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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.


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.


1 Comment

  1. […] Karamazov game, where two brothers have a chance to help a third. (See here for the math, and here for a math-lite exposition.) It goes like this: There’s a famous rule in evolutionary theory […]


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