Anyone interested in a deeper dive into separating equilibria (including the math!) would do well to read "The Art of Strategy" by Avinash Dixit and Barry Nalebuff. I found the book to be a highly accessible introduction to a wide variety of game theoretic concepts; detailed enough to establish a solid foundation without getting quagmired in opaque technical proofs.
Counterpoint: if you know that you are not in a pooling equilibrium and a pooling equilibrium is unfeasible, it makes no sense to not fight as hard as you can. Your signal may be uncredible but what can you do? You have to send what signals you can. In fact, the smart defector will be more likely to not defend themselves, because they're more likely to be aware of the first-order countersignaling effect and try to go for a second-order play. In other words, fraudsters or honest, the heuristic of "distrust people with ineffective denials" screens out the inept and little else.
And that’s why honest people are never touchy about the matter of being trusted. They know that in order to distinguish themselves from dishonest people, they have to pay a price that’s too expensive for dishonest people to pay.
Never, really? What a miracle that no honest person fails to understand pooling equilibria!
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One of my favorite passages from Atlas Shrugged is this one, when Cherryl is beginning to have second thoughts about her marriage to James Taggart:
The logic here might be worth explaining in case it's not obvious. One might object: if you're honest (and therefore deserve to be trusted), shouldn't you be touchy about people incorrectly not trusting you? Not trusting you is a mistake that harms your interests and the other's. That's terrible! Why wouldn't you be touchy about it?
The problem is that in order to be trusted, it's not enough to be trustworthy; the other needs to know that you're trustworthy. You could try telling them, "Hey, you can trust me," but that doesn't work if a dishonest person could just as easily say the same thing.
A solution, if there is one, has to take the form of saying something that a dishonest person couldn't just as easily say.
Economists call the kind of situation in which it's possible for one type of agent to say something that a different type couldn't just as easily say a separating equilibrium. To understand separating equilibria, imagine two types of owners of spherical cats shopping for cat insurance on a frictionless plane: those with healthy cats, and those with sick cats. Insurance companies on the frictionless plane can't tell the difference between healthy and sick cats, and therefore face an adverse selection problem where the very fact that someone is willing to buy insurance implies that they're less profitable to insure in expectation (because sick cats are in more need of cat insurance).
Depending on some math that we don't have time for, there can be situations in which the adverse selection problem is solved by the different types of cat owners having incentives to buy different policies. Sick cats face a greater fraction of possible worlds in which they file a claim than healthy cats, so making those worlds worse for the policyholder by decreasing the policy benefit payout decreases the expected value of the policy more for sick cats than healthy cats.
As a corollary, the owners of healthy cats are willing to accept a smaller discount on premium to take out a smaller-benefit policy rather than a larger-benefit one. We end up in a situation where the healthy-cat owners pay a lower premium for a lower-benefit policy and the sick-cat owners pay a higher premium for a higher-benefit policy. The choice of policy becomes a signal that distinguishes the cats. There's nothing technically stopping the owners of sick cats from buying the lower-benefit policy—but it's not worth their while; they need the comprehensive coverage for their sick cat. That's our separating equilibrium.
But depending on some more math that we don't have time for, there are other situations in which we instead get a pooling equilibrium in which everyone buys the same policy. The insurance company treating everyone in a pooling equilibrium the same amounts to treating healthy cats quantitatively more like sick cats and vice versa, because the company has no way to tell the difference. All the cats are mixed together and can't be distinguished in the fog of the market.
There is a crucial asymmetry. Owners of sick cats would prefer that the insurance company falsely believe that their cats are healthy, whereas owners of healthy cats want their cats to be seen as they are. Pooling equilibria generally benefit the former at the expense of the latter. The existence of sick cats makes healthy cats shoulder the burden of more risk: the healthy-cat owners would prefer to transfer more risk to the insurance company by buying a policy with higher benefits, but the insurance company can't sell it to them, because then the sick-cat owners would buy it, too.
(As it turns out, there can be pooling equilibria that are better for the owners of healthy cats than some separating equilibria—but that's only because the cost they'd pay in reduced benefits-per-unit-premium to separate themselves from the owners of sick cats isn't worth it. They'd be even better off if cats never got sick.)
Sometimes, there's nothing you can say. Earlier in Atlas Shrugged, Ellis Wyatt confronts Dagny Taggart, Operating Vice-President of the Taggart Transcontinental railroad. A competing railroad that Wyatt's oil business depends upon has just been strangled by anti-competitive regulation, and Wyatt has come to Dagny (as "the only one who's got any brains in this rotten outfit") to deliver an "ultimatum." If Taggart Transcontinental expects to profit from the lack of competition without providing reliable transportation, Wyatt will exact revenge: "it is now in your power to destroy me; I may have to go; but if I go, I'll make sure that I take all the rest of you along with me." Dagny is pained to be so distrusted by Wyatt—the regulatory capture wasn't her doing—but knows better than to object:
The reason Dagny can't plead innocence or take offense is because, as an honest businessperson trapped in a pooling equilibrium with corrupt ones, she can see that the words of someone in her position are cheap talk that isn't credible to Wyatt. Not getting defensive about the accusation isn't much evidence—a corrupt executive with acting skills could maintain the same composure—but being touchy about it would be discrediting, because only a corrupt executive has an incentive to try to discourage the scrutiny that would separate the types. Dagny's non-defensiveness can't itself win Wyatt's trust, but it's enough to slightly soften Wyatt's aggression, because it at least clarifies that she knows that she'll be judged on whether the trains actually run.
And that's why honest people are never touchy about the matter of being trusted. They know that in order to distinguish themselves from dishonest people, they have to pay a price that's too expensive for dishonest people to pay. Sometimes the price might be very expensive (Dagny didn't deserve to be treated that way; the healthy cat owners are bearing more risk than they should have to), but it's the only way things could be; the hidden Bayesian structure of the universe is not kind. Those who try to wriggle out of paying the price should understand what signal they're sending.