Natural selection favors those genes that are statistically associated with higher individual fitness (Fisher 1930; Price 1970). Genes can be associated with higher individual fitness in two ways: either by directly enhancing the reproductive success of their bearer (direct fitness) or by enhancing the reproductive success of other individuals who carry copies of the same genes (indirect fitness; Fisher 1930; Hamilton 1964). Hamilton (1964, 1970, 1975, 1996) argued that natural selection would lead organisms to appear designed to maximize their inclusive fitness, which is the sum of direct and indirect fitness (Grafen 2006a).
There are two basic mechanisms by which indirect fitness effects may arise (Hamilton 1964, 1975). First, individuals could aid genealogical kin who, owing to coancestry, are expected to share genes in common. This is the most common route by which indirect benefits could accrue, and so the indirect consequences are often termed “kin selection” (Maynard Smith 1964). Second, individuals may provide benefits to those who share a particular gene in common, irrespective of their genealogical relationship. Dawkins (1976) illustrated this mechanism by imagining a gene that causes its bearer to grow a green beard and to behave altruistically toward other individuals with green beards; hence, such genes have become known as “greenbeards.” However, the fundamental requirement is an assortment mechanism that directs nepotistic behavior toward other carriers of the gene, rather than a phenotypic marker per se (Hamilton 1975). Furthermore, the greenbeard effect can operate by harming nonbearded individuals as well as by helping bearded individuals (Gardner and West 2004, 2010), and it can occur even when the beard and behavior phenotypes are encoded by separate genes, provided these are in linkage disequilibrium (Haig 1997; Gardner and West 2010).
There has been disagreement over whether greenbeards are “outlaws” that conflict with other genes in the individual's genome. Intragenomic conflict occurs when different loci in the same individual are selected in different directions (Burt and Trivers 2006). Many authors have suggested that greenbeards lead to intragenomic conflict because the genetic relatedness between actor and beneficiary can be much greater at the greenbeard locus than at other loci in the genome, and so altruistic helping could be favored at the greenbeard locus when it is disfavored at other loci (Alexander and Borgia 1978; Ridley and Grafen 1981; Dawkins 1982; Haig 1996; Okasha 2002; Helantera and Bargum 2007; West et al. 2007b; Bourke 2011). However, Ridley and Grafen (1981) showed that a modifier gene, arising at another locus and fully suppressing the greenbeard phenotype, is only favored by selection in those contexts that disfavor the original greenbeard gene itself. This is because, although full suppression of the greenbeard phenotype saves “modified” individuals the cost of the social behavior, it also prevents them from benefiting from the behavior of fellow greenbeards.
The generality of this argument that greenbeards do not lead to intragenomic conflict remains unclear. Ridley and Grafen (1981) noted that a modifier gene that suppresses the behavior but retains the beard would always be favored, suggesting scope for intragenomic conflict (see Okasha 2002; Helantera and Bargum 2007). However, Gardner and West (2010) argued that this does not lead to conflict, given that a “falsebeard” allele, arising at the greenbeard locus and having exactly the same effect as the partial suppressor, would also always be favored (hence selection acts identically at the greenbeard and modifier loci). These issues remain obscure while framed verbally, especially in light of additional complications raised by real-world examples of greenbeard genes. For example, different individuals may express the beard and the behavior (i.e., class structure; see Keller and Ross 1998), and there may be genealogical relationship (kinship) between social partners (Ridley and Grafen 1981; Haig 1996; Rice et al. 2008, 2009; Gardner and West 2010).
Here, we formally analyze if, when, and why greenbeards cause intragenomic conflict. Our first aim is to formally model Gardner and West's (2010) verbal argument that a modifier arising elsewhere in the genome, fully or partially suppressing the greenbeard phenotype, will be favored or disfavored in accordance with a gene that arises at the greenbeard locus and gives rise to an identical phenotype. Specifically, we determine whether there is a mathematical analogy (isomorphism) between: (1) a two-locus model involving a greenbeard locus and a modifier locus, and (2) a one-locus model of greenbeard, nonbeard, and falsebeard alleles. Our second aim is to examine the impact of class structure and genealogical relationship between social partners. Our analyses also allow us to formally classify greenbeard social behavior and to clarify conceptual links with other evolutionary phenomena, including sexual antagonism and reciprocity.