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Maintaining Cooperation with Indirect Reciprocity in a Private Assessment Model

By Lina Sorg

Reciprocity—the practice of exchanging goods or services between parties for mutual benefit—forms the basis of human cooperation. “The term ‘reciprocity’” means that you cooperate and are helped back by either the beneficiary themselves or a third party,” Hisashi Ohtsuki of the Graduate University for Advanced Studies, SOKENDAI said. “It’s like an exchange.” Individuals who engage in direct reciprocity—which is responsible for dyadic (two-person) cooperation—decide whether to cooperate with another person based solely on their own experiences. Simply put, it follows the logic of “You scratch my back and I’ll scratch yours.” “This is quite common in a friendship,” Ohtsuki said. “You help your friend and your friend helps you back.” In contrast, indirect reciprocity follows the logic of “You scratch my back and someone will scratch yours.” This type of exchange—which comprises many individuals who consider both their own experiences and the experiences of others—explains large-scale cooperation within human society.

Richard Alexander coined the term indirect reciprocity in a 1987 book titled The Biology of Moral Systems [1]. Social status is important in this context, as the involved parties make their decisions based on information about who is “good” and “bad.” Indirect reciprocity can sustain cooperation if people in the population share the same opinion of a focal individual, but it is unclear whether the same is true when opinions are privately held. During a minisymposium presentation at the 2024 SIAM Conference on the Life Sciences, which took place last week in Portland, Ore., Ohtsuki examined the feasibility of indirect reciprocity as an evolutionarily stable cooperative strategy even when people have differing opinions of the same individual.

Ohtsuki used a toy model with three individuals—Alice, Bob, and Chris—to demonstrate the basic concept of indirect reciprocity. If Bob cooperates with Chris and Alice observes this act, she then determines that Bob is a good person and subsequently decides to help Bob. “There’s a type of repeated behavior and evaluation process going on here,” Ohtsuki said. This behavior reflects a straightforward technique called the discriminator (DISC) strategy, during which one distinguishes between a good and bad person (often based on reputation) and responds accordingly by either cooperating or defecting. 

Figure 1. Visualization of public assessment, during which a single observer forms an opinion that influences the entire public. Figure courtesy of Hisashi Ohtsuki.
The actual process of evaluating others, however, is less clear-cut. The route of assigning a reputation to someone is often called a social norm because of its relevance to moral judgment thinking. For example, if Bob cooperates with Chris and Alice knows about this interaction, the most clear-cut way to assign a reputation to Bob is for Alice to assume that he is good. But if Bob defects from Chris, Alice will think that Bob is bad because he failed to cooperate. This type of social norm—which regards those who cooperate with someone as good and those who defect as bad—is known as scoring. Though it initially seems relatively straightforward, scoring does not maintain cooperation with indirect reciprocity because it fails to justify the punishers. If Bob did not help Chris because Chris is a wrongdoer, Alice’s use of scoring concludes that Bob must be bad — even though his defection is not necessarily bad behavior. “In reality, we tend to think that Bob is still a good person because he refused to help a bad person,” Ohtsuki said.

As such, one cannot sufficiently evaluate Bob’s behavior based on his actions alone; background information is necessary to truly understand the interaction. Social norms should therefore consider not only action \(A\) (Bob’s action), but also background information \(X\) (Chris’ reputation) and \(Y\) (Bob’s reputation before the interaction). A map \(f\) hence accounts for these three sources of information in order to assign Bob with a reputation. What kind of social norm (i.e., map \(f\)) maintains cooperation as an evolutionarily stable strategy and maximizes the level of cooperation within the population?

Ohtsuki then introduced two types of models for reputation assessment: public and private assessment models. During public assessment, a single observer makes an assessment that influences the public. For instance, Alice might observe an interaction between Bob and Chris, assign Bob a new reputation, and tell everyone else about this assignment. If Bob helps Chris, Alice will decide that Bob is good and spread the word to everyone in her network (see Figure 1). During private assessment, however, everyone conducts their own evaluations; each observer independently observes the interaction and assigns individual assessment reputations.

Next, Ohtsuki explained that the effect of noise and error cause a discrepancy in model outcomes that would otherwise not exist in a perfect world. If an observer mistakenly assigns Bob with a bad reputation during public assessment, then everyone else also believes that Bob is bad. This error effectively creates a new state of consensus — a feature that works favorably in that everyone has the same opinion, which sustains cooperation. But during private assessment, the observers commit errors independently; errors accumulate over time, much disagreement arises, and consensus is easily broken. 

Figure 2. Four second-order social norms: stern judging (SJ), simple standing (SS), shunning (SH), and scoring (SC). When Bob cooperates with or defects on a good Chris (CG or DG), all four norms assign Bob with the same good or bad reputation. But when Bob cooperates with or defects on a bad Chris (CB and DB), all of the norms do not share the same consensus. Figure courtesy of Hisashi Ohtsuki.

To investigate whether cooperation can be established under private assessment, Ohtsuki assumed infinitely repeated random matching — including collective utilization of the DISC strategy, zero costs or benefits associated with defection, and the existence of an action error. He then examined four second-order social norms that account for Bob’s action \(A\) and background information \(X\) (i.e., Chris’ reputation): stern judging (SJ), simple standing (SS), shunning (SH), and the aforementioned scoring (SC). When Bob cooperates with or defects on a good Chris, all four social norms agree on a respective good or bad reputation for Bob. But when Bob cooperates with or defects on a bad Chris, it less clear whether his behavior is good or bad. In response, consensus between the social norms is split as to whether Bob deserves a good or bad rating (see Figure 2).

Ohtsuki thus generated a corresponding image matrix, ran a simulation, and studied all four social norms separately to make sense of the ratings. As was established earlier, SC solely accounts for the action and ignores background information; it is too naïve and fails to justify the punishers. Despite accounting for the background, SJ is too complex and leads to an unstoppable accumulation of errors. SH is only moderately complex but yields too many bad reputations, which threatens sustained cooperation. That leaves SS, which is moderately complex and assigns a good reputation most of the time; as such, it is the best social norm for cooperation under private assessment. “Simplicity is very much the key,” Ohtsuki said. “It is the best policy in an error-prone world to sustain cooperation.”

[1] Alexander, R. (1987). The biology of moral systems. Piscataway, NJ: Aldine Transaction.

Lina Sorg is the managing editor of SIAM News.
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