaultinthenthgame,sotheplayer mighttrytodefaultin‘n–1’thgame.Andthisinformationis availabletoPlayerBwhointurnwilltrytodefaultat‘n–2’th gameandsoon.Ultimately,theplayerswillturntodefaulting intheveryfirstgame.Itbecomesalmostsimilartocentipede games. And turns out to be that defaulting is the optimal strategytobeattheotherperson

to it and ensure that the outcome is not the worst one. Similarly,intheinfiniterepetitivePrisoner’sDilemmagame, defaultingissoughtwheneverit’sknownthattheparticular game is the last game or whenever the other player starts defaulting.Forthepurposeofthispaperthefocuswillbeon single shot Prisoner’s Dilemma. The iterative model was introducedjusttogivesensetowhatwouldhappeninthe longrunandifit’sanydifferent,thefinaloutcome,fromthe single-shotPrisoner’sDilemma.

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5. PROPOSITION

Theconceptsdefinedanddiscussedabovewillbeusedinthe latterpartofthepapertodiscussmoreexamplesandcase studies on Prisoner’s Dilemma. I personally believe in confessingisthebeststrategyeventhoughit’snotthebest outcome from the given available outcomes. In the below paragraphsthepaperwilltrytoputupthepropositionand discuss why confessing is the best argument and ‘Not Confessing’isnot.Thepaperwillcitemultipleexamplesto provetheargument.Firstofmanyexamplesis–

Therearetwofirmsinthemarket,AandB.Boththefirms are operating with equal market share. One day firm A comes up with a strategy to reduce the prices of their productandalsoinvestsonadvertisingtomarkettheirnew strategy of reduced pricing. If A only continues with this strategy,itwouldbenefitandwouldsubstantiallyincrease revenueanditsmarketshare.ButifBalsofollowsthesuit anddecreasesthepriceandinvestsinmarketingthesame strategy, it would be able to retain their customers and revenue. Since, both the firms exploit the reduced pricing strategytheyultimatelyretaintheirmarketshareandmay ormaynothavethesamerevenue.But,ontheotherhandif boththefirmssticktotheiroriginalplan,theywillcontinue tomakeprofits.Andthelaststrategyistheoptimalstrategy forthefirmsanditworksonthegrouprationalityreasoning.

The problem with group rationality in this approach is, companies tend to work on maximizing their own profitabilityandareleastconcernedabouttheother.Infact, theywouldlikeforotherfirmtobesecondtothemandtake theirmarketshare.Hence,someonewillalwaysdefaultand try to gain more revenue and market share. Hence, the optimalgroupstrategyisnotalwaysgoodfortheindividual. Itisreallyhardinthereal lifetogettoa commonground thatisreallygoodforeveryone.Someonesomewherewill always have this thought and it is safe to assume that everyone will have this thought to maximize their profitability and act as a rational individual. The act of individualrationalitywhenmadebyeveryoneelseinvolved in the same act results in ‘Tragedy of commons’ and as a result of which everyone involved gets hit by some point, maybesomeordinalvalueasanabsolutevalueishardtobe given to this outcome. It’s very common in all kind of businessandorganizationstendtoactinthismanner.

Reduce Price & PromoteFirm-A

Reduce Price & PromoteFirm-B

Don’t reduce Price & don’t promoteFirm-B

Revenue (may) decreaseandsame marketshare

Firm A gains Revenue & Market share

Don’treducePrice& don’tpromoteFirm -A

Firm B gains Revenue & Market share

Same revenue and market share as before

It’sevidentfromtheabovematrixthatcolludingandcoming to agreement is better but it’s hard to retain the same agreement for the longest period of time. For the longest timeinmylifeIbelievedthatAppleisanorganizationthat doesn’tfollowtherulesofthegameanddoesn’tfollowthe suit.Itsetstherulesandplaysitalone.Appleproductswere considered premium (are also considered premium now) notonlybecauseofpricingbutalsobecauseoftheservice theyprovided.Appleneverwentintoapricewarorreduced theirpriceoftheproducts.Applenevergaveanydiscounts inanyholidayormassshoppingevents.Ialwayswondered whatmadethemdosoandhowaretheyabletosustainfor thislongwithoutheadinginthedirectionwhereotherfirms were headed. I was really amazed. Prisoner’s Dilemma or other decision theory paradoxes are also game theory strategies that many firms deploy. Some firms have been successful in their strategy and have changed the game. Philosophically, we might see something as a paradox whereasinreallifeitmightjustbearationalchoicebasedon one’sprofitandfuturevision.Butuntilnow,thistimewhen Applereleasedtheirnewphones,theywereinforashock. Theirpricingstrategy didn’t work out.Apple thoughtthat theyhavebeenthemasterofthegameandcancontinueto beso.Andwereveryconfidentwhentheylaunchedthenew modelsinSeptemberofthisyearonlytolearnthehardway thattheycannotcontinuetoavoidthemarket.Withevery yeariterationApplediscontinuethepreviousyear’smodel andtheyfollowedthesamestrategythisyeartoo.Butafter seeing the sales of this year iphones, Apple has to change their decision and has started production of the previous model(IphoneX).Basically,thisisasgoodasreducingthe priceoftheiPhonebecauseusersarenotreadytobuythe costlier model. In a latest update, Apple has reduced the priceoftheirphonesintheJapanesemarket

Theaboveexamplealsoprovesthestrategyofcooperating and/orconfessing.Tosurviveandnotbeinapositiontoget surprised onemust be readytoadopttothe environment andmakearationaldecision.Manysectorsoftheeconomy have two main rivals. In the U.S., for example, the fierce rivalrybetweenCoca-Cola andPepsiCoinsoft drinks,and HomeDepotversusLowe’sinbuildingsupplies,hasgiven rise to numerous case studies in business schools. Other fierce rivalries include Starbucks versus Tim Horton’s in Canada, and Apple versus Samsung in the global mobile phonesector.

The proposition to defect is a better proposition because thereisnolegalbindingtoanyonetoholdtheirendofthe bargain. Anyone, any organization can defect at any given point and the cost will be left to bore by the other party involved in the dilemma. Cooperation is a trust game and trustisverycostlyinrealworld.Twostrangerscooperating and trusting each other is not normal. Organizations are competitorsandcooperationwillonlycomeintoplaywhen that is the only best strategy. Organizations always try to outplayandoutsmarttheircompetitors.Inaboveexamples,

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cooperatingwasn’tthebeststrategyallthetime.IfFirmA knows that the other firm B will cooperate, then it makes sense financially and strategically to defect and get the marketshareandmorerevenue.FirmBwillfollowthesuit andtheindustrywillshifttothenewnormal.Itmighttake time,butthepricewarwillendandtheorganizationswill shifttoanewnormalwherethepriceswillbestable,market sharewillbestableandorganizationswilltrysomethingelse togainthenextwaveofmarketshare.Unlessthereisalegal agreement it is nearly impossible for firms to cooperate. Cooperationwillhappenwhentheotheroptionsyieldlossor nottheequalnumberofgains.

Researchers often use the single-trial Prisoner’s Dilemma whentheywanttostudyhowpeopleapproachoneanother intheabsenceofahistoryofinteractionandintheabsence of a future of interaction. Hence, these choices are not influenced by considerations regarding the past (e.g., retaliation)orthefuture(e.g.,adoptingastrategytoobtain cooperation).Inthesesituations,intuitionoftheotherplaya veryimportantrole.Anyinformationthatisrelevanttoone’s expectationsabouttheother’spossiblechoiceisuseful,at leastwhenone’sownchoicedependsonwhattheotheris going to do. For example, people expect much more cooperationfromanotherpersonwhoisperceivedashonest than from another who is perceived as dishonest. Also, people may also derive expectations from stereotypical information.

5. REGRET IN PRISONER”S DILEMMA

Regret has a very important role in everyone’s life, the thingspeopledo,thewaypeopleact,thewaypeoplemake decisions and evaluate their options. Regret makes the beingsdothingsthatotherwisesomemightnotdoitatall. Regretisafeelingofsadness,repentance,ordisappointment oversomethingthat'shappenedorbeendone…especiallya loss or missed opportunity. Regret is not just a simple emotion, the way the definition above implies. It also involveswhatphilosopherscallaconditionalintent:IfIhad ittodoover,Iwoulddothingsdifferently.Problemis,that leadstoaparadox.Decisionanalystsfeelthattheaxiomsare so logical and reasonable that the situations in which the decisionmakersviolatetheaxiomsare‘paradoxes’.

Thereissomethingcalledsymmetricregret.Notalldecision problemsyieldasymmetricregretoutcome.Toputthingsin perspectivehereisanexample–Afarmerhasacropwhose valueatharvesttime,twomonthsfromnowisuncertain.He isgettinganoffernowfromabusinessmanforhiscropata fixedpriceperbushelofthegrainthatwillbeproducedat the harvest. Whatever maybe the market condition irrespective of that, the businessman is ready to offer the farmerafixedperbushelpriceforthegrain.Tosimplifythe example, the price per bushel of grain at the time of the harvestwillbeeither$5or$9withequalprobability.The currentofferfromthebusinessmanthatfarmerwillreceive is $7 per bushel of grain. This offer is irrespective of any

conditionsanditwillnotchangeatthetimeoftheharvest. Nowweknowthequantitativevalueofthegrainperbushel atthetimeofharvest.Themaximumworthitcanattainand theminimum.Thecurrentofferisfixedanddoesn’thaveto dowithanymarketcondition.Atraditionaldecisionanalysis showthatthefixedpriceofferfromthebusinessmanhasno riskandguaranteesmoneywhereaswaitingdoeshaverisks. Itcaneithermakethefarmerricherthanthecurrentofferor poorer,dependinguponthemarketcondition.

Thefarmerhasa decisiontomake.It’sdifficultforhim to predictthepriceandatthesametimehedoesn’twanttodo somethingthatmakeshimregretlatter.Howwillthefarmer make a rational decision? What tools can he use to make decision? If the farmer sells the grains right now to the businessmanat$7perbushelofgrainandthepriceofthe grainatthetimeoftheharvestbecomes$9thenthefarmer willregretforthe$2loss.Ontheotherhand,ifthefarmer sellsthecropsnowtothebusinessmanat$7perbushelof grain and the price of the grain at the time of the harvest becomes $5 then the farmer will be happy that he saved himselfalossof$2perbushelofgrain.Theothercondition wherethefarmerdoesn’tsellthecropstothebusinessman now. At the time of the harvest is $9, the farmer will be happythathetooktherightdecision.Ifthepriceatthetime oftheharvestis$5,thenthefarmerwillregretthatwhyhe didn’t sell off the grains when he had the offer from the businessman.So,eithersideofthedecisionhehasaregret value of $2. Hence, this is symmetrical regret condition, whichitmakesitevenhardertosolve.Butnotallproblems indecisiontheoryaresymmetrical.

Regretisnotoftenusedbythedecisiontheoriststosolvea problem.Butifitisusedintheproblemsthenmostofthe problemscanbemadesimplerandaidthedecisionmakers tocometoaconclusionfasterandefficiently.Let’stakeup theprisoner’sdilemmaproblemandapplyregretfunctionto itandseeifwecangettoasolutionfasterandalsolet’stry toprovetheoverallstandof‘defaulting’.

ConfessB Don’tConfessB

ConfessA -10,-10 -1,-20

Don’tConfessA -20,-1 -2,-2

Let’sassignaregretvaluetoeachoftheoption

ConfessB Don’tConfessB

ConfessA a1,b1 a3,b2

Don’tConfessA a2,b3 a4,b4

ForA–a2>a1>a4>a3

AndforB–

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b2>b1>b4>b3

As mentioned above, the regret value for A is the highest whenitgets20yearsinprison.It’ssecondhighestwhenit gets10yearsinprison,thirdhighestwhenitgets2yearsin prisonandleastwhenitgets1yearinprison.Similarly,forB the regret value is the highest when it gets 20 years in prison.It’ssecondhighestwhenitgets10yearsinprison, thirdhighestwhenitgets2yearsinprisonandleastwhenit gets1yearinprison.Theleastregretvaluecomeswhenthe individualgetstheminimumprisonsentence.Andhowdoes thathappen?Ithappensviabetrayingtheotherprisoner.But the striking thing to notice here is the ordinal value or philosophicalvalueofthisregretwhichisafeeling,issame forboththeprisoners.Ifwecancompoundthisaffectwith theotherprisoner’ssentencewhichiswaylowerthanthatof the prisoner who cooperated, the regret feeling gets exponentiallycompounded.Thisfeelingofregretistoomuch to be comprehended by just writing here or discussing it. Thisoptionofbetrayingtoeachotherissameandavailable toboth.Ifforonceaprisonergetsbetrayedandhegetsthe same option again, he will never again cooperate and will defaultalways.Thetimetospendinprisonistoomuchand somethingtoanalyzehereis,theprisonwhoeventhinksof cooperating doesn’t get anything by doing so. There will always be a chance that the other prisoner might default. Thischance,howeverlowit might be will always create a senseofregretwhichwillbehigherthanthechanceofthe otherprisonernotdefaulting.Let’ssaythatthechancethat PrisonerAwilldefaultis0.3.so,thechancethattheprisoner Awillcooperateis0.7.Andthisinformationisavailableto PrisonerB.theprobabilityofPrisonerAdefaultingisenough toinduceasenseofregretinthePrisonerB.Andthisregret willbemorethan0.7wheninducedintheprisonerB.Thisis howtheprisonerBshouldthinkanddefault.

Goingbyabovelogicandexplanationitseemsrationalfor prisonertobetray.Andboththeprisonerswillthinkalike, becausetheinformationavailabletobothofthemissame. There is no extra information available to either of the prisoners. For them to behave and think alike is not a surprise.Thiseliminatestheleastregretfuloption.Thenext oneistocooperate.Ifbothofthemcooperate,theygetthe secondleastregretvalue.Butthatisthecatch,it’sthesecond leastregretvalue.Oncetheprisonersagreetocooperateand are certain that the other will cooperate, one will try to betrayandgettotheleastregretvalue.Butagain,thesame informationisavailabletotheotheraswell andtheother willalsothinktobetrayandensurethathisregretvalueis theleast.Again,ifonekeepshisendofthedealanddoesn’t defaultwhereastheotherbetraysandgethisregrettothe least value, the one who cooperated will never ever cooperateifgiventhesameoptionispresentedagain(canbe thoughtofiteratedPrisoner’sDilemma).Andifthesameis repeatedwithdifferentpairsofprisoners,cooperatingmight notbeeasyandsimpleinthelongrunforlimitedrepetitive gameoronetimegame.Thislogicalsoentailstodefaulting.

As described above, the second-least regret option is also ruledout.Thisbringsthethirdoption,i.e.10yearsinprison for each of the prisoners. And this seems to be the most optimaloutcomeforboththeprisoners.Givenallthelogic and information, no one could have one better. Because withoutanexternalagreementit’sveryhardforprisonersto ensure that the other will keep their end of the bargain. Defaultingensuresthattheprisonerwillgetamaximumof 10 years of prison which is much better than 20. If the prisonerisnotdefaulting,thenthereisaconstantfearofthe other person. The dependency on someone else makes it even harder and the regret value shoots up. To minimize one’sregretandmaximizetheunsaidutility,prisonershould default.

Now the paper will examine how regret plays out in the secondexamplediscussedindetailabove.

Reduce Price & PromoteFirm-A Don’treducePrice & don’t promote Firm-A

Reduce Price & PromoteFirm-B Revenue (may) decrease and same market share

Don’treducePrice & don’t promote Firm-B

Firm A gains Revenue&Market share

Applyingregretfunctiontothematrix:

Firm B gains Revenue&Market share

Samerevenueand market share as before

Reduce Price & PromoteFirm-A Don’treducePrice & don’t promote Firm-A

Reduce Price & PromoteFirm-B a1,b1 a3,b2

Don’treducePrice & don’t promote Firm-B

a2,b3 a4,b4

ForA–a3>a1>a4>a2

ForB–b3>b1>b4>b2

Asmentionedabove,theregretvalueforAishighestwhen its revenue decreases along with the market share. This situationhappenswhenAdoesn’tdefaultandcooperatebut B defaults and doesn’t cooperate. The regret is second highestwhentherevenuedecreasesbutthemarketshare remainsthesame.Eventhoughtheregretseemstobehigh butinthesubsequentparagraphitwill beshownthatthe firmisbetteroffdefaulting.Regretisthirdhighestwhenthe firm A doesn’t default and cooperates and the revenue & marketshareremainsthesameandleastwhenfirmAgets more Revenue and market share. Similarly, for B highest

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when its revenue decreases along with the market share. ThissituationhappenswhenBdoesn’tdefaultandcooperate butAdefaultsanddoesn’tcooperate.Theregretissecond highestwhentherevenuedecreasesbutthemarketshare remainsthesame.Eventhoughtheregretseemstobehigh butinthesubsequentparagraphitwill beshownthatthe firmisbetteroffdefaulting.Regretisthirdhighestwhenthe firm B doesn’t default and cooperates, and the revenue & marketshareremainsthesameandleastwhenfirmBgets more Revenue and market share. The least regret value comes when the individual gets the maximum benefit as comparedtotheotherfirm.Andhowdoesthathappen?It happensviabetrayingtheotherfirm.Butthestrikingthing tonoticehereistheordinalvalueorphilosophicalvalueof thisregretwhichisafeeling,issameforboththefirms.Ifwe cancompoundthisaffectwiththeotherfirm’sbenefitwhich is way higher than that of the firm that cooperated, the regretfeelinggetsexponentiallycompounded.Thisfeelingof regretistoomuchtobecomprehendedbyjustwritinghere or discussing it. This option of betraying to each other is sameandavailabletoboth.Ifforonceafirmgetsbetrayed anditgetsthesameoptionagain,thatfirmwillneveragain cooperateandwilldefaultalways.Thelossofrevenueistoo much, companies and go bankrupt, share prices might plummet,theircustomersmightleavethem,andmultitude of other things might happen. There will always be a probability that the other firm might default. This chance, howeverlowitmightbewillalwayscreateasenseofregret whichwillbehigherthantheprobabilityoftheotherfirm not defaulting. Let’s say that the chance that Firm A will defaultis0.3.so,thechancethatthefirmAwillcooperateis 0.7. And this information is available to Firm B. The probabilityoffirmAdefaultingisenoughtoinduceasense ofregretinthefirmB.Andthisregretwillbemorethan0.7 wheninducedinthefirmB.ThisishowthefirmBshould thinkanddefault.

Goingbyabovelogicandexplanationitseemsrationalfor firmstobetray.Andboththefirmswillthinkalike,because theinformationavailabletobothofthemissame.Thereis no extra information available to either of the firms. For them to behave and think alike is not a surprise. This eliminates the least regretful option. The next one is to cooperate. If both of them cooperate, they get the second leastregretvalue.Butthatisthecatch,it’sthesecondleast regret value. Once the firms agree to cooperate and are certainthattheotherwillcooperateandnotdefault,onewill trytobetrayandgettotheleastregretvalue.Butagain,the sameinformationisavailabletotheotherfirmaswelland theotherfirmwillalsothinkalikewiththesameavailable informationandtrytobetrayandensurethatitsregretvalue istheleast.Again,ifonekeepsitsendofthedealanddoesn’t defaultwhereastheotherbetraysandgettheregretvalueto the least value, the one who cooperated will never ever cooperateagainifgiventhesameoptioninthefuture.Andif the same game is repeated with different pairs of firms, cooperatingmightnotbeeasyandsimpleinthelongrunfor

limitedrepetitivegameorone-timegames.Thislogicalso entailstodefaulting.

As described above, the second-least regret option is also ruledout.Thisbringsthethirdoption,i.e.samemarketshare for both the firms and revenue might remain the same or decrease a bit because of lowering the prices of their offerings. But the revenue might also remain the same because the firms might get more sales because of lower pricesandpossiblymorerevenue.Thisoptionasitisturns outtobethemostoptimaloutcomeforboththefirms.Given allthelogicandinformation,noonecouldhavedonebetter. Because without an external agreement it’s very hard for firms to ensure that the other will keep their end of the bargain.Reducingthe price oftheirofferings ensuresthat thefirmswillretainthesamemarketshare,theircustomers will be with them, they will have a company to run. If the firmisnotloweringtheirprice,thenthereisaconstantfear of the other firm’s strategy. The dependency on someone elsemakesitevenharderandtheregretvalueshootsup.To minimizeone’sregretandmaximizetheunsaidutility,firms shoulddefault.

6. CONCLUSION

Utilitytheoryisoftencriticizedbecauseitfailstopredict actual behavior for some straightforward comparisons amongalternativeswithuncertainconsequences.Itisquite evidentandeasytounderstandwhydecisionmakersmaybe skeptical of expected utility analysis as a predictive tool whenitapparentlyfailsevenforsomesimplecomparisons. Theanalyst’sonlydefenseofthetheoryhasbentooffera careful explanationofwhycertainofthedecisionmaker’s responsesarenotalignedwiththeaxioms.Thisexplanation usuallyleavesthedecisionmakerunimpressed.Thispaper shows that some of the paradoxical behavior of decision makers is consistent with a desire to avoid post decision regret.

Regret is very important feeling and philosophically if thoughtaboutitinretrospectitcreatesafeelingthatisnot desired.Regretcomesafterthebeinghastakenadecision andhasreceivedtheoutput.Theoutputreceivedisn’tthe desiredoutcomeoftheactionperformedbythebeingand this causes to produce a feeling of regret in the being. Philosophically it impacts the being the most and all the future actions taken by the being, regret has a very importantroletoplayinit.Asseenintheaboveexamples, even though the best outcomes were something lese and ideally most desirable outcome but the fear of regret, the feeling thatthe beingsare notcomfortable with made the beingstoactinawaythatwillminimizetheirregret.Andif seen in retrospect, minimizing regret was the best way to think about it. Regret is a very powerful tool in Decision Theoryandtosomeextentispracticaltoo.Itcanchangethe course of one’s decision making approach. The paper analyzes many examples and shows how regret plays an

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importantroleindecisionmaking.Whatisone’sobjective outcomewhenregretisappliedversuswhenitisnot.Regret makes decision makers more rational and pragmatic. It makesthedecisionmakerstothinkaboutalltheoutcomes philosophically and not just objectively. Philosophy has a deep relation withdecision making.Peopleget influenced toomuchbyobjectivityoftheoutcomesandtendtoneglect rational philosophical decision theory that has a vital influence on the decision-making process and how beings feelaftertheoutcome.

Prisoner’sDilemmaonesuchgamethathasmanyoutcomes andtheoutcomeschangedependingontheotherperson’s choice.Justthinkaboutthebeautyofthegame–eventhough thedecisiononemakesisindependent,buttheoutcomehas aninfluenceofthedecisionmadebytheotherperson.It’sa beautiful game. Anyone involved in the game wants his outcometobemaximized.Buttomaximizeone’soutcomeit hastobeinsyncwiththeotherperson’sdecision.Itgivesa sense of collaboration and communication. And again, the game is beautiful because even after collaboration and communicationit’sseenthatpeoplearen’tabletomaximize theiroutcome.Cooperationwithoutaccountabilityisabig askfromanybeing.Andit’sverypracticalasshowninthe paper.Thetwoexamplesdiscussedinthepapershowshow theprisonersandthefirmswillperformwithandwithout collaboration depending upon how the other prisoner or firm performs. Whenever the prisoner would think to cooperate the very immediate thought would be what if the other prisoner doesn’t cooperate and defaults? The entireprisontimeononeselfisalongtimethattheprisoner has to go through. More than this, the thought of other prisonerdefaulting,willmakeprisonerA’sthoughtprocess to lean towards defaulting. And when analyzing all the optionsavailable,outcomesofeachoption,itturnsoutthat defaultingisbetterthananythingelse.Thepaperprovesthe pointthatboththeprisonersarebetterofiftheydefault.It’s seenthatwithoutanyexternalagreementit’sveryhardfor anyoneofthemtotrusttheotherandnotdefault.Theprice topayfornotdefaultingistoohigh.Thepaperanalyzesa secondexamplewheretwofirmscompetingwitheachother inamarketforhighermarketshareandrevenue.Thefirms cancontinuetopricethesameandretaineverythingasitis. Butthat’sveryrisky.Becausetechnologiesevolve,market evolve,customersevolve,andanyonecanaccess amarket fromanywhere.Anynewfirmcanenteranynewmarketand disrupt it. All these are feasible examples. And the regret involvedwithanyofthesehappeningandthefirmlossesout becauseitwastoobusyplayingthepartofcooperationwill be too high. Not to forget the objective loss will be higher too.Withthisexample,thepaperpickeduptheprisoner’s dilemmagameandappliedtoareal-lifescenario.It’sclear thatcooperatingeventhoughcanbebetterbutinthelong run, firms were better off by defaulting or in this case reducingthepriceoftheirproductsand/orofferings.

Hence, the paper successfully analyzed and presented the points to showcase how defaulting is better in Prisoner’s Dilemma.

REFERENCES

[1] https://en.wikipedia.org/wiki/Prisoner%27s_dilemma.

[2] https://plato.stanford.edu/entries/prisoner-dilemma/

[3] http://www.gametheorystrategies.com/2011/06/01/tr agedy-of-the-commons/

[4] https://1000wordphilosophy.com/2014/04/24/theprisoners-dilemma/

[5] https://www.investopedia.com/articles/investing/110 513/utilizing-prisoners-dilemma-business-andeconomy.asp

[6] https://www.open.edu/openlearn/history-thearts/culture/philosophy/the-prisoners-dilemma-detail

[7] https://pileusblog.wordpress.com/2010/04/27/theethics-of-the-prisoners-dilemma/

[8] http://psychology.iresearchnet.com/socialpsychology/decision-making/prisoners-dilemma/

[9] https://www.philosophytalk.org/blog/logic-regret