Signaling Math

It has been two years since I posted a summary of how signaling works; recent discussions suggest maybe I should try again.  Be warned; this time I'll use more math.

Consider authors who must choose a level of emotion e for their writing, given their propaganda factor p, which says how much they care about persuading readers, relative to informing readers.  Assume that authors prefer to be perceived by readers has having low propaganda p, and that everyone knows that m is the maximum possible value of p.  

Assume authors maximize a utility,

U(e;p) = -0.5(e-p)2 – E[p|e] ,

where E[p|e] is the reader estimate of author propaganda p, after having observed author emotion e.  The first term says emotion is more useful for propaganda authors, but the second term says using more emotion may tip off readers to such author intentions.

If readers already knew author propaganda p, signaling would not be an issue, and authors would just choose e = p.   However, if readers do not fully know author propaganda factor p, then if readers always make exactly the rational inference from observing emotion levels e, and if authors always exactly maximize their utility given this reader behavior (and if we use standard game theory refinements), then the equilibrium satisfies

p = e + 1 – exp(e-m) .

The worst possible author with p = m chooses e = p, just as if signaling were not an issue.  But all other authors choose e < p, asymptotically approaching e = p – 1.  The choice of emotion e fully reveals propaganda p, and everyone but the worst possible type p = m uses less emotion than they would if signaling were not an issue.

So when I suggest that the reason engineers, lawyers, accountants, and academics try to avoid emotion is that they want to be believed by skeptical readers, it is not enough to argue that someone concerned only with imparting as much info as possible to trusting readers would use lots more emotion than such authors do.  The whole point is that reader trust cannot be assumed.

(Note the equilibrium above applies for any info readers have about author propoganda p, as long as their posterior given that info has support up to m.)

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  • http://elder-gods.org/~larry/ Larry D’Anna

    That was *not* math.

  • http://profile.typekey.com/halfinney/ Hal Finney

    Looks like math to me! Actually I’m getting kind of bogged down in the math. What happens to a person who has p = 0, someone who is just interested in informing? He chooses e = -1? What does that mean?

    And what are the units here, how can the number 1 have absolute meaning? Is there some implicit natural unit for p and e? This may be related to the addition in the utility function of a function of e^2 and a function of e, again seeming to violate unit conventions.

  • John Maxwell

    Can anyone recommend any good books on game theory that will help me understand the math behind the second half of this blog post?

  • John Maxwell

    @Hal Finney:

    If you look at Robin’s utility function, both terms are negative. This function tells you how much an author with a propaganda factor p has to *lose* by playing any given emotional level e.

    If readers know an author’s propaganda level to begin with, E[p|e] will be constant. In this special case the author will always let e=p, so that the first term will do no damage to his utility. If everyone has heard me speak a million times and knows that I am mildly biased, I will mildly play emotions up every time. There’s no incentive for me to pretend objectivity, because my bias is known.

    The unit’s aren’t terribly important. I can make some up if you want. Let’s say that each of p and e range from 0 to 1. A value of 1 for p means that all the author is concerned about is converting his audience to his perspective: he tells them whatever he thinks will bring them around. Then m = 1. A value of 1 for e indicates that the author is choosing to go for his audience’s emotions with the best of his ability.

  • http://elder-gods.org/~larry/ Larry D’Anna

    Hal: That’s the point. I looks like math. One might even go so far as to say Robin is attempting to *signal* rigor. But it isn’t real math. It’s just some blatantly ridiculous assumptions with a silly little calculation based on them.

  • kebko

    “It’s just some blatantly ridiculous assumptions with a silly little calculation based on them.”
    Maybe Robin’s trying to get on Obama’s economics team. Is there a multiplier effect in here anywhere?

  • http://peco.wordpress.com peco

    How does Robin get from the first equation to the second one? I don’t know enough game theory to see how he does it.

  • http://macroethics.blogspot.com nazgulnarsil

    Consider authors who must choose a level of emotion e for their writing, given their propaganda factor p

    wait, I could have become a professor of economics saying stuff like this?

    BRB, 72 PhD’s

  • http://www.hagiograffiti.com Manuel Mörtelmaier

    @Larry:

    You have indeed made a very sharp observation here. In fact, even if equations are consistent with ZFC, with all the symbols solidly defined in the main text, they can still just “[]..look like math..[]”, and not be “[]..real math.”.

    That’s if they don’t have Equalia. Then they’re just Zombie Equations!

    Unfortunately there’s no way to ever find out about that; you’d have to be the equation in order to know…

  • Felix

    I have to assume that *everyone* knows that m is the maximum of p? Does that mean that everyone has read your definition that m is the maximum of p, or does it mean something, well, meaningful? And what does it mean that e=p? Levels of emotion and levels of propaganda are calculated in the same units?

    I’m confused by this math. I don’t think the question is whether this is math or not, but whether it’s useful or useless math.

  • http://profile.typepad.com/6p010537043dce970c/ Wei Dai

    Peco, I’ve forgotten most of the game theory math I’d learned, so I don’t know how to solve this game, but it’s not hard to verify Robin’s solution. First suppose that the reader uses p = e + 1 – exp(e-m) to determine what to believe after seeing e. Then the author has to maximize U(e;p) = -0.5(e-p)^2 – E[p|e] = -0.5(e-p)^2 – e + 1 – exp(e-m). The derivative of U(e;p) with respect to e is p + exp(e-m) – e – 1, so solving dU(e;p)/de=0 for e gives e = p – 1 + exp(e-m) as an optimum. Then, given what the author does, the reader’s supposed belief after seeing e is clearly rational.

    John Maxwell, my class used Game Theory by Fudenberg and Tirole, which I liked. But that was many years ago, so I don’t know if it’s still the best text.

    Robin, you should warn the reader (of this blog) not to put too much faith into intuitions derived from this math. You only presented the one-shot game, whereas in a realistic situation the game will be a repeated one. For example, in a repeated game, it seems unlikely that an author would fully reveal his propaganda factor p in the first round, and therefore the reader’s prior probability distribution about the author’s type should make a bigger difference than in this one-shot game.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Wei’s calculation verifying the equilibrium is correct.

    Wei, if we repeat the game but between every repetition there is even a small chance that propaganda p has changed (up to max m), we get this same equilibrium repeated each time.

    Felix, yes, everyone knows m is the max, which means p can’t be larger than m.

    John is right, the units don’t matter. I could have instead used U(e;p) = -0.5(e-a*p)^2 – b*E[p|e] , and got a slightly more complex equilibrium equation, but I was going for max simplicity here.

  • http://elder-gods.org/~larry Larry D’anna

    Manuel: Heh.

  • http://profile.typekey.com/halfinney/ Hal Finney

    I figured out more about how the math works. We are looking for a formula that gives e as a function of p for writers to use, and the inverse formula for p as a function of e, for readers to use. The second one is more convenient to solve for. When readers see emotion level e, they infer propaganda intention p from this formula p(e). That will then be the same as the second term E[p|e] in the utility function: U = -0.5*(e-p)^2 – p(e). To maximize this we take the derivative with respect to e and set it to zero:

    0 = (p-e) – dp/de; or dp/de = p-e. (This BTW is why the 0.5 was used, it cancels the exponent 2 when we take the derivative and gives us a nice simple form.)

    Google is amazing. I googled for “dy/dx = y – x” and found this page: Solve First Order Differential Equations which tells exactly how to solve it. You get Robin’s solution, although there are a couple of constants of integration k1 and k2 which are introduced, and the general solution is:

    p = e + 1 + k2 * exp(e – k1)

    You can verify that this satisfies the differential equation. We also want that as e ranges from [0,m] that p will stay within that range (assuming m is also the max for p) and this leads to the choices Robin made, k1=m and k2=-1.

    As for my question of what do you do if p=0, I think ultimately the answer is that you can’t signal it. Even if you write purely without emotion, e=0, people will still assume that you have an agenda. And that is consistent with our real-life experience.

    At the same time I think readers know that when they see unemotional writing, that is consistent with p anywhere from 0 to 1 (ignoring the small exponential term). So they really don’t know the writer’s intentions, and their prior assumptions come into play. Therefore they don’t infer p=1 from e=0, they infer a probability distribution on [0,1] and the average p value will probably be somewhat lower. So the model slightly breaks down for very unemotional writing, but I imagine that this is just a small effect.

  • http://profile.typepad.com/6p010537043dce970c/ Wei Dai

    Wei, if we repeat the game but between every repetition there is even a small chance that propaganda p has changed (up to max m), we get this same equilibrium repeated each time.

    Robin, this makes no sense, and you should be asking yourself what is going on. Why would the reader let a single observation override everything else he knows about the author? Clearly readers do not behave like this in real life, and authors do manage to establish reputations which they make use of in later writings. Perhaps you’re getting this outcome because you’re making the assumption that e can be communicated with infinite precision and noise-free, both of which are unrealistic. What happens if you make e discrete and add a random noise factor?

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Wei, adding a small noise to seeing e only changes the equilibrium a small amount. On realism, why must you tell your lover each day that you love her; even if you yell at her instead, shouldn’t she have figured our your love from all the other day’s she’s known you? Why must a businessman show up in a suit everyday; even if he shows up in a swimsuit one day won’t everyone know he’s still serious from all his previous suit-wearing days?

    Hal, silly me, I just worked it out by hand. Btw, I wasn’t assuming any particular lower bound for p or e.

  • http://profile.typepad.com/6p010537043dce970c/ Wei Dai

    Wei, adding a small noise to seeing e only changes the equilibrium a small amount.

    What happens in the repeated game, when the chance that p has changed is also small? Even in the one-shot game, this makes sense only if “small noise” is defined relative to the amount of uncertainty in the reader’s prior belief about the author.

    On realism, why must you tell your lover each day that you love her; even if you yell at her instead, shouldn’t she have figured our your love from all the other day’s she’s known you? Why must a businessman show up in a suit everyday; even if he shows up in a swimsuit one day won’t everyone know he’s still serious from all his previous suit-wearing days?

    Point taken, but I note that in some relationships the lovers do not have to reaffirm their love every day, and some businessmen manage to get away with not wearing suits every day. There’s an interesting interplay between the amount of noise in the signal, the amount of prior information you have about the signaler, and in a repeated game, the probability that the signaler’s type has changed in each round, which is missing in your model. I think Eliezer’s point was that he has established enough authorial reputation to get away with using emotion once a while, and your model doesn’t really address that.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Wei, even with a small chance of large changes, small errors in observing e won’t change much. Large observing errors, or large choice errors, can make more difference.

  • http://profile.typekey.com/halfinney/ Hal Finney

    Trying to pull back a bit from the math tar pit, what about that first term, -0.5(e-p)^2? This says that, ignoring signaling effects, a writer will be best served by matching his emotion level to his propaganda level. Robin justifies this by saying, “emotion is more useful for propaganda authors”. That is no doubt true, but is it the end? Isn’t it arguably the case that emotion is useful for objective authors too? I took that to be Eliezer’s point, that stories and similar emotional touches help to communicate, whether your goal is to persuade or to inform.

    I tried modifying Robin’s model slightly, to have this term peak at e = p+1. So (neglecting signaling) even people who are pure informers with p=0 would use e=1, a degree of emotion in getting their point across. And then more propaganda oriented writers would use still more emotion. So the equation becomes U = -0.5(e-p-1)^2 – E[p|e]. I actually solved this the long way, but it turns out the solution is a trivial modification of Robin’s, basically just moving the axes: p = e – exp(e-m). It’s the same as Robin’s but without the +1. And neglecting the exponential term as we focus around p=0, we find that the equation is simply p = e.

    At first I thought, a-ha, no signaling! But then I realized I was wrong, and in fact IMO this is an even better demonstration of Robin’s point. Pure informers would like to write with e=1, to make their points easier to understand. But signaling considerations force them to write with e=0, even knowing that their work will be less readable. The result is technical writing as dry as dust, as we see today.

  • http://profile.typepad.com/6p010537043dce970c/ Wei Dai

    Robin, I had read “small noise” as small chance of (possibly large) noise, instead of zero chance of large noise, which I see now is what you actually meant. As you say, large noise should make more difference.

  • Grant

    Can some of the critics explain why Robin’s math is so terrible? Yes its very simplified, but its based on a very simplified model which, as I understand it, was not intended to be realistic.

  • Douglas Knight

    Grant,
    What’s good about it? RH doesn’t indicate why he bothered to write this post instead of just linking to old one. That was a greatly superior exposition of signaling, though this one may teach it other ways.

    One thing that is valuable about this example, as a supplement to the previous one, is that the prior only enters through its maximum; the outcome of the previous model depends on the mix of productive and unproductive workers. Wei Dai and I (on another thread) were surprised at that. Models are particularly valuable for demonstrating that phenomena are possible.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Wei, I did not mean a zero chance of large noise.

    Douglas, that previous example equilibrium also depends only on the support of the distribution of types. That is standard for separating equilibria.

  • Douglas Knight

    It’s true that the separating equilibrium does not depend on the distribution of types, but which states are equilibria does depend. When the proportion of productive workers is high, no schooling is also an equilibrium.

  • http://profile.typekey.com/paultopia/ Paul Gowder

    The first term in the utility function seems to suggest that emotion is costly for the author? Why should that be?

  • Cyan

    Paul, the first term says that a mismatch between the level of written emotion and the propaganda factor encoding the author’s desire to persuade is costly.

  • http://profile.typekey.com/paultopia/ Paul Gowder

    Oh. Hah, some how my brain edited the exponent out. Doh.

  • Brian Macker

    Wow, more pseudo-math coming out of economics.

  • http://profile.typekey.com/halfinney/ Hal Finney

    I think what Brian and other critics are implicitly suggesting is that having equations is a sign of competence and rigor; hence, it may be expected that authors will try to use equations to signal these qualities, even when they are not as fully present as the use of equations would suggest. Just as the math in this article suggests that writers will use less emotion than they would like to, for the same reason writers will use more equations than they would like to. And just as readers complain about the dryness of technical writing, so readers will complain about excessive use of equations. Therefore it could be argued that by complaining about the equations, critics are actually validating the argument presented in the equations!

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Hal, yes, one must use math excessively to credibly signal competence and rigor. But the complaints are not yet articulate enough to determine their source. It is not clear to me whether my point could have been made effectively with less math, and since it seemed I should give a fuller explanation if signaling at some point, this seemed a good opportunity.

  • Cyan

    But the complaints are not yet articulate enough to determine their source.

    Can this math-esque model be tested experimentally? How could one measure e or p in the wild? How can reducing inherently massively multidimensional quantities to two scalars be justified? Is there any reason to expect that such a reduction would yield an accurate description of what goes on in human brains to even first order?

    I’m not quite as skeptical as Brian Macker, but I can’t uncritically adopt this math as an accurate representation for what actually goes on in the real world. Suggestive, sure — conclusive, no. (Grant thinks it’s not intended to be realistic, but I don’t see anything in the post that would prompt that inference.)

  • http://profile.typepad.com/6p010537043dce970c/ Wei Dai

    Hal wrote: And just as readers complain about the dryness of technical writing, so readers will complain about excessive use of equations.

    I think people are complaining that there is too little, not too much, math. Robin’s post gave the impression of blind reliance on an oversimplified model, which people are understandably wary of, given the recent news stories about how oversimplified risk modeling was a major cause of the current economic crisis.

    Robin wrote: Wei, I did not mean a zero chance of large noise.

    Robin, in that case I’m not sure what you mean. You can clear this up by giving us the model you have in mind, with noise and repetition, and its equilibrium solutions.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Cyan and Wei, your complaints seem to be against the very idea of economics modeling; every model is “oversimplified” in the sense of neglecting relevant details. I don’t know how I gave you the impression that I have “blind reliance” on my model as an “accurate representation.” Perhaps you wanted a generic disclaimer, of the sort that could go on any model, that it may not exactly correspond to reality?

  • http://profile.typepad.com/6p010537043dce970c/ Wei Dai

    Robin, your model has to compete with other tools that the reader has for understanding the world, including his or her own highly evolved native social intelligence. By “oversimplified” I meant that the initial model lacks enough relevant details that it seemed unlikely to perform well in practice relative to those other tools. If you were presenting the model as a starting point for further research or as a pedagogical tool, that would be one thing, but you were apparently applying it to a real world situation, and giving people (Eliezer) advice based on it.

    As for economics modeling in general, I am more skeptical of it than I used to be. It’s a lot of fun to do in the armchair, but can be pretty dangerous in the real world, where it’s easy to miss a relevant detail with serious consequences, or have one’s model misapplied by others in inappropriate settings. Generic disclaimer of course don’t do much good (although given human nature I think they’re still better than nothing). More useful would be specific disclaimers about what the model assumes, guidelines on when it is likely or not likely to give sensible results, and rationales behind any technical modeling choices such as the specific form of utility function and whether variables are continuous or discrete (e.g. whether there is reason to believe that the model is insensitive to these choices).

  • Cyan

    Robin, my questions aren’t rhetorical — if your model has the value you seem to ascribe to it, then there should be good answers for each one. (E.g., PCA can be a sound way of arguing that massively multidimensional quantities can be summarized by a small set of scalars.) I’d be happy to have my queries answered because it would mean I could place some trust in your model to help me understand the world. As it stands, I can’t.

    every model is “oversimplified” in the sense of neglecting relevant details.

    The question is whether it’s too simplified to be of any practical use.

    I don’t know how I gave you the impression that I have “blind reliance” on my model as an “accurate representation.”

    You feel it’s accurate enough to support the point you’re making in the penultimate paragraph. I’m not so sure.

  • http://profile.typekey.com/robinhanson/ Robin Hanson

    Wei, it seems to me that you are just saying that your intuition disagrees with my model. Duly noted, but not exactly a stinging criticism.

    Cyan, this wouldn’t be much harder to test that most social science signaling hypotheses. But since you don’t seem to know much social science, I don’t see the point in outlining to you how I’d go about that if I had the funding and time to do so.

    I continue to get this sort of flak when I post on social science; most commenters here seem to consider friendly AI theory more well established than social science.

  • Cyan

    The point isn’t so much to tell me the answers — it’s that until someone does the study, the model is an hypothesis, and gives no particular support to the reality of any conclusions one might want to draw from it.

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