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Beating Hamilton’s Rule

Here’s a link to my blog, a post introducing my work on socially enforced nepotism.

Neoteny

Under construction

Weak One-Drop Rule (Brazil)

The United States has a one-drop rule with regard to who’s black: even one drop of African ancestry makes someone African American. But the one-drop rule is missing in Brazil. That’s the story, anyway. But in reality things are more complicated. In the United States, according to a recent Pew survey, most whites and Hispanics actually think of Barack Obama (white American mother, black African father) as mixed, although most African Americans think he’s just black. And in Brazil (at least in Bahia) I found that many people follow a weak one-drop rule. People I interviewed overwhelmingly stated that the child of a white man and a black woman, or a black man and a white woman, would belong to some mixed race category like moreno or mulato ­– so no strong one-drop rule. But on further inquiry, many people said that two mixed parents could more easily have a black child than a white child. My interviewees were more likely to say that a black and a mixed parent could have a black child than that a white and a mixed parent could have a white child. And they were more likely to say that two black parents could have a mixed child than two white parents. The consistent theme here is a weak one-drop rule: while mixed race individuals are not simply classified as black, they are treated as closer to blacks than to whites in their hereditary potential.

Why some Brazilians buy into the weak one-drop rule is unclear. Is black ancestry seen positively, as stronger than white? Is black ancestry seen negatively, as an impurity that can’t be erased? (Note that the pattern described above is not consistent with a theory of recessive and dominant traits, nor did anybody bring up this possibility.) These are topics I hope to address in future research.

PDF: looks and living kinds

Optimality Theory

Optimality Theory (OT) is a theory of grammar, originally developed in the field of phonology (speech sounds), which I’ve found useful in explaining how people categorize their kin.

One way to understand OT is to compare it with theories of optimization in economics and behavioral ecology. In these theories, people are typically faced with the problem of finding the best outcome in the face of tradeoffs and constraints, where the best outcome maximizes some quantity (utility, fitness). In OT too people are trying to find the best grammatical output in the face of constraints. But in OT constraints are tradeoffs between constraints are handled by ranking. Isaac Asimov’s Laws of Robotics work this way:

  1. A robot may not injure a human being or, through inaction, allow a human being to come to harm.
  2. A robot must obey the orders given it by human beings, except where such orders would conflict with the First Law.
  3. A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws

You follow rule the first rule if at all possible, the second rule as long as this doesn’t violate the first rule, and so on. (This is like alphabetical order, where you sort by the first letter, then by the second letter if the first letter is tied, and so on. Economists call this a lexicographic preference.) In OT, there are two types of constraints, corresponding to two imperatives of communication: give as much relevant information as possible, and keep difficult, non-prototypical forms to a minimum.

Here’s an example of how this works for English kin terms

UNDER CONSTRUCTION

 

 

Wife’s Sister

Describe Distance

Describe Affinity

Minimize Spouse’s Sibling

Wife

*!

 

 

Sister

 

*!

 

Spouse’s Sister 

 

 

*

This very simple example can be extended with more constraints and different inputs to account for why sister-in-law and grandfather are complex, multi-part words, while sister and father are simple, why English has one word for Mother’s Sister and Father’s Sister (aunt) and another one for Mother, and why siblings but not cousins are distinguished by sex.

And there is a further payoff. OT is meant to account not just for grammar in one language, but for variation of grammar across cultures. In the example above, consider what happens if we change the constraint ranking: we can get a language that classifies a Wife’s Sister as a Wife, or as a Sister. In an analysis involving more kin terms and more constraints, we may find, however, there are some Input-Output mappings that can’t be produced by any constraint ranking. These correspond to logically possible but empirically non-existent terminologies.

Finally, there may be one more payoff to the OT analysis of kinship. In my published work I consider kin terminology. But in my latest work (under review), I show that the same machinery that produces kin terminology can also generate marriage rules: dividing kin into those one should, can’t, and may marry. Some scholars have proposed the existence of a human moral grammar faculty, comparable to the faculty for linguistic grammar. My recent research suggests that linguistic grammar and moral grammar use some of the same psychological machinery. This implies that Claude Lévi-Strauss was onto something when he wrote, “linguists and sociologists do not merely apply the same methods but are studying the same thing”

PDF: human kinship grammar

The Brothers Karamazov Game

The Brothers Karamazov Game is a simple game that illustrates the concept of group nepotism, or socially enforced nepotism. Socially enforced nepotism involves multiple individuals acting jointly to help their mutual kin. It contrasts with individual nepotism where one individual acts on her own to help one or more of her kin. Most presentations of the concepts of kin selection, inclusive fitness, and Hamilton’s Rule stick with individual nepotism. But socially enforced nepotism may be especially relevant to understanding human kinship, where people often help their kin not because they like them so much, but because it’s expected of them, as a moral obligation.

 

So imagine a game (in the game theory sense) played by three brothers: Dostoevsky’s brothers Karamazov — Ivan, Alyosha, and Dmitri – let’s say*. Suppose that Dmitri often finds himself in trouble. He turns sometimes to Ivan for help, and sometimes to Alyosha. He is never in a position to repay either of his brothers. We can ask an evolutionary question: under what conditions will a gene for kin altruism spread, a gene that leads Ivan and/or Alyosha to help Dmitri, thereby raising Dmitri’s fitness, but lowering their own?

 

We consider two cases. The first case involves individual nepotism. Ivan and Alyosha decide on their own whether to help Dmitri, each acting without regard for what the other does on other occasions.

Individual Nepotism in the Brothers Karamazov Game

In this case, altruism toward kin is governed by Hamilton’s Rule: Ivan should help Dmitri as long as the cost to himself is less than ½ the benefit to Dmitri. This is because Ivan shares half his genes by recent common descent with Dmitri, so a gene for altruism sitting in Ivan’s body will help copies of itself to spread as long as Hamilton’s Rule, C  <  ½ B,  is satisfied. The same applies symmetrically to Alyosha.

 

But now consider a second case. In this case, Ivan makes an offer to Alyosha. He will give extra help to Dmitri provided Alyosha does the same.

Slide5

Socially enforced nepotism in the Brothers Karamazov Game

When we work through the math we get a different answer: a gene for conditional nepotism will spread as long as the cost to Ivan is less than (1+6p)/(2+8p) times the benefit to Dmitri, where p is the frequency of the gene in question. (Again, the same applies symmetrically to Alyosha.) When p = 0, this quantity is 1/2, when p = 1, this quantity is 7/10. In other words, a gene for extra nepotism, over and above regular old Hamiltonian nepotism, won’t have any special advantage when it’s at near-zero frequency, but if it can get to higher frequency, it will spread at the expense of a gene for individual nepotism. We can put it this way: the effective coefficient of relatedness is greater when kin work together instead of separately (between .5 and .7, versus .5). Socially enforced nepotism can result in Ivan and Alyosha both treating Dmitri as if he were closer kin than a brother. The principle applies broadly: altruism toward kin is a public good: you get more of it when you enforce collective contributions. 

PDF: group nepotism

*In Dostoevsky’s novel, Dmitri is an older half brother to Ivan and Alyosha. For our game, we make all the brothers full brothers.

 

Mathematical note

It takes a certain amount of algebra to derive the above results. However, it’s not too hard to work out the answers for the special cases of = 0 and = 1 (assuming diploid inheritance). Below I show how we can also use inclusive fitness theory to get approximately the right answer and then discuss why the inclusive fitness result is not exactly correct.

 

Suppose Ivan tries to use inclusive fitness theory to calculate how much extra help he should give to Dmitri when this is part of a package deal where Alyosha also provides extra help. His calculation goes like this: Half the time, he pays a cost C to provide a benefit B to Dmitri (who shares half his genes). The rest of the time, Alyosha (who shares half his genes) pays a cost C to provide a benefit B to Dmitri (who shares half his genes). Ivan comes out ahead in inclusive fitness terms as long as (-C + ½ B) + (- ½ C + ½ B) > 0, which implies C < 2/3 B. In other words, the effective coefficient of relatedness when Ivan and Alyosha work together is 2/3.

 

However, in deriving the inclusive fitness answer we snuck in the assumption that Alyosha shares half of Ivan’s genes by recent descent. This assumption is correct (on average) if all Ivan knows about Alyosha is that Alyosha is his brother. However, if Ivan knows that Alyosha is his brother, and that Alyosha has agreed to give extra help, then the probability of the two being identical by recent descent at the locus for altruism may no longer be ½. 

 

People who work on kin selection are aware that inclusive fitness calculations based on genealogical coefficients of relatedness don’t always give the right answers. The lesson I take from this exercise is that we do have to work out answers explicitly, but that inclusive fitness calculations may give qualitative insights even when they are not exactly correct.