costly signalling

Thinking about basics again, and it seems like a framework for the basics of costly signalling that is slightly more general than the average textbook version might be of some value.

The basics that an agent has some piece of information that it would like to credibly communicate, and has available a set of possible actions, some of which would directly lead to a lower payoff, but especially if the piece of information were false; as long as my gain from being believed exceeds the cost if my message is true, but is less than the cost if my message is false, then I can credibly and profitably use those actions to communicate my information so that other agents will behave in a way that helps me recoup my signaling cost.[1]

There are a variety of things I might like to incorporate into this, and what I'm particularly contemplating right now is something mechanism designish: if a designer can change the set of actions available and/or their costs, which such changes will improve welfare?  I think that the most interesting thing to note that requires a moment's thought but not a deep analysis is that, while reducing the costs of signalling seems like a good idea, everything falls apart if it becomes cheap to signal the information when it's false — unless the reduction in cost fully compensates you for being unable to communicate credibly, at least.  The clearest beneficial case, then, would be one in which you can make signalling cheaper when it's true, but without reducing the cost of sending a false signal.

I might want the information to be continuous, or at least richer than binary.  In that case, you're likely to get "partially pooling equilibria", such that if the agent wants it to be believed that a parameter is large, the agent behaves with some randomness, with some overlap in behavior between situations in which the parameter is small and when it's in-between, ultimately leading observers to make a higher guess for the value of the parameter when they see a "higher value" sort of signal, but not putting full confidence in it.  The mechanism designer then is likely to face a choice in which a lower cost of signalling in general makes the signals less informative, resulting in some knock-on inefficiency that has to be weighed against the direct cost.

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[1] You could also have the cost of signalling be the same, regardless of truth, but the benefits of being believed higher when it's true; again, the sign of the net benefit should be positive if it's true and negative if it's false.

finance conventions

I've asserted at various times that finance is easy, so they have to invent strange conventions to make it hard.[1]  In his Tuesday column, Matt Levine gave an example, sort of:

The difference is that if you buy a $100 Venezuela 9.25 percent bond a day before the semiannual interest payment is due, and the price is $20, then if it trades clean you pay the seller $20 for the bond plus like $4.60 of accrued interest, while if it trades flat you just pay the seller the $20.

This is correct in some sense, but the emphasis is not what I think a person not steeped in finance conventions would find natural; the way I would put it is

The difference is that if you buy a $100 Venezuela 9.25 percent bond a day before the semiannual interest payment is due, and you want to agree to a price of like $24.60, then if it trades clean you call the price $20 with like $4.60 of accrued interest, while if it trades flat you just call the price $24.60.

The effect of "accrued interest" is to smooth out price drops; for a bond trading at par, the day before a $2 payment, you'll pay $102 (more or less), while the next day you'll pay $100 (because you aren't getting the $2 payment, the seller is), and if it trades "clean" then, by convention, you call it $100 on both days. Stock traders just accept that the day a stock goes "ex-dividend" the price drops, and I think in a day when traders are sitting in front of computers, it's more straightforward to call the price the price instead of adopting weird rules to make it seem to behave differently from how it actually does.

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[1] The hardest parts of finance, though, are law.  Conventions are second.

information and interaction

A point that I've made, but that has perhaps been better illustrated by Borges, is that extra information is less information; if you have 4MB of data, from which you need to find the 1k you want, you have, on some level, less information than if you just had the 1k.  (Maybe 12 bits less?  I don't know.)  As a related matter, if I need information from you, we may well be able to transmit it efficiently if we can go back and forth a bit than if not.  If I send one of 2ⁿ messages indicating a broad category, and you respond with one of 2^m responses to help me clarify my next request, and that request is l bits, and the final answer is k bits, then we've exchanged a total of n+m+l+k bits; if I had to send a single request, I would need to send l bits for each of the 2^m responses you might send to my initial message (plus perhaps the n bits as well); my request is 2^ml bits, which is huge. If you know I need the information, but have to send it without my request, that's 2^2mlk bits you have to send me to make sure I get what I want.

I kind of got to thinking about this in the context of the Mars rover, for which two-way communication is possible, but with latency.  If the latency doubles, to the extent that analogues for n and l are appreciable, you've basically just halved the rate of information transmission; the ability to recover from that latency by transmitting extra information on spec is basically negligible.