Contact and linksPhone: +31-20-5255293 Email: C.M.vanVeelen@uva.nl Check also: the evolution and games site by Julián García and me. This site also contains a tutorial on the use of the Price equation MusicI play in the Philharmonic Funk Foundation and the Ulysses Ensemble. Index number theory
(2009)
The apples and oranges theorem for price indices
Economics Letters
103, 12-14
Link to article
The question is whether or not there is a generally meaningful way to compute price indices. I explore the duality between price and quantity indices, and present an impossibility result that is analogous to the one for quantity indices. (2008)
A note on different approaches to index number theory
American Economic Review
98, 1722-1730
Link to article
Link to web appendices
In the literature, two approaches to index numbers are distinguished: the axiomatic approach and the economic approach. In this note we discuss the way in which these two approaches differ and how that affects what the numbers mean. The difference is regularly described as one between an approach that does and an approach that does not assume that quantities arise from optimizing behaviour. We argue that a more accurate description is that the difference lies in whether or not optimizing agents, or representative consumers, are assumed to optimize the same utility function. It is exactly this distinction that sets the (different) limitations of both approaches for constructing a meaningful indicator of real income. (2007)
Multilateral indices: conflicting approaches?
Review of Income and Wealth
53, 372-378
Link to article
This short paper focusses on an apparent conflict between two results from different approaches to the problem of finding multilateral index numbers. The impossibility theorem of Van Veelen (2002) is an axiomatic result that rules out the existence of a multilateral index that satisfies four modest requirements. This also implies that no bilateral index can consistently be generalized to a multilateral setting. Adopting a revealed preference approach, Dowrick and Quiggin (1997) however construct a multilateral extention of Fisher's ideal index, which preserves a range of desirable properties. This note shows what it is that drives the divergence between those two results. It also gives implications for practical use of results from either approach. (2002)
An impossibility theorem concerning multilateral international comparison of volumes
Econometrica
70, 369-375
Link to article
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 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. Evolutionary game theory and theoretical biology
(2018)
Can Hamilton's rule be violated?
eLife
7
e41901
Link to article
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. (2018)
No strategy can win in the repeated prisoner’s dilemma: linking game theory and computer simulations.
Frontiers in Robotics and AI
5
102
Link to article
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. (2018)
Predictably angry — facial cues provide a credible signal of destructive behavior
Management Science
64
3352-3364
Link to article
Evolutionary explanations of anger as a commitment device hinge on two key assumptions. The first is that it is predictable, ex ante, whether someone will get angry when feeling that he or she has been badly treated. The second is that anger is associated with destructive behavior. We test the validity of these two assumptions. We collected photos of responders in an ultimatum game before they were informed about the game that they would be playing, and we filmed responders with webcams during play. We then showed pairs of photos consisting of one responder who rejected and one responder who accepted (a low offer) to an independent group of observers. We find that observers are better than chance at detecting who rejected the low offer; they do 10% better than random guessing would. We also find that anger at receiving a low offer is associated with rejection (2018)
Fisher’s fundamental theorem of natural selection, dynamic sufficiency, and the necessity of higher moments
Journal of Theoretical Biology
437
299-304
Link to article
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. (2017)
A general evolutionary framework for the role of intuition and deliberation in cooperation
Nature Human Behaviour
1
0152
Link to article
In the experimental and theoretical literature on social heuristics, the case has been made for dual-process cooperation. Empirical evidence is thought to be consistent with the idea that people tend to be nice before thinking twice. A recent theoretical paper moreover suggests that this is also the type of dual process one would expect from evolution. In ‘Intuition, deliberation, and the evolution of cooperation’ by Bear and Rand1, natural selection never favours agents who use deliberation to override the impulse to defect, while deliberation can be favoured if it serves to undermine cooperation in interactions without future repercussions. Here we show that this conclusion depends on a seemingly innocuous assumption about the distribution of the costs of deliberation, and that with different distributions, dual-process defectors can also evolve. Dual-process defectors intuitively defect, but use deliberation to switch to cooperation when it is in their self-interest to do so (that is, when future repercussions exist). The more general model also shows that there is a variety of strategies that combine intuition and deliberation with Bayesian learning and strategic ignorance. Our results thereby unify and generalize findings from different, seemingly unrelated parts of the literature. (2017)
Hamilton's rule
Journal of Theoretical Biology
414
176-230
Link to article
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 (2016)
In and out of equilibrium I: Evolution of strategies in repeated games with discounting
Journal of Economic Theory
161
161-189
Link to article
In the repeated prisoner's dilemma there is no strategy that is evolutionarily stable, and a profusion of neutrally stable ones. But how stable is neutrally stable? We show that in repeated games with large enough continuation probabilities, where the stage game is characterized by a conflict between individual and collective interests, there is always a neutral mutant that can drift into a population that is playing an equilibrium, and create a selective advantage for a second mutant. The existence of stepping stone paths out of any equilibrium determines the dynamics in finite populations playing the repeated prisoner's dilemma. (2014)
A simple model of group selection that cannot be analyzed with inclusive fitness
Journal of Theoretical Biology
360
279-289
Link to article
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. (2014)
Evil green beards: Tag recognition can also be used to withhold cooperation in structured populations
Journal of Theoretical Biology
360
181-186
Link to article
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. (2013) Human cooperation among kin and close associatesmay require enforcement of norms by third parties Cultural Evolution: Society, Technology, Language and Religion 45-60 Peter. J. Richerson and Morten H. Christiansen (eds.) MIT Press (2013)
Interpretations arising from Wrightian and Malthusian fitness under strong frequency dependent selection
Ecology and Evolution
3
1278-1280
Link to article
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. (2012)
Direct reciprocity in structured populations
Proceedings of the National Academy of Sciences
109
9929-9934
Link to article
Perspective in Science by Thom Sherratt and Gilbert Roberts
Christopher Jensen’s blog.
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. (2012)
Robustness against indirect invasions
Games and Economic Behavior
74
382-393
Link to article
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. (2012)
Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics
Journal of Theoretical Biology
299
64–80
Link to article
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. (2012)
Multi-player games on the cycle
Journal of Theoretical Biology
292
116-128
Link to article
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 the 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. (2011)
The replicator dynamics with n players and population structure
Journal of Theoretical Biology
276,
78-85
Link to article
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 even the value of inclusive fitness becomes meaningful. (2011)
A rule is not a rule if it changes from case to case (a reply to Marshall's comment)
Journal of Theoretical Biology
270,
189–195
Link to article
In order to circumvent the disagreement about the Price equation and focus on the issue of the predictive power of inclusive fitness for group selection models, I derive Queller's and Marshall's rule without the Price equation. Both rules however need a translation step in order to be able to link them to the group selection model in Van Veelen (2009). Queller's rule applies to games with 2 players and 2 strategies, and is general. Marshall's rule on the other hand applies only to a small subset of 3-player games. His rule is correct, but for other, similarly small subsets we would get other rules. This implies that if we want a rule that applies to all symmetric games with 3 players and 2 strategies, it will have to use a vector of dimension 2 that represents population structure. More in general: for group selection models with groups of size n, a correct and general prediction will need to use a vector of dimension n-1 that represents population structure. (2010)
It takes grouping and cooperation to get sociality
Journal of Theoretical Biology
264,
1240–1253
Link to article
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. (2009)
Group selection, kin selection, altruism and cooperation: when inclusive fitness is right and when it can be wrong
Journal of Theoretical Biology
259,
589-600
Link to article
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. (2009)
Evolution in games with a continuous action space
Economic Theory
39,
355-376
Link to article
Allowing for games with a continuous action space, we investigate how evolutionary stability, the existence of a uniform invasion barrier, local superiority and asymptotic stability relate to each other. This is done without restricting the populations of which we want to investigate the stability to monomorphic population states or to strategies with finite support. (2009) Does It Pay to be Good? Competing Evolutionary Explanations of Pro-Social Behaviour The Moral Brain. Essays on the Evolutionary and Neuroscientific Aspects of Morality 185-200 Verplaetse, J.; Schrijver, J. de; Vanneste, S.; Braeckman, J. (eds.) Springer (2007)
In Love and War; altruism, norm formation, and two different types of group selection
Journal of Theoretical Biology
249, 667-680
Link to article
We analyse simulations reported in "The co-evolution of individual behaviors and social institutions" by Bowles, Choi & Hopfensitz (2003) in the Journal of Theoretical Biology 223, 135-147 and begin with distinguishing two types of group selection models. The literature does not provide different names for them, but they are shown to be fundamentally different and have quite different empirical implications. The working of the first one depends on the answer to the question "is the probability that you also are an altruist large enough", while the other needs an affirmative answer to "are our interests enough in line". The first one therefore can also be understood as a kin selection model, while the working of the second can also be described in terms of direct benefits. The actual simulation model is a combination of the two. It is also a Markov chain, which has important implications for how the output data should be handled. (2007)
Hamilton's missing link
Journal of Theoretical Biology
246, 551-554
Link to article
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. (2006)
Why kin and group selection models may not be enough to explain human other-regarding behaviour
Journal of Theoretical Biology
242, 790-797
Link to article
Models of kin or group selection usually feature only one possible fitness transfer. The phenotypes are either to make this transfer or not to make it and for any given fitness transfer, Hamilton's rule predicts which of the two phenotypes will spread. In this article we allow for the possibility that different individuals or different generations face similar, but not necessarily identical possibilities for fitness transfers. In this setting, phenotypes are preference relations, which concisely specify behaviour for a range of possible fitness transfers (rather than being a specification for only one particular situation an animal or human can be in). For this more general setup, we find that only preference relations that are linear in fitnesses can be explained using models of kin selection and that the same applies to a large class of group selection models. This provides a new implication of hierarchical selection models that could in principle falsify them, even if relatedness - or a parameter for assortativeness - is unknown. The empirical evidence for humans suggests that hierarchical selection models alone are not enough to explain their other-regarding or altruistic behaviour. (2005)
On the use of the Price equation
Journal of Theoretical Biology
237, 412-426
Link to article
This paper distinguishes two categories of questions that the Price equation can help us answer. The two different types of questions require two different disciplines that are related, but nonetheless move in opposite directions. These disciplines are probability theory on the one hand and statistical inference on the other. In the literature on the Price equation, this distinction is not made. As a result of this, questions that require a probability model are regularly approached with statistical tools. In this paper we examine the possibilities of the Price equation for answering questions of either type. By spending extra attention on mathematical formalities, we avoid that the two disciplines get mixed up. After that, we look at some examples, both from kin selection and from group selection, that show how the inappropriate use of statistical terminology can put us on the wrong track. Statements that are 'derived' with the help of the Price equation, are therefore in many cases not the answers they seem to be. Going through the derivations in reverse can however be helpful as a guide how to build proper (probabilistic) models that do give answers. Reviews and commentaries
(2012) Review of "A cooperative species: Human reciprocity and its evolution" by Samuel Bowles and Herbert Gintis Journal of Economic Literature 50(3) 797-803 Link to article (2011) Selection for positive illusions Nature 477, 282-283 Link to article (2010) Call for a return to rigour in models Nature 467, 661 Link to article
Working papers
(2012)
In and out of equilibrium II: evolution in repeated games with discounting and complexity costs
PDF-file
We explore evolutionary dynamics for repeated games with small, but positive complexity costs. To understand the dynamics, we extend a folk theorem result by Cooper (1996) to continuation probabilities, or discount rates, smaller than 1. While this result delineates which payoffs can be supported by neutrally stable strategies, the only strategy that is evolutionarily stable, and has a uniform invasion barrier, is All D. However, with sufficiently small complexity costs, indirect invasions - but now through 'almost neutral' mutants - become an important ingredient of the dynamics. These indirect invasions include stepping stone paths out of full defection. (2002)
Altruism, fairness and evolution: the case for repeated stochastic games
PDF-file
This paper is an effort to convince the reader that using a stochastic stage game in a repeated setting - rather than a deterministic one - comes with many advantages. The first is that as a game it is more realistic to assume that payoffs in future games are uncertain. The second is that it allows for strategies that make an evolutionary approach possible, while folk theorem strategies do not allow for such an analysis. But the most important feature is that such a setting allows for equilibrium strategies that look very much like human behaviour; altruism and fairness will be shown to feature in a natural way in equilibrium. Other linksThe week before my thesis defense, Witho Oost interviewed me for the IKON radio program 'de andere wereld van zondagavond' . Around the same time Martijn van Calmthout wrote an article in the science section of the Volkskrant on my thesis and how it was received at the Vrije Universiteit. In 2004 VPRO's Noorderlicht had a summer science series called 'de Zomerdenktank'. Primatologist Frans de Waal, theologist and physicist Wim Drees and I were invited for the episode on altruism, morality and evolution. |