Cooperation through uncertainty

1 Defective kin altruism

The classic selfish-gene kin-altruism model tells me I should value people according to their genetic relatedness to me, as measured by the probability \(r\) that a random gene in them is a copy of the same gene in me. Assuming no inbreeding and a simple pedigree, \(r\) is \(1/2\) for full siblings or children, \(1/4\) for half-sibs, \(1/8\) for first cousins, and so on. “Costs” and “values” are measured in units of expected gene copies, a.k.a. fitness. I should be willing to pay a cost \(C\) in my own reproduction to give a benefit \(B\) to a relative if and only if \(rB > C\) (Hamilton 1964a, 1964b).

OK, let’s assume I am given \(B\) and \(C\) — can I get \(r\)? Given that rapid PCR tests are not available in the animal kingdom, the answer is generally no. Instead, assuming that I am solving for the merciless logic of kin selection, I need to guess \(r\) from context cues, if I am to be an optimal evolutionary actor.

I suspect that solving this inference problem too precisely is not helpful, even by evolution’s own logic. Here I will unpick it and see if my suspicions pan out.

The problem (to help or not to help) is a classic classification problem with all the annoyances that entails — false positives, false negatives, AUC, ROC curves, and so on, even for an ideal learner. Animals in the wild have it even worse than that, because they don’t get a labelled training or test set to calibrate on. In practice they use surrogates for \(r\) — for example, under phenotype matching, an animal learns a template from its family (the charmingly named “armpit effect”) — and treats others as kin to the degree they match it (Lacy and Sherman 1983).

We can analyse the cue dynamics as a signal detection problem (Finally! My MSc is relevant again!). A classic seems to be Reeve’s acceptance threshold (Reeve 1989). Given some correlate of \(r\), call it \(d\), accept anyone closer than \(d^{*}\) as kin and help them; reject the rest.

Derivation: I have noisy \(d\), so the most I can do is apply the rule in expectation: \[\text{help} \iff \mathbb{E}[r \mid d]\,B > C \iff \mathbb{E}[r \mid d] > C/B.\] then \(d^{*}\) is the cue value where \(\mathbb{E}[r\mid d]\,B\) crosses \(C\). But \(d^{*}\) is a cutoff on the cue, not on \(r\), which is why on its face it does not look like \(rB>C\).¹

Reeve runs a two-class classifier version of this. Sort the world into kin (relatedness \(r_k\)) and non-kin (\(r\approx 0\)); observing the cue gives me a posterior \(q = P(\text{kin}\mid d)\). The per-encounter payoffs, relative to doing nothing, are \(r_k B - C\) for helping a relative, \(-C\) for helping a stranger, and \(0\) for sitting on my hands, so helping pays in expectation when \(q\,r_k B - C > 0\): \[q > \frac{C}{r_k B}, \qquad\text{equivalently}\qquad \underbrace{q\,r_k}_{\mathbb{E}[r\mid d]} > \frac{C}{B}.\]

Because the distributions overlap, every choice of \(d^{*}\) buys one error with another — a permissive threshold accepts non-kin and wastes \(C\) on them, a stringent one rejects kin and forgoes \(r_kB-C\) for each one. That is, we face the standard classification trade-off between false positives and false negatives, and the optimal \(d^{*}\) is the one that maximises expected fitness.

Even in this simple setting, we need to consider base rates to choose \(d^{*}\): the prior probability that a random individual it meets is kin, and the costs, to choose the optimal threshold.

When kin are common — inside the natal nest, say — and snubbing a relative costs more than feeding a stranger, the optimal rule tends to permissive: accept almost everyone and tolerate the acceptance errors. When kin are rare and altruism is expensive, the rule turns choosy.

Fitness-maximising recognisers are necessarily imperfect. Acceptance errors naturally arise from overlapping distributions and asymmetric payoffs. Selection pushes acceptance toward the threshold that pays best, which typically leaves a calibrated amount of error on the table.

Decision theory in signal detection is something of a big deal amongst evolutionary biologists (Sherman, Reeve, and Pfennig 1997; Pfennig and Sherman 1995), and the acceptance-threshold model has since been carried well beyond kin, to e.g. host defences against brood parasites, anywhere an animal must sort a stream of individuals into accept and reject under uncertainty (Suarez et al. 2020; Scharf et al. 2020).

2 Evolution should invent a better kin detector

TBD

3 Evolution of ambiguous signalling

There is a whole mini-field in the evolution of ambiguous signalling and whether it might be an evolutionarily viable strategy (Fröhlich, Jäger, and Achimova 2025; Galeazzi and Rich 2026; Mühlenbernd 2020; O’Connor 2015; Santana 2014; Smaldino, Flamson, and McElreath 2018).

4 Incoming

Related, converse: common knowledge problems.

5 References