Fisher's fundamental theorem of natural selection, dynamic sufficiency, and the necessity of higher moments.

The "average effects" of alleles in Fisher's Fundamental Theorem of Natural Selection are meant to capture what having an allele does to fitness. These average effects however are generally not constant, because the way they are determined implies that they may depend on the composition of the current population. This can easily be mistaken for dynamic insufficiency. In a recent paper by Smerlak and Youssef (2017), both of these issues are moreover treated as related to the question whether or not there is a way around needing all higher moments for determining the long run behavior of a dynamical system. In this note I will argue that these are three unrelated issues.

Hamilton's rule.

This paper reviews and addresses a variety of issues relating to inclusive fitness. The main question is: are there limits to the generality of inclusive fitness, and if so, what are the perimeters of the domain within which inclusive fitness works? This question is addressed using two well-known tools from evolutionary theory: the replicator dynamics, and adaptive dynamics. Both are combined with population structure. How generally Hamilton's rule applies depends on how costs and benefits are defined. We therefore consider costs and benefits following from Karlin and Matessi's (1983) "counterfactual method", and costs and benefits as defined by the "regression method" (Gardner et al., 2011). With the latter definition of costs and benefits, Hamilton's rule always indicates the direction of selection correctly, and with the former it does not. How these two definitions can meaningfully be interpreted is also discussed. We also consider cases where the qualitative claim that relatedness fosters cooperation holds, even if Hamilton's rule as a quantitative prediction does not. We furthermore find out what the relation is between Hamilton's rule and Fisher's Fundamental Theorem of Natural Selection. We also consider cancellation effects - which is the most important deepening of our understanding of when altruism is selected for. Finally we also explore the remarkable (im)possibilities for empirical testing with either definition of costs and benefits in Hamilton's rule.

Direct reciprocity in structured populations.

Reciprocity and repeated games have been at the center of attention when studying the evolution of human cooperation. Direct reciprocity is considered to be a powerful mechanism for the evolution of cooperation, and it is generally assumed that it can lead to high levels of cooperation. Here we explore an open-ended, infinite strategy space, where every strategy that can be encoded by a finite state automaton is a possible mutant. Surprisingly, we find that direct reciprocity alone does not lead to high levels of cooperation. Instead we observe perpetual oscillations between cooperation and defection, with defection being substantially more frequent than cooperation. The reason for this is that "indirect invasions" remove equilibrium strategies: every strategy has neutral mutants, which in turn can be invaded by other strategies. However, reciprocity is not the only way to promote cooperation. Another mechanism for the evolution of cooperation, which has received as much attention, is assortment because of population structure. Here we develop a theory that allows us to study the synergistic interaction between direct reciprocity and assortment. This framework is particularly well suited for understanding human interactions, which are typically repeated and occur in relatively fluid but not unstructured populations. We show that if repeated games are combined with only a small amount of assortment, then natural selection favors the behavior typically observed among humans: high levels of cooperation implemented using conditional strategies.

A simple model of group selection that cannot be analyzed with inclusive fitness.

A widespread claim in evolutionary theory is that every group selection model can be recast in terms of inclusive fitness. Although there are interesting classes of group selection models for which this is possible, we show that it is not true in general. With a simple set of group selection models, we show two distinct limitations that prevent recasting in terms of inclusive fitness. The first is a limitation across models. We show that if inclusive fitness is to always give the correct prediction, the definition of relatedness needs to change, continuously, along with changes in the parameters of the model. This results in infinitely many different definitions of relatedness - one for every parameter value - which strips relatedness of its meaning. The second limitation is across time. We show that one can find the trajectory for the group selection model by solving a partial differential equation, and that it is mathematically impossible to do this using inclusive fitness.

Evil green beards: Tag recognition can also be used to withhold cooperation in structured populations.

Natural selection works against cooperation unless a specific mechanism is at work. These mechanisms are typically studied in isolation. Here we look at the interaction between two such mechanisms: tag recognition and population structure. If cooperators can recognize each other, and only cooperate among themselves, then they can invade defectors. This is known as the green beard effect. Another mechanism is assortment caused by population structure. If interactions occur predominantly between alike individuals, then indiscriminate cooperation can evolve. Here we show that these two mechanisms interact in a non-trivial way. When assortment is low, tags lead to conventional green beard cycles with periods of tag based cooperation and periods of defection. However, if assortment is high, evil green beard cycles emerge. In those cycles, tags are not used to build up cooperation with others that share the tag, but to undermine cooperation with others that do not share the tag. High levels of assortment therefore do not lead to indiscriminate cooperation if tags are available. This shows that mechanisms that are known to promote cooperation in isolation can interact in counterintuitive ways.

Multi-player games on the cycle.

In multi-player games n individuals interact in any one encounter and derive a payoff from that interaction. We assume that individuals adopt one of two strategies, and we consider symmetric games, which means the payoff depends only on the number of players using either strategy, but not on any particular configuration of the encounter. On the cycle we assume that any string of n neighbouring players interacts. We study fixation probabilities of stochastic evolutionary dynamics. We derive analytical results on the cycle both for linear and exponential fitness for any intensity of selection, and compare those to results for the well-mixed population. As particular examples we study multi-player public goods games, stag hunt games and snowdrift games.

Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics.

It is often suggested that any group selection model can be recast in terms of inclusive fitness. A standard reference to support that claim is "'Quantitative genetics, inclusive fitness, and group selection" by Queller (1992) in the American Naturalist 139 (3), 540-558. In that paper the Price equation is used for the derivation of this claim. Instead of a general derivation, we try out a simple model. For this simple example, we find that the result does not hold. The non-equivalence of group selection and kin selection is therefore not only an important finding in itself, but also a case where the use of the Price equation leads to a claim that is not correct. If results that are arrived at with the Price equation are not correct, they can typically be repaired by adding extra assumptions, or explicitly stating implicit ones. We give examples with relatively mild and with less mild extra assumptions. We also discuss why the Price equation is often referred to as dynamically insufficient, and we try to find out what Price's theorem could be.

The replicator dynamics with n players and population structure.

The well-known replicator dynamics is usually applied to 2-player games and random matching. Here we allow for games with n players, and for population structures other than random matching. This more general application leads to a version of the replicator dynamics of which the standard 2-player, well-mixed version is a special case, and which allows us to explore the dynamic implications of population structure. The replicator dynamics also allows for a reformulation of the central theorem in Van Veelen (2009), which claims that inclusive fitness gives the correct prediction for games with generalized equal gains from switching (or, in other words, when fitness effects are additive). If we furthermore also assume that relatedness is constant during selection - which is a reasonable assumption in a setting with kin recognition - then inclusive fitness even becomes a parameter that determines the speed as well as the direction of selection. For games with unequal gains from switching, inclusive fitness can give the wrong prediction. With equal gains however, not only the sign, but also even the value of inclusive fitness becomes meaningful.

The evolution of morality and the role of commitment.

A considerable share of the literature on the evolution of human cooperation considers the question why we have not evolved to play the Nash equilibrium in prisoners' dilemmas or public goods games. In order to understand human morality and pro-social behaviour, we suggest that it would actually be more informative to investigate why we have not evolved to play the subgame perfect Nash equilibrium in sequential games, such as the ultimatum game and the trust game. The 'rationally irrational' behaviour that can evolve in such games gives a much better match with actual human behaviour, including elements of morality such as honesty, responsibility and sincerity, as well as the more hostile aspects of human nature, such as anger and vengefulness. The mechanism at work here is commitment, which does not need population structure, nor does it need interactions to be repeated. We argue that this shift in focus can not only help explain why humans have evolved to know wrong from right, but also why other animals, with similar population structures and similar rates of repetition, have not evolved similar moral sentiments. The suggestion that the evolutionary function of morality is to help us commit to otherwise irrational behaviour stems from the work of Robert Frank (American Economic Review, 77(4), 593-604, 1987; Passions within reason: The strategic role of the emotions, WW Norton, 1988), which has played a surprisingly modest role in the scientific debate to date.

It takes grouping and cooperation to get sociality.

Cooperation and grouping are regularly studied as separate traits. The evolution of sociality however requires both that individuals get together in groups and that they cooperate within them. Because the level of cooperation can influence selection for group size, and vice versa, it is worth studying how these traits coevolve. Using a generally applicable two-trait optimization approach, we provide analytical solutions for three specific models. These solutions describe how cooperative associations of non-relatives evolve, and predict how large and how cooperative they will be. The analytical solutions help understand how changes in parameter values, such as the group carrying capacity and the costs of cooperation, affect group size and the level of cooperation in equilibrium. Although the analytical model makes a few simplifying assumptions-populations are assumed to be monomorphic for grouping as well as for cooperative tendencies, and group size is assumed to be deterministic-simulations show that its predictions are matched quite closely by results for settings where these assumptions do not hold.

The group selection-inclusive fitness equivalence claim: not true and not relevant.

The debate on (cultural) group selection regularly suffers from an inclusive fitness overdose. The classical view is that all group selection is kin selection, and that Hamilton's rule works for all models. I claim that not all group selection is kin selection, and that Hamilton's rule does not always get the direction of selection right. More importantly, I will argue that the paper by Smith (2020; Cultural group selection and human cooperation: a conceptual and empirical review. Evolutionary Human Sciences, 2) shows that inclusive fitness is not particularly relevant for much of the empirical evidence relating to the question whether or not cultural group selection shaped human behaviour.

The problem with the Price equation.

In this paper, I will argue that the generality of the Price equation comes at a cost, and that is that the terms in it become meaningless. There are simple linear models that can be written in a Price equation-like form, and for those the terms in them have a meaningful interpretation. There are also models for which that is not the case, and in general, when no assumptions on the shape of the fitness function are made, and all possible models are allowed for, the regression coefficients in the Price equation do not allow for a meaningful interpretation. The failure to recognize that the Price equation, although general, only has a meaningful interpretation under restrictive assumptions, has done real damage to the field of social evolution, as will be illustrated by looking at an application of the Price equation to group selection. This article is part of the theme issue 'Fifty years of the Price equation'.

The cancellation effect at the group level.

Group selection models combine selection pressure at the individual level with selection pressure at the group level. Cooperation can be costly for individuals, but beneficial for the group, and therefore, if individuals are sufficiently much assorted, and cooperators find themselves in groups with disproportionately many other cooperators, cooperation can evolve. The existing literature on group selection generally assumes that competition between groups takes place in a well-mixed population of groups, where any group competes with any other group equally intensely. Competition between groups however might very well occur locally; groups may compete more intensely with nearby than with far-away groups. We show that if competition between groups is indeed local, then the evolution of cooperation can be hindered significantly by the fact that groups with many cooperators will mostly compete against neighboring groups that are also highly cooperative, and therefore harder to outcompete. The existing empirical method for determining how conducive a group structured population is to the evolution of cooperation also implicitly assumes global between-group competition, and therefore gives (possibly very) biased estimates.

Group selection, kin selection, altruism and cooperation: when inclusive fitness is right and when it can be wrong.

Group selection theory has a history of controversy. After a period of being in disrepute, models of group selection have regained some ground, but not without a renewed debate over their importance as a theoretical tool. In this paper I offer a simple framework for models of the evolution of altruism and cooperation that allows us to see how and to what extent both a classification with and one without group selection terminology are insightful ways of looking at the same models. Apart from this dualistic view, this paper contains a result that states that inclusive fitness correctly predicts the direction of selection for one class of models, represented by linear public goods games. Equally important is that this result has a flip side: there is a more general, but still very realistic class of models, including models with synergies, for which it is not possible to summarize their predictions on the basis of an evaluation of inclusive fitness.

Can Hamilton's rule be violated?

How generally Hamilton's rule holds is a much debated question. The answer to that question depends on how costs and benefits are defined. When using the regression method to define costs and benefits, there is no scope for violations of Hamilton's rule. We introduce a general model for assortative group compositions to show that, when using the counterfactual method for computing costs and benefits, there is room for violations. The model also shows that there are limitations to observing violations in equilibrium, as the discrepancies between Hamilton's rule and the direction of selection may imply that selection will take the population out of the region of disagreement, precluding observations of violations in equilibrium. Given what it takes to create a violation, empirical tests of Hamilton's rule, both in and out of equilibrium, require the use of statistical models that allow for identifying non-linearities in the fitness function.

No Strategy Can Win in the Repeated Prisoner's Dilemma: Linking Game Theory and Computer Simulations.

Computer simulations are regularly used for studying the evolution of strategies in repeated games. These simulations rarely pay attention to game theoretical results that can illuminate the data analysis or the questions being asked. Results from evolutionary game theory imply that for every Nash equilibrium, there are sequences of mutants that would destabilize them. If strategies are not limited to a finite set, populations move between a variety of Nash equilibria with different levels of cooperation. This instability is inescapable, regardless of how strategies are represented. We present algorithms that show that simulations do agree with the theory. This implies that cognition itself may only have limited impact on the cycling dynamics. We argue that the role of mutations or exploration is more important in determining levels of cooperation.

Interpretations arising from Wrightian and Malthusian fitness under strong frequency dependent selection.

Fitness is the central concept in evolutionary theory. It measures a phenotype's ability to survive and reproduce. There are different ways to represent this measure: Malthusian fitness and Wrightian fitness. One can go back and forth between the two, but when we characterize model properties or interpret data, it can be important to distinguish between them. Here, we discuss a recent experiment to show how the interpretation changes if an alternative definition is used.

Hamilton's missing link.

Hamilton's famous rule was presented in 1964 in a paper called "The genetical theory of social behaviour (I and II)", Journal of Theoretical Biology 7, 1-16, 17-32. The paper contains a mathematical genetical model from which the rule supposedly follows, but it does not provide a link between the paper's central result, which states that selection dynamics take the population to a state where mean inclusive fitness is maximized, and the rule, which states that selection will lead to maximization of individual inclusive fitness. This note provides a condition under which Hamilton's rule does follow from his central result.