# Does It Pay To Be Nice? Game Theory and Evolution

DoesIt Pay To Be Nice? Game Theory and Evolution

Itpays to be nice. This is not with reference to the warm pleasantfeeling, but to the payment in terms of evolution by succeedinggenetically. One cannot avoid being nice. Rather, humans and otherpopulations that interact together find it necessary to co-operatewith one another. According to Professor Martin Nowak, co-operationis inevitable and it is in every aspect of life that people engagein. He links it to the evolution process. Various methods can beapplied in defining success such as book keeping and playing ofgames. This paper, through the IB Mathematics exploration andapplication of the Game theory and evolution, answers the questiondoes it pay to be nice?

Thereis a possibility of studying competition and co-operation in amathematical way. Co-operation is an important condition, aftermutation and natural selection, which is necessary for evolution totake place. As evolution intends to come up with new creativetechniques, for instance the emergence of the first cell, the aspectof co-operation plays a very critical role (Skyrms, 52). Variousscholars such as Professor Nowak and his colleagues initiated andcarried out investigation on various evolutionary games. The studyfor each game had a basis on the ways in which different populationsinteract in the real world. Their conclusions showed co-operation asthe most successful behavior from one game to another. The judgmentwas based on mathematical and evolutionary terms.

Theuse of mathematics in the description of human behavior, for instanceco-operation, seems illogical. However, it is paramount to understandit is possible as a result of the merging of two important aspects inmathematics. One of these aspects is the mathematics of evolution. Itwas first introduced by Charles Darwin in 1859 and could onlydescribe it in words (Martin & Ariel, 98). Over many yearsafterwards, this description remained that way even after revolutionson the definition. However, today things are different because thisevolutionary theory has a firm base on mathematical aspects.According to Nowak, describing the theory of evolution verbally couldnot have achieved the level of precision associated to the use ofmathematical aspect in describing it. He compares the lack ofapplication of mathematical aspects of evolution theory to failure ofphysics to apply newton’s equation in studying the solar system.

Nowakand his colleagues as well based their works and studies on the gametheory. It is an important area of mathematics founded by John VonNeumann and Oskar Morgenstern who were a mathematician and aneconomist respectively. This took place in the mid-19^{th}century and helped revolutionize economics and made it possible todescribe and therefore understand the strategies used in decisionmaking in an environment deemed to be competitive. In game theory, aperson considers a game consisting of two players with the outcomesbeing cost or benefit (Skyrms, 132). For instance, between player Aand B, the outcomes depend on the decisions of both players A and B.This phenomenon is different from the previously used one in that,the interaction between these two players is very vital and nooutcome should be considered in seclusion. According to Nowak, thisensures that, it is not an optimization problem for either player Aor B.

Takingfor instance player A and B are playing a game and choices have to bemade simultaneously. The choices are either co-operating ordefecting. The possible outcomes would therefore be: both A and Bdefect, A defects but B co-operates, B defects while A co-operates,and both player A and B co-operate. Another good example of gametheory is the prisoner’s dilemma. In this case, two prisoners aretaken into custody but held in different rooms. In the interrogationprocess, it is made clear that if they testify in betrayal of theirpartner, they would be released but their partner would be imprisonedfor 10 years. If both prisoners testify against each other, eachwould get a five year imprisonment. However, if both prisoners failto testify, each would get a six months imprisonment (Bolton et al,18). In such an instance, the best strategy for the first prisonerwould be to testify and either walk free of get five yearsimprisonment. This is because in case he fails to testify and theother prisoner does, he would have to spend his ten years in prisonand allow the other prisoner walk free. This results in aninteresting moral dilemma, whereby it would be best for bothprisoners to testify against each other and have them put in prisoneven if they were innocent.

Inthe above example of prisoner’s dilemma, it does not leave anyoption for co-operation between the prisoners if they are to act in arational manner. However, there are certain situations under whichco-operation is significantly successful. The case of prisoner’sdilemma does not display any essence of co-operation in it, notunless there are some mechanisms put in place to ensure co-operation.A simple mechanism that allows for this co-operation is what isreferred to as direct reciprocity (Rasmussen, 73). In this case, onedoes not participate in the prisoner’s dilemma once. One keepsplaying the game making a decision on whether the current outcomewould be the last after attaining some probability. A good example isin the case of dice rolling and then deciding that the outcome withan even number would be the last. In a case where it a one off game,the outcome from the first roll would be the final but it wouldresult in a defect. This will be similar for all outcomes thusresulting in defecting all the time a game is played. The principlein direct reciprocity is that the underlying strategy stipulates theoutcome depends on the past behavior of the opponent. For instance,let *a*be the probability that you co-operated last time and you will stillco-operate this time and *b*be the probability that you defected last time but you co-operatethis time. The strategy in this case I will not dwell on what you didbut a reaction to what you did. The defining parameters here are *a*and*b*.

Anexploration of the prisoner’s dilemma was carried out in the late19^{th}century by Robert Axelrod, a political scientist. He invited peopleto present strategies that would compete as players instead of thetwo players in the prisoner’s dilemma. In his theory, each playerwould represent a strategy and would play with all other players withthe winner being the one possessing highest payoff when thetournament comes to an end (Maynard, 82). In this situation, theundisputed champion was tit for tat. A player in such a tournamentwill always co-operate at the first meeting. After that a player willalways repeat the other player’s last move. Such that if in thelast game the other player co-operated, you co-operate and defect ifhe defected. This is a tit for tit strategy as Nowak (58), describesit and in this case, *a*wouldbe equated to one and*b *tozero. Still in the late 19^{th}century, Nowak and his supervisor, Karl Sigmund explored the resultsin the tournaments and from them, which led to introduction ofcritical and important ideas from evolution. Mutation was one ofthese ideas. In this case, the old tournaments applied the artificialstrategies instead of generating strategies through random mutationthrough natural selection. Nowak (91), then improvised this bybeginning with a randomized distribution of strategies by selectingvalues of *a*and*b*forevery initial strategy in a random manner. The reproductive successwould be the pay off in each round with the proposition that, themore successful a player was, the more the offspring produced toparticipate in the next game. The strategy for each offspring had abasis on that of their parents with random mutations of *a*and*b *takingplace. Nowak (95), also noted that in such a strategy, tournamentsdid not allow for occurrence of errors. He proposed that errors canoccur under two conditions: doing one thing right and accidentlyengage in another thing usually referred to as the trembling hand andremembering the past incorrectly, referred to as fuzzy mind. Fuzzymind results in an event whereby two players have differentinterpretations about the past. This is a normal phenomenon in humanbeings for two parties to have conflicts and blame each other forstarting it. Nowak (101), rectified this problem by introducing anerror possibility by assigning each player a probability that theywould make an error.

Nowaktested his mathematical evolutionary model of the repeated prisoner’sdilemma by running it on a computer and observing the results. Thefirst observation was emergence of always defect. Despite what firstplayer does, the other one will always defect. It is irrationallyobvious that the defectors emergevictorious in a game beginning witha population randomly strategized (Bolton et al, 58). This simplyimplies that if one is engaging in a random play, it is always fortheir best to defect. However, these defects did not stay at the topfor a long time. The population after a while shifted from alwaysdefect to the tit for tat strategy for winning that was put forth byAxelrod in his original tournament. More interestingly, this strategydid not stay for long. It lasted for the next few generations.Actually, what it did was to act like a propeller to the nextstrategy and disappeared. What followed was a generous tit for tatstrategy which made use of the idea of forgiveness. In this strategy,if a player co-operated in the last play, the other player wouldco-operate in the current play, implying *a*= 1. Again, if instead the player defected, the other one would stillco-operate with a given probability. The result here is that, thesecond player will always co-operate if the first one co-operated andsometimes co-operate even when the first player defected. This iswhat resulted in the strategy of forgiveness. The probability offorgiveness in this case was the probability *b*foreach strategy.

Thiscase is evidently different from Axelrod’s strategy where tit fortat reigned. The system continued with evolution process with thesociety on the other hand becoming more and more co-operative to apoint until it was dominated by co-operative players. Nowak, (105)stipulates that this kind of a society that always co-operatesresults to the invasion of always defect. Thus after few occurrencesof mutations, the process starts all over again. This creates abeautiful scenario of co-operation and defection implying thatco-operation will never fully dominate or be a fully stable strategy.This in turn results to a mathematical situation of oscillations ashumans evolve with co-operation in existence for some time, it facesdestruction, it is restored and so forth.

Playingwith regard to the direct reciprocity results in an increase inco-operation as per Nowak, (110). In this strategy, playing with acertain player for a second time depends on how they played with youin the previous games. The concern here comes in whereby you havenever played with the current play in all your previous games andthat after playing with them, you might never meet in future. In sucha case, decision is based on the way a certain player played againstthe other players in the previous games. This means that the decisionis based on one’s reputation. This results in a more complexstrategy referred to as indirect reciprocity. In this case, a numberof games are played in a particular round with players who are pairedrandomly so that one acts as a donor and the other one as therecipient. The donor makes a choice to either help the recipientthrough co-operation and thus giving them a benefit which is at acost to the donor or may chose not to help them and defect. At anypoint, the benefit is always greater than the cost. The priority ofthis strategy is not placed on the immediate payoff resulting fromeach game but on building and improving the donor’s reputation toincrease the future successes of participating in a game (Vincent,154). This is measured by the number of off springs which in turnhave a dependence on the total costs and benefits incurred in aparticular round. According to Nowak, this is helping out a personwhich should not necessarily mean one ought to expect the helpreturned by that particular person. Rather, look at the benefit ofimproving and building a good reputation.

Thestrategies applied in indirect reciprocity when playing are a socialnorm and a rule (Dugatkin, 187). The social norm helps players inmaking a decision on how to judge the reputation of the other playersand on how best to interpret their actions. The social norms may bevery easy such as “all players are good or bad” or they may behardly noticeable by making a consideration on the behavior of aplayer together with the reputation previous opponents. For instance,a player is said to be good if they helped the players said to havegood reputation but declined to help those said to have a badreputation. Most simple approaches apply the use of social norm forplayers. In case of evolution of these social norms, it results in amore mathematically challenging problem.

Themathematical definition of social norm states that the player’sreputation changes, in our eyes as we observe them play with otheropponents (Bolton et al, 89). For instance, a player’s reputationis set at a number, p, which is equated to zero according to ourassessment up until the time we observe them play. Their reputationis deemed to increase by a unit every time they play as long as wecan observe them help but decreases every time we observe that theyhave not helped. In this case, p, is an integer. A more complexsituation arises in this strategy if a player’s reputationincreases when they are observed helping or not helping players witha high reputation. The action rule stipulates how a player acts in agame. This is with reference to the probability strategy of thedirect reciprocity, on that this time a player’s decision as adonor to help a recipient will depend on how the player perceivestheir reputation and not on the previous experiences they have. Forexample, a player may decide to help a recipient with a reputationlarger than a certain set value.

Inthis strategy, co-operative strategies are viewed as those thatresult in help for the recipient even in cases where the donor has noinformation about the reputation of the recipient, say zero. In thiscase, a donor would be said to be co-operative if his or herjudgement on the reputation of the donor is placed at a value notmore than zero. This implies that the donor does not only help theplayers with a positive reputation but also the players participatingfor the first time or those he had never observed them play. Thestrategy is thus more co-operative at lower values of the recipient’sreputation. In instances where the judgment of the donor on therecipient’s reputation is greater than zero, then it would behighly unlikely for them to help a recipient with zero reputation,for instance, in the case of a player participating for the veryfirst time. In this strategy, a donor who always defects for allplayers would set the level of the recipient’s reputation atinfinity.

Afterrunning tournaments based on indirect reciprocity, Nowak, (203)discovered that after allowing for mutation and errors, thepopulation would circulate from unbending defectors to unconditionalco-operators through co-operative strategies back and forth. The mostsuccessful strategies in terms of how long they dominated in thepopulation were the ones which showed co-operation and discriminationbasing on the reputation of their opponents. The indirect reciprocityhelps explain various altruistic behaviors such as charity actions. Agood instance is the case of a university approaching a donor torequest for donation. The donor might not agree on the basis ofvarious interactions with the university but would do that to get agood reputation from the society and be seen as a charitable andvaluable member in the society. In this case, one can deduce thatgiving to others does not only make us feel good about ourselves butalso have the society and the community regard to us highly, which inturn increases our reputation and might be of great help and benefitto us in the future.

Althoughthere is vast evidence that animals as well engage in direct andindirect reciprocity, humans have advanced it to a new level, whichis attributed on the development of language. The animals’ versionof indirect reciprocity is more complex because animals are not in aposition to share experiences with that of the past behavior ofothers. Rather, they observe the behaviors directly. This explainswhy only human beings have grown to experience full blown indirectreciprocity. The co-operative force of indirect reciprocity in humanis not the only driving force for the evolution of the human languagebut also helped in the development of human’s understanding of thesocial complexity (Vincent, 145). Indirect reciprocity helped in theselection of human language and social intelligence. In as much asthe games that involve direct and indirect reciprocity exhibitco-operation, it is also evident in other various evolutionary games.In games where defecting would result in a win in a mixed population,co-operation is successful in cases where players have restriction tointeract with their neighbors. This is from the belief that neighborshelp each other.

Variousresearches indicate that the key insight of the evolutionary gametheory is based on the fact that behaviors involve interactionsbetween various organisms in a certain population and the success ofa particular organism has a dependence on how it interacts with otherorganisms. How an individual organism fits in a population cannot bedetermined solitary but it is evaluated with regard to the populationit lives in (Bourke, 52). A good example is the case of a certainspecies of the beetles. Every beetle’s fitness in a certainenvironment is determined by the extent to which it is in a positionto cater for food and thus supply of nutrients. Introducing aparticular mutation resulting in an increase in their body sizeproduces two different species of beetles. The big beetles have aproblem of fitness as they strive to divert more nutrients from whatthey eat. The beetles here compete for food and it is most likely thelarge bodied ones have more effectiveness in claiming a larger share.In this situation, there are three possible outcomes: if thecompeting beetles are of the same size, they get equal food share, ifthe large beetles compete with the small ones, they will get agreater share of food, and in the two cases, the large beetles willhave less fitness benefit of the food they get become some of it willbe used in the maintenance of their metabolism.

Supposingin the above case a beetle is paired with other beetles competing forfood during his lifetime. The overall fitness of a beetle will beequated to the average fitness basing on the experiences it has withthe various interactions with other beetles (Bourke, 59). Thisoverall fitness will in turn dictate its reproductive success, whichrefers to its strategy into the next future off springs. In this typeof setting, any strategy is evolutionary stable as long as the wholepopulation is using the strategy. Any other organism using adifferent strategy will die in the long run. Representing thismathematically, let’s say the whole population is applying astrategy *s*anda smaller population is using a different strategy *t.*Thefitness of this small population should be less than that of thelarge population. Because it is fitness that translates intoreproduction success, the evolutionary process of the smallpopulation will shrink with time and eventually die. In such a case,an organism’s fitness in a population is said to be the expectedpayoff that result from interactions with other organisms in thepopulation. We can also deduce that the strategy *t*invadesstrategy *s*ata certain level *x.*Thestrategy *s*issaid to be evolutionary stable if there exists a number, say *y,*tothe extent that whenever any strategy, *t,*attacksstrategy *s,*atall levels, level *x*isalways less than *y.*Suchthat any organism applying *s*hasmore fitness than the one applying strategy *t*(Bourke,115)*.*Taking a mathematical example, the large population uses *x*fractionwhile the small one will use the remaining 1-*x.*assumingthe small population receives a payoff of 5 with a probability of1-*x*,the large beetle will receive a payoff of 1 with probability *x*.the expected payoff will be 5(1-*x*)+ 1(*x*)– 5-4*x*

Onecan easily deduce that in the event of applying mathematics ofevolution results in co-operation. These different mechanisms: directand indirect reciprocity, spatial game, evolutionary stability, andgroup selection have an influence on how people, animals, andorganisms leading to the emergence of co-operation. Various scholars,a good example being Nowak, have this conviction that co-operation isnot just an outcome of evolution but a necessary condition forevolution to take place. It is evident that co-operation plays a verycritical role in the evolutionary process. This explains Nowak’sargument on co-operation being the third and an important pillar ofevolutionary process right after natural selection and mutation.Co-operation is not only important in the process of evolution butalso for survival. The biggest challenge being faced today is on howstability of the intelligent life on this planet can be maintained.The solution to this lies in co-operation. People of all walks oflife in the world should co-operate now and in the future if thischallenge is to be solved. Participating in evolutionary games withthe people we have past experiences with is not sufficient anymore.Every action we undertake in the current generation will have animpact in the next generations to come. Those generations will haveto pay for our actions and that is why it is paramount for us to actin ways that will not implicate the future generations.Unfortunately, this is a great challenge being faced today.

Conclusion

Theunderstanding of the mathematical game theory and evolution is veryimportant in helping human beings become good co-operators. This isbecause it will be of help and result in a global rational analysisin the circumstances under which mathematics can be helpful in theidentification of problems and thus their solutions. It is hopefulthat the evolution of the strategies that will apply clear andstraight forward mathematical arguments will soon be discovered andhelp promote this aspect of co-operation. This will not only be forthe benefit of the current generation but also the future ones. Allthese aspects revolve around being nice, and thus answering thequestion, it does pay to be nice.

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