For a given population of potential trustees--actors or objects in whom others may seek to place trust--a positive reputation system is a formalized or institutionalized procedure or process by which a particular positive reputation is acquired or lost. Positive reputation systems are common in modern society. Examples include awarding of certifications, awards, credentials, and positive reviews. This study mathematically derives effects of two general characteristics of such systems--how easy it is to get a reputation and how effective the reputation is at discriminating between cooperators and noncooperators--in the context of a third factor, the proportion of cooperators in the population. Some findings are as follows. The gain in confidence from a potential trustee having a reputation is a U-shaped function of reputation ease, with maximum depending on proportion of cooperators in the population. For potential trustees with a positive reputation, and trustors seeking and able to make deals with a limited number of trustees with positive reputations, the reputation is worth more the harder it is to get. However, when reputation effectiveness is moderate to high, the worth of a potential trustee with a positive reputation compared to the worth of a potential trustee without one becomes maximal when reputation is easy. The study also suggests a way to model continuous positive reputation systems--ones in which positive reputations are acquired gradually, as through amassing references.
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