Matthijs van Veelen

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.

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.

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.

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.

An impossibility theorem concerning multilateral international comparison of volumes

This paper considers the problem how to make real income comparisons between more than two countries or across more than two years. First, four basic requirements are formulated that we think a sensible way of comparing should satisfy. The Apples and Oranges theorem then shows that no method for making multilateral comparisons could ever have those four minimal properties. Going from bilateral to multilateral real income indices will therefore always come at a cost; at least one of these very modest requirements will have to be violated. Since the problem under consideration can be seen as a dual one, this impossibility theorem has serious consequences. After all, a method to compare price levels internationally implies a method to compare real income levels, and the same goes for its intertemporal equivalent; if we could find a consistent way of computing inflation, we would also have a consistent measure of real income growth. The theorem therefore implies that there is a limit to the meaning of both international price level and international income level comparisons as well as to the significance of reported inflation rates and growth rates. In all these cases, we should be aware which method is used and which requirement(s) that particular method fails to satisfy.

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.

Robustness against indirect invasions

Games that have no evolutionarily stable strategy may very well have neutrally stable ones. (Neutrally stable strategies are also known as weakly evolutionarily stable strategies.) Such neutrally, but not evolutionarily stable strategies can however still be relatively stable or unstable, depending on whether or not the neutral mutants it allows for – which by definition do not have a selective advantage themselves – can open doors for other mutants that do have a selective advantage. This paper defines robustness against indirect invasions in order to be able to discern between those two very different situations. Being robust against indirect invasions turns out to be equivalent to being an element of a minimal ES set, where this minimal ES set is the set that consists of this strategy and its (indirect) neutral mutants. This is useful, because we know that ES sets are asymptotically stable in the replicator dynamics.

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.

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.

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

Fitness is the central concept in evolutionary theory. It measures a phenotypes 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.