From 707aa2e2ca738fc1a4bd2a970d44cc84950a046f Mon Sep 17 00:00:00 2001 From: "Zack M. Davis" Date: Tue, 3 Mar 2026 15:16:15 -0800 Subject: [PATCH] edits to "Trust Requires ..." draft --- trust_requires_separating_equilibria.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/trust_requires_separating_equilibria.md b/trust_requires_separating_equilibria.md index d95a403..3ce07d4 100644 --- a/trust_requires_separating_equilibria.md +++ b/trust_requires_separating_equilibria.md @@ -1,28 +1,28 @@ # Trust Requires Separating Equilibria -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: +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](https://www.lesswrong.com/posts/vfjptEJ2oahLqRyZz/justice-cherryl): > It was his sudden, angry "so you don't trust me?" snapped in answer to her first, innocent questions that made her realize she did not—when the doubt had not yet formed in her mind and she had fully expected that the answers would reassure her. She had learned, in the slums of her childhood, that honest people were never touchy about the matter of being trusted. 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 honest and trustworthy; the other needs to _know_ that you're honest and 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. +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](https://www.lesswrong.com/posts/ybG3WWLdxeTTL3Gpd/communication-requires-common-interests-or-differential). Economists call the kind of situation in which this is possible a "separating equilibrium". +A solution, if there is one, has to take the form of [saying something that a dishonest person couldn't just as easily say](https://www.lesswrong.com/posts/ybG3WWLdxeTTL3Gpd/communication-requires-common-interests-or-differential). Economists call the kind of situation in which this is possible 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 people with sick cats are in more need of cat insurance). +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. Owners of sick cats expect to face a greater fraction of possible worlds in which they file a claim than owners of 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 the owners of sick cats than healthy cats. That is to say, 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 owners of healthy cats pay a lower premium for a lower-benefit policy and the owners of sick cats pay a higher premium for a higher-benefit policy. The choice of policy becomes a signal that distinguishes what kind of cat the owner has; neither type has an incentive to send the signal of the other. That's our separating equilibrium. +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. -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 owners of healthy cats quantitatively more like the owners of 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. +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; neither type has an incentive to send the signal of the other. That's our separating equilibrium. -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. (As it turns out, there are 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.) +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. -[TODO RESEARCH: um, except the point of the separating equilibrium is that it's _not_ true that everyone would prefer the insurance company think their cat is healthy?! What is the correct explanation of the asymmetry that makes sick dishonest and healthy honest?] +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 an actuarially fair policy with higher benefits, but the insurance company can't sell it to them, because then the sick-cat owners would buy it, too. -[TODO RESEARCH: so, um what does the math say about which equilibrium we get? I'm expecting an answer of the form, when the healthy can send a signal by accepting the lower-benefit-per-premium policy, but when is that possible? The textbook was not overwhelmingly clear on what parameters that depends on.] +(As it turns out, there are 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.) ---- -- 2.53.0