(This is a sample cover image for this issue. The actual cover is not yet available at this time.)
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
Author's personal copy Journal of Theoretical Biology 307 (2012) 117–124
Contents lists available at SciVerse ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
On the evolution of coarse categories Friederike Mengel a,b,n a b
School of Economics, University of Nottingham, University Park Campus, Nottingham NG7 2RD, United Kingdom Department of Economics (AE1), Maastricht University, PO Box 616, 6200 MD Maastricht, The Netherlands
H I G H L I G H T S c c c
We provide an evolutionary explanation for why people use coarse categories. We do not impose any cost whatsoever on reasoning. Coarse categories evolve in a process of noisy cultural transmission.
a r t i c l e i n f o
abstract
Article history: Received 9 June 2011 Received in revised form 4 April 2012 Accepted 20 May 2012 Available online 31 May 2012
We compare the evolutionary fitness of different cultures (or populations), where we think of culture as partitioning a set of decision situations into categories of situations treated the same. Information about optimal behavior in each category is passed on via a process of noisy cultural transmission. We show that coarse partitions (distinguishing less situations) can provide higher evolutionary fitness even if there are no explicit costs to holding finer partitions. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Categorization Evolution
1. Introduction The importance of categorical reasoning for human cognition has long been recognized by scholars in psychology, neuroscience, the cognitive sciences and in other fields.1 Allport (1954) famously noted that ‘‘the human mind must think with the aid of categories. We cannot possibly avoid this.’’ and Harnad (2005) argues that categorization is the single most fundamental cognitive process. Stereotyping and prejudice are often treated as inevitable consequences of categorization. See e.g. Allport (1954), Hamilton (1981), Tajfel (1969) or the review by Fiske (1998). Other authors have argued that categorization can explain the emergence of cultural differences. If we think of institutions as creating game forms, then extrapolation across games within different sets of institutions can create those seemingly inconsistent patterns of behavior which are often referred to as cultural differences (Bednar and Page, 2007). While the empirical relevance of categorization and its important implications are widely acknowledged, much less progress has been made in understanding why it is evolutionary successful
to categorize.2 The answer seems easy if there are no costs to categorization in the sense that lumping two situations together in one category does not force a suboptimal decision in either of them (see e.g. Harnad, 2005). If, however, there are substantial costs to categorization such as a reduction in decision making quality in some problems, it is much less clear why categorization should be evolutionary successful. In this note we propose a model that explains why coarse categories can lead to higher evolutionary fitness in this case. Unlike previous studies, we abstract from reasoning costs and the like, which – one could argue – are subject to evolutionary pressures themselves.3 We think of culture as defining a partition of a set of decision problems and are interested in which culture (partition) will lead to higher evolutionary fitness. Information about how to behave in each category (or equivalence class) of situations is culturally transmitted subject to errors.4 We are interested in environments where errors are not vanishing, but substantial. To model selection in our environment we rely on the quasi-species
2
Exceptions are Samuelson (2001) or Mengel (2012) among others. See Rubinstein (1998) for an introduction to the automaton literature or Samuelson (2001), Fryer and Jackson (2008) or Mengel (2012) among others. 4 The reader can find a discussion of cultural evolution (and how it compares to genetic evolution) in the books by Cavalli-Sforza and Feldman (1981) or Boyd and Richerson (1985, 2005). See also Bisin and Verdier (2001), Henrich and GilWhite (2001) or Mengel (2008) among many others. 3
n Correspondence address: School of Economics, University of Nottingham, University Park Campus, Nottingham NG7 2RD, United Kingdom. E-mail addresses:
[email protected],
[email protected] 1 For surveys see Cohen and Lefebvre (2005) or Murphy (2002).
0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.05.016
Author's personal copy 118
F. Mengel / Journal of Theoretical Biology 307 (2012) 117–124
model from evolutionary biology. This model is often used to study the mutation selection process of genomic sequences in error-prone environments.5 The quasi-species equation coincides with the wellknown replicator equation in error-free environments, but has received less attention so far in evolutionary game theory and in models of cultural evolution compared to the latter. Cultural information, however, tends to mutate rapidly and hence it seems natural to study problems of cultural evolution in a context where errors are non-vanishing. Hence one contribution of this paper consists in applying the quasi-species equation to a problem of cultural evolution. We also show that the noisiness of cultural transmission matters, since the evolutionary selection obtained by relying on the quasi-species equation differs in many cases from the one that would be obtained via the Replicator Dynamics. We focus on the analytically more tractable cases with either two or three situations that a decision-maker has to distinguish. Via examples we also develop some intuition for why the fitness advantage of categorization increases with the number of situations the decision-maker faces. The intuition is as follows. Starting from k situations any additional situation creates at least 2k additional action profiles in the fine partition and hence possibilities for mutation. Hence, the marginal cost of relying on the fine partition increases the more decision situations there are. The marginal benefit of being able to distinguish one additional situation is lower, on the other hand, the more situations there are, because the least frequent situation (which would be the first that a coarser partition will neglect) will have a lower contribution to overall fitness if there are more situations.
2. The basic model Denote by S ¼ fs1 ,s2 g a set of two decision situations. For each decision situation s A S there is a unique optimal action, which we denote by in ðsÞ. All other actions ia in ðsÞ are suboptimal and each action is optimal for at least one situation. The following payoff function pi ðsÞ (showing the payoff obtained when choosing action i in situation s) formalizes this idea ( ) 1 if i ¼ in ðsÞ pi ðsÞ ¼ 0 else
We are interested in selection in a world where errors in replication (cultural transmission) play a serious role. Evolution in such error-prone environments can be modeled using the quasispecies equation (Eigen et al., 1989). A quasi-species is defined as the equilibrium mutant distribution that is generated by a mutation– selection process. Selection among action profiles does not act on a single mutant but on the quasi-species as a whole. The most prominent examples of such evolution in the literature are probably the human immunodeficiency viruses HIV-1 and HIV-2.7 In the limit where there are no errors in replication the quasi-species equation corresponds to the replicator equation. With positive errors, though, the frequency of a given action profile will not only depend on its replicative value but also on the likelihood with which it is produced by erroneous replication of other profiles as well as their frequencies. To avoid confusion we will usually indicate an action profile (prescribing actions for multiple situations) by (i) as opposed to a single action (a singleton profile) which we denote by i. For the coarse partition there are two (21) possible profiles and for the fine partition four (22) possible profiles. Denote by i(s) the action prescribed by action profile (i) in situation s and let eij be the probability that cultural transmission of action profile (i) results in action profile (j). For now we assume that eij ¼ e, 8ia j and eii ¼ 1ðn1Þe, where n denotes the number of possible action profiles. We also assume that e o 14. This ensures that eii 4 eij ,8i aj or in other words this ensures that each profile is more likely to be accurately replicated than to mutate into any other given profile. We will often place further restrictions on e, which we will discuss in each case below and we will discuss alternative assumptions on errors in Section 4. For any given partition K, let pðiÞ ðKÞ be the frequency of action ! profile (i) and denote by p ðKÞ ¼ ðpð1Þ ðKÞ, . . . ,pðnÞ ðKÞÞ the vector of frequencies of the different profiles in the population using partition K. Note that n¼2 for the coarse partition and n¼4 for the fine P partition. Denote by pðiÞ ðKÞ ¼ s A S f s piðsÞ ðsÞ the fitness (payoff) of action profile (i) under partition K (where fs is the frequency of ! situation s) and let p ðKÞ ¼ ðpð1Þ ðKÞ, . . . , pðnÞ ðKÞÞ be the fitness vector of action profiles. The average fitness of the population is then given ! ! b ðKÞ. by the inner product p ðKÞn p ðKÞ ¼: p The quasi-species equation is given by
pðiÞ ðKÞ ¼
n X
b ðKÞpðiÞ ðKÞ, pðjÞ ðKÞpðjÞ ðKÞeji p
8ðjÞ ¼ 1, . . . ,n
j¼1
Hence any action i chosen in situation s yields payoffs of 1 if it is optimal (i.e. if i ¼ in ðsÞ) and payoffs of 0 otherwise. A decisionmaker can either distinguish the two situations s1 and s2, i.e. use a fine partition KF ¼ ffs1 g,fs2 gg of S or not distinguish them, i.e. use the coarse partition KC ¼ fs1 ,s2 g of S. In each period decision makers face a decision problem randomly drawn from S. Decision situation s1 occurs with frequency f and consequently situation s2 occurs with frequency 1 f. Information about which action to choose in each category is culturally transmitted, but there is noise (errors) in transmission. For the fine partition an action profile consisting of one action for each situation is transmitted. For the coarse partition only one action is transmitted since the two situations are not distinguished. Our model admits an interpretation both in terms of horizontal and vertical cultural transmission. Under horizontal transmission an individual could learn (subject to errors) action profiles from others with a probability proportional to their payoff. Under vertical transmission information about action profiles could be passed on (subject to errors) from one generation to the next.6
Hence action profile (i) is obtained by replicating any profile (j) at rate pðjÞ ðKÞ times the probability that replication of (j) generates b ðKÞ to ensure that the total (i). Each profile is removed at rate p population size remains constant. In the limiting case of error free replication (eij ¼ 0, 8i aj), the quasi-species equation becomes the replicator equation. Now to solve the quasi-species equation let us combine fitness and errors to derive the mutation selection matrix WðKÞ ¼ ½wji ¼ ½pðjÞ ðKÞeji . Then we can rewrite ! ! b ðKÞ! p ðKÞ p ðKÞ ¼ p ðKÞWðKÞp with equilibrium ! b ðKÞ! p ðKÞ p ðKÞWðKÞ ¼ p Hence the average fitness of the population (culture) with partition K is the largest eigenvalue of the matrix W and the left-hand eigenvector associated with this eigenvalue (with the proper P normalization pðiÞ ðKÞ ¼ 1) provides the equilibrium structure of the quasi-species. It is important to note that there is only
5
See e.g. Nowak (2006) for a good introduction. For models of horizontal and vertical transmission see e.g. Boyd and Richerson (2005), Bisin and Verdier (2001), Henrich and Boyd (1998), Mengel (2008) or Boyd and Richerson (1995) among many others. 6
7 See e.g. Nowak et al. (1990), Fontana and Schuster (1987), Arnaout et al. (1999), Wodarz and Nowak (1999), Fontana (2002) or the summary in the textbook by Nowak (2006).
Author's personal copy F. Mengel / Journal of Theoretical Biology 307 (2012) 117–124
119
evolutionary selection of action profiles for an exogenously given partition. The fitness of a partition (or culture) K is determined by the largest eigenvalue of the matrix WðKÞ. In the following we will compare this average fitness across different partitions K. At the end of Section 4 we also discuss an example where different partitions compete within one population.
3. Basic result We want to understand under which conditions it can provide higher fitness to a population to rely on the coarse partition, i.e. not to distinguish the two decision situations. In principle one would expect the fine partition to be optimal since it allows the decision maker to choose the optimal action for each decision situation. Intuitively the advantage of using the finest partition is biggest whenever both situations occur equally often. If one situation is much more frequent than the other, using the coarse partition hurts less since – if the action which is optimal in the very frequent situation is chosen with the coarse partition – then the fine partition can improve on the coarse partition ‘less often’. Our main result states that there is an error threshold s.t. whenever errors in cultural transmission are sufficiently frequent, then the coarse partition yields higher average fitness for a population. b ðKF Þp b ðKC Þ o 0. b Result 1. There exists b e ðf Þ o 14 s.t. 8e 4 be ðf Þ : p e ðf Þ is 1 b ðKF Þp b ðKC Þ is decreasing in e, 8e o 12. decreasing in 9f 29 and p The fact that b e ðf Þ o 14 ensures that, irrespective of the partition, a cultural offspring is more likely to learn the action profile of its parent than any other profile. The intuition behind the proposition is as follows. On the one hand the coarse partition protects the decision maker in environments where there is a non-trivial chance of errors because it ensures that she gets it right in at least one of the decision situations. With the fine partition on the other hand there is a profile which ‘‘gets it right’’ all the time. This profile will be evolutionary more successful than any other profile under our assumptions, i.e. it will have the highest population share (see below). There is also a profile which gets it wrong all the time but this profile will have the lowest population share. Under the Replicator Dynamics the ‘‘best profile’’ (that ‘‘gets it right’’ all the time) would eventually be the unique profile present in the population, implying that the fine partition will always lead to higher fitness. This is not necessarily the case in an environment where errors play a serious role. If only the best profile did occasionally mutate, then the finest partition would still have the highest fitness. However, because mutations with the fine partition create an ensemble of action profiles present in the population all of which mutate, the coarse partition can have higher fitness even if errors are small. Hence the quasi-species equation conveys in a natural way the idea that cultural information represented by the fine partition is more complicated to transmit. b ðKF Þp b ðKC Þ as a function Fig. 1 illustrates the fitness difference p of e and f. It can be seen that if one of the two situations is much more important than the other (f close to zero or one), then the coarse partition usually does better than the fine partition (the fitness difference is negative). The fine partition does best when both situations are equally important (f¼0.5). In terms of errors, the fine partition does best when errors are small. (As discussed above the range where e 4 0:5 is not very intuitive.) The following example provides a numerical illustration of the intuition behind Result 1. It also illustrates that the required error rate is not very high (in the example around 10%). Example 1. Let us consider a simple example where S ¼ f0; 1g. Situation s¼0 occurs with probability f 0 ¼ 34 and has optimal
Fig. 1. Fitness difference (on the vertical axis) between fine partition and coarse partition as a function of E (on the x1-axis)and f (on the x2-axis).
action ið0Þ ¼: 0 and s ¼1 has optimal action ið1Þ ¼: 1 and f 1 ¼ 14. The fitness of action 0 in the coarse partition is pð0Þ ðKC Þ ¼ f 0 n1 þf 1 n0 ¼ 34. Similarly pð1Þ ðKC Þ ¼ 14. In the fine partition we have pð00Þ ðKF Þ ¼ 14, pð01Þ ðKF Þ ¼ 1, pð10Þ ðKF Þ ¼ 0 and pð11Þ ðKF Þ ¼ 34. We also assume that be ðf Þ o 14. This ensures that irrespective of the partition a cultural offspring is more likely to learn the action profile of its parent than any other profile. The mutation selection matrix for the coarse partition is 0 1 3ð1eÞ 3e B 4 4 C C WðKC Þ ¼ B @ e ð1eÞ A 4
4
The largest eigenvalue and hence the average fitness of a population with the coarse partition is given by 1e 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 12e þ 4e 4 2 and the left hand eigenvector for this eigenvalue is given by ðp0 ,1p0 Þ where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1e 12e þ 4e2 þ 3 e 3e rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ¼ 1e 12e þ 4e2 1þ þ 3e 3e For the population with the fine partition we have 0 1 3ð13eÞ 3e 3e 3e B 4 4 4 4 C B C B e ð13 e Þ e e C B C WðKF Þ ¼ B C B 0 0 0 0 C B C @ e e e ð13eÞ A 4 4 4 4 with the largest eigenvalue being bound above by pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 8ð514e þ 3 320e þ44e Þ Hence whenever
e4
pffiffiffi 1 7 pffiffiffi 3 þð16 213Þ1=3 1=3 16 ð16 213Þ
! 0:114
the population using the coarse partition has higher evolutionary fitness. At this value of e the frequencies of the different action profiles are approximately ðp0 ,1p0 Þ ð0:81,0:19Þ for the coarse partition and for the fine partition ðp00 ,p01 ,p10 ,p11 Þ ð0:28,0:48,0:12,0:12Þ. The example illustrates also that the result is not simply due to the fact that the error probability is so high that there is no
Author's personal copy 120
F. Mengel / Journal of Theoretical Biology 307 (2012) 117–124
adaptation anymore. In both populations (under both partitions) the action profile with the highest fitness also has the highest population share.
partition) decreases very rapidly with the number of different situations.
4. Discussion and additional results 3.1. Generalization and limits Assume next that each individual error is less likely under the fine partition, i.e. that any two given profile mutates into each other not with probability e, but rather with probability ae, where a A ½0; 1. The case a ¼ 1 is the case studied above. If a ¼ 13 the overall mutation probability under the coarse and fine partition is the same and if a A ð13,1Þ, then the overall mutation probability is higher under the fine partition, while each individual error is less likely. Hence the mutation selection matrix under the fine partition is given by 0 1 ð13aeÞf aef aef aef B C ae 13ae ae ae B C C W a ðKF Þ ¼ B B C 0 0 0 0 @ A aeð1f Þ aeð1f Þ aeð1f Þ ð13aeÞð1f Þ
Result 2. If a r 13 in W a ðKF Þ, then the population relying on the fine partition has higher fitness whenever e o 0:25. The proof of this Result can be found in the Appendix. Hence if
a is small enough, then the finest partition has higher fitness for reasonable error rates. Fig. 2 illustrates how the critical value of e (that leads to higher fitness of the coarse partition) depends on a and f. The fine partition has higher fitness the smaller a and the further f is from 12. The coarse partition has higher fitness only if the overall mutation probability for a fine profile is at least as high as that of a coarse profile ða 4 13Þ. It seems to make sense, though, that in particular very complicated profiles, such as e.g. (100010010101) are wrongly transmitted with at least somewhat higher overall probability than a simple profile such as 1. In the next section we will see that the critical error rate (ensuring that coarse categorization provides higher fitness than the finest
In this section we study first a case where errors are similaritybased in the sense that more ‘‘similar’’ profiles (that differ in less coordinates) are more likely to mutate into each other than more distant profiles. We also illustrate why categorization tends to provide higher fitness gains the more decision situations there are and we show an example with evolving partitions. 4.1. A model with non-linear (similarity-based) errors We want to see whether we can obtain a similar result in a model where the overall probability of mistakes is not linear in the number of profiles and where not all mutations are equally likely. In particular we will assume that mutation probabilities are similarity-based, i.e. that profiles that differ in less coordinates are more likely to mutate into each other. If each coordinate mutates with probability e, then the probability for sequence (00) to mutate into (01) or (10) is e but to mutate into ð11Þ the probability is e2 . (01) mutates into (00) or (11) with probability e but mutates into (10) with probability e2 only, etc. For the coarse partition this assumption makes no difference. (1) mutates into (0) (and vice versa) with probability e as above. For the fine partition the mutation selection matrix will now look as follows: 0 1 ð12ee2 Þf ef ef e2 f B C B C e 12ee2 e2 e C W sb ðKF Þ ¼ B B C 0 0 0 0 @ A e2 ð1f Þ eð1f Þ eð1f Þ ð12ee2 Þð1f Þ In order to ensure that a cultural offspring is more likely to learn the action profile of p itsffiffiffiffiffiffiparent than any other profile, we need to assume that e rð 133Þ=2. Result 2 shows that the coarse partition can still provide higher fitness in this case, while at the same time the action profile with the highest fitness has the highest population share under both partitions. sb Result 3. If errors are similarity pffiffiffiffiffiffi based as modeled in W ðKF Þ, there exist pairs ðe,f Þ, where e oð 133Þ=2 and f A ð0; 1Þ, under which a population relying on the coarse partition has higher fitness than a population relying on the fine partition.
The proof for Result 3 can be found in the Appendix. There it is also shown that the type (action profile) with the highest fitness can have the highest population share in such a case. Hence Result 1 is robust to the introduction of non-linear errors and all qualitative features are preserved. This is also illustrated by Fig. 3. Fig. 3 is the analog of Fig. 1 and depicts the fitness difference in the case of similarity based errors. The interpretation is very similar to the case of Fig. 1. If one of the two situations is much more important than the other (f close to zero or one), then the coarse partition usually does better than the fine partition (the fitness difference is negative). The fine partition does best when both situations are equally important (f¼0.5). In terms of errors, the fine partition does best when errors are small. 4.2. Three situations, three actions and similarity based errors (3S3A) Fig. 2. Critical value of E as a function of a and f (illustrative only). The case a o 0:33 is studied in Result 2, where the fine partition has higher fitness irrespective of E o 0:25. Below the dashed line the fine partition has higher fitness and above the coarse partition. The case a ¼ 1 is studied in Result 1 and the case a ¼ 1 and f ¼ 34 in the Example.
To gain some insight into how the fitness difference changes as the number of situations increases, we will look at some cases with more than two situations. We start with a model with three situations and three different optimal actions. Hence in this
Author's personal copy F. Mengel / Journal of Theoretical Biology 307 (2012) 117–124
121
a good comparison, since the critical error level encountered in Section 3 is such that selection with the finest partition will no longer favor the fittest genotype (action profile). What is the intuition for this result? Note that if there are k situations already, any additional situation creates at least 2k additional action profiles in the fine partition and hence possibilities for mutation.8 Hence, using the fine partition becomes more costly the more decision situations there are. The marginal benefit of being able to distinguish one additional situation is lower, on the other hand, the more situations there are, because typically the least frequent situation (which would be the first that a coarser partition will not distinguish) will have a lower contribution to overall fitness if there are more situations. Fig. 3. Fitness difference (on the vertical axis) with similarity based errors as a function of E (on the x1-axis) and f (on the x2-axis).
Result 4. Assume e o0:117. If e o0:05, then the finest partition always has higher fitness than the coarsest partition. Partition K1 has higher fitness than the finest partition whenever e 40:004.
Table 1 Ranking of partitions in terms of fitness, where binary relation ‘‘4 ’’ is to be read as ‘‘has strictly higher fitness than’’ and ‘‘¼ ’’ as ‘‘has the same fitness as’’.
In the following we will look at a problem where in all situations either action 0 or 1 is optimal. Clearly, in this case there is no need for any partition to have more than two categories. We will analyze this situation first using our assumption on errors outlined in Section 3 and then we will study evolutionary selection of partitions within this framework, both via examples only.
f 1 ¼ 34, f 2 ¼ f 3 ¼ 18
e ¼ 0:03 e ¼ 0:01 e ¼ 0:005 e ¼ 0:001
K1 4 K3 ¼ K2 4 KF 4 KC K1 4 K3 ¼ K2 4 KF 4 KC K1 4 KF 4 K3 ¼ K2 4 KC KF 4 K1 4 K3 ¼ K2 4 KC
model S ¼ fs1 ,s2 ,s3 g. We denote the frequency of situation si by fi and we say that action i is optimal in situation si where i ¼ 1; 2,3. With three situations there are five possible partitions. We denote the coarsest partition again by KC and the finest partition by KF . We will label the three intermediate partitions K1 , K2 , K3 according to the situation they place in a singleton set (i.e. K1 ¼ ff2; 3gf1gg). The finest partition allows for 27 ¼33 possible action profiles, the intermediate partitions for 9 ¼32 profiles and the coarsest partition allows for three possible action profiles. We will also assume that errors are similarity based, as discussed above. This will mean that e.g. profile (212) mutates into (222) with probability e, into (322) with probability e2 and into (123) with probability e3 . We will assume that e o 0:117 to ensure that a cultural offspring is more likely to learn the action profile of its parent than any other profile. To be able to compare our results to those obtained before we will assume again that f 1 ¼ 34 and f 2 ¼ f 3 ¼ 18. Under the Replicator Dynamics (e ¼ 0), the coarsest partition will have fitness 0.75, the intermediate partitions will lead to average fitness of 0.875 and the finest partition will lead to fitness of the population of 1. The reason simply is that under Replicator Dynamics the fittest profile will be selected for. Under the coarse partition this will mean ‘‘getting it right’’ in the most important situation f1, for the intermediate partitions it will mean ‘‘getting it right’’ in f1 and in one of the less important situations and the finest partition allows agents to always ‘‘get it right’’. With errors strictly bound away from zero, though, we get a different picture. Table 1 shows that for e r 0:03 the finest partition has higher fitness than the coarsest partition. However, some categorization proves to be evolutionary fit, since partition K1 leads to higher fitness than KF even for errors as low as 0.005. Now if we compare the critical error rate that ensures that some categorization yields higher fitness than relying on the finest partition, we can see that it is much lower in this case compared to the case with two actions studied above. The same will be true if we focus on the critical rate which renders the coarsest partition KC more fit than the finest partition. The latter, however, does not seem
4.3. Three situations, two actions (3S2A) Let us consider a model with three situations s1 ,s2 ,s3 , where we assume that action 0 is optimal in both situations s1 and s2, but action 1 is optimal in situation s3. Again we denote the coarsest partition again by KC and the finest partition by KF . We will label the three intermediate partitions K1 , K2 , K3 according to the situation they place in a singleton set (i.e. K1 ¼ ff2; 3gf1gg). We focus on homogeneous errors in this example. The coarsest partition allows for two possible profile, the intermediate partitions for four and the finest partition allows for eight possible profiles. Hence to ensure that irrespective of the partition a cultural ‘‘offspring’’ is more likely to learn the action profile of its ‘‘parent’’ than any other profile we need to assume e o0:125. Table 2 shows the ranking of the different partitions or populations in terms of their average fitness for three different error rates. A number of things are noteworthy about these rankings. Note that for e ¼ 0:08 the coarsest partition has higher fitness than the finest partition. This error rate is below the critical error rate encountered in Example 2, which is natural since the number of places where the fine partition can mutate has increased. This is in spite of the fact that the distribution of situations is closer to the uniform distribution now, which tends to benefit the fine partition. Result 5. (i) Under model3S2A partition K3 has higher fitness than KF irrespective of the value of e. (ii) The critical error rate ensuring that KC has higher fitness than KF is below (0.08) if f 1 ¼ 12 ,f 2 ¼ f 3 ¼ 14 and is strictly below0.04 if f 1 ¼ 34 ,f 2 ¼ f 3 ¼ 18. 4.4. Evolving partitions Next assume that e ¼ 0:08, but let us assume now that different partitions are competing within a population. For simplicity let us also assume that evolutionary selection of action 8 How many additional profiles are created depends on how many different actions there are. For two different actions 2k additional profiles are created. For k optimal actions the number increases from kk to ðk þ 1Þk þ 1 , etc.
Author's personal copy 122
F. Mengel / Journal of Theoretical Biology 307 (2012) 117–124
Table 2 Ranking of partitions in terms of fitness, where binary relation ‘‘ 4’’ is to be read as ‘‘has strictly higher fitness then’’ and ‘‘ ¼’’ as ‘‘has the same fitness as’’. f 1 ¼ 12, f 2 ¼ f 3 ¼
1 4
e ¼ 0:12 e ¼ 0:08 e ¼ 0:04 f 1 ¼ 34, f 2 ¼ f 3 ¼
KC 4 K3 4 K1 4K2 4KF K3 4 KC 4 K1 4K2 4KF K3 4 KF 4 KC 4 K1 4K2 1 8
e ¼ 0:12 e ¼ 0:08 e ¼ 0:04
KC 4 K3 4 K1 4K2 4KF KC 4 K3 4 K1 4K2 4KF K3 4 KC 4 K1 4K2 4KF
profiles is infinitely faster than that of partitions s.t. we can take the fitness of each partition as given. There are different ways to model the evolutionary competition of partitions. Under symmetric errors we would expect the population shares in the equilibrium quasi-species to reflect the ranking found in Table 2, i.e. we would expect the following ranking p3 4pC 4 p1 4 p2 4 pF , where pi is the population share of partition Ki . To make the problem more interesting we make the following two assumptions on how partitions mutate into each other:
transmitted, coarse partitions can have higher evolutionary fitness. Finally coarse partitions are favored if cultural information is more complex, i.e. if there are more decision situations. A question we have not addressed is what happens if partitions and action profiles evolve within the same population and at the same speed. As long as action profiles of different length never mutate into each other we expect results should be robust. To the best of our knowledge the idea that coarse partitions may be evolutionary more fit because they are more robust to errors in cultural transmission is new to the literature. In Samuelson (2001) and Mengel (2012) coarse partitions may evolve because there is some cost (albeit small) to holding finer partitions. Fryer and Jackson (2008) partly characterize optimal partitions in a model of categorical thinking and show some nonmonotonicity. Mohlin (2009) studies optimal categorization and shows that the optimal number of categories is determined by a trade-off between (a) decreasing the size of categories in order to enhance category homogeneity and (b) increasing the size of categories in order to enhance category sample size and hence be able to make better predictions. Also in Al Najjar (2009) coarse decision making arises to mitigate the problem of over-fitting the data.
Appendix A. Additional results
a category is split into two proper subsets (of any size) with probability Z;
A.1. Asymmetric errors
two categories are merged into one with probability Z. Those assumptions ensure that neither fine nor coarse partitions mutate more easily per se. They imply that KC and KF mutate into Ki , i ¼ 1; 2,3 with probability Z and vice versa, but KC mutates into KF only with probability Z2 and partitions Ki , i ¼ 1; 2,3 mutate into each other only with probability Z2 . Result 6. Assume e ¼ 0:08. If Z ¼ 0:1, then p3 4 pC 4pF 4p1 4p2 in the equilibrium quasi-species, but if Z ¼ 0:01, then p3 4 pC 4 p1 4p2 4pF . The Replicator Dynamics (Z ¼ 0) preserves the ranking found in Table 2, but if Z is non-vanishing then this is not necessarily the case. In particular the finest partition can have a higher population share in this case, since the partition with the highest replicative value K3 mutates more easily into KC and KF compared to partitions K1 and K2 . In spite of this result the partition with the highest average fitness after selection of action profiles will also have the highest population share after evolutionary selection of partitions (at least for the error ranges considered). 4.5. Discussion Overall our results show that in error-prone environments, where we model evolutionary selection using the quasi-species equation, populations relying on coarser partitions may have higher evolutionary fitness. Coarse partitions are favored when some situations are much more important than others, since in these cases it hurts less to bundle the less frequent situation together with the more frequent situation (and choose the optimal action corresponding to the more frequent situations). Coarser partitions are also favored when errors are higher (though they need not be linear in the number of action profiles). If errors are normalized, though, such that the overall wrong transmission of a complicated profile such as (100010010101) is not more likely than that of a simple profile such as (0), then the fine partition cannot be beaten. As soon as it is even slightly more likely though that complicated profiles are erroneously
One might also consider a model where profiles (00) and (11), being less complex, mutate with lower overall probability than profiles (01) or (10). In particular one may think that those profiles mutate with the same overall probability than profiles (0) and (1) under the coarse partition, while (01) and (10) have higher probabilities to mutate. This would lead to the following mutation–selection matrix for the fine partition 0 1 e e e f f f ð1eÞf B C 3 3 3 B C B C e 13e e e a C B W ðKF Þ ¼ B C 0 0 0 0 B C @e A e e ð1f Þ ð1f Þ ð1f Þ ð1eÞð1f Þ 3 3 3
Result A.1. If errors are asymmetric as modeled in W a ðKF Þ, the population relying on the coarse partition can have higher fitness while at the same time e o emax . Proof. The maximal eigenvalue under the coarse partition for ðKC Þ ðe,1Þ ¼ 1e and the maximal eigenvalue under the fine f¼1 is lmax partition is pffiffiffi 2 3e ðKF Þ lmax ðe,1Þ ¼ 12e þ 3 ðKF Þ ðKC Þ lmax ðe,1Þ o lmax ðe,1Þ, 8e 40. Since the maximal eigenvalue is con-
tinuous in the matrix entries (Demmel, 1997) there exists ðKF Þ ðKC Þ ðe,f Þ oð0:25,1Þ s.t. lmax ðe,f Þ o lmax ðe,f Þ, which completes the proof. & Appendix B. Proofs
b ðKF Þp b ðKC Þ is Lemma 2. For any e 40 the fitness difference p maximized at f¼1/2. b ðKF Þp b ðKC Þ ¼ p01 f þ p00 ð12f Þp10 ð1f Þ þ Proof. Note that p p0 ð12f Þ: If we take derivatives of this expression with respect
Author's personal copy F. Mengel / Journal of Theoretical Biology 307 (2012) 117–124
b ðKF Þp b ðKC Þ is increasing in f whenever to f we find that p p0 o
123
B.2. Proof of Result 2
1 p00 p11 þ 2 2
and decreasing otherwise. But now we know that in equilibrium p00 _p11 ’pð00Þ _pð11Þ 3f _12 3p0 _12. Hence f ¼ 12 maximizes the fitness difference. &
Proof. The maximal eigenvalue under the coarse partition for ðKC Þ f¼1 is lmax ðe,1Þ ¼ 1e and the maximal eigenvalue under the fine partition is 1ð2e=3Þ 4 1e. We also know from Lemma 2 that f¼1 minimizes the fitness difference between the fine and the coarse partition, which completes the proof. &
B.1. Proof of Result 1
B.3. Proof of Result 3
b ðKC Þ is decreasing in b ðKF Þp Proof. First we will show that p b ðKC Þ is given by e, 8e o 12. p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ðKC Þ ¼ 12ð1e þ ð12f Þ2 eð28f þ8f 2 Þ þ e2 Þ p
Proof. Denote the maximal eigenvalue associated with the coarse
with derivative 0
2
ðK Þ
CÞ lðK max ð0:5,f Þ ¼ 0:5.
ðK Þ
C We furthermore know that lmax ðe,f Þ is continuous 1 and decreasing in e A ½0,0:5 and increasing in f 2. (It is continuous in e A ½0,0:5 since the determinant of WðKC Þ ¼ f ð1f Þð12eÞ is
ðK Þ
F ð0,f Þ ¼ 1 continuous and decreasing in e.) We also know that lmax
1
and that
1B ð12f Þ þ e C @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA A ½1; 0 2 2 2 2 ð12f Þ 2ð12f Þ e þ e
ðKF Þ lmax ð
e,f Þ is continuous and decreasing in e A ½0,0:5. ðK Þ
F Furthermore we know that lmax ðe,1Þ ¼ 1ee2 which is below CÞ lðK max ð
b ðKF Þ is given by the largest eigenvalue of the mutation–selection p matrix and hence solves the following equation: 0 1 ð13eÞf ef ef ef B C e 13e e e B C ! ! C¼p p ðKF ÞB B C b ðKF Þ p ðKF Þ 0 0 0 0 @ A eð1f Þ eð1f Þ eð1f Þ ð13eÞð1f Þ
From this set of equations we get b ðKF Þð14eÞÞp01 ¼ ðp b ðKF Þð14eÞð1f ÞÞp11 ðp
ðK Þ
C C ðe,f Þ. We know that lmax ð0,f Þ ¼ maxff ,1f g and partition by lmax
e,1Þ ¼ 1e irrespective of the value of e. Hence in this case the coarse partition has higher fitness. Since the maximal eigenvalue is a continuous function of the matrix this will also be true for some f o 1, pffiffiffiffiffiffi while at the same time e oð 133Þ=2 (Demmel, 1997). Note that, since we know only that the two functions intersect for e A ½0,0:5 we do not know whether the type with the highest fitness will have the highest population share. We can show via two examples (where we assume f a1 to illustrate robustness) that it is possible in principle that the type with the highest fitness will have the highest population ðK Þ
ð1Þ
or ð14eÞðp01 ð1f Þp11 Þ b ðKF Þ ¼ p p01 p11 Taking derivatives we find 1 0 @p01 @p11 4ðp p Þ ðp ð1f Þp Þ þ e ð1f Þ 01 11 01 11 B C @e @e B C B C @ðp p Þ @ A 01 11 ð14eÞ ðp01 ð1f Þp11 Þ @e b ðKF Þ @p ¼ @e ðp01 p11 Þ2 Furthermore b ðKF Þ @p b ðKC Þ @p @e @e 0 1 @p01 @p @ðp p Þ 4 1 þ e ð1f Þ 11 01 11 B C @e @e @e Cð1Þ o0 ¼B @ A 1
b ðKF Þp b ðKC Þ ¼ 1ð1f Þ 40, whereas at e ¼ 0:25 : and at e ¼ 0 : p b ðKC Þ o 0 whenever f a 12. Since both derivatives are conb ðKF Þp p tinuous in e and negative for all values of E there exists b e A ð0, 14Þ s.t. b b b p ðKF Þ o p ðKC Þ, 8e 4 e . From Lemma 2 it follows that the upper bound on b e ðf Þ can be found by looking at the uniform case. In this case the spectrum of the mutation selection matrix for the fine partition is given by n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o sKF ¼ 0, 12 ð14eÞ, 14ð38e 7 18e þ 32e2 Þ and the maximal eigenvalue associated with the coarse partition is given by lmax ¼ 12 which exceeds the maximal element from sKF whenever e Z0:25. This completes the proof. &
F share. Let us assume that f ¼ 34 and that f ¼ 56 in turn. lmax ð0:4, 34 Þ
0:49o 0:53 ¼
CÞ 3 lðK max ð0:4, 4Þ
and
ðKF Þ lmax ð0:4, 56
ðK Þ
C Þ 0:49 o 0:55 ¼ lmax ð0:4, 56Þ.
ðKF Þ lmax ð0:4, 34Þ 0:49
is given The left hand eigenvector associated with by ðp00 ,p01 ,p10 ,p11 Þ ð0:23, 0:28,0:22,0:26Þ and the left hand ðK Þ
F ð0:4, 56Þ 0:49 is given by eigenvector associated with lmax ðp00 ,p01 ,p10 ,p11 Þ ð0:22,0:31, 0:2,0:25Þ. Hence in both cases the type with the highest fitness does (a) not vanish from the population (critical error rate) and (b) even has the highest population share. &
B.4. Proof of Result 4 Proof. To prove the result we again simply compute the eigenvalues of the different matrices. The largest eigenvalue corresponding to the 3 3 matrix associated ffi with the coarsest pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 partition is given by 16 ð7 þ 169e2 110e þ 2513eÞ.
KC K1 K2 K3 KF
e ¼ 0:12
e ¼ 0:08
e ¼ 0:04
0.666 0.596 0.533 0.664 0.501
0.693 0.641 0.590 0.786 0.589
0.720 0.693 0.664 0.886 0.745
(0.787) (0.675) (0.642) (0.724) (0.496)
(0.816) (0.739) (0.704) (0.803) (0.605)
(0.845) (0.806) (0.780) (0.891) (0.758)
The table shows the eigenvalues corresponding to the 9 9 matrices associated with the intermediate partitions and the 27 27 matrix associated with the finest partition. We are not reproducing these matrices here for space reasons, but – denoting the three actions by 1,2,3 – we note that under the assumption of similarity based errors profile (112) mutates e.g. into profile (212) with probability e, into (222) with probability e2 and into (321) with probability e3 . For intermediate partitions profiles are of length two and mutate either with probability e or e2 . &
Author's personal copy 124
F. Mengel / Journal of Theoretical Biology 307 (2012) 117–124
B.5. Proof of Result 5 Proof. The coarsest partition has largest eigenvalue pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lmax ¼ ðð1eÞ=2Þ þ 12e þ4e2 =4. The closed form solutions for the eigenvalues of the 4 4 and 8 8 matrices associated with the intermediate and finest partitions are very long expressions. The following table shows the maximal eigenvalues for each of these matrices rounded to three digits after the comma.
KC K1 K2 K3 KF
e ¼ 0:03
e ¼ 0:01
e ¼ 0:005
e ¼ 0:001
0.7053 0.8016 0.7792 0.7792 0.7474
0.7350 0.8494 0.8408 0.8408 0.8312
0.7425 0.8621 0.8577 0.8577 0.8602
0.7485 0.8724 0.8715 0.8715 0.8812
In brackets are the maximal eigenvalues for the case where f 1 ¼ 34, f 2 ¼ f 3 ¼ 18. &
B.6. Proof of Result 6 Proof. The model described leads to the following mutation– selection matrix, where the first row corresponds to KC , the second-to-fourth row to K1 , K2 , K3 , and the last row to KF 0 B B B B B B @
0:69ð13ZZ2 Þ
0:69Z
0:64Z 0:59Z
Bisin, A., Verdier, T., 2001. The economics of cultural transmission and the evolution of preferences. J. Econ. Theory 97 (2), 298–319. Boyd, R., Richerson, P., 1985. Culture and the Evolutionary Process. University of Chicago Press. Boyd, R., Richerson, P., 1995. Why does culture increase human adaptability? Ethol. Sociobiol. 16, 125–143. Boyd, R., Richerson, P., 2005. The Origin and Evolution of Cultures. Oxford University Press. Cavalli-Sforza, L., Feldman, M., 1981. Cultural Transmission and Evolution. Princeton University Press, NJ. Cohen, H., Lefebvre, C. (Eds.), 2005. Handbook of Categorization in Cognitive Science. Elsevier. Demmel, J.W., 1997. Applied Numerical Linear Algebra. SIAM. Eigen, M., McCaskill, J., Schuster, P., 1989. The molecular quasi-species. Adv. Chem. Phys. 75, 149–263. Fiske, S.T., 1998. Stereotyping, prejudice, and discrimination. In: Gilbert, F.S., D.T., Lindzey, G. (Eds.), Handbook of Social Psychology, Chapter 25. Oxford University Press. Fontana, W., 2002. Modeling evo-devo with RNA. BioEssays 24, 1164–1177. Fontana, W., Schuster, P., 1987. A computer model of evolutionary optimization. Biophys. Chem. 26, 123–147. Fryer, R., Jackson, M., 2008. A categorical model of cognition and biased decision making. BE—J. Theor. Econ. 8 (1). (Article 6). Hamilton, D. (Ed.), 1981. Cognitive Processes in Stereotyping and Inter-Group Behavior. Erlbaum, Hillsdale, NJ. Harnad, S., 2005. To cognize is to categorize: cognition is categorization. In: Handbook of Categorization in Cognitive Science. Elsevier. Henrich, J., Boyd, R., 1998. The evolution of conformist transmission and the emergence of between-group differences. Evol. Hum. Behav. 19, 215–242. Henrich, J., Gil-White, F., 2001. The evolution of prestige: freely conferred deference as a mechanism for enhancing the benefits for cultural transmission. Evol. Hum. Behav. 22, 165–196. Mengel, F., 2008. Matching Structure and the Cultural Transmission of Social Norms. J. Econ. Behav. Organ. 67, 608–623. Mengel, F., 2012. Learning across games. Games Econ. Behav. 74 (2), 601–620. Mohlin, E., 2009. Optimal Categorization. Mimeo.
0:69Z
0:69Z
0:69Z2
0:64ð12Z2Z2 Þ
0:64Z2
0:64Z2
0:64Z
0:59Z2
0:59ð12Z2Z2 Þ
0:59Z2
0:59Z
0:78Z
0:78Z2
0:78Z2
0:78ð12Z2Z2 Þ
0:78Z
0:58Z2
0:58Z
0:58Z
0:58Z
0:58ð13ZZ2 Þ
If Z ¼ 0:1, the left hand eigenvector ðpC ,p1 ,p2 ,p3 ,pF Þ ð0:19,0:14,0:12,0:38,0:16Þ. &
is
given
by
References Al Najjar, N., Pai, M., 2009. Coarse Decision Making. Mimeo. Allport, G., 1954. The Nature of Prejudice. Addison-Wesley. Arnaout, R., Loyd, A., o’Brien, T., Goedert, J., Nowak, M., 1999. A simple relationship between viral load and survival time in HIV-1 infection. Proc. Natl. Acad. Sci. U.S.A. 96, 11549–11553. Bednar, J., Page, S., 2007. Can game theory explain culture? Rationality Soc. 19 (1), 65–97.
1 C C C C C C A
Murphy, G.L., 2002. The Big Book of Concepts. MIT Press. Nowak, M., 2006. Evolutionary Dynamics: Exploring the Equations of Life. Belknap/Harvard. Nowak, M., May, R., Anderson, R., 1990. The evolutionary dynamics of HIV quasispecies and the development of immunodeficiency disease. AIDS 4, 1095–1103. Rubinstein, A., 1998. Modeling Bounded Rationality. MIT Press. Samuelson, L., 2001. Analogies adaptation and anomalies. J. Econ. Theory 97 (2), 320–366. Tajfel, H., 1969. Cognitive aspects of prejudice. J. Soc. Issues 25, 79–97. Wodarz, D., Nowak, M., 1999. Evolutionary dynamics of HIV-1 induced subversion of the immune response. Immunol. Rev. 168, 75–89.
