MModdelliing andd RReciiprocall AAllttruiism TThhee  uussee  ooff  ggaammee  tthheeoorreettiicc   simulation as a way to model  rreecciipprrooccaall  aallttrruuiissmm AA fformall ddeffiiniittiion off a moddell An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of loggical or qquantitative relationshipps between them. Models in this sense are constructed to enable reasoning within an idealized logical framework about these processes and are an important component of scientiffic thheories. Iddeallizeddhhere means thhat thhe moddell may make explicit assumptions that are known to be false (or incomplete) in some detail. Such assumptions may be jjuussttiiffiieedd oonn tthhee ggrroouunnddss tthhaatt tthheeyy ssiimmpplliiffyy tthhee mmooddeell wwhhiillee, at the same time, allowing the production of acceptably accurate solutions, as is illustrated below. The Use of Models Models are useful when there is some undderstanddiing off thhe probbllem, bbut iiddeas abbout how to analyze the data are limited. Models help •define the pproblem •organize thought about the problem •understand data •test understanding •make further ppredictions Procedures for Modeling: Begin with the simplest model or one that has fewest assumptions and the fewest variables for tthhe ffollllowiing reasons: •Few assumptions allows the model to be more ggeenneerraall, oorr ccoovveerr aa ggrreeaatteerr nnuummbbeerr ooff ccaasseess (e.g., players do not have to be rational, tthheeyy ssiimmppllyy ffoollllooww rruulleess ooff tthhuummbb)).. •If a model with few variables is moderately successful then more variables can be added to make it more precise and realistic (if complex models fail it is difficult to determine which variable is problematic). TThhee MMooddeelleerr''ss DDiilleemmmmaa MMooddeellss ccaannnnoott ssiimmuullttaanneeoouussllyy mmaaxxiimmiizzee the following important qualities : precision (make accurate predictions) generality ((apply to a large number off cases)) realism ((cappture all the variables that determine the outcome) Thee Baassicc Proobbleem: How can cooperation evolve in a world of egoists unconsttraiinedd bby a centtrall autthhoriitty?? SSuchh a siittuattiion assumes that individuals are selfish or have an incentive to be selfish. Answering this question will allow us to specify the conditions under which reciprocal altruism can evolve. “TThhee pprriissoonneerr'ss ddiilleemmmmaa iiss ssiimmppllyy aann aabbssttrraacctt ffoorrmmuullaattiioonn ooff some very common and interesting situations in which what is best for each person individually leads to mutual defection, whhereas everyone woulldd hhave bbeen bbetttter offff wiitthh muttuall cooperation.” P. 9 The Evolution of Cooperation, R. Axelrod TThhe PPriisoner’’s DDiillemma The basic idea is that if you are a prisoner planning to escape with some fellow prisoners. You have the choice of bbeeiinngg ffaaiitthhffuull ttoo tthheemm aanndd bbeenneeffiittiinngg ffrroomm tthheeiirr ppllaann, oorr yyoouu can betray them and earn what may be a very considerable reward from the authorities. You also must consider that this choice will have occurred to the others, which means that you stand in danger of special retribution if you are betrayed first. This kind of pproblem can be abstracted into a little ggame and an associated payoff matrix. TThhee PPrriissoonneerr’’ss DDiilleemmmmaa PPaayyooffff MMaattrriixx Column Player Cooperate Defect R=3, R=3 S=0, T=5 Reward for Sucker's payoff Cooperate MMuuttuuaall aanndd tteemmppttaattiioonn ttoo cooperation defect Row pplaayyeer T=5, S=0 P=1, P=1 Temptation to Defect Punishment for defect and mutual defection ssuucckkeerr''ss ppaayyooffff If the only winning strategy is to defect, how can cooperation emerge? The only way in which cooperation can (cid:132) emergge is if there will be future interactions. TThhaatt iiss,, tthhee ggaammee wwiillll bbee ppllaayyeedd aaggaaiinn aanndd (cid:132)(cid:132) again. TThhee ggrreeaatteerr tthhee nnuummbbeerr ooff iitteerraattiioonnss tthhee (cid:132)(cid:132) more likely cooperation will evolve Best Strategy (more or less): TTiitt-ffor-ttatt Tit-for-tat is based on a simple algorithm: (cid:132) Coopperate on first meetingg (cid:132) Do what ever your opponent did to you in the (cid:132) last round Other strategies do well under other (cid:132) ccoonnddiittiioonnss Win, stay; loose, shift (a simple Pavlovian (cid:132) model))