How would a reputation system in a gift economy look like? Part of me thinks it would lead to a perversion of the system.

Asked differently: in a system where money does not hold any power, what would status/reputation be?

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We need only look at forums back in the 2000-2010 era. Your rep was essential to people trusting what you said. However, it did open the door to sockpuppetting. This is where bad actors make multiple accounts and reinforce each other's points. This is easily sniffed out but it's a low-risk way to pervert a forum.