Tragedy of the Commons doesn't apply only to "public" goods but to shared resources. Liquidity in the trading books is a shared resource. If you prefer, you can also see it as a prisoner dilemma: if not all the whales stop cashing out, the ones who keep doing so are the ones who keep damaging the price but get the benefit of cashing out earlier. This creates a positive incentive for being selfish.
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I was just using public good as a way to say that it was open to many participants, when I think the whale situation should be more studied as more of an oligopoly. I like the prisoner dilemma approach better. However I see it just as another possibility to what CAN happen. There are many possible outcomes in economics and I think that both are equally as likely. It depends on what kind of people the actors are.