On Randomly Changing Conformity Bias in Cultural Transmission Downloaded by Guest on September 30, 2021 Downloaded by Guest on September 30, 2021 X If 2) ,2, 1, 1

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Downloaded by guest on September 30, 2021 a Denton K. Kaleda cultural in transmission bias conformity changing randomly On PNAS biases. 14) (13, other and/or frequency-dependent name 12) with baby (8, However, consistent 11). more (10, appear names distributions motifs, baby pottery and Neolithic citations, (9), breeds suggested patent dog been of choices has for (8), account performing frequency-dependent copying to random children Unbiased as known young (7). transmission, by discrimination exhibited numerical been Anti- has (7). tasks conformity discrimination numerical and (6), discrimination frequency, population its is than variant less or common (4). greater more respectively rate a a and (3). at when of population conformity adopted occur terms the include which in in biases anticonformity, variants couched char- cultural “frequency-dependent” is and the (2). 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Akçay, Jeremy display Erol biases by animals reviewed anticonformist 2021); nonhuman 15, April and review for Humans (sent 2021 6, July Feldman, W. Marcus by Contributed eateto ilg,Safr nvriy tnod A935 and 94305; CA Stanford, University, Stanford Biology, of Department uashv xiie ofriyi etlrtto 5,line (5), rotation mental in conformity exhibited have Humans aindnmc fatatwoetasiso rmone from popu- transmission whose finite trait the a studied of (1) dynamics Feldman lation and avalli-Sforza n 01Vl 1 o 4e2107204118 34 No. 118 Vol. 2021 f“oemdl”smldb ahidvda.W also We individual. each by sampled models” “role of | tcatclclstability local stochastic a r Liberman Uri , n t oee,dnmcproperties dynamic However, . b | convergence n acsW Feldman W. Marcus and , | t .Under ). b colo ahmtclSine,TlAi nvriy e vv sal69978 Israel Aviv, Tel University, Aviv Tel Sciences, Mathematical of School ulse uut2,2021. 20, August Published doi:10.1073/pnas.2107204118/-/DCSupplemental.y at online information supporting contains article This 1 BY-NC-ND) (CC .y Attribution-NonCommercial- 4.0 Commons NoDerivativesCreative License distributed under is article access open This interest.y competing no declare Max authors The R.M., and Durham; Anthropology.y of Evolutionary for University Institute J.R.K., Planck Pennsylvania; of University E.A., research, Reviewers: performed research, paper. designed y M.W.F. the wrote and and U.L., K.K.D., contributions: Author vr otpeiu ahmtclmdl fcnomt have conformity conformity time-dependent, of coefficients. than turnover models rather convex constant, mathematical How- incorporated a previous bias. indicative show direct most names or 1930) ever, function frequency-dependent baby to positive turnover male of (1880 indicative but concave from decades names earlier bias, a baby from frequency-dependent show US the negative popular 2010 to of 12, to ref. 1950s In 1960 the Aschian. than from other declined and, also has humans has of States, It studies (16). 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ANTHROPOLOGY EVOLUTION Cavalli-Sforza and Feldman (ref. 3, chap. 3) and Boyd “non-” or “anticonformist” states over time (40, 42, 47, 50, 51), and Richerson (ref. 4, chap. 7) studied models of frequency- to our knowledge, random temporal variation in the confor- dependent transmission of a cultural trait with two variants. Boyd mity coefficients themselves has not been modeled previously. and Richerson (4) incorporated conformist and anticonformist In reality, the degree to which groups of individuals conform bias through a conformity coefficient denoted by D. In their may change over time, as illustrated by the finding that young simplest model, if the frequency of variant A is p and that of vari- children anti-conformed while older children conformed in a ant B is 1 − p, then the frequency of variant A in the offspring discrimination task (7); thus, it seems reasonable to expect generation, p0, is that different generations may also exhibit different levels of conformity. Indeed, generational changes have occurred for p0 = p + Dp(1 − p)(2p − 1), [1] Aschian conformity (29) and possibly in frequency-dependent copying of baby names (12). Our stochastic model may there- where D > 0 entails conformity (A increases if its frequency is fore produce more realistic population dynamics than previ- 1 p > 2 ), D < 0 entails anticonformity, D = 0 entails random copy- ous deterministic models, and comparisons between the two ing, and −2 < D < 1. In this model, each offspring samples the can suggest when the latter is a reasonable approximation to cultural variants of n = 3 members of the parental generation the former. (hereafter, role models). Sampling n > 3 role models requires We also allow the number of role models, nt , to vary over different constraints and, if n > 4, there are multiple conformity time. Agent-based conformity models have incorporated tem- coefficients (Eq. 19). poral (43) and individual (43, 45, 46) variation in the number Many subsequent models have built upon Boyd and Rich- of sampled individuals, whereas here, all members of genera- erson’s (4) simplest model (Eq. 1). These have incorpo- tion t sample the same number nt of role models. Causes of rated individual learning, information inaccuracy due to envi- variation in nt are not explored here, but there could be sev- ronmental change (30–34), group selection (35), and other eral. For instance, different generations of animals may sample transmission biases, including payoff bias (36), direct bias, different numbers of role models due to variation in popula- and prestige bias (37). Other models, which include a sin- tion density. In humans, changes in the use of social media gle conformity coefficient and preserve the essential features platforms or their features may cause temporal changes in of Eq. 1, incorporate individual learning, environmental vari- the number of observed individuals. For example, when Face- ability (32, 38), group selection (39), and multiple cultural book added the feature “People You May Know,” the rate of variants (38). new Facebook connections in a New Orleans dataset nearly In agent-based statistical physics models, the up and down doubled (52). spins of an electron are analogous to cultural variants A and In the stochastic model without selection, regardless of the B (40, 41). Individuals are nodes in a network and choose fluctuation in the conformity coefficient(s), if there is confor- among a series of actions with specified probabilities, such as mity on average, the population converges to one of the three independently acquiring a spin, or sampling neighboring individ- equilibria present in Boyd and Richerson’s (4) model with con- uals and adopting the majority or minority spin in the sample. n ∗ formity (D(j ) > 0 for 2 < j < n in Eq. 19). These are p = 1 The number of sampled role models can be greater than three ∗ ∗ 1 (fixation on variant A), p = 0 (fixation on variant B), and p = 2 (42, 43). (Anti)conformity may occur if all (42–47), or if at (equal representation of A and B). However, their stability least r (40, 48), sampled individuals have the same variant. properties may differ from those in the deterministic case. In In contrast, Boyd and Richerson’s (4) general model (Eq. 19) Boyd and Richerson’s (4) model with random copying, every allows, for example, stronger conformity to a 60% majority initial frequency p0 is an equilibrium. Here, with random copy- of role models and weaker conformity or anticonformity to a ing expected and independent conformity coefficients, there is 95% majority (in humans, this might result from a perceived ∗ 1 convergence to p = 0, 2 , or 1. In this case, and in the case difference between “up-and-coming” and “overly popular” ∗ 1 with conformity expected, convergence to p = 0, 2 , or 1 also variants). holds with stochastic variation in the number of role models, In Boyd and Richerson’s (4) general model, individuals sam- nt . With either stochastic or constant weak selection in Boyd n n ple role models, which is more realistic than restricting and Richerson’s (4) simplest model (Eq. 1) and random copying 1 ∗ to 3 (as in Eq. ); individuals may be able to observe more expected, there is convergence to a fixation state (p = 0 or 1). n > 4 than three members of the previous generation. With , Finally, with anticonformity in the deterministic model or anti- different levels of (anti)conformity may occur for different sam- conformity expected in the stochastic model, nonconvergence j n ples of role models with one variant. In addition to the can occur. example above with 60 and 95% majorities, other relationships between the level of conformity and the sample j of n are pos- Generalizing the Deterministic Model sible. For example, the strength of conformity might increase Here we generalize Eq. 1 to allow the coefficient D to change as the number of role models with the more common variant randomly at each generation. If t denotes the generation number increases. In a recent exploration of Boyd and Richerson’s (4) (t = 0, 1, 2, ... ), then Eq. 1 becomes general model, we found dynamics that departed significantly from those of Eq. 1 (49). If conformity and anticonformity occur n for different majorities j of n role models (i.e., j > 2 ), polymorphic equilibria may exist that were not possible with Eq. 1. pt+1 = pt + Dt pt (1 − pt )(2pt − 1). [2] In addition, strong enough anticonformity can produce nonconvergence: With as few as 5 role models, stable cycles in variant frequencies may arise, and with as few as 10 role models, chaos In Eq. 2 we assume that the random variables Dt for t = is possible. Such complex dynamics may occur with or without 0, 1, 2, ... are identically distributed with −2 < Dt < 1 for n = 3 2 selection. role models, E(Dt ) = d, Var(Dt ) = σ , and Dt are independent Here, we extend both Boyd and Richerson’s (4) simplest of pt in each generation t. We focus on the possible convergence (Eq. 1) and general (Eq. 19) models to allow the conformity of pt to equilibrium as t → ∞. coefficient(s) to vary randomly across generations, by sampling Eq. 1 entails that σ2 = 0, and we now summarize the known them from probability distributions. Although some agent-based results for this model. If the conformity coefficients are constant models allow individuals to switch between “conformist” and over time, i.e., Dt ≡ d 6= 0 for all t, then

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E p ∗ t 0 1 2 < . . . d = (p (p t x p t p | and , − +1 0 = Eq. 0, ≤ (p . t p = p t 0 ≤ t t p ` (1 2 1 t 0 +1 p −−−→ p t = ) t ≤ t ) | E +1 1 t , →∞ < {p 0 p 2 1 oa equilibrium. an to − sapolymorphic a is > ) p p (D p 0 | ≤ t t ≥ nal that entails < 0 then 1, E t p = ) | p ∗ t p p ≤ −−−→ 0 } p t 2 h process the 1, < t ≤ ∗ t 1 = t t 0, (p p t ∞ →∞ ) t )]. < ) 1. p =0 1 = t ) if > mle that implies p p 1 < ∞ 0 D ` < < p 0 t and , sthere Is 2. ihprob- with < d p . converges. t t ) hnif then 0, < n a and 1, p t E < p (p 2: r also are > changes If . t 1 2 for ∞ then 1, (p then , t n if and 0 2 1 − p t with and and +1 1 2 0 ∗ ` [3] [4] 1 2 ≤ = < = ) | iia if similar variable dom p p 2) 1) 4. Result values? these of each takes p Var(D Now if applies argument similar A 1. probability with p Proof: h rbblte f0and 0 of probabilities the p that implies which 3. Result Proof: 1)] By probability hrfr,since Therefore, If he aus0 ,and 1, 0, values three 12 But rbblt 1. probability 0 E ∞ 0 ∞ t ∗ < so , d Applying If eut3 Result o all for 0 = If If )(2p ≤ spsil fadol if only and if possible is −−−→ p 0 6= p p t d eut2, Result Var(p 0 →∞ t 0 ∞ 2 2 2 1 P E 1 2 , 0 = < (1 rmEq. From If < hnEq. then , < ∞ (1 p (p (p n yEq. by and Var(p P If ∗ 2 1 E − t p d If 1 2 p E eueEq. use we , − ∞ t − E hnalso then , E 0 nal htterno variable random the that entails 1 = 1; ,1 2, 1, 0, = (p +1 t 0 (p = < d = p +1 (p t Var(p 2, Result < [p 1) (p p < = p = ) t ∞ namely, = t P = ) E p ) ∞ σ p t t +1 ∞ and , = ) 2 t t 2 1 +1 1, ≤ 0 ∞ 2 1 2 −−−→ +1 +1 )=1 = 0) = E (D E (2p ) t , (1 dE E < σ 2(1 = ) →∞ 2 p σ E | E hsi hscase, this in Thus 0. since 2, = ) (D 8 (2p 2 ae w aus0and 0 values two takes E p ∞ t | | 2 1. − p (p and , (p t implies hnEq. then 0, = ) t p p 5, p [ > . . . ∞ p 2 [Var(p = ) t − p 0 = t t t E 2 p t 0, = ) ∞ t 0 ∗ = ) = ) ∞ p ) P (1 (1 ) 2 1 Eq. 0, ∞ 2 1) (p ≤ ∞ = and , −−−→ − o all for (1 − p (p t orsodn otecntn equilibria constant the to corresponding − osethat see to )(2p − or − →∞ 2 t t , D p Var(D p 2 1 1) + ) ∞ p − + t 2p ∞ E p t then −−−→ p htaetepoaiiisthat probabilities the are What . t t then 2 1 , 0 p t +1 ) 2 ∞ t →∞ ), sidpnetof independent is or 0 = E p ∞ 11 ∞ E [p 0 ≤ ) −−−→ r si Eq. in as are ≥ dE ∞ E t , 2 ) t (p (D →∞ | ∞ = − (2p 2 2 1 mle that implies ,1 2, 1, 0, = p (p 0 t )(2p p (2p https://doi.org/10.1073/pnas.2107204118 p ∞ (1 )p t [ t ∞ hrfr rmEq. from Therefore 1)]. ihpoaiiy1 and 1, probability with E ∞ 2 1 ihpoaiiy1 ec,Eq. Hence, 1. probability with p )]+ ∞ )p t = ) t t 2 2 P − (1 p or ∞ (1 − P Var(p (1 ) (1 ∞ t ∞ p (p mle that implies (1 p (p Var p − − ∞ 1) 2 − p p − − − ∞ ∞ = and ∞ ∞ 0 ∞ − (1 2 p 1) p If . p . . . )(2p )=0. 1)]= p (1 ∞ )=2p = 1) = 2 1 ∞ 1 2 ∞ 1 = t [E p t = + 2 − )(2p 1 = ) n as and , ) p t )(2p h ruetis argument The 13. − )(2p ) 2 0 Therefore . )(2p ∞ 0 = Var(p (p 2 1 p E (2p ∞ and < p 2 = ) ∞ p 1 = t [p ∞ a aeol the only take can t +1 − t ec h ran- the Hence . ∞ p . ∞ ) t − , t 2 1 t )(2p 2 0 (1 − o if Now 0. = 1)] − E . (2p − < | − t < 1)]. PNAS p 0 ). E p (p 1) 1), 0 − )=0 = 1) p − t )=0 = 1) 2 1 . ∞ (p ∞ )] 0 t 2 then , p +1 1. . < ∞ t − E − p | )(2p 1. 7 = ) ∞ | (p )=0 = 1) 1) f12 of 3 p (1 [10] [13] [14] [11] [12] with t t with 2 0 p = ) = ) [9] [5] [6] [7] [8] t p ∞ − ≤ − 0 . ,

ANTHROPOLOGY EVOLUTION If d = E(Dt ) 6= 0, the analysis is more complicated because in 3A), it is necessary that E[Dt (2)] < 0, in which case the dynamics this case it is not clear what E(p∞) is. We address this case using are similar to Eq. 1 with anticonformity (D < 0). With n = 3 and the notion developed by Karlin and Liberman (53) of stochastic identically distributed conformity coefficients, there is always ∞ local stability (SLS) of a constant equilibrium. Let {xt }t=0 be convergence of the population to an equilibrium. a sequence of random variables satisfying xt+1 = f (xt ) for t = If Dt (2) are not identically distributed, nonconvergence can ∗ 0, 1, 2, ... . Then x is a constant equilibrium of this sequence if occur with n = 3 role models. In Fig. 4A, Dt (2) alternates x ∗ satisfies x ∗ = f (x ∗). between two fixed values, producing a stable, two-generation Definition: SLS. A constant equilibrium x ∗ is said to be SLS if for cycle in the frequency of A. This cycle is confined to a region 1 1 any ε > 0 there is a δ > 0 such that in (0, 2 ) or ( 2 , 1), depending on whether the initial frequency p0 1 is below or above p = 2 , respectively. [Recall that Observation 1 ∗ ∗ |x0 − x | < δ =⇒ P lim xt = x ≥ 1 − ε. [15] does not rely on the assumption of identically distributed Dt (2).] t→∞

∞ More Role Models. The above results apply when the number of In our case, the sequence {pt }t=0 determined by Eq. 2 has three ∗ ∗ ∗ 1 role models is n = 3. Do such results for global convergence hold constant equilibria: p = 0, p = 1, and p = 2 . Following ref. ∗ n 53, to determine whether a constant equilibrium p is SLS we for any number of role models? To address this, we follow 2 examine the linear approximation of the transformation in Eq. 2 Boyd and Richerson (4) and Eq. above is replaced by “near” p∗. The linear approximations of Eq. 2 near p∗ = 0 and ∗ p = 1 for small νt are both n ! X Dt (j ) n j n−j pt+1 = pt + p (1 − pt ) . [19] ν = ν (1 − D ), [16] n j t t+1 t t j =0 ∗ 1 while that near p = 2 for small ηt is Eq. 19 with the subscript t removed from Dt (j ) is the gen- Dt ηt+1 = ηt (1 + 2 ). [17] eral form of Boyd and Richerson’s (4) deterministic model. The sum contains the probability of each possible sample j In Eq. 16, the frequency of a rare variant increases under anti- of n role models with variant A (a binomial, due to random conformity (Dt < 0), whereas in Eq. 17, the frequency of a 1 sampling) multiplied by the scaled conformity coefficient for common variant (ηt > ) increases under conformity (Dt > 0). D (j ) 2 that sample, t . For any generation t = 0, 1, 2, ... the vector Thus, we have the following: n Dt = [Dt (0), Dt (1), ... , Dt (n)] of conformity coefficients Dt (j ) j = 0, 1, 2, ... , n Result 5. has the following properties for : ∗ 1) If E [log(1 − Dt )]< 0, then both p = 0 and ∗ p = 1 are SLS. If E [log(1 − Dt )] > 0, then Dt (0) = Dt (n) = 0, Dt (n − j ) = −Dt (j ), [20a] P (limt→∞ pt = 0)= P (limt→∞ pt = 1)= 0. Dt ∗ 1 2) If E log(1 + 2 ) < 0, then p = 2 is SLS, whereas if Dt 1 E log(1 + 2 ) > 0, then P limt→∞ pt = 2 = 0. and for j = 1, 2, ... , n − 1, −j < Dt (j ) < n − j . [20b] x As both functions y = log(1 − x) and y = log(1 + 2 ) are concave (for x < 1 and x > −2, respectively), we apply Jensen’s Eq. 20b ensures that probabilities of acquiring A and B are in inequality to obtain (0, 1). The left-hand side of Eq. 20a entails that if a sample contains A(B) at frequency 1, an individual acquires A(B) with E [log(1 − Dt )] < log [E(1 − Dt )], [18] probability 1, and the right-hand side entails “equal treatment” Dt Dt E log(1 + 2 ) < log E(1 + 2 ) , of A and B. If an individual samples j of type A and n − j of type B, the amount by which it (anti)conforms to A, namely 2 since −2 < Dt < 1 with probability 1 and σ = Var(Dt ) > 0. Dt (j ), must be equal and opposite to the amount by which it Applying inequalities in Eq. 18 and Result 5, we have the (anti)conforms to B, −Dt (j ), for all probabilities to sum to 1. following: Therefore, when A rather than B is present in n − j role models, its conformity coefficient Dt (n − j ) must equal that for B in the Result 6. same scenario: −Dt (j ). Therefore, Eq. 19 is equivalent to ∗ ∗ 1) If d = E(Dt ) ≥ 0, then p = 0 and p = 1 are SLS. ∗ 1 2) If d = E(Dt ) ≤ 0, then p = is SLS. n ! 2 X Dt (j ) n h j n−j n−j j i ∗ pt+1 = pt + pt (1 − pt ) − pt (1 − pt ) , Therefore, if d = E(Dt ) = 0, all three constant equilibria, p = n j ∗ ∗ 1 j =k 0, p = 1, and p = are SLS. Hence p∞ can take the three 2 [21] values as stated in Result 4. The dynamics with n = 3 role models are illustrated in Figs. 1 n n+1 where k = 2 + 1 if n is even and k = 2 if n is odd. A, D, and G; 2A; and 3A. Dt is denoted by Dt (2) because it cor- Assume that the conformity coefficients change randomly at responds to the case where j = 2 of 3 role models have the same each generation t, and for any t = 0, 1, 2, ... , Dt is a vector of variant [the reason for this Dt (j ) notation becomes clear with random variables such that the Dt s are identically distributed and n > 4 and is described under Eq. 19]. Consistent with Observa- independent of the population state pt . The entries of Dt , Dt (j ), 1 1 1 1 tion 1, runs that start in (0, 2 ) or ( 2 , 1) remain in (0, 2 ) or ( 2 , 1), need not be identically distributed. Since probabilities of differ- ∗ 1 respectively. Equilibria are p = 0, 2 , and 1. It is possible that ent samples j of n, with corresponding Dt (j ), depend on pt , it is all three equilibria are SLS with E[Dt (2)] = 0, E[Dt (2)] < 0, or possible that higher or lower pt corresponds to greater or smaller ∗ 1 E[Dt (2)] > 0 (Fig. 1 A, D, and G, respectively). For p = 2 to differences between pt+1 and pt . be unstable (Fig. 2A), it is necessary that E[Dt (2)] > 0. In this Assume first that E(Dt ) = 0; i.e., E(Dt (j )) = 0 for all t = case, the dynamics are similar to those of deterministic Eq. 1 0, 1, 2, ... and j = 0, 1, 2, ... , n. In this case we have the with conformity (D > 0). For p∗ = 0 and 1 to be unstable (Fig. following result:

4 of 12 | PNAS Denton et al. https://doi.org/10.1073/pnas.2107204118 On randomly changing conformity bias in cultural transmission Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 hn if Then, for sabuddmrigl n h atnaeconvergence martingale the and martingale that bounded implies theorem a is ihprobability with 0.5. SD and −0.05 truncation) (before mean with ( on distribution normal 2, 1, 0, Proof: nrnol hnigcnomt isi utrltransmission cultural in bias conformity changing randomly On al. et Denton probability with D probability with with equilibria although shown, are 1. Fig. 2) 1) 7. Result t (4) Let probability If, probability process the If edn admvariables, random pendent E ∼ j [D naddition, in E U ,2, 1, = . . . If tcatclclsaiiyof stability local Stochastic t σ ,0] (−4, (D (j j 2 )] Eq. , D = h eno log[1 of mean the , t = ) D t Var . . . t 1. 1. ihprobability with (k sidpnetof independent is 5 4 2 3 3 1 {p 0, 19 (B) . and and D , ), (D t t D n hnsatn rmayiiilstate initial any from starting then G) ( 0.5. SD and 0.05 truncation) (before mean with 2) (−3, on distribution normal truncated a from sampled is (3) } mle that implies D t t ∞ − t D D n =0 (k ,1 ihma bfr truncation) (before mean with 1) −2, t (j t t lim (k = (4) (2) 1 ), )) 4, converges; 1), + = = D t [σ →∞ D etepstv aineof variance positive the be ihprobability with −3 ihprobability with −1.5 t 5adS ..(F 0.5. SD and −0.05 t (k 0 p* (3) − 5 1 = then . . . . p D = 1), + ∼ p* D E t t σ (n 0, t U = n , = (3) (p ,1 ihprobability with 1) [0, D p − 2 1 0 = p htis, that 0, ∞ a aelne ob ece;hwotnec qiiru a ece sin is reached was equilibrium each often how reached; be to longer take may 1 , p ∼ t . . . t ∞ n h eno log of mean the and 1)], +1 (n t 1 2 U ae h values the takes .Ec ltsos2 uswt initial with runs 25 shows plot Each 1. , ihpoaiiy1. probability with , ,2 ihprobability with 2) [0, and | − D as p t t 1) )= lim (n 5 1 ) D E . ) n p 3 1 t D t r independent are (D = →∞ o any for ( . t 0 = (0) t Hence . (3) H 5, t ) = ) D = n p t t = ihprobability with 1.5 ,1 ihma bfr truncation) (before mean with 1) (−4, on distribution normal truncated a from sampled is (4) 5adS ..(E 0.5. SD and −0.05 D 0, 4 3 = 0 t 0 4, t and (n r inde- are ≤ p 2 1 {p for D ∞ 1 , h t p 0]. = ) 5 3 (3) D 1 t D 0 } and + with with t t ≤ t ∞ t = (3) (j =0 = D 9wt probability with 0.99 1 ) t D 2 ∼ (2) t (3) U i ,0] (−3, for admvrals Eq. that variables, random ic Var(p have Since ∼ Var(p + U 3 5 ) n ,0] (−3, n and = n X p j Var(p = = =k ihprobability with −1 0 .(A) 3. = 4, n X j D t =k +1 −1 n σ 5(e)ad2 uswith runs 25 and (red) 0.25 t D (3) ihprobability with j 2 2 t t ,1) (−3, on distribution normal truncated a from sampled is (3) = ) n +1

n σ = = j 2 2 n 3 4 t j 5wt probability with −2.85 | +1

Var(p 3, ! and p n t j D 2 = ) ) ! E t (2) D 2 t 1 4 t Var (3) h (C . ∼ ) p h p t U j = (E 5 2 ) (1 t j ,1 ihprobability with 1) [0, (D) . ihprobability with −2.5 (1 n 21 (p − = − 5, t p n +1 p t mle o any for implies = D ) t https://doi.org/10.1073/pnas.2107204118 n ) t | 3, p (4) n −j p 0 5 2 −j D t = . ∼ ))+ t − 2 ssmldfo truncated a from sampled is (2) 5(le.Egt generations Eighty (blue). 0.75 − U p eto A section Appendix, SI [ ,1 ihprobability with 1) 0, p t n E t n −j −j (Varp (1 3 2 (1 1 4 ( . − and − I ) n p n t = p t t D +1 PNAS ) = t ,1 2, 1, 0, = t j 3, ) (2) i 5, j 2 | i D . p 2 D ∼ t t (2) t | )) (4) U . we , ,0] (−2, = f12 of 5 5 4 along = [23] [22] 0.99 and . . . 0.9

ANTHROPOLOGY EVOLUTION 1 Fig. 2. Stochastic local stability of p* = 0, 1 and instability of p* = 2 . Each plot shows 25 runs with initial p0 = 0.499 (red) and 25 runs with p0 = 0.501 (blue). The ﬁrst 80 generations are shown, although equilibria may take longer to be reached; how often each equilibrium was reached is in SI Appendix, h Dt (2) i section A, along with E[Dt (j)], the mean of log[1 − Dt (n − 1)], and the mean of log 1 + 2 for n = 3. (A) n = 3, Dt (2) ∼ U[0, 1) with probability 0.9 and Dt (2) ∼ U(−2, 0] with probability 0.1. (B) n = 4, Dt (3) ∼ U[0, 1) with probability 0.925 and Dt (3) ∼ U(−3, 0] with probability 0.075. (C) n = 5, Dt (4) ∼ U[0, 1) with probability 0.98 and Dt (4) ∼ U(−4, 0] with probability 0.02. Dt (3) ∼ U[0, 2) with probability 0.55 and Dt (3) ∼ U(−3, 0] with probability 0.45. (D) n = 4, 3 1 3 1 Dt (3) = 0.99 with probability 4 and Dt (3) = −2.99 with probability 4 .(E) n = 5, Dt (4) = 0.99 with probability 4 and Dt (4) = −3.99 with probability 4 . 3 1 Dt (3) = 0.1 with probability 4 and Dt (3) = −0.1 with probability 4 .

∗ 1 Now as pt −−−→ p∞ with probability 1, by the Lebesgue con- near the constant equilibrium p = given in ref. 49, t→∞ 2 ` ` equation B8: vergence theorem E(pt ) −−−→ E(p∞) for any ` = 1, 2, ... , and t→∞ Eq. 23 implies that  n−1 !  0 n−2 X D(j ) n ν = ν 1 + 1 ((2j − n)) = νφ . [25]  2 n j  n j =k n−1 !2 2 2 X σj n h j n−j n−j j i E p∞(1 − p∞) − p∞ (1 − p∞) = 0. Ref. 49’s table S1 shows that depending on n, φn can be negative, n2 j ∗ j =k in which case log φn is not defined and conditions for SLS of p = 1 [24] 2 cannot be determined. Also, Observation 1 does not hold. With n > 3 role models and E(Dt ) =6 0, there need not be j n−j n−j j convergence to an equilibrium (Fig. 5). Below we determine con- Hence p∞(1 − p∞) = p∞ (1 − p∞) for j = k, k + ditions for convergence to an equilibrium with n = 4 role models 1, ... , n − 1 and p∞ = 0, p∞ = 1, or p∞ = 1 − p∞ in which case 1 1 and, subsequently, a general number n of role models. p∞ = 2 . Thus, with probability 1, p∞ takes values 0, 2 , 1. ∗ ∗ The following result concerning the SLS of p = 0 and p = 1 Example: The Case of n = 4 Role Models. Recall that the recursion holds for E(Dt ) = 0 as well as E(Dt ) 6= 0. with fixed conformity coefficients is the same for n = 3 and 4 (4). However, with random conformity coefficients the case n = 4 Result 8. exhibits different evolutionary dynamics from those with n = 3 role models. When n = 4, we have five conformity coefficients ∗ ∗ Dt (j ) for j = 0, 1, ... , 4, Eq. 20a and inequality in Eq. 20b, 1) If E [log (1 − Dt (n − 1))] < 0, then p = 0 and p = 1 are SLS. 2) If E [log (1 − Dt (n − 1))] > 0, then P (limt→∞ pt = 0) = 0 Dt (0) = Dt (2) = Dt (4) = 0, Dt (1) = −Dt (3), and P (limt→∞ pt = 1)= 0. [26] − 3 < Dt (3) < 1.

The proof is similar to that of Result 5 based on the linear Hence, from Eq. 21 with n = 4 and k = 3, we have approximation of Eq. 19 near p∗ = 0 and p∗ = 1 in equation B2 of Denton et al. (49) and the assumption that Dt (n − 1) < 1 pt+1 = pt + Dt pt (1 − pt )(2pt − 1), [27] by Eq. 20b. If E (Dt (n − 1))> 0, then using Jensen’s inequality, ∗ ∗ E [log (1 − Dt (n − 1))]< 0, so p = 0 and p = 1 are SLS. where Dt are identically distributed and correspond to Dt (3) We are unable to obtain corresponding results for the in Eq. 26. Therefore in this case [1 + 2Dt pt (1 − pt )] with ∗ 1 1 SLS of p = 2 . Consider the linear approximation of Eq. 19 −3 < Dt < 1 can be negative, and by Eq. 3, pt+1 − 2 and

6 of 12 | PNAS Denton et al. https://doi.org/10.1073/pnas.2107204118 On randomly changing conformity bias in cultural transmission Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 nrnol hnigcnomt isi utrltransmission cultural in bias conformity changing randomly On al. et Denton (C generations. Now Eq. of when convergence convergence global be global d still of there proof Will hold. the not in does factor key a was A 10 for ran 4. Fig. and 0.05 probability reached. equilibrium The (blue). 3. Fig. ..(D) 0.9. n p (A) . = = t Since 5, − 3 E p D n efcson focus we Q (D 2 1 t n t = xc yls ahpo hw 5rn ihinitial with runs 25 shows plot Each cycles. Exact tcatclclsaiiyof stability local Stochastic (1 (4) t eto A section Appendix, SI = 6 +1 a aedfeetsgs Therefore signs. different have can t 3, p ) eeain;n qiirawr ece.Dsrpin ftednmc,eatiiilfeunis and frequencies, initial exact dynamics, the of Descriptions reached. were equilibria no generations; − ∼ 5, p t D 6 0 = = +1 t U D p t +1 2 lentsbetween alternates (2) ) ,1 ihpoaiiy04and 0.4 probability with 1) [0, t t Q (4) n = ) − with − = t = 2 1 5, 2D + 1 2 1 D Q 1 4 ihprobability with −3.9 t D − n t and (2) 2 p* t (p = (4) = 4 = Q ∼ = t = U t n Eq. and , p ? 2 1 p t ,0] (−2, 9and −3.99 t t − srahdi vr ae lhuhol h rt8 eeain r hw n hseulbimmytk ogrt be to longer take may equilibrium this and shown are generations 80 ﬁrst the only although case, every in reached is 1 4 − reports − − 1 2 2 1 p* 2 1 ) Q 2 ihpoaiiy09.(B) 0.95. probability with 2 5ad09eeyohrgnrto.(B) generation. other every 0.9 and −1.15 Then . = t a aedfeetsgs from signs, different have can 1+2D + [1 28 E 2 1 D 2 [ D t , n ntblt of instability and seuvln to equivalent is (3) 4 3 t (j D , h eno log[1 of mean the )], D t = t (4) t t ,1 2, 1, 0, = (4) vr hr eeainadi h te generations other the in and generation third every −2.1 p ∼ which 1, Observation t = (1 U ihprobability with 0.9 ,0] (−4, − p p 0 t . . . < )] p* ihpoaiiy0.6. probability with 2 2 1 {p . n = rd n 5rn with runs 25 and (red) = t ,1 ahpo hw 5rn ihinitial with runs 25 shows plot Each 1. 0, } n 4, − when [28] [29] 3 = D D t t (n (3) , 4 1 − ∼ , D n h eno log of mean the and 1)], U t σ Since 9. Result process the to applies result following The Proof: {Q then (3) ,1 ihpoaiiy02 and 0.25 probability with 1) [0, 2 n = D t = = } t Q (3) ihprobability with 0.9 4, t ∞ Var(D Q D =0 ∞ ert Eq. Rewrite D p t ∼ t 0 +1 t sidpnetof independent is (3) a aeol w values, two only take can If ovre ihprobability with converges > U ,2 ihpoaiiy01and 0.1 probability with 2) [0, = = 2 1 t d ehave we ), bu) ihygnrtosaeson lhuhsimulations although shown, are generations Eighty (blue). Q 9eeytidgnrto and generation third every −2.99 = t E 4D + 1 (D 29 h t 1 ) as + 4 3 > and , D t D p t 0, t (4) 2 0 (2) p E 4 1 = t D hnfrany for then [D i D = − t n oas of also so and 0 rd n 5rn with runs 25 and (red) 0.001 for t (3) t (3) 9and 0.99 (j Q )] https://doi.org/10.1073/pnas.2107204118 ∼ n t = r eotdin reported are Q = U ihprobability with −1.9 4 + ∞ 1, ,0] (−3, D .(A) 3. t 0 = (3) D n if and D t t (3) 2 ∼ 0 n or ihpoaiiy07.(C 0.75. probability with {Q ≤ U = D = Q 4 1 ,0] (−3, t Q 0.5. (3) Q − t 3, Q t if , ∞ } 0 section Appendix, SI Q D ∞ = t ∞ ≤ =0 t = d t (2) 9i h other the in 0.99 lim = ihprobability with PNAS 4 1 = : 2 4 1 ∼ . h process the E . U 4 1 p (D . ,1 with 1) [0, t 0 →∞ | = t f12 of 7 ) 0.999 [30] and Q t ) ,

ANTHROPOLOGY EVOLUTION Fig. 5. Stochastic ﬂuctuation. Each plot shows 25 runs with initial p0 = 0.49 (red) and 25 runs with p0 = 0.51 (blue). Eighty generations are shown, although 6 simulations ran for 10 generations; no equilibria were reached. Mean frequencies and E[Dt (j)] are reported in SI Appendix, section A. (A) n = 5, Dt (4) = −3.9 1 1 1 1 with probability 2 , Dt (4) = −3.4 with probability 2 , Dt (3) = −2.9 with probability 2 , and Dt (3) = −2.4 with probability 2 .(B) n = 5, Dt (4) = −0.6 with 1 1 1 1 probability 2 , Dt (4) = −0.8 with probability 2 , Dt (3) = 1.85 with probability 2 , and Dt (3) = 1.95 with probability 2 .

E (Q | Q ) 1 1 d σ2 t+1 t As 0 ≤ − Q∞ ≤ and d 1 + + < 0, we have [31] 4 4 4 4 1 2 2 1 2 = Qt 1 + 4d 4 − Qt + 4 σ + d 4 − Qt .

If d > 0, then E (Qt+1 | Qt )≥ Qt for t = 0, 1, 2, ... . Thus 1 h d σ2 i ∞ 1 0 ≤ 4E Q∞ 4 − Q∞ d 1 + 4 + 4 ≤ 0. [36] {Qt }t=0 is a submartingale and since 0 ≤ Qt ≤ 4 , we can apply the martingale convergence theorem to deduce that with ∞ probability 1, {Qt }t=0 converges to Q∞. Also applying the ` ` 1 Lebesgue convergence theorem, E Qt −−−→ E Q∞ for any Inequalities in Eq. 36 imply that E Q∞ − Q∞ = 0 and t→∞ 4 1 1 ` = 1, 2, ... . Q∞ 4 − Q∞ = 0 so that Q∞ = 0 or Q∞ = 4 with probabil- Thus Eq. 31 entails that ity 1. Dynamics with n = 4 role models are illustrated in Figs. 1 B, E, and H; 2 B and D; and 3B. The conformity coef- 1 E(Q∞) = E(Q∞) + 4dE Q∞ − Q∞ ficient, Dt , is denoted by Dt (3) (Eqs. 26 and 27). Because 4 [32] 2 2 1 2 Observation 1 no longer holds with n > 3, runs that start in + 4 σ + d E Q∞ 4 − Q∞ . 1 1 1 (0, 2 ) (red) or ( 2 , 1) (blue) can cross the p = 2 line (e.g., 1 Fig. 1B). Fig. 1 B, E, and H shows examples in which all Since d > 0, we deduce from Eq. 32 that E Q∞ − Q∞ = 0 ∗ 1 4 three equilibria p = 0, , and 1 are SLS with E[Dt (3)] = 0, 1 2 1 2 and E Q∞ 4 − Q∞ = 0. But 0 ≤ Q∞ ≤ 4 with probability 1. E[Dt (3)] < 0, and E[Dt (3)] > 0, respectively. Fig. 2 B and D Q 1 − Q = 0 Q ∗ 1 Hence ∞ 4 ∞ with probability 1 and ∞ can take shows that p = 2 can be unstable with either E[Dt (3)] > 0 or 1 E[Dt (3)] < 0 E[Dt (3)] < 0 only the two values Q∞ = 0 or Q∞ = 4 . . However, is necessary for instability 1 2 1 of p∗ = 0 and 1 (Fig. 3B and Result 8). Here, the dynam- If Q∞ = 0, then limt→∞ 2 − pt = 0 and limt→∞ pt = 2 . 1 ics are similar to those of Eq. 1 with anticonformity (D < If Q∞ = 4 , then limt→∞ pt (1 − pt ) = 0. Therefore for t large, ∗ 1 ∗ ∗ 0), apart from fluctuations across p = before reaching this either pt is near p = 0 or pt is near p = 1. But if d > 0, 2 ∗ ∗ 1 equilibrium. both p = 0 and p = 1 are SLS. Therefore, if Q∞ = 4 , either Removing the assumption that Dt (3) are identically dis- limt→∞ pt = 0 or limt→∞ pt = 1. ∞ tributed, Fig. 4B shows an example in which Dt (3) takes one fixed Conclusion. If E (Dt )> 0, then the process {pt } converges t=0 value every third generation and another fixed value in the other with probability 1. If p∞ = limt→∞ pt , then p∞ can assume only 1 generations. This system produces an exact six-generation cycle the three values 0, 1, 2 . in the frequency pt . Unlike in the case of n = 3 (Fig. 4A), this The situation is more complicated when d = E(Dt ) < 0. Eq. 31 cycle occurs around p = 1 because Observation 1 does not hold. can be rewritten as E (Qt+1 | Qt )= Qt (1 + ut ) with 2

1 2 2 1 2 ut = 4d 4 − Qt + 4 σ + d 4 − Qt . [33] Convergence for Any Number of Role Models. Result 9 can be gen- eralized to apply for any number n of role models. Recall that 1 1 Since 0 ≤ 4 − Qt ≤ 4 , we have with n role models we have

1 h d σ2 i ut ≤ 4 − Qt d 1 + + . [34] 4 4 4 n ! X Dt (j ) n h j n−j n−j j i 2 2 2 pt+1 = pt + pt (1 − pt ) − pt (1 − pt ) , 4d + d + σ ≤ 0 4d + E(Dt ) ≤ 0 n j If , or equivalently, if , then j =k ut ≤ 0 and E (Qt+1 | Qt )≤ Qt , making the process a bounded [37] supermartingale so that Qt −−−→ Q∞ with probability 1. For t→∞ where k = n + 1 if n is even and k = n+1 if n is odd. Denton et D ∈ [−3, 0] σ2 ≤ −d 2 − 3d 2 2 example, this occurs if t so that by al. (49) showed that for any j ≥ k the Bhatia–Davis inequality (54). Therefore, if ut ≤ 0, a similar argument to the proof of Result 9 (Eq. 32) gives

1 2 2 1 2 j n−j n−j j 4dE Q − Q + 4 σ + d E Q − Q = 0. pt (1 − pt ) − pt (1 − pt ) ∞ 4 ∞ ∞ 4 ∞ [38] [35] = pt (1 − pt )(2pt − 1)Gj (pt (1 − pt )),

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Result distributed identically with models role 1. least at when Moreover, any for that shows algebra Simple ntedtriitcmdlwt togatcnomt (49). anticonformity strong with model deterministic the in with model stochastic the with model deterministic the in those to lar either E n in given probability and even is where nrnol hnigcnomt isi utrltransmission cultural in between bias conformity changing cycle randomly On runs al. blue et Denton and red the 4B, Fig. frequencies. produced, different in is cycle gen- six-generation unlike other exact an the but dis- 4B, in Fig. value identically in fixed As another erations. not and generation therefore third every are 4C, Fig. (and In tributed). values fixed between hl if while ∗ ∗ .Fg 3 Fig. 8). (Result 1 and ntedtriitcmdl tbeplmrhceulbi with equilibria polymorphic stable model, deterministic the in [D [D [D 5 = p Fg 1 (Fig. (4) eaenwrayt rv ovrec eutfrany for result convergence a prove to ready now are We tbecce a loocrwe h coefficients the when occur also can cycles Stable with dynamics illustrate To of proof The As 6= 0 = p t k .Ti sas osbewt both with possible also is This E). t t t t E k G D (1 E (4)] (4)], (4)] 1 2 < n (1 [D r hw nFg.1 Figs. in shown are j t E n a eSSwhile SLS be may 1 and [D G p a cu fiue1o e.4) i.5B Fig. 49). ref. of 1 (figure occur can − (3) ≥ (p 0 section Appendix, SI − [D n ∗ t j < < t and C, p (3)] nbt Eqs. both in 3, t E If 0, = (4)], saplnma ntevariable the in polynomial a is sodand odd is (1 p Moreover, 1. t k t orsodt h ae where cases the to correspond 0 0 ) [D (3)] t p and F, E n = ) − sncsaybtntsfcetfrisaiiyof instability for sufficient not but necessary is and D < −k n 6= t [ 1 2 −k E (3)] n D p eut10 Result (3) 0 < n a cu with occur can 1 and , +1 2 t 1 2 n − j 1, [D t )=[p = )) E rif or − ial,nnovrec sas osbein possible also is nonconvergence Finally, . (j = 0 5 = > p and > t × [D if I )] p (3)] t or n k epciey.Nt htaltreequilib- three all that Note respectively). , D nohrcsswith cases other In 0. k t 0 −k n " n if , t > −k (3)] E t 2j = hr r two are there E if 1 2 sodd, is C ntsoni i.1.Fxto states Fixation 1). Fig. in shown (not (4) E < 0 (1 40a t X −n [D ssmlrt h ro of proof the to similar is i . n p [D =0 (1 n p (1 and [D o all for E ∗ 0 ∞ > see and even is − t −1 and 1 + t 2 = − [ (4)] − ntsoni i.2.Tecondition The 2). Fig. in shown (not (3)] t C, B. and D or 0, p lim = 4]= (4)] n D (1 p p p h process the t t 1 2 , ) (4)], t t and F, ∗ D > E < hw xmlsi which in examples shows k )] k ) > − sSSwith SLS is 40b = k t 0 [p = [D n 4 ≤ E 0 (3) t hs yaisaesimi- are dynamics These 0. p [p = −j ≤ →∞ D 1 2 oemdl,eape with examples models, role t E [D and t j ) h oe of power the (4)], −1 p susal if unstable is E (4) t i < [D D ahtk n xdvalue fixed one take each E k t t p (1 D t I (4)] [D t p (1 ≤ t 2j [D n = 2 ; t t (j E t {p − (3)] . , and −n , t E 1 D − t p [ ) n 2 (4)], where n then D 4]= (4)] p t > t [D j C (4) −1i } parameters: 1 + (1 p t noprtsthese incorporates t 5 = < 4 = )] E t ∞ 0 t (3)] t =0 E and )] − (3)] n [D 2 E and 0 < p then , [ , n −1 D k eut9 Result # ∞ p ovre with converges [D and 5A) (Fig. D E −1 0 t 2 r3 respec- 3, or > [p E ≥ t t = (4)] < 3 E; ) (2p and [ 3]= (3)] (4), a assume can E t t [D 0 D (2p 0. (3)] (1 for n 2 0 [D t Fg 2 (Fig. D t t < 3]=0, = (3)] 1 + (4)] rboth or D D − t − t t C p > (3)] n 0 − n is and D [40b] [40a] n (3) (3) D p ∗ cycle 1), p [39] > t 0 t if and and 1). 0 = > ≥ ∗ (3) (4) )] or = > < < 4. C n is 0 3 fe eeto,tefeunyof frequency the selection, of after fitness the that assume we ants Selection. Random Weak B. section Appendix, of proofs the (where 12. Result 2) 1) 11. Result models eralize sueta eeto swa othat so weak is Eq. approximate selection that Assume ues lim case, this In generation? each at frequency the randomly change can models in described erties Models. Role of Number Random itiue admvrals Then before, variables. random as distributed that, and variables random tributed all for that now Suppose each For 1) t n 19 Given 2) For p ,1 2, 1, 0, = 0 = ' rmEqs. From o n generation any For (n and probability D If, p If k admvariables random for constraints the to subject are . . . 5, 4, 3, of pendently t ,1, 0, →∞ |s t ∞ n for and p 1+ (1 A n ≥ D = E t t t , | (k ihprobability with and + 21) k n p D eut 7 Results frl oesas odwe h ubro role of number the when hold also models role of 3) naddition, in n [D ml,w aetefloigresult: following the have we small, n t t p t 2 1 t t t n 0 = (0) n +1 s j = t X 2 ), +1 . . . h rosof proofs The . t =k n t . . . )p scoe admya ahgeneration. each at randomly chosen is = −1 n (n B p t If D n (j j = t t t 2 , if p t t +1 1+(1 [1 + h ofriycoefficients conformity the , 42a t p n noteBy n ihro 4 oe with model (4) Richerson and Boyd the into ,2, 1, = 1. eut 7 Results N ) D h ubro oemodels role of number the , eed on depends ) ∞ p p t 1 + 0 D n E (k | t t = n r admvralsidpnetof independent variables random are ' − p hc stesm o all for same the is which , n +1 n t n t + 43 ihprobability with and and t t [D t t t (j sodd), is rmapoaiiydsrbto ntevalues the on distribution probability a from , eut 7 Results 1+ (1 s 1+ (1 (n 1), + 0 = ] if X t j + 1 ftevariant the of − ) by n n =0 p . . . t t t

n t 2 42b 0, = ) 10 (j t p 1. t (where given n 1+(1 + [1 s s sp and D ,1 2, 1, 0, = )(2p , j oicroaentrlslcino vari- on selection natural incorporate To o all for ) t see and even is . . . )p t n t )p otecs hr h ubro role of number the where case the to n ! | n ehv h olwn eut htgen- that results following the have we then t 1+(1 + [1 p t (t n eut 11 Results , t 1+(1 + [1 (j t and , t h t epciey n r ie in given are and respectively, 10, −1, − , D 1+(1 + [1 ]> ,1 2, 1, 0, = p A ) n t D j − j n n

1) p k t (1 otegoa ovrec prop- convergence global the Do t is 0 10 t ,1, 0, = and , n ∞ t A https://doi.org/10.1073/pnas.2107204118 −j (n D n p . . . t j − = 1 − t + 1 t (j − A ]−sp sgvnby given is o n xdcntn number constant ﬁxed any for sue h values the assumes t )(2p ! for h ofriycoefﬁcients conformity the and p < − = ) j n − p k 2 ete write then we p p )(2p tgeneration at t ,1 , . . . 1, 0, = t t D and s t D . . . )(2p p j ) 1) 1 + = . . . (1 t n n t 2 −D eaiet for 1 to relative n j p t t )(2p − 1+(1 + [1 t −j where , n ), ∞ = − (j − , 12 t t | 1)D 2 − +1 s n ; n s k if ) − p | 1)D t a sueteval- the assume can t t t t n r independent are t < 1)D (n , r eysmlrto similar very are , − r dnial dis- identically are ) p ssal ecan We small. is t if then D t k n n t − D n n t ] t t scoe,inde- chosen, is 1)D 2 ] t t t −j t n − nt n 1, + −j . ] p . r identically are − t t PNAS p )(2p j (0), see and even is lim t (1 sodd), is t ) t j 1 + +1 ] ,1, 0, . . . . − t →∞ p B − s(Eqs. as p | t D , o any for They . Then . t n 1) n 1 2 [42b] [41b] n [42a] [41a] ) f12 of 9 t j t p [43] [44] [45] D then with 3 = (1), i − t . ] SI 2 = 1 . ,

ANTHROPOLOGY EVOLUTION ∗ 1 Result 13. For any t = 0, 1, 2, ... , suppose that st and Dt are inde- are three equilibria at p = 0, 2 , and 1, with convergence to one of pendent of each other, both are independent of pt , E(Dt ) = 0, and these. Others have implemented qualitatively different formula- E(st ) 6= 0. Then with probability 1, limt→∞ pt = p∞ and p∞ = 0 tions of conformity that also include a single, constant conformity or p∞ = 1. coefficient and have found more complex dynamics (56, 57). 2 Proof: Let E(st ) = µ and Var(Dt ) = σ . Then, as A model of “strong anticonformity” was explored in ref. 56, where individuals acquire a variant that is sampled at a frequency 2 [1 + (1 − pt )(2pt − 1)Dt ] of 0 with probability 1 (i.e., the left-hand side of Eq. 20a does 2 2 2 [46] not hold). In that model, stable cycles of variant frequencies can = 1 + 2(1 − pt )(2pt − 1)Dt + (1 − pt ) (2pt − 1) Dt , occur with n = 3 role models. In refs. 56 and 57, a model of conformity was studied in which there is one cultural variant of Eq. 45 implies that interest, such as an item of learned information. Conformity or 2 2 anticonformity occurs if this variant is present in the majority of E (pt+1 | pt )' pt 1 + µ(1 − pt ) 1 − pt (1 − pt )(2pt − 1) σ . sampled role models, but if the variant is present in a minority of [47] role models, conformity or anticonformity to the opposite type 2 2 (absence of information) does not occur (i.e., the right-hand side Since −2 < Dt < 1, we have σ = E Dt < 4, and as 0 ≤ pt ≤ 1 we have of Eq. 20a does not hold). In ref. 57, social and individual learn- 2 2 1 ers are in environments subject to change, and stable cycles in pt (1 − pt )(2pt − 1) σ < · 1 · 4 = 1. [48] 4 variant frequencies as well as chaos may occur. Finally, ref. 58 Thus if µ > 0, then E (pt+1 | pt )≥ pt , and if µ < 0, then 1 ∞ included a modified version of Eq. in a susceptible, infectious, E (pt+1 | pt )≤ pt , and the process {pt }t=0 is either a bounded susceptible (SIS) epidemic-type model and found polymorphic submartingale or a bounded supermartingale. The martingale ∗ 1 equilibria with p 6= 2 . They also found that under some condi- convergence theorem then ensures that limt→∞ pt = p∞ with tions, a polymorphic equilibrium and a fixation state could be probability 1. stable simultaneously. We can also classify p∞. For example, if µ = E(st ) > 0, then We previously showed that complex dynamics including cycles, Eq. 47 implies that ∗ 1 chaos, and stable polymorphic equilibria with p 6= 2 are possible under Boyd and Richerson’s (4) general model of conformity 2 2 E(pt+1) ≥ E(pt ) +µ Ê pt (1 − pt ) 1 − pt (1 − pt )(2pt − 1) σ specified by Eqs. 19 and 20 (49). This model includes confor- [49] mity coefficients that are fixed over time but can differ in sign and magnitude depending on the number of role models sam- for any 0 < µˆ < µ. Since pt −−−→ p∞ and 0 ≤ pt ≤ 1, for t→∞ pled that carry the same variant. For example, in ref. 19, fruit ` ` all t = 0, 1, ... , E(pt ) −−−→ E(p∞) for any ` = 1, 2, ... . Then flies that observed 60% of conspecifics copulating with pink (or t→∞ from Eq. 49, green) males showed a greater than 60% chance of copulating with the more commonly chosen male (i.e., conformist bias), E p (1 − p ) 1 − p (1 − p )(2p − 1)2σ2 ≤ 0. [50] whereas those that observed 83% of conspecifics copulating with ∞ ∞ ∞ ∞ ∞ pink (or green) males showed a less than 83% chance of copu- But Eq. 48 entails that lating with the more commonly chosen male (i.e., anticonformist bias). Including both conformity and anticonformity that depend 2 2 on the numbers of A and B in the sample of role models is p∞(1 − p∞) 1 − p∞(1 − p∞)(2p∞ − 1) σ ≥ 0, [51] ∗ 1 necessary to allow polymorphic equilibria with p 6= 2 , but is not necessary to produce cycles or chaos. Cycles or chaos can which means that p∞(1 − p∞) = 0 with probability 1. We there- ∗ occur when n ≥ 5 or n ≥ 10, respectively, and anticonformity is fore infer that with probability 1, either p∞ = p = 0 or p∞ = p∗ = 1. It should be pointed out that Result 13 also applies to the sufficiently strong (49). The present analysis generalizes previous models by allowing case when st = s = µ is constant over time. Observe that both p∗ = 0 and p∗ = 1 are constant equilibria the conformity coefficient(s) to vary over time. Temporal variation in conformity may account for differences in baby name even though st and Dt are random variables. Following ref. 49, the SLS of these equilibria is determined as follows: turnover in recent versus older decades (12), and individuals’ ten- dency to exhibit a related bias, namely Aschian conformity, has ∗ decreased over time (29). Thus, conformity coefficients may be p = 0 is SLS if E {log[(1 + st )(1 − Dt )]}< 0 better regarded as random variables than fixed parameters. At 1 − D [52] p∗ = 1 is SLS if E log t < 0. each generation, t, conformity coefficient(s) are sampled from a 1 + st continuous or discrete probability distribution. With n ≥ 5 role models there are multiple conformity coefficients in the model Thus, using Jensen’s inequality, if E(Dt ) ≥ 0 and E(st ) ≤ 0, both (Eq. 19), which may be sampled from the same or from differ- ∗ E [log(1 + st )] and E [log(1 − Dt )] are negative and p = 0 is ent probability distributions, whereas with n = 3 or 4 there is a h (1−Dt ) i SLS. If st are small for t = 0, 1, 2, ... , then E log ≈ single conformity coefficient. All members of generation t display (1+st ) ∗ the same levels of conformity, Dt . We have not considered indi- E {log[(1 − Dt )(1 − st )]}, and p = 1 is SLS if E(Dt ) ≥ 0 and vidual variation in conformity, which may be important in real E(st ) ≥ 0. If st ≡ s is constant, then if E(Dt ) ≥ 0, populations. We assume that Dt is sampled independently of the population frequency, pt [although the probability of sampling j p∗ = 0 is SLS if − 1 ≤ s ≤ 0 of n role models with conformity coefficient Dt (j ) does depend on ∗ [53] p = 1 is SLS if s ≥ 0. pt ]. Association between Dt and pt , such as increasing all conformity coefficients Dt (j ) with increasing pt , could be incorporated Discussion in future models to determine whether these relationships affect Previous analytical models of conformity have included constant the presence or stochastic local stability of equilibria. rather than variable conformity coefficient(s). Many of these In addition, we analyzed the role of variation in the number of include a single conformity coefficient and dynamics that follow role models, nt , which is always finite and greater than or equal Eq. 1 (4, 30–33, 35–37) or a similar formulation (32, 38, 39, 55). to 3. In agent-based models of conformity on networks, the num- These, dynamics are represented by an S-shaped curve and there ber of sampled role models can vary by individual (43, 45, 46)

10 of 12 | PNAS Denton et al. https://doi.org/10.1073/pnas.2107204118 On randomly changing conformity bias in cultural transmission Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 ntbeand unstable if anteed expected, is copying role random of number time-varying 1 or Fig. fixed models. a with holds result vergence here model, conformity epciey oevr with Moreover, respectively. 1 Fig. hand, (n E (i.e., where unequal equilibria and be cannot nonzero there are hence, frequencies other; the the increase to to variant relative one cause can (anti)conformity in fluctuations for variables random independent if ever, initial any [D and variant uncommon an or common 0 a favor not do viduals model. stochastic the conformity to with approximation reasonable 1 equation (4) Richerson’s expected, models, role of number the whether true is n This ensured. also is t librium (variant ant (p nrnol hnigcnomt isi utrltransmission cultural in bias conformity changing randomly On al. with et Denton variant common A occur. also can nonconvergence mity, conformity of model (4) Richerson’s anticonformity. than and Boyd with consistent while unstable be with SLS be can 1 across fluctuations the from (apart p model anticonformity 3 (4) Fig. of with model dynamics stochastic expected, the In anticonformity conformity. with dynamics ministic all hold also varying. selection randomly than for rather results constant, these is of selection all if Finally, SLS. are 1 and variant favor variant to favor to selec- weak expected if is a expected, tion conformity to or convergence copying random case, either this and In (p other. state the fixation to relative average, on as itive, of one to gence frequency initial p with 1 population (Fig. A SLS be can if equilibria three all model stochastic the [i.e., In expected attraction. conformity with of domains two the separates (e.g., transmission information pop- of a modes in media). individuals in social of changes dispersion or role the of in ulation of members changes numbers individu- example, to different all For due models encounter time. for role given may same of a generations the number at different is population the but the model, time, in present als over the change In can (43). models time over and all 2 1 t ∗ ∗ ,1, 0, = (s < o all for sfie rsohsial aibe niei h deterministic the in Unlike variable. stochastically or fixed is , ntedtriitcmdlwt admcpig hr indi- where copying, random with model deterministic the In nbt eemnsi n tcatcmdl ihanticonfor- with models stochastic and deterministic both In [D anticonformity with model deterministic the In With [D conformity with (4) model deterministic the In n > 4 = = (p 1 = n 2 n t 2 (p p > ) 1 2 < 2 1 < < 0 6= r esnbyapoiae yBy n Richerson’s and Boyd by approximated reasonably are ) ) 3, < j E ro ocnegnei h case the in convergence to prior j n nrae o1adtefeunyo nucmo vari- uncommon an of frequency the and 1 to increases . . . p .A ntemdlwt ofriyepce,ti con- this expected, conformity with model the in As 1). E A j < p < 2 1 ovrec to convergence , ∗ [D ∗ 3 = n (s n ) ∗ xd and fixed) nEq. in 0) = = n ,cnegnet n fteequilibria the of one to convergence ], n D t 3 = t = 3 = erae o0 hs tbeeulbi are equilibria stable Thus, 0. to decreases (j ) nEq. in 1 2 p and nEq. in 0 6= oemodels, role p A–C 1 2 ]=0 = )] ∗ 0 ∗ qa rqece of frequencies equal and , p Fg 1 (Fig. B or = edntsprt h oan fattraction: of domains the separate not need 0 = sa qiiru.I h tcatcmdl how- model, stochastic the In equilibrium. an is ∗ ,teei ocag nvratfrequencies variant in change no is there 19], p , E nal htoevrati eetvl favored, selectively is variant one that entails n 0, = ∗ hw htif that shows n 1 2 p ,cnegnet neulbimi guar- is equilibrium an to convergence 19], hw htaltreequilibria three all that shows 3 = 0 = ∗ o all for r1 cuswt rbblt .With 1. probability with occurs 1) or p 4 = ,tefeunyo omnvariant common a of frequency the 19], sgoal tbe poiet h deter- the to opposite stable, globally is p 0 = ∗ H ∗ 2 1 0 = p r4and 4 or .O h te ad fcnomt is conformity if hand, other the On ). ,arsl more result a 2D), (Fig. SLS are 1 and r1wt rbblt .I hsmodel, this In 1. probability with 1 or , = 4) nwihcase which in (49), ∗ sSS n if and SLS; is p = E j 1 2 (variant ∗ p n (D n all and = ∗ 1 2 E ,By and Boyd 2A), (Fig. unstable is 4 = 0, = A, [D edntb ntbe n thus and unstable, be not need t n 1 2 n ekslcinwith selection weak and 0, = ) E ,4 3, = t p and osntocr hsi intu- is This occur. not does j (j [D ∗ 2 1 B > p t )] 1 = n r l SLS. all are 1 and , A t 0 ,1, 0, = E (2)] xd,weesteequi- the whereas fixed), > n 2 < r5(epciey and (respectively) 5 or , and [D hnteei conver- is there then , G E sSS fi sexpected is it if SLS; is 0 n 2 1 (s < t for and 4 = (3)] (p (D t B . . . 0 both 0, = ) p A 0 n 2 or .O h other the On ). ∗ > susal and unstable is > < and , H < (n 0 = 0 p p E 0) 0, 2 1 .Moreover, ). ∗ < ∗ j ) 3 = [ 0, = < 0, = D p (j (j rvdsa provides p n are 1 and a reach can D ∗ t n < ) ) and ) (3)] = t > < (j n all and p p 2 1 1 2 1 2 n 1 2 (j r1 or , ∗ ∗ 0 0 ) and , < and 1 = 0 = 3 = can = ) are for for 0, B p distributed with identically sible of case the where distributed tically model. conformity equilibria: with 49, ref. and of 1 figure In samples different for possible that such mity frequency aefralidvdasi ouain(1 6,wihw have we conformity which in 56), (41, variation population individual-level a Incorporating the in assumed. be individuals not all poly- need for these conformity same of of levels stability Finally, local conformity equilibria. stochastic morphic in allow or variation eliminate stochastic would incorporating deter- whether to interesting mine be would it models, multipopulation infinite, of number (D given the a for in model, variation how models, explore role cycling could fixed, research with tional distributed examples identically in of only dis- multivariate removed assumption same the the and from generation tribution, each at sampled was with cycle frequency two-generation exact an with equilibria to polymorphic close stable frequencies the average around occurred fluctuation model, with equilibria morphic 5B case stochastic gous around around cycle two-generation a illustrates 49 p ref. in 2 Figure that on. such so favored strongly be then a rdc upiigplmrhcequilibria polymorphic surprising sizes population produce infinite can with model conformity Richerson’s deterministic and (4) Boyd into migration dynamics. incorporating population example, For alter greatly also may polymorphic Migration of equilibria. frequencies elimination in variant resulting populations, in infinite than fluctuation populations stochastic Finite greater conformity. experience in variation individual-level them, and among migration with subpopulations sample populations, infinite smaller the as population-level individuals, of com- indicator sampled reliable variant variation. less a 10 a as perceived share of be individuals might 7 sampled to 100 be of pared might 70 (anti)conformity example, if For stronger important. be and between exist relationships known any include coefficients conformity In of contexts. vector discernment the also analysis, mental might our similar conformity of decreased in much (4) time tasks Richerson over and discrimination decrease Boyd line (29), in time in over trends conformity just example, deterministic Aschian For models. reveal as new also stimulating time, may over conformity be research to Such which conformity priate. in hence variation and temporal estimated of generations, over amount group age the fixed allowing a of members by exhibited mity coefficients, conformity for and constant only possible Boyd is with in nonconvergence than where model complex (4) more are Richerson’s dynamics These 4B). (Fig. 0.1929 ∗ = hsfr u icsinhsprandt ae ihiden- with cases to pertained has discussion our far, Thus with Finally, te xesoso u nlsscudmdlfiierte than rather finite model could analysis our of extensions Other confor- of level the explore could research empirical Future t = esesohsi utainaon vrg frequencies average around fluctuation stochastic see we ) 1 2 D ssmldfo rbblt itiuin u ed not do we but distribution, probability a from sampled is 2 1 rsetvl) With (respectively). 3 1. = (3) D (if with p t ∗ p nta ylsbtenfie aus ee niein unlike Here, values. fixed between cycles instead p = p p 0 n t n 0 ∗ < n > 2 1 3 = > 0 = 5 = 9 t ssoni i.5A. Fig. in shown is sascae ihtelvlo ofriy nour In conformity. of level the with associated is , 2 1 p 2 1 n cnomt) hr r w tbepolymorphic stable two are there (conformity), 2 1 or ) t .1927 ntesohsi oe with model stochastic the In . +1 a esrnl ifvrddet anticonfor- to due disfavored strongly be may > oemdl.Fg 4A Fig. models. role or D n ohcnomt n niofriyare anticonformity and conformity both 4, E t p t swl below well is hsasmto srmvdi i.4, Fig. in removed is assumption This . p [D = 0 = 0 and p n < t 4]= (4)] ∗ .8072 an , n 6= 2 1 n 1 h yl cusaoeo below or above occurs cycle the , 4 = 5 = − j D 2 1 n of t eentfudi h stochastic the in found not were p −0.7 (if https://doi.org/10.1073/pnas.2107204118 ylscnocraround occur can cycles , dmninlcnomt vector conformity -dimensional , ∗ itiuin ih eappro- be might distributions n D 0 = p p p oemdl ihoevariant. one with models role D 4 = (4) t 0 +2 = n t and > . ocnegnei pos- is nonconvergence , p ee nteanalo- the in Here, 8073. 3 = 1 2 swl above well is ∗ 2 1 hsrr ain may variant rare this ; hw neapeof example an shows .Tu,atog poly- although Thus, ). 6= −0.7 E D n hr fteinitial the if where , [D t 2 1 ≥ and D t rmteconstant the from n 5 3]=1 = (3)] (p (anticonformity) t 5 = (49). Fg ) Addi- 4). (Fig. ∗ PNAS n 6= t hc may which , fluctuation , p 2 1 nFig. in .9, ) = | 4) In (49). D 1o 12 of 11 1 2 p t and , (D = p was = t 1 2 )

ANTHROPOLOGY EVOLUTION could produce more realistic dynamics, as well as enable a more age, only fixation states p∗ = 0 or 1 are reached as t → ∞. With direct application to empirical research on individual biases. anticonformity expected, nonconvergence is possible under dif- In conclusion, if conformity coefficients vary over time, evo- ferent conditions from those in previous deterministic models lutionary dynamics are seen that are not possible with constant (49). Finally, when conformity coefficients are not identically dis- conformity coefficients. In the constant conformity model, at tributed, exact cycles can arise with as few as n = 3 role models. least one equilibrium must be unstable (49), whereas here all Thus, temporal fluctuations in the patterns of conformity can equilibria can be stochastically locally stable simultaneously. We cause significant departures from the conclusions that have been find that irrespective of whether n is constant or time vary- drawn from models with constant levels of conformity. n ing, if conformity is expected [E[Dt (j )] > 0 for all 2 < j < n] or random copying is expected [E[Dt (j )] = 0 for all j and Dt (j ) Data Availability. There are no data underlying this work. p∗ = 0, 1 are independent], convergence to an equilibrium 2 , or ACKNOWLEDGMENTS. This research was supported in part by the Morrison 1 is ensured. If random copying is expected, n = 3, and there is Institute for Population and Research Studies at Stanford University and the weak selection that favors one variant over the other on aver- Stanford Center for Computational, Evolutionary, and Human Genomics.

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