| Best-fitting scenarios across adult Homo, and resulting predicted life history for H. sapiens. a, Best adult fitting scenarios across Homo (figure modified with permission from figure 8.1 of ref. 1 ). Pie charts and plots respectively show the challenge combination and shape of EEE versus skill that yielded the best adult fit using the same Q and R parameters (Extended Data Figs. 6, 7). b, Life history with the challenge combination yielding the best adult fit for H. sapiens. Resulting life periods are indicated above the body mass plot. Vertical lines are ages at which the growth strategy changes suddenly; within childhood, they occur when brain growth begins and terminates. 

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... the value for the metabolic cost of memory B k is in information units, the model makes predictions for skill level in such units. In particular, the resulting predicted skill level for the best fitting scenario for H. sapiens in Fig. 4b in the main text is x § k (ø a ) = 3.92 TB (Extended Data Fig. 10e). For comparison, we contrast this value with available, preliminary estimates of the human brain's information storage capacity. Neuropil in the rat hippocampus is estimated to sustain 4.7 bits of information per synapse 37 and the human neocortex is estimated to have 0.15 quadrillion synapses 38 , which would very roughly suggest 587.5 TB of storage in the human ...

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... work 1 showed that varying ' 0 affects body and brain mass early in ontogeny but has little effect on their adult values. We then took the ten P-parameter combinations that yield the best adult fit with H. sapiens across the six cases considered. All such 10 combinations occurred in the case of exponential competence with submultiplicative cooperation (in which case, ' 0 = 0.6). For each of these 10 combinations, we obtained uninvadable growth strategies with ' 0 2 {0.4, 0.45, 0.5} to find a ' § 0 and a combination of P that yielded a solu- tion maximizing ontogenetic fit °E[D(ø)]. The resulting best ontogenetic fit occurred for the same parameter combination P § but with the value ' § 0 = 0.5 ( Fig. 4b in the main ...

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... the model considers females only, we use only female values when available. We used observed adult val- ues of brain and body sizes for H. sapiens [body (51.1 kg) and brain (1.31 kg) for females 15 Here we describe how we identified the value of maternal provisioning at birth (' 0 ) that locally maximized ontogenetic fit in Fig. 4b of the main text after having maximized adult fit. In our exhaustive search across the P-parameters, we identified the P § -parameter combination that maximized adult fit °D(ø a ) for H. ...

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... solutions were consistently replicated with the software and hardware specified in the Supplementary Information. Replication using other software or hardware may require modification of the solver setup to obtain stable costate estimation as described in the Supplementary Information section 5. Analyses leading to Fig. 3 and 4 can be replicated without optimal control software using the data deposited in Zenodo with the identifier https://doi.org/10.5281/zenodo. 1197479 ...

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... For instance, P 1 = 1 denotes that individuals face only ecological challenges, whereas P 1 = P 2 = 0.5 denotes that individuals only face ecological and cooperative challenges and with equal proportions. We define the growth-metabolic rate as the rate of heat release by a resting individual due to tissue production. Moreover, we define the growth strategy as the fraction of the growth-metabolic rate due to the production of each tissue throughout life. Thus, the growth strategy generates an ontogenetic profile of brain and body size. We consider that the growth strategy evolves by natural selection, and study its evolution using standard evolutionary-invasion analysis; that is, we consider the increase in frequency by selection of rare genetic mutations that control the growth strategy. There is a stable monomorphic female brain size in the population when rare mutants of the growth strategy cannot invade the population; that is, such resident growth strategy is 'uninvadable' 31,32 . We obtain an uninvadable growth strategy using evolutionary-invasion analysis for function-valued traits, since the growth strategy is a function of time (age). Because skill level depends (though not exclusively) on brain size due to energy conservation principles 24 , the evolution of brain size causes the evolution of skill level. Accordingly, a cooperating partner's skill level and the difficulty of competitive challenges are evolving environments, which constitute the ulti- mate distinction between ecological and social challenges in our analysis. This evolving environment implements the notion that sociality can yield evolu- tionary arms races in cognition as proposed by social hypotheses [8][9][10] . Energy-extraction efficiency. An important quantity in the model is the individual's EEE, defined as the rate of energy extraction divided by the rate of energy extraction if the individual is maximally successful at energy extraction. We model the individual's EEE as a function of her skill level and that of cooperating or competing peers. To do this, we consider two mathematical functions commonly used in contest models: a 'power competence' function that allows for strongly decelerating EEE as the indi- vidual gains skills when she is young, and an 'exponential competence' function that allows for weaker deceleration ( Fig. 2b and Extended Data Fig. 1c). We also let the skills of cooperating partners interact in an additive, multiplicative or submultiplicative (geometric mean) way (the geometric mean is a good descriptor of the average skill in the pair if peers have disparate skill levels). Additionally, we assume that if a sufficiently young individual fails to overcome a challenge, then she can extract energy from an environment facilitated by her mother. Parameters. The model has 4 basic parameters, collectively denoted by P, that specify the proportion of each social challenge, and the effects of which we study here; 13 further parameters, collectively denoted by Q, that measure the metabolic costs of the brain and other tissues, the size of the brain and other tissues at birth and the demography of the population, for which empirical estimates are available; and a final 9 parameters, collectively denoted by R, that measure skill metabolic costs, maternal provisioning, mutation size and how skill level affects energy extraction, for which we use reasonable values given the available data ( Fig. 2c; Supplementary Information 4). For example, R parameters include the metabolic cost of memory and the values we use for this (in megajoules per year per terabyte) fall within an empirically estimated range for resting energy consumption for stored motor patterns in cerebellum Purkinje cells in rats 33 . The exact values used for R are chosen within such reasonable ranges as they yield a high ontogenetic fit between predicted and observed body and brain mass in H. sapiens when there are only ecological challenges (that is, P 1 = 1; Extended Data Fig. 3g, h). This approach is a rea- sonable starting point given that the fundamental constraint for a large brain is thought to be the metabolic costs of brain, which are incorporated in the estimated Q parameters. The values chosen for the R parameters mean that the difficulty of ecological challenges is high but not exceedingly so, memory is metabolically expensive (although in the low end of the empirically estimated range), and skills are moderately effective at overcoming the challenges. Using these Q and R parameter values, it was previously shown that ecological chal- lenges alone can generate adult brain and body sizes of ancient human scale: of late H. erectus scale with strongly decelerating EEE and of Neanderthal scale with weakly decelerating EEE 24 . Here we use the same Q and R parameter values to study the effects of the social-challenge parameters P. Key equations. We assume that the population is large and mostly constituted by individuals with a resident growth strategy and by vanishingly rare individuals with a mutant growth strategy. At age t, a focal mutant individual has a mass of tissue i (for i ∈ {b, r, s} for brain, reproductive and somatic) of x i (t) (in kilograms) and a skill level of x k (t) (in terabytes). The growth rate of tissue i ∈ {b, r, s} ...

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... find that increasing the proportion of cooperative challenges decreases both adult absolute brain size (hereafter 'brain size') and adult relative brain size (hereafter 'encephalization quotient' , which is the adult brain size divided by expected brain size for a given body size 2 ; Fig. 3a-c and Extended Data Fig. 4). By contrast, increas- ing the proportion of between-individual competitive challenges increases brain size when EEE is weakly decelerating with skill ( Fig. 3a), but decreases brain size when EEE is strongly decelerating (Fig. 3b and Extended Data Fig. 4). However, although between- individual competition increases brain size with weakly decelerating EEE, the result is larger brains and smaller bodies than those of modern humans ( Fig. 3a and Extended Data Fig. 4). Between- individual competition also decreases body mass as it increases the difficulty of energy extraction and thus limits the energy available for body growth; consequently, between-individual competition increases the encephalization quotient, either because brain size increases and body size decreases or because brain size decreases and body size decreases more strongly than brain size. Increasing the proportion of between-group competition generally decreases brain size, but increases the encephalization quotient, because body size decreases more strongly than brain size (Fig. 3a, b) ...

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... find that increasing the proportion of cooperative challenges decreases both adult absolute brain size (hereafter 'brain size') and adult relative brain size (hereafter 'encephalization quotient' , which is the adult brain size divided by expected brain size for a given body size 2 ; Fig. 3a-c and Extended Data Fig. 4). By contrast, increas- ing the proportion of between-individual competitive challenges increases brain size when EEE is weakly decelerating with skill ( Fig. 3a), but decreases brain size when EEE is strongly decelerating (Fig. 3b and Extended Data Fig. 4). However, although between- individual competition increases brain size with weakly decelerating EEE, the result is larger brains and smaller bodies than those of modern humans ( Fig. 3a and Extended Data Fig. 4). Between- individual competition also decreases body mass as it increases the difficulty of energy extraction and thus limits the energy available for body growth; consequently, between-individual competition increases the encephalization quotient, either because brain size increases and body size decreases or because brain size decreases and body size decreases more strongly than brain size. Increasing the proportion of between-group competition generally decreases brain size, but increases the encephalization quotient, because body size decreases more strongly than brain size (Fig. 3a, b) ...

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... find that increasing the proportion of cooperative challenges decreases both adult absolute brain size (hereafter 'brain size') and adult relative brain size (hereafter 'encephalization quotient' , which is the adult brain size divided by expected brain size for a given body size 2 ; Fig. 3a-c and Extended Data Fig. 4). By contrast, increas- ing the proportion of between-individual competitive challenges increases brain size when EEE is weakly decelerating with skill ( Fig. 3a), but decreases brain size when EEE is strongly decelerating (Fig. 3b and Extended Data Fig. 4). However, although between- individual competition increases brain size with weakly decelerating EEE, the result is larger brains and smaller bodies than those of modern humans ( Fig. 3a and Extended Data Fig. 4). Between- individual competition also decreases body mass as it increases the difficulty of energy extraction and thus limits the energy available for body growth; consequently, between-individual competition increases the encephalization quotient, either because brain size increases and body size decreases or because brain size decreases and body size decreases more strongly than brain size. Increasing the proportion of between-group competition generally decreases brain size, but increases the encephalization quotient, because body size decreases more strongly than brain size (Fig. 3a, b) ...

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... determine if any combination of social-challenge parameters P yields an accurate prediction of adult brain and body sizes of H. sapiens and closely related species, we obtained solutions exhaustively across the P parameter space while holding the other parameters (Q and R) fixed ( Fig. 3d, e; Supplementary Information 5). We find near-perfect adult fits across Homo species (Fig. 4a and Extended Data Figs. 6-8). A near-perfect adult fit for H. sapiens occurs with a large proportion of ecological challenges (approximately 60%), a moderate proportion of cooperative challenges (around 30%), a small proportion of between- group competitive challenges (around 10%), and an approximately complete absence of between-individual competitive challenges (around 0%) (Figs. 3e, 4a and Extended Data Figs. 6, 9). In the resulting reconstruction for Homo, ecological challenges increase brain size whereas social challenges decrease it (Extended Data Fig. 4), the pro- portion of ecological challenges tends to increase from early to late Homo species, and a steep increase in encephalization quotient from Homo ergaster to Homo heidelbergensis is due to a transition from strongly to weakly decelerating EEE (Fig. 4a). The adult best-fit eco- social scenario for H. sapiens also yields a predicted life history that closely matches the species' life-history timing ( Fig. 4b and Extended Data Fig. 10). The resulting ontogenetic fit is high for body size, but lower for brain size early in ontogeny (Fig. 4b), perhaps caused in part by our use of a power-law relationship between resting metabolic rate and body mass that underestimates resting metabolic rate early in the ontogeny 24 . With the adult brain size resulting from the best-fitting scenario for H. ...

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... determine if any combination of social-challenge parameters P yields an accurate prediction of adult brain and body sizes of H. sapiens and closely related species, we obtained solutions exhaustively across the P parameter space while holding the other parameters (Q and R) fixed ( Fig. 3d, e; Supplementary Information 5). We find near-perfect adult fits across Homo species (Fig. 4a and Extended Data Figs. 6-8). A near-perfect adult fit for H. sapiens occurs with a large proportion of ecological challenges (approximately 60%), a moderate proportion of cooperative challenges (around 30%), a small proportion of between- group competitive challenges (around 10%), and an approximately complete absence of between-individual competitive challenges (around 0%) (Figs. 3e, 4a and Extended Data Figs. 6, 9). In the resulting reconstruction for Homo, ecological challenges increase brain size whereas social challenges decrease it (Extended Data Fig. 4), the pro- portion of ecological challenges tends to increase from early to late Homo species, and a steep increase in encephalization quotient from Homo ergaster to Homo heidelbergensis is due to a transition from strongly to weakly decelerating EEE (Fig. 4a). The adult best-fit eco- social scenario for H. sapiens also yields a predicted life history that closely matches the species' life-history timing ( Fig. 4b and Extended Data Fig. 10). The resulting ontogenetic fit is high for body size, but lower for brain size early in ontogeny (Fig. 4b), perhaps caused in part by our use of a power-law relationship between resting metabolic rate and body mass that underestimates resting metabolic rate early in the ontogeny 24 . With the adult brain size resulting from the best-fitting scenario for H. ...

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... determine if any combination of social-challenge parameters P yields an accurate prediction of adult brain and body sizes of H. sapiens and closely related species, we obtained solutions exhaustively across the P parameter space while holding the other parameters (Q and R) fixed ( Fig. 3d, e; Supplementary Information 5). We find near-perfect adult fits across Homo species (Fig. 4a and Extended Data Figs. 6-8). A near-perfect adult fit for H. sapiens occurs with a large proportion of ecological challenges (approximately 60%), a moderate proportion of cooperative challenges (around 30%), a small proportion of between- group competitive challenges (around 10%), and an approximately complete absence of between-individual competitive challenges (around 0%) (Figs. 3e, 4a and Extended Data Figs. 6, 9). In the resulting reconstruction for Homo, ecological challenges increase brain size whereas social challenges decrease it (Extended Data Fig. 4), the pro- portion of ecological challenges tends to increase from early to late Homo species, and a steep increase in encephalization quotient from Homo ergaster to Homo heidelbergensis is due to a transition from strongly to weakly decelerating EEE (Fig. 4a). The adult best-fit eco- social scenario for H. sapiens also yields a predicted life history that closely matches the species' life-history timing ( Fig. 4b and Extended Data Fig. 10). The resulting ontogenetic fit is high for body size, but lower for brain size early in ontogeny (Fig. 4b), perhaps caused in part by our use of a power-law relationship between resting metabolic rate and body mass that underestimates resting metabolic rate early in the ontogeny 24 . With the adult brain size resulting from the best-fitting scenario for H. ...

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... determine if any combination of social-challenge parameters P yields an accurate prediction of adult brain and body sizes of H. sapiens and closely related species, we obtained solutions exhaustively across the P parameter space while holding the other parameters (Q and R) fixed ( Fig. 3d, e; Supplementary Information 5). We find near-perfect adult fits across Homo species (Fig. 4a and Extended Data Figs. 6-8). A near-perfect adult fit for H. sapiens occurs with a large proportion of ecological challenges (approximately 60%), a moderate proportion of cooperative challenges (around 30%), a small proportion of between- group competitive challenges (around 10%), and an approximately complete absence of between-individual competitive challenges (around 0%) (Figs. 3e, 4a and Extended Data Figs. 6, 9). In the resulting reconstruction for Homo, ecological challenges increase brain size whereas social challenges decrease it (Extended Data Fig. 4), the pro- portion of ecological challenges tends to increase from early to late Homo species, and a steep increase in encephalization quotient from Homo ergaster to Homo heidelbergensis is due to a transition from strongly to weakly decelerating EEE (Fig. 4a). The adult best-fit eco- social scenario for H. sapiens also yields a predicted life history that closely matches the species' life-history timing ( Fig. 4b and Extended Data Fig. 10). The resulting ontogenetic fit is high for body size, but lower for brain size early in ontogeny (Fig. 4b), perhaps caused in part by our use of a power-law relationship between resting metabolic rate and body mass that underestimates resting metabolic rate early in the ontogeny 24 . With the adult brain size resulting from the best-fitting scenario for H. ...

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... determine if any combination of social-challenge parameters P yields an accurate prediction of adult brain and body sizes of H. sapiens and closely related species, we obtained solutions exhaustively across the P parameter space while holding the other parameters (Q and R) fixed ( Fig. 3d, e; Supplementary Information 5). We find near-perfect adult fits across Homo species (Fig. 4a and Extended Data Figs. 6-8). A near-perfect adult fit for H. sapiens occurs with a large proportion of ecological challenges (approximately 60%), a moderate proportion of cooperative challenges (around 30%), a small proportion of between- group competitive challenges (around 10%), and an approximately complete absence of between-individual competitive challenges (around 0%) (Figs. 3e, 4a and Extended Data Figs. 6, 9). In the resulting reconstruction for Homo, ecological challenges increase brain size whereas social challenges decrease it (Extended Data Fig. 4), the pro- portion of ecological challenges tends to increase from early to late Homo species, and a steep increase in encephalization quotient from Homo ergaster to Homo heidelbergensis is due to a transition from strongly to weakly decelerating EEE (Fig. 4a). The adult best-fit eco- social scenario for H. sapiens also yields a predicted life history that closely matches the species' life-history timing ( Fig. 4b and Extended Data Fig. 10). The resulting ontogenetic fit is high for body size, but lower for brain size early in ontogeny (Fig. 4b), perhaps caused in part by our use of a power-law relationship between resting metabolic rate and body mass that underestimates resting metabolic rate early in the ontogeny 24 . With the adult brain size resulting from the best-fitting scenario for H. ...

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... determine if any combination of social-challenge parameters P yields an accurate prediction of adult brain and body sizes of H. sapiens and closely related species, we obtained solutions exhaustively across the P parameter space while holding the other parameters (Q and R) fixed ( Fig. 3d, e; Supplementary Information 5). We find near-perfect adult fits across Homo species (Fig. 4a and Extended Data Figs. 6-8). A near-perfect adult fit for H. sapiens occurs with a large proportion of ecological challenges (approximately 60%), a moderate proportion of cooperative challenges (around 30%), a small proportion of between- group competitive challenges (around 10%), and an approximately complete absence of between-individual competitive challenges (around 0%) (Figs. 3e, 4a and Extended Data Figs. 6, 9). In the resulting reconstruction for Homo, ecological challenges increase brain size whereas social challenges decrease it (Extended Data Fig. 4), the pro- portion of ecological challenges tends to increase from early to late Homo species, and a steep increase in encephalization quotient from Homo ergaster to Homo heidelbergensis is due to a transition from strongly to weakly decelerating EEE (Fig. 4a). The adult best-fit eco- social scenario for H. sapiens also yields a predicted life history that closely matches the species' life-history timing ( Fig. 4b and Extended Data Fig. 10). The resulting ontogenetic fit is high for body size, but lower for brain size early in ontogeny (Fig. 4b), perhaps caused in part by our use of a power-law relationship between resting metabolic rate and body mass that underestimates resting metabolic rate early in the ontogeny 24 . With the adult brain size resulting from the best-fitting scenario for H. ...

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... 0 0 r Thus equation (11) poses a differential game problem: it is a 'game' between mutant and resident because the mutant's payoff R u v ( ( , )) 0 depends on the resident strat- egy, it is 'differential' because it depends on differential equations (equations (1) and (2)), and it is 'evolutionary' rather than a typical differential game because only the mutant's payoff is maximized rather than both the mutant and resident's pay- offs. Because equation (11) involves maximization with respect to functions u ( ) rather than points, this maximization poses an optimal control problem. We solve this problem numerically by finding a best response to the resident (optimal con- trol problem), setting the best response as the resident, and iterating until conver- gence to a point at which the mutant and resident strategies are indistinguishable to a chosen extent. To do so, we use the software GPOPS 35 . Figure specifications. For Fig. 3a, b, plots are around only ecological challenges; that is, for a given plot, the remaining two P j 's are set to zero. For social challenges, the arrows in Fig. 3c describe the qualitative effect determined in Fig. 3a, b of increasing the proportion of a social challenge as the proportion of ecological challenge decreases; for ecological challenges, the arrows describe the quali- tative effect of increasing the environmental difficulty α as found in Extended Data Fig. 3g, h. The patterns in Fig. 3a-c also hold around the best-fitting P* for H. sapiens; that is, when for a given plot, the remaining two P j 's are set to the values of P* (Extended Data Fig. 4b, c). The 'missing' dots in Fig. 3d are P j combinations that did not converge to an uninvadable growth strategy (for example, owing to cycling solutions, suggesting possible evolutionary branching (female dimorphism) in brain size) or that were unreachable from lack of convergence of nearby runs (Supplementary Information 5). For Fig. 3a-e, cooperation is submultiplicative and for Fig. 3d, e, competence is exponential (see Extended Data Fig. 4 for all cases). Figure 4a shows the hominin species for which we find a near-perfect adult fit (that is, for which the best adult fit is greater than the chosen threshold of −D(τ a ) = −0.05; Extended Data Figs. 6-8). For Fig. 4a, cooperation is submultiplicative (respectively additive) for weakly (respectively strongly) decelerating EEE. In Fig. 4b, dots are the values for an average H. sapiens female as previously reported 21 . The resulting life periods in Fig. 4b are defined as 'childhood' , when there has not been allocation to production of reproductive tissue from birth; 'adolescence' , when there is allocation to production of somatic and reproductive tissues; and 'adulthood' , when there is only allocation to production of reproductive tissue. The EEE from maternal provisioning at birth (part of the R parameters) in Fig. 4b is slightly smaller than its benchmark value to improve ontogenetic fit further without affecting adult fit (ontogenetic fit is −E(D(τ)) = −0.22 using ϕ = . ...

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... 0 0 r Thus equation (11) poses a differential game problem: it is a 'game' between mutant and resident because the mutant's payoff R u v ( ( , )) 0 depends on the resident strat- egy, it is 'differential' because it depends on differential equations (equations (1) and (2)), and it is 'evolutionary' rather than a typical differential game because only the mutant's payoff is maximized rather than both the mutant and resident's pay- offs. Because equation (11) involves maximization with respect to functions u ( ) rather than points, this maximization poses an optimal control problem. We solve this problem numerically by finding a best response to the resident (optimal con- trol problem), setting the best response as the resident, and iterating until conver- gence to a point at which the mutant and resident strategies are indistinguishable to a chosen extent. To do so, we use the software GPOPS 35 . Figure specifications. For Fig. 3a, b, plots are around only ecological challenges; that is, for a given plot, the remaining two P j 's are set to zero. For social challenges, the arrows in Fig. 3c describe the qualitative effect determined in Fig. 3a, b of increasing the proportion of a social challenge as the proportion of ecological challenge decreases; for ecological challenges, the arrows describe the quali- tative effect of increasing the environmental difficulty α as found in Extended Data Fig. 3g, h. The patterns in Fig. 3a-c also hold around the best-fitting P* for H. sapiens; that is, when for a given plot, the remaining two P j 's are set to the values of P* (Extended Data Fig. 4b, c). The 'missing' dots in Fig. 3d are P j combinations that did not converge to an uninvadable growth strategy (for example, owing to cycling solutions, suggesting possible evolutionary branching (female dimorphism) in brain size) or that were unreachable from lack of convergence of nearby runs (Supplementary Information 5). For Fig. 3a-e, cooperation is submultiplicative and for Fig. 3d, e, competence is exponential (see Extended Data Fig. 4 for all cases). Figure 4a shows the hominin species for which we find a near-perfect adult fit (that is, for which the best adult fit is greater than the chosen threshold of −D(τ a ) = −0.05; Extended Data Figs. 6-8). For Fig. 4a, cooperation is submultiplicative (respectively additive) for weakly (respectively strongly) decelerating EEE. In Fig. 4b, dots are the values for an average H. sapiens female as previously reported 21 . The resulting life periods in Fig. 4b are defined as 'childhood' , when there has not been allocation to production of reproductive tissue from birth; 'adolescence' , when there is allocation to production of somatic and reproductive tissues; and 'adulthood' , when there is only allocation to production of reproductive tissue. The EEE from maternal provisioning at birth (part of the R parameters) in Fig. 4b is slightly smaller than its benchmark value to improve ontogenetic fit further without affecting adult fit (ontogenetic fit is −E(D(τ)) = −0.22 using ϕ = . ...

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... 0 0 r Thus equation (11) poses a differential game problem: it is a 'game' between mutant and resident because the mutant's payoff R u v ( ( , )) 0 depends on the resident strat- egy, it is 'differential' because it depends on differential equations (equations (1) and (2)), and it is 'evolutionary' rather than a typical differential game because only the mutant's payoff is maximized rather than both the mutant and resident's pay- offs. Because equation (11) involves maximization with respect to functions u ( ) rather than points, this maximization poses an optimal control problem. We solve this problem numerically by finding a best response to the resident (optimal con- trol problem), setting the best response as the resident, and iterating until conver- gence to a point at which the mutant and resident strategies are indistinguishable to a chosen extent. To do so, we use the software GPOPS 35 . Figure specifications. For Fig. 3a, b, plots are around only ecological challenges; that is, for a given plot, the remaining two P j 's are set to zero. For social challenges, the arrows in Fig. 3c describe the qualitative effect determined in Fig. 3a, b of increasing the proportion of a social challenge as the proportion of ecological challenge decreases; for ecological challenges, the arrows describe the quali- tative effect of increasing the environmental difficulty α as found in Extended Data Fig. 3g, h. The patterns in Fig. 3a-c also hold around the best-fitting P* for H. sapiens; that is, when for a given plot, the remaining two P j 's are set to the values of P* (Extended Data Fig. 4b, c). The 'missing' dots in Fig. 3d are P j combinations that did not converge to an uninvadable growth strategy (for example, owing to cycling solutions, suggesting possible evolutionary branching (female dimorphism) in brain size) or that were unreachable from lack of convergence of nearby runs (Supplementary Information 5). For Fig. 3a-e, cooperation is submultiplicative and for Fig. 3d, e, competence is exponential (see Extended Data Fig. 4 for all cases). Figure 4a shows the hominin species for which we find a near-perfect adult fit (that is, for which the best adult fit is greater than the chosen threshold of −D(τ a ) = −0.05; Extended Data Figs. 6-8). For Fig. 4a, cooperation is submultiplicative (respectively additive) for weakly (respectively strongly) decelerating EEE. In Fig. 4b, dots are the values for an average H. sapiens female as previously reported 21 . The resulting life periods in Fig. 4b are defined as 'childhood' , when there has not been allocation to production of reproductive tissue from birth; 'adolescence' , when there is allocation to production of somatic and reproductive tissues; and 'adulthood' , when there is only allocation to production of reproductive tissue. The EEE from maternal provisioning at birth (part of the R parameters) in Fig. 4b is slightly smaller than its benchmark value to improve ontogenetic fit further without affecting adult fit (ontogenetic fit is −E(D(τ)) = −0.22 using ϕ = . ...

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... 0 0 r Thus equation (11) poses a differential game problem: it is a 'game' between mutant and resident because the mutant's payoff R u v ( ( , )) 0 depends on the resident strat- egy, it is 'differential' because it depends on differential equations (equations (1) and (2)), and it is 'evolutionary' rather than a typical differential game because only the mutant's payoff is maximized rather than both the mutant and resident's pay- offs. Because equation (11) involves maximization with respect to functions u ( ) rather than points, this maximization poses an optimal control problem. We solve this problem numerically by finding a best response to the resident (optimal con- trol problem), setting the best response as the resident, and iterating until conver- gence to a point at which the mutant and resident strategies are indistinguishable to a chosen extent. To do so, we use the software GPOPS 35 . Figure specifications. For Fig. 3a, b, plots are around only ecological challenges; that is, for a given plot, the remaining two P j 's are set to zero. For social challenges, the arrows in Fig. 3c describe the qualitative effect determined in Fig. 3a, b of increasing the proportion of a social challenge as the proportion of ecological challenge decreases; for ecological challenges, the arrows describe the quali- tative effect of increasing the environmental difficulty α as found in Extended Data Fig. 3g, h. The patterns in Fig. 3a-c also hold around the best-fitting P* for H. sapiens; that is, when for a given plot, the remaining two P j 's are set to the values of P* (Extended Data Fig. 4b, c). The 'missing' dots in Fig. 3d are P j combinations that did not converge to an uninvadable growth strategy (for example, owing to cycling solutions, suggesting possible evolutionary branching (female dimorphism) in brain size) or that were unreachable from lack of convergence of nearby runs (Supplementary Information 5). For Fig. 3a-e, cooperation is submultiplicative and for Fig. 3d, e, competence is exponential (see Extended Data Fig. 4 for all cases). Figure 4a shows the hominin species for which we find a near-perfect adult fit (that is, for which the best adult fit is greater than the chosen threshold of −D(τ a ) = −0.05; Extended Data Figs. 6-8). For Fig. 4a, cooperation is submultiplicative (respectively additive) for weakly (respectively strongly) decelerating EEE. In Fig. 4b, dots are the values for an average H. sapiens female as previously reported 21 . The resulting life periods in Fig. 4b are defined as 'childhood' , when there has not been allocation to production of reproductive tissue from birth; 'adolescence' , when there is allocation to production of somatic and reproductive tissues; and 'adulthood' , when there is only allocation to production of reproductive tissue. The EEE from maternal provisioning at birth (part of the R parameters) in Fig. 4b is slightly smaller than its benchmark value to improve ontogenetic fit further without affecting adult fit (ontogenetic fit is −E(D(τ)) = −0.22 using ϕ = . ...

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... 0 0 r Thus equation (11) poses a differential game problem: it is a 'game' between mutant and resident because the mutant's payoff R u v ( ( , )) 0 depends on the resident strat- egy, it is 'differential' because it depends on differential equations (equations (1) and (2)), and it is 'evolutionary' rather than a typical differential game because only the mutant's payoff is maximized rather than both the mutant and resident's pay- offs. Because equation (11) involves maximization with respect to functions u ( ) rather than points, this maximization poses an optimal control problem. We solve this problem numerically by finding a best response to the resident (optimal con- trol problem), setting the best response as the resident, and iterating until conver- gence to a point at which the mutant and resident strategies are indistinguishable to a chosen extent. To do so, we use the software GPOPS 35 . Figure specifications. For Fig. 3a, b, plots are around only ecological challenges; that is, for a given plot, the remaining two P j 's are set to zero. For social challenges, the arrows in Fig. 3c describe the qualitative effect determined in Fig. 3a, b of increasing the proportion of a social challenge as the proportion of ecological challenge decreases; for ecological challenges, the arrows describe the quali- tative effect of increasing the environmental difficulty α as found in Extended Data Fig. 3g, h. The patterns in Fig. 3a-c also hold around the best-fitting P* for H. sapiens; that is, when for a given plot, the remaining two P j 's are set to the values of P* (Extended Data Fig. 4b, c). The 'missing' dots in Fig. 3d are P j combinations that did not converge to an uninvadable growth strategy (for example, owing to cycling solutions, suggesting possible evolutionary branching (female dimorphism) in brain size) or that were unreachable from lack of convergence of nearby runs (Supplementary Information 5). For Fig. 3a-e, cooperation is submultiplicative and for Fig. 3d, e, competence is exponential (see Extended Data Fig. 4 for all cases). Figure 4a shows the hominin species for which we find a near-perfect adult fit (that is, for which the best adult fit is greater than the chosen threshold of −D(τ a ) = −0.05; Extended Data Figs. 6-8). For Fig. 4a, cooperation is submultiplicative (respectively additive) for weakly (respectively strongly) decelerating EEE. In Fig. 4b, dots are the values for an average H. sapiens female as previously reported 21 . The resulting life periods in Fig. 4b are defined as 'childhood' , when there has not been allocation to production of reproductive tissue from birth; 'adolescence' , when there is allocation to production of somatic and reproductive tissues; and 'adulthood' , when there is only allocation to production of reproductive tissue. The EEE from maternal provisioning at birth (part of the R parameters) in Fig. 4b is slightly smaller than its benchmark value to improve ontogenetic fit further without affecting adult fit (ontogenetic fit is −E(D(τ)) = −0.22 using ϕ = . ...

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... 0 0 r Thus equation (11) poses a differential game problem: it is a 'game' between mutant and resident because the mutant's payoff R u v ( ( , )) 0 depends on the resident strat- egy, it is 'differential' because it depends on differential equations (equations (1) and (2)), and it is 'evolutionary' rather than a typical differential game because only the mutant's payoff is maximized rather than both the mutant and resident's pay- offs. Because equation (11) involves maximization with respect to functions u ( ) rather than points, this maximization poses an optimal control problem. We solve this problem numerically by finding a best response to the resident (optimal con- trol problem), setting the best response as the resident, and iterating until conver- gence to a point at which the mutant and resident strategies are indistinguishable to a chosen extent. To do so, we use the software GPOPS 35 . Figure specifications. For Fig. 3a, b, plots are around only ecological challenges; that is, for a given plot, the remaining two P j 's are set to zero. For social challenges, the arrows in Fig. 3c describe the qualitative effect determined in Fig. 3a, b of increasing the proportion of a social challenge as the proportion of ecological challenge decreases; for ecological challenges, the arrows describe the quali- tative effect of increasing the environmental difficulty α as found in Extended Data Fig. 3g, h. The patterns in Fig. 3a-c also hold around the best-fitting P* for H. sapiens; that is, when for a given plot, the remaining two P j 's are set to the values of P* (Extended Data Fig. 4b, c). The 'missing' dots in Fig. 3d are P j combinations that did not converge to an uninvadable growth strategy (for example, owing to cycling solutions, suggesting possible evolutionary branching (female dimorphism) in brain size) or that were unreachable from lack of convergence of nearby runs (Supplementary Information 5). For Fig. 3a-e, cooperation is submultiplicative and for Fig. 3d, e, competence is exponential (see Extended Data Fig. 4 for all cases). Figure 4a shows the hominin species for which we find a near-perfect adult fit (that is, for which the best adult fit is greater than the chosen threshold of −D(τ a ) = −0.05; Extended Data Figs. 6-8). For Fig. 4a, cooperation is submultiplicative (respectively additive) for weakly (respectively strongly) decelerating EEE. In Fig. 4b, dots are the values for an average H. sapiens female as previously reported 21 . The resulting life periods in Fig. 4b are defined as 'childhood' , when there has not been allocation to production of reproductive tissue from birth; 'adolescence' , when there is allocation to production of somatic and reproductive tissues; and 'adulthood' , when there is only allocation to production of reproductive tissue. The EEE from maternal provisioning at birth (part of the R parameters) in Fig. 4b is slightly smaller than its benchmark value to improve ontogenetic fit further without affecting adult fit (ontogenetic fit is −E(D(τ)) = −0.22 using ϕ = . ...

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... 0 0 r Thus equation (11) poses a differential game problem: it is a 'game' between mutant and resident because the mutant's payoff R u v ( ( , )) 0 depends on the resident strat- egy, it is 'differential' because it depends on differential equations (equations (1) and (2)), and it is 'evolutionary' rather than a typical differential game because only the mutant's payoff is maximized rather than both the mutant and resident's pay- offs. Because equation (11) involves maximization with respect to functions u ( ) rather than points, this maximization poses an optimal control problem. We solve this problem numerically by finding a best response to the resident (optimal con- trol problem), setting the best response as the resident, and iterating until conver- gence to a point at which the mutant and resident strategies are indistinguishable to a chosen extent. To do so, we use the software GPOPS 35 . Figure specifications. For Fig. 3a, b, plots are around only ecological challenges; that is, for a given plot, the remaining two P j 's are set to zero. For social challenges, the arrows in Fig. 3c describe the qualitative effect determined in Fig. 3a, b of increasing the proportion of a social challenge as the proportion of ecological challenge decreases; for ecological challenges, the arrows describe the quali- tative effect of increasing the environmental difficulty α as found in Extended Data Fig. 3g, h. The patterns in Fig. 3a-c also hold around the best-fitting P* for H. sapiens; that is, when for a given plot, the remaining two P j 's are set to the values of P* (Extended Data Fig. 4b, c). The 'missing' dots in Fig. 3d are P j combinations that did not converge to an uninvadable growth strategy (for example, owing to cycling solutions, suggesting possible evolutionary branching (female dimorphism) in brain size) or that were unreachable from lack of convergence of nearby runs (Supplementary Information 5). For Fig. 3a-e, cooperation is submultiplicative and for Fig. 3d, e, competence is exponential (see Extended Data Fig. 4 for all cases). Figure 4a shows the hominin species for which we find a near-perfect adult fit (that is, for which the best adult fit is greater than the chosen threshold of −D(τ a ) = −0.05; Extended Data Figs. 6-8). For Fig. 4a, cooperation is submultiplicative (respectively additive) for weakly (respectively strongly) decelerating EEE. In Fig. 4b, dots are the values for an average H. sapiens female as previously reported 21 . The resulting life periods in Fig. 4b are defined as 'childhood' , when there has not been allocation to production of reproductive tissue from birth; 'adolescence' , when there is allocation to production of somatic and reproductive tissues; and 'adulthood' , when there is only allocation to production of reproductive tissue. The EEE from maternal provisioning at birth (part of the R parameters) in Fig. 4b is slightly smaller than its benchmark value to improve ontogenetic fit further without affecting adult fit (ontogenetic fit is −E(D(τ)) = −0.22 using ϕ = . ...

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... weakly decelerating EEE and additive cooperation, between- group competition increases both brain size and the encephalization quotient (Extended Data Fig. 4). Moderately frequent between- individual or between-group competition can lead to no allocation to brain and body growth (blue dots in Fig. 3a and Extended Data Fig. 4; see also Extended Data Fig. 5a, d); additionally, moderately frequent between-group competition in the presence of substantial cooperation can lead to arms races in brain size, which fail to yield stable, large brains (for example, because of cycling in brain size or eventual collapse to no allocation into brain growth (Extended Data Fig. 5)). This is because energy extraction becomes exceed- ingly difficult in the presence of large-brained competitors such that investments in brain or body growth do not pay off and the individual instead invests in early ...

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... weakly decelerating EEE and additive cooperation, between- group competition increases both brain size and the encephalization quotient (Extended Data Fig. 4). Moderately frequent between- individual or between-group competition can lead to no allocation to brain and body growth (blue dots in Fig. 3a and Extended Data Fig. 4; see also Extended Data Fig. 5a, d); additionally, moderately frequent between-group competition in the presence of substantial cooperation can lead to arms races in brain size, which fail to yield stable, large brains (for example, because of cycling in brain size or eventual collapse to no allocation into brain growth (Extended Data Fig. 5)). This is because energy extraction becomes exceed- ingly difficult in the presence of large-brained competitors such that investments in brain or body growth do not pay off and the individual instead invests in early ...

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... previously published data 21 for parameter estimates, our results suggest that adult human-sized brains and bodies may result from ecological challenges as drivers of brain expansion, with coop- eration and between-group competition decreasing brain and body size and between-group competition increasing the encephalization quotient by decreasing body size more strongly than brain size ( Fig. 3a and Extended Data Fig. 4b). In this eco-social scenario, between- individual competition is unimportant, as it does not lead to human-sized brains and bodies. Cooperation decreases brain size, because it allows individuals to rely on their partners' skills and thus decrease their own investment into costly brains (cooperation invites cheating), which is consistent with diminished brain sizes in cooperatively breeding birds 26 and mammals 27 , including primates 28 . For instance, among mole rats, naked mole rats are the most specialized in cooperative breeding and have the smallest relative brain size 29 (however, allomaternal care and brain size are positively associated in mammals 30 , but allomaternal care constitutes cooperation targeted at young, which vanishes in adult- hood as opposed to the peer-cooperation studied here). Similarly, between-group competition can decrease brain size probably because between-group competition involves cooperation between group mem- bers, allowing individuals to rely on their partners' skill. The result that exceedingly frequent competition decreases absolute and relative brain size may be relevant to the observed decreased brain size in ceta- ceans with the largest group sizes 19 . Cooperation can also decrease body size in our model, because when brain size is disfavoured so too can be body size. This is because a consequence of our model is that a key reason to grow somatic tissue is to make energy available for brain growth: increasing the mass of inexpensive somatic tissue can increase the energy available for tissue (and brain) growth due to the physical constraint imposed by the power-law relationship between resting met- abolic rate and body mass 24 ...

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... where x ˆ k is the asymptotic skill level in adulthood; equation (5) in the Methods). By comparison, current rough estimates 25 suggest a human-neocortex storage capacity of approximately 600 TB (Supplementary Information ...

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... our assessment fails to support social hypotheses as expla- nations for the evolution of human brain size, and is more consistent with ecological hypotheses. Our results suggest causal interpretations that differ from some current thinking on the evolution of human cog- nition. Specifically, we obtained an eco-social scenario that involves a substantial proportion of cooperation (30% against nature and 10% against others), which could shape cognition towards cooperation. This would help to explain aspects of human cognition that facilitate cooperation 11 , even if cooperation has not been a driver of human brain expansion. Additionally, because our analysis suggests that brain expansion in Homo has not been driven by peer cooperation or com- petition, our results indicate that social complexity may have had a more limited role in human brain size expansion than is commonly thought-although we emphasize that our analysis is an illustrative starting point and future extensions are encouraged (see Supplementary Information 9). Therefore, our results highlight the fundamental ques- tion of why ecological challenges would have favoured substantial brain figure 8.1 of ref. 1 ). Pie charts and plots respectively show the challenge combination and shape of EEE versus skill that yielded the best adult fit using the same Q and R parameters (Extended Data Figs. 6, 7). b, Life history with the challenge combination yielding the best adult fit for H. sapiens. Resulting life periods are indicated above the body mass plot. Vertical lines are ages at which the growth strategy changes suddenly; within childhood, they occur when brain growth begins and terminates. expansion in humans but less so in other taxa 10,15 . One clue is suggested by our finding that H. sapiens-sized brains and bodies can be obtained only under weakly decelerating EEE (Figs. 3, 4 and Extended Data Fig. 6a): in other words, only when young individuals can maintain a substantial rate of increase in their efficiency of energy extraction as they acquire skills. One possibility is that culture (or cumulative culture) facilitates weakly decelerating EEE if learning from the pool of skills in the population allows individuals to maintain a relatively high rate of increase in EEE as their skill level increases when young. More specifically, the evolution of progressively elaborated social learn- ing, teaching and language 11-14 may have enabled young individuals to continue gaining skills with age, possibly promoting less strongly decelerating EEE. In this respect, our results are consistent with aspects of various cultural hypotheses for brain evolution 13,14 and an explicit account of the effect of culture on EEE could help to address whether culture (or cumulative culture) has enabled ecological challenges to drive brain expansion in humans in ways that have not occurred in other ...

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... the allocation to the growth of reproductive tissue during adolescence increases ( * u r between t m and t a ) and adolescence shortens. In the central column, the increased allocation to the growth of reproductive tissue increases the mass of reproductive tissue, but brain mass does not change with B r for B r ≥ 70 MJ kg −1 y −1 . In the right column, as the mass of reproductive tissue increases, body mass increases slightly, which is more noticeable for B r ≤ 100 MJ kg −1 y −1 . An exceedingly small B r (<70 MJ kg −1 y −1 ) disrupts the predicted life history, which with B r = 60 MJ kg Fig. 4 | Effects of challenge types on brain size. a, b, Outer rows are for the cooperation cases that were considered; outer columns are for the competence cases. a, Around the pure ecological scenario (that is, in a given plot for P j as P 1 decreases, the remaining two P j 's are set to zero). b, Around the best fitting scenario for H. sapiens (that is, in a given plot for P j as P 1 decreases, the remaining two P j 's are set to the best fitting P* found in Fig. 3d. c, Summary of the qualitative effects of challenge types on brain size. For social challenges, the direction of the arrows is taken from a, b. For ecological challenges, the direction of the arrow is taken from Extended Data Fig. 3g as the environmental difficulty α increases. A dash (−) indicates an approximately invariant relationship and a dot (·) indicates insufficient data points for identifying a relationship. The arrows in Fig. 3c are taken from this summary, in which, for social challenges, the arrows are those of submultiplicative cooperation. AC, additive cooperation; EC: exponential competence; MC, multiplicative cooperation; SC, submultiplicative cooperation. Fig. 5 | Typical results when there is convergence to no brain growth or when there is no convergence to an uninvadable growth strategy. a-e, Adult values over best-response iterations for cases of no brain growth or no convergence to an uninvadable strategy. a, Amplifying cycle leads to no brain growth. b, Stable cycle. c, Arms race that ends when the solver warns that the optimal control problem (OCP) may be infeasible. This might arise if the best response to the last iteration necessarily involves a substantially different growth strategy, which is not allowed in the optimization as the best response is constrained to be sufficiently similar to that in the previous iteration. It is possible that such substantially different best response involves either no brain growth (for example, as seen under purely ecological challenges when the environmental difficulty is exceedingly high 24 ( Supplementary Information 4.4)) or substantially more allocation to brain growth (which appears unlikely given the energetic constraints). d, A short arms race in encephalization quotient that leads to no brain growth. e, Amplifying cycle that ends when the solver warns that the OCP may be infeasible. a, c, e, g, i, k, For the top left plot, as P 1 increases, P 2 decreases, whereas for the remaining plots as P 2 , P 3 and P 4 increase, P 1 decreases; for a given plot, the remaining P j are set to the corresponding P* shown in Fig. 4a ...

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... the allocation to the growth of reproductive tissue during adolescence increases ( * u r between t m and t a ) and adolescence shortens. In the central column, the increased allocation to the growth of reproductive tissue increases the mass of reproductive tissue, but brain mass does not change with B r for B r ≥ 70 MJ kg −1 y −1 . In the right column, as the mass of reproductive tissue increases, body mass increases slightly, which is more noticeable for B r ≤ 100 MJ kg −1 y −1 . An exceedingly small B r (<70 MJ kg −1 y −1 ) disrupts the predicted life history, which with B r = 60 MJ kg Fig. 4 | Effects of challenge types on brain size. a, b, Outer rows are for the cooperation cases that were considered; outer columns are for the competence cases. a, Around the pure ecological scenario (that is, in a given plot for P j as P 1 decreases, the remaining two P j 's are set to zero). b, Around the best fitting scenario for H. sapiens (that is, in a given plot for P j as P 1 decreases, the remaining two P j 's are set to the best fitting P* found in Fig. 3d. c, Summary of the qualitative effects of challenge types on brain size. For social challenges, the direction of the arrows is taken from a, b. For ecological challenges, the direction of the arrow is taken from Extended Data Fig. 3g as the environmental difficulty α increases. A dash (−) indicates an approximately invariant relationship and a dot (·) indicates insufficient data points for identifying a relationship. The arrows in Fig. 3c are taken from this summary, in which, for social challenges, the arrows are those of submultiplicative cooperation. AC, additive cooperation; EC: exponential competence; MC, multiplicative cooperation; SC, submultiplicative cooperation. Fig. 5 | Typical results when there is convergence to no brain growth or when there is no convergence to an uninvadable growth strategy. a-e, Adult values over best-response iterations for cases of no brain growth or no convergence to an uninvadable strategy. a, Amplifying cycle leads to no brain growth. b, Stable cycle. c, Arms race that ends when the solver warns that the optimal control problem (OCP) may be infeasible. This might arise if the best response to the last iteration necessarily involves a substantially different growth strategy, which is not allowed in the optimization as the best response is constrained to be sufficiently similar to that in the previous iteration. It is possible that such substantially different best response involves either no brain growth (for example, as seen under purely ecological challenges when the environmental difficulty is exceedingly high 24 ( Supplementary Information 4.4)) or substantially more allocation to brain growth (which appears unlikely given the energetic constraints). d, A short arms race in encephalization quotient that leads to no brain growth. e, Amplifying cycle that ends when the solver warns that the OCP may be infeasible. a, c, e, g, i, k, For the top left plot, as P 1 increases, P 2 decreases, whereas for the remaining plots as P 2 , P 3 and P 4 increase, P 1 decreases; for a given plot, the remaining P j are set to the corresponding P* shown in Fig. 4a ...

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... the allocation to the growth of reproductive tissue during adolescence increases ( * u r between t m and t a ) and adolescence shortens. In the central column, the increased allocation to the growth of reproductive tissue increases the mass of reproductive tissue, but brain mass does not change with B r for B r ≥ 70 MJ kg −1 y −1 . In the right column, as the mass of reproductive tissue increases, body mass increases slightly, which is more noticeable for B r ≤ 100 MJ kg −1 y −1 . An exceedingly small B r (<70 MJ kg −1 y −1 ) disrupts the predicted life history, which with B r = 60 MJ kg Fig. 4 | Effects of challenge types on brain size. a, b, Outer rows are for the cooperation cases that were considered; outer columns are for the competence cases. a, Around the pure ecological scenario (that is, in a given plot for P j as P 1 decreases, the remaining two P j 's are set to zero). b, Around the best fitting scenario for H. sapiens (that is, in a given plot for P j as P 1 decreases, the remaining two P j 's are set to the best fitting P* found in Fig. 3d. c, Summary of the qualitative effects of challenge types on brain size. For social challenges, the direction of the arrows is taken from a, b. For ecological challenges, the direction of the arrow is taken from Extended Data Fig. 3g as the environmental difficulty α increases. A dash (−) indicates an approximately invariant relationship and a dot (·) indicates insufficient data points for identifying a relationship. The arrows in Fig. 3c are taken from this summary, in which, for social challenges, the arrows are those of submultiplicative cooperation. AC, additive cooperation; EC: exponential competence; MC, multiplicative cooperation; SC, submultiplicative cooperation. Fig. 5 | Typical results when there is convergence to no brain growth or when there is no convergence to an uninvadable growth strategy. a-e, Adult values over best-response iterations for cases of no brain growth or no convergence to an uninvadable strategy. a, Amplifying cycle leads to no brain growth. b, Stable cycle. c, Arms race that ends when the solver warns that the optimal control problem (OCP) may be infeasible. This might arise if the best response to the last iteration necessarily involves a substantially different growth strategy, which is not allowed in the optimization as the best response is constrained to be sufficiently similar to that in the previous iteration. It is possible that such substantially different best response involves either no brain growth (for example, as seen under purely ecological challenges when the environmental difficulty is exceedingly high 24 ( Supplementary Information 4.4)) or substantially more allocation to brain growth (which appears unlikely given the energetic constraints). d, A short arms race in encephalization quotient that leads to no brain growth. e, Amplifying cycle that ends when the solver warns that the OCP may be infeasible. a, c, e, g, i, k, For the top left plot, as P 1 increases, P 2 decreases, whereas for the remaining plots as P 2 , P 3 and P 4 increase, P 1 decreases; for a given plot, the remaining P j are set to the corresponding P* shown in Fig. 4a ...