Honest Signals

FIRST, have a look at this lovely intro by Carl Bergstrom - great stuff: http://octavia.zoology.washington.edu/handicap/honest_intro_01.html. Incidentally Veblen's book "Theory of the Leisure Class" (1899) can be read online, e.g. search for it on the VUW library and you'll see the link. Short list of examples where costly signaling might occur:
  • BEGGING FOR FOOD. Offspring signal hunger to parent (eg. birds: signal is chirping)
  • DETERING PREDATORS
    • prey signal escape ability to predators (eg. impala: stotting, babblers, meercats: alarm calls)
    • prey signal poison/sting/other invisible defence to predators (eg. frogs, bees)
  • INTIMIDATING RIVALS. Males signal fighting ability to each other (eg. deer: antlers, birds: food gifting)
  • ATTRACTING MATES. Males signal 'quality' (genetic or life-history) to females (eg. peacocks: tails, humans: cars, haircuts, conversation)
  • ATTRACTING VECTORS. Plants signal food to pollinators? (eg. flowering plants: colour)???
  • WEALTH AND STATUS (subsumed by the above possibilities I think, e.g. attracting mates and intimidating rivals)
    • conspicuous leisure (sports, tans)
    • conspicuous consumption (cars, haircuts, lawns)
    • higher education (Veblen: signals membership of leisure class) (Spense: signals prediction to employers).

A general model

MF: this is my attempt to get some clarity for myself about this, for kicking around with the SocialEvolutionReadingGroup

Imagine two populations: signalers and receivers. Each signaler has a hidden quality ($q$) that only they know about, and a signal level $ s = s(q)$. Receivers respond to this signal at a level $ r = r(s) $. These functions are heritable and affect the fitness of individuals, so they are subject to evolutionary pressures.

An individual signaler has fitness that depends on its $q$, its $s(q)$, and the resulting attention $r(s)$ it gets from receivers: $ w = w(q,s,r) $, but remember that $r$ is a function of $s $ so we can treat this as just $ w = w(q,s) $ if we want to. Signaling brings benefits via the attention it generates, but (i) an animal might not be able to realise these benefits fully, and (ii) signaling might also have direct costs. So let's say an individual signaler's fitness is the benefit minus the cost, and that the benefit is a combination of the potential boon from receivers $r(s) $, but weighted by the signaler's ability $u(q)$ to utilize that boon. I've written a $q$ there to indicate the ability to make the most of things could depend on ones underlying quality.

\[ w = w_0 + r(s)u(q) - c(s,q) \]

We'll assume that $ \frac{d r(s)}{d s} \geq 0 $ i.e. $\; r(s)$ increases monotonically with $s$, meaning receivers discriminate positively in favour of signalers. Also assume $ 0 < u(q) < 1$ and $ \frac{du}{dq} > 0$.

In the super-simple case, $u(q) =1 $ (ones quality doesn't affect ability to make the most of the receivers) and there are no costs, leaving $w = w_0 + r(s)$. Now imagine plotting the fitness $w(q,s)$ as a function of $q$ and $s(q)$. If $w = w_0 + r(s)$ the "default" fitness (due solely to the direct effect of receivers) is a surface that rises in the $s(q)$ direction but doesn't vary with $q$. Obviously if this was all there was to it, the best thing for a signaler to do would be to signal maximally, regardless of $q$. Of course this wouldn't be "honest" signaling, and the reactions of receivers would evolve away from this: they'd start to ignore the signal as it conveys no useful information to them. What they're interested in is $q$ (but note I'm not going into what their fitness function is - simply assuming it's such that it pays them to respond to high $q$ if they can manage to infer it).

We need a shorthand for the "optimal" signal level to adopt for a given $q$. Denote the value for which $ \frac{d w}{ds} \mid_q = 0 $ by $s^\star(q)$. Perfectly honest signaling happens when $s^\star(q) = q$. For this to be true, we need $w(q,s)$ to have a "ridge" running along the diagonal $s=q$. But more generally this ridge has to run in the right direction: $s^\star(q)$ has to rise with $q$.

When is honest signaling an ESS?

CONDITION 1: $s^\star(q)$ exists.

CONDITION 2: $ \frac{d^2 w}{ds^2} < 0 \,\,\, \forall q $. This ensures $s^\star(q)$ is a maximum, not a minimum! [This might be stronger than necessary - it might be enough to have $ \frac{d^2 w}{ds^2} \mid_{s^\star(q)} < 0 $. But whatever.] Together these two ensure there's an optimal signal level to adopt, for any given $q$.

CONDITION 3:
\[ \frac{d s^\star(q)}{dq} > 0 \]
ie. the optimal signal level goes up with $q$.

Simplifications

This is (maybe) true, but too general to be of much use. One simplification is to focus on solutions that give "perfect" honest signaling in the sense that $s^\star = q$. In addition, there seem to be two obvious simplifications that we could make in order to generate some concrete possibilities here.
  1. We could ignore $u(q)$ and focus on the direct costs of signaling, so that $ w(q,s) = w_0 + r(s) - c(s,q) $. We might anticipate (especially if we've read the literature...) that $ \frac{dc}{ds} > 0 $ (ie. signals have a cost) but $ \frac{dc}{dq} < 0 $ (that cost is lower for high-q individuals). But this alone isn't enough to give a "ridge" (see below).

    Example 1: suppose we say the benefits are linear so that $r(s)=s$. That means the cost function can be anything for which $ \frac{\partial^2 c}{\partial s^2} > 0 $ and $ \frac{\partial c}{\partial s} \mid_{s=q} \; = \; 1 $, so a candidate cost function is $c(s) = \exp(s-q) $, i.e. $w = w_0 + s - e^{s-q} $.

    Example 2: suppose we say the benefits are logarithmic, $r(s)=\log(s) + \mathrm{const}$. That means the cost function can be anything for which $ \frac{\partial^2 c}{\partial s^2} > -\, \frac{1}{s^2} $ and $ \frac{\partial c}{\partial s} \mid_{s=q} \; = \; \frac{1}{s} $, so a candidate cost function is $c(s) = \frac{s}{q} $ for example. That's a fitness of $ w = w_0 + \log(s) - \frac{s}{q} $.

  2. Alternatively we could make the direct costs simple and just dependent on the size of the signal (but not on $q$), and instead focus on the ability $u(q)$ to make the most of the responses to one's signal. In that case the true "cost" of having low-q could be an inability to make the most of the potential benefits from signaling: $ w = w_0 + r(s) u(q) - c(s) $.

    Example 3: $ w = w_0 + s e^q - e^s $. This is Example 1, times $ e^q $.

    Example 4: $ w = w_0 + \log(s) q - s $. This is Example 2, times $ q $.

Examples

Summary table of the above examples:
Example benefit cost in words
1 $s $ $ e^{s-q} $ costs are exponential in signal strength, and exponentially worse if you're of low $q$
benefits scale linearly with signal strength
2 $ \log(s) $ $ s/q $ signals are costly, and REALLY costly if you are low quality
benefits are logarithmic in signal strength: bigger and bigger signals bring less and less improvement
3 $ s e^q $ $ e^s $ cost of signaling is exponential and their impact merely linear, but that impact can be exploited by an amount that's exponential in $q$
4 $ \log(s) \; q $ $ s $ cost of signaling is linear
the direct impact of those signals is logarithmic, and is exploitable $\propto q$

On the plots below, the contours indicate fitness (ie. benefit minus cost), going from blue (low) to red (high). The black dots indicate the true maxima in the y-direction for each x, as a check.

out.png

The fourth is kind of note-worthy, in that nett fitness actually goes down with quality. That sounds weird, but remember that "quality" here refers to the value of that individual's traits / genes to the receiver. Maciek pointed out this is close to the case of hungry offspring signaling to a parent.

notes

  • It's not really a ridge!! Surface's slope is actually monotonic in $q$ for all these examples (I think).
  • Notice there are plenty of possibilities that involve costly signals, and have those costs greater for low-$q$ individuals, but which DON'T generate honest signaling. For example, with linear benefits $r(s) \propto s$ and costs $ c(q,s) = s (1-q)$, the second derivative isn't negative so there's no maximum. There can't be any honest signalling equilibrium for such a system. So the costly, preferentially high-$q$ signal story is neither necessary (since direct costs don't have to be higher for low-$q$ agents) nor sufficient (since you can have both yet still have no ridge).
  • Think about effect of noise in the q-to-s mapping, and costs of reception (requires model of receivers, which I've been avoiding). Aim to describe efficacy of cheap, noisy signals versus expensive accurate ones. "Speed dating".
  • Think about one population, in which everyone is both a signaler and a receiver. I think this constrains the (joint) fitness equations a lot. But not sure how to combine the fitnesses from these two activities. Add? Multiply? Zogrify?

what about the receivers? and what is q, really?

The model has completely ignored receivers so far, simply assuming them to be paying attention to the signals and "rewarding" those with strong signals over those without.

Denote fitness of the j-th receiver by (capital) $W_j$.

Noted above: $q_i$ is said to be a "quality" of the i-th signaler. Firstly, the word is potentially misleading in implying that more $q$ is better for the signaler. This isn't the case, e.g. hungry chicks in a nest: the "quality" here is their hunger, which is bad for chicks! My Example 4 above is like this too (in fact it's quite a good fit to the hungry chick scenario, as Maciek Wojnar pointed out [caveat: kin relationships between chicks make this case messy though]).

So the fitness $w_i$ could increase OR decrease with $q_i$, so what is it, "really" (at the gene's eye view)?

It's "the thing that receivers are trying to learn about by attending to the signal", which means it's the change in their fitness that they get by saying "Yes" rather than "No" to the i-th signaler.

So the DEFINITION of $q_i$ is going to be in terms of a gradient of $W_j$ (or difference in the strict YES/NO case). Think of a world in which things act or don't act towards each other. Denote the degree to which the j-th receiver ACTS TOWARDS the i-th signaler by $a_{i \leftarrow j} = a(q_i)$. Consider the definition
\[ q_i = \frac{\partial W_j}{\partial a_{i \leftarrow j}} \]

But ouch, doesn't that mean $q_i$ needs a $j$ dependence? THat really messes with the notion that it's a quantity bound to the signaler alone...

sketch of signal-dependent coalition formation model (doesn't fly yet)...

Suppose the k-th receiver has "alliances" with signalers, where $a_{ik}$ is strength of alliance with i-th signaler. The value of such alliances is monotonic in number of alliances:
\[ \mathrm{benefit} = f(\sum_j a_{jk} ) \]
but with each alliance comes a risk $p_{i}$ that the i-th signaler will "betray" receiver k. The prob that none of the alliances that k enters into ends up in betrayal is the product of the probs that each of them individually doesn't betray (ie. all it takes is one weak link), or
\[ \prod_j p_j^{a_{jk}} \]
So the value to receivers of their alliances is
\[ V_k \;\; = \;\; \prod_j p_j^{a_{jk}} \;\; f(\sum_{j^\prime} a_{j^\prime k} ) \]
Now, if we choose the mathematically convenient exponential for function $f$, then we get
\[ V_k \;\; = \;\; \prod_j (e p_j)^{a_{jk}} \]
and if we say that fitness is $\log V$ then
\[ W_k \;\; = \;\; \sum_j a_{jk} \log(e p_j) \]
and therefore we can derive the quantity that receivers are trying to learn about to be this:
\[ q_i \;\; = \; \log(e p_i) \]

That's negative for $p_i < \mathrm{threshold} $ (in this case it's $1/e$ but I guess that's tunable), meaning "risky" individuals are bad for you, but "loyal" ones are good for you. That's a start...

i.e. what I'm doing is defining a fitness for receivers, and using that to 'derive' what they should attend to in deciding which signalers to ally themselves with In this case I've made it a "weakest link" model, suitable for trusting scenarios where anyone can ruin it for everyone, if they want to. Here, whether they want to is a genetically controlled trait.

DOES THAT MAKE ANY SENSE as a model of coalition formation via signaling?

NO PAYOFF HERE FOR DEFECTION: ALL P WILL GO TO 1... GROUP SIZE THEN GROWS TO ENTIRE POPULATION: NO OUT-GROUP LEFT. BORING...

Super-simple version of all this obfuscation....

Maybe it's a red herring to worry so much about the details of surfaces. Maybe it's sufficient to talk about the simple game:
  low $q$ high $q$
signal $\alpha$ $\beta$
no signal $\gamma$ $\delta$

where $\alpha$ etc are the payoffs (fitnesses) from signalling (or not) in each case. For "honest" signaling to be preferred by signalers, we need just these two conditions:
  • $\gamma > \alpha$, and
  • $\beta > \delta $

And to make it pay for group members to use $q$ as a "badge of membership" (if they could only know it...), we need
  • high $q \;\; \longrightarrow $ including this signaler in the group would be *good* for those group members with clout, and
  • low $q \;\; \longrightarrow $ including this signaler in the group would be *bad* for those group members with clout

VariousGames
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