rogers_paradox.Rmd

---
title: 'Rogers'' paradox: An agent-based model in R'
author: "Alex Mesoudi"
output: 
  pdf_document:
    fig_caption: yes
    fig_height: 5
    fig_width: 8
date: '`r format(Sys.time(), "%d %B, %Y")`'

---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Background

Rogers (1988) presented a model exploring the evolution of social learning relative to individual learning, in a changing environment. The result, since known as 'Rogers' paradox', is that a population containing social learners can never have higher fitness than a population with no social learners. While not inherently paradoxical, this runs counter to the common claim that social learning underpins our species' unusual evolutionary success.

In the model, agents exhibit one of two behaviours. One behaviour is 'correct' in the current environment, the other is 'incorrect'. Correct behaviour results in higher fitness than incorrect behaviour. Periodically, the environment changes such that previously correct behaviours are no longer correct.

Agents can be either *individual learners*, who directly sample the environment and determine the correct behaviour with a fixed probability, or *social learners*, who select another agent at random and copy their behaviour. Individual learning is assumed to be more costly than social learning.

In a changing environment, a frequency-dependent equilibrium exists between social learners (aka information scroungers) and individual learners (aka information producers), similar to producer-scrounger dynamics in feeding behaviour. When rare, social learners can copy adaptive behaviour from individual learners without bearing the higher cost of individual learning. They therefore do better than individual learners and increase in frequency. But when the environment changes, social learners are left copying out-of-date information from other social learners. Individual learners now do better, because they can detect the environmental change and identify the new correct behaviour.

At this equilibrium, social and individual learners must have equal fitness, by definition. Hence, social learning does not increase overall population fitness, relative to a population entirely composed of individual learners.

## Model outline

There exist $N$ agents who reproduce each generation. Generations are non-overlapping such that $N$ new agents are created each generation to replace the $N$ previous agents. 

Agents are either individual or social learners, and can have either correct or incorrect behaviour given the current state of the environment. 

Each generation, individual learners acquire the correct behaviour with probability $p$. Social learners choose one of the previous generation at random (irrespective of whether they are social or individual learners) and copy their behaviour. 

The first generation is entirely made up of individual learners. Agents reproduce asexually in proportion to their fitness, which is determined by whether they have the correct behaviour (+$b$) or not (-$b$), as well as intrinsic costs to individual ($c$) and social ($s$) learning (where $c>s$). There is a small mutation rate $\mu$ for learning strategy such that offspring sometimes have different learning styles to their parent. This is how social learning is introduced into the population. 

The environment shifts with probability $u$ each generation. The simulation is run for $t_{max}$ generations with multiple independent runs. 

In Rogers' model the environment switched between one of two states. However this can sometimes lead to odd dynamics under high rates of environmental change, as agents can copy behaviour from the previous generation that was then incorrect, but is now correct. Instead the current model uses more realistic never-repeating environments, such that with probability $u$ the environment switches to an entirely novel state.

## Parameters

Symbols are the same as in Rogers' original paper, where possible.

* $w$, the baseline fitness given to all agents, fixed at $w=1$
* $b$, the fitness bonus/penalty for being in the right/wrong environment, fixed at $b=0.5$ (so agent fitness is $w+b$ when behaviour matches the environment, or $w-b$ if it doesn't)
* $c$, where $bc$ = the cost of individual learning. Constrained such that $b(1+c)<1$, to avoid negative fitnesses
* $s$, where $bs$ = the cost of social learning. Constrained again to $b(1+s)<1$
* $u$, the probability on each generation of an environmental shift ($0<u<1$)
* $N$, the number of agents ($N>0$)
* $\mu$, the probability of a new agent having a different learning strategy to its parent (i.e. the mutation rate) ($0< \mu <1$)
* $p$, the probability of an individual learner learning the correct behaviour. NB this was always 1 in Rogers' original model ($0.5<p<1$)
* $t_{max}$, the maximum number of generations per run
* $runs$, the number of simulation runs 

## Basic model output

Figure 1 shows an illustrative case of Rogers' paradox, with a moderately changing environment and reasonable cost of individual learning. The top graph shows the proportion of the population who are social learners. The rest are individual learners. The bottom graph shows the overall population fitness. The horizontal line in the bottom figure shows the expected fitness of a population of 100% individual learners. Lines are mean values calculated from multiple runs, with grey areas indicating ranges (maximum and minimum values for all runs).

For the parameter values shown in Figure 1, one can see that social and individual learners co-exist (top figure) and the population as a whole never exceeds the pure individual learning fitness (bottom figure). This is Rogers' paradox.

```{r echo=FALSE, message=FALSE, fig.cap = "Illustrative run showing Rogers' paradox"}
# install & load packages---------------------------
library(ggplot2)
library(Rmisc)

# initialise agent data frame
InitialiseAgent <- function (N) {
  # create vectors of length N:
  # 'learning' style is initially all individual, 'behaviour' and 'fitness' are all NA
  learning <- rep("individual", N)
  behaviour <- rep(NA, N)
  fitness <- rep(NA, N)
  
  # combine them into a data frame to return, ensuring that none become factors
  data.frame(behaviour, learning, fitness, stringsAsFactors = FALSE)
}

# environmental change function
EnvironmentalChange <- function(E, u) { 
  # generate 1 random number between 0 and 1. If less than u, change the environment
  if (runif(1) < u) {  
    E <- E + 1 } # increment environment to a new novel value
  E  # return E
}

# initialise output data frame, to store results
InitialiseOutput <- function(t.max) {
  timestep <- vector("numeric",t.max)  # timestep
  n.SL <- vector("numeric",t.max)  # number of social learners
  w.SL <- vector("numeric",t.max)  # fitness of social learners
  W <- vector("numeric",t.max)  # total absolute fitness of population
  data.frame(timestep, n.SL, w.SL, W)  # return this as 'output'
}

#learning function
LearningStage <- function (agent, p, E, agent.previous) {
  
  # individual learners:
  individual.learners <- agent$learning == "individual"  # vector of individual learners
  success <- runif(nrow(agent))  # vector with N random numbers each between 0 and 1
  agent$behaviour[individual.learners & success < p] <- E  # if agent is an individual learner, and p exceeds success, then they learn the correct behaviour
  agent$behaviour[individual.learners & success >= p] <- E - 1  # otherwise, they learn incorrect behaviour, always correct behaviour minus one
  
  # social learners:
  social.learners <- agent$learning == "social"  # vector of social learners
  # continue only if there are social learners
  if (sum(social.learners) > 0) {  
    # a vector containing randomly chosen demonstrators, one for each social learner. With replacement, so that demonstrators can be picked more than once. Note they can potentially copy themselves, but for large N this is rare
    demonstrators <- sample(1:nrow(agent), sum(social.learners), replace = TRUE)  
    # now copy the behaviour of demonstrators into behaviour of social learners
    agent$behaviour[social.learners] <- agent.previous$behaviour[demonstrators]  
  }
  
  agent  # return agent data frame
}

#  functions for getting fitness & number of learners, plus overall mean fitness W
GetSLnumber <- function (agent) {
  sum(agent$learning == "social") / nrow(agent) } # return n.SL as a proportion of N

GetW <- function (agent) {
  sum(agent$fitness, na.rm = TRUE) } # return the total fitness of all agents

GetSLfitness <- function (agent) {
  if (sum(agent$learning == "social") > 0) {  # if there is at least one social learner
    w.SL <- mean(agent$fitness[agent$learning == "social"])  # get their mean fitness
  } else {
    w.SL <- 0  # otherwise return zero
  }
  w.SL
}

#  calculate fitness of agents
CalculateFitness <- function (agent, w, E, b, c, s) {
  # start with baseline fitness w
  agent$fitness <- w  
  
  # for those agents with behaviour matched to the environment, add b
  agent$fitness[agent$behaviour == E] <- agent$fitness[agent$behaviour == E] + b  
  # for those agents with behaviour not matched to the environment, subtract b
  agent$fitness[agent$behaviour != E] <- agent$fitness[agent$behaviour != E] - b  
  
  # impose cost b*c on individual learners
  agent$fitness[agent$learning == "individual"] <- agent$fitness[agent$learning == "individual"] - b*c  
  # impose cost b*s on social learners
  agent$fitness[agent$learning == "social"] <- agent$fitness[agent$learning == "social"] - b*s  
  agent  # return updated agent
}

#  produce new generation from previous
ReproductionStage <- function (agent, agent.previous, mu) {
  
  # from now on, agent denotes new generation, agent.previous denotes previous
  
  agent$behaviour <- NA  # reset agent's behaviour and fitness
  agent$fitness <- NA
  
  # to determine learning, need to get probability of individual learner being a parent, weighted by relative fitness
  W <- GetW(agent.previous)  # get overall total fitness of previous generation
  # get probability of new agent being an individual learner, weighted by fitness
  if (sum(agent.previous$learning == "individual") > 0) {
    prob.IL <- sum(agent.previous$fitness[agent.previous$learning == "individual"]) / W 
  } else {
    prob.IL <- 0
  }

  # vector with random numbers each between 0 and 1, to determine probability of being an individual learner
  success.IL <- runif(nrow(agent.previous))  
  # vector with random numbers each between 0 and 1, to determine probability of mutation
  success.mu <- runif(nrow(agent.previous))  
  
  # if parent is an ind learner and no mutation, then they're an ind learner
  agent$learning[success.IL < prob.IL & success.mu >= mu] <- "individual"  
  # if parent is a social learner and no mutation, then they're a social learner
  agent$learning[success.IL >= prob.IL & success.mu >= mu] <- "social"  
  # if parent is an ind learner plus mutation, then they're a social learner
  agent$learning[success.IL < prob.IL & success.mu < mu] <- "social"  
  # if parent is a social learner, plus mutation, then they're an ind learner
  agent$learning[success.IL >= prob.IL & success.mu < mu] <- "individual"  
  
  agent  # return agent
}

# update output data frame
GetOutput <- function(output, t, agent) {
  output$timestep[t] <- t
  output$n.SL[t] <- GetSLnumber(agent)
  output$w.SL[t] <- GetSLfitness(agent)
  output$W[t] <- GetW(agent) / nrow(agent)
  output
}

#  run the simulation once
RunSimulation <- function (N, t.max, u, p, w, b, c, s, mu, output) {
  
  # check parameters, to avoid negative fitnesses
  if (b*(1+c) > 1 || b*(1+s) > 1) {
    stop("Invalid parameter values: ensure b*(1+c) < 1 and b*(1+s) < 1")
  }
  
  #initialise agent data frame
  agent <- InitialiseAgent(N)
  
  #  initialise environment E
  E <- 0
  
  for (t in 1:t.max) { # cycle thru timesteps
    
    #  update agent as a result of learning
    agent <- LearningStage(agent, p, E, agent.previous)
    
    #  update agent's fitness as a result of fitness calculations
    agent <- CalculateFitness(agent, w, E, b, c, s)
    
    #  get variables needed for the plots later on
    output <- GetOutput(output, t, agent)
    
    # copy this generation into agent.previous, to serve as demonstrators to next gen's social learners
    agent.previous <- agent
    
    #  create new generation of agents from agent.previous
    agent <- ReproductionStage(agent, agent.previous, mu)
    
    #  possible environmental change
    E <- EnvironmentalChange(E, u)
    
  }  # end of timestep-for
  
  output  # return output data frame
}

# simulation loop, repeat 'runs' times and append output to output dataframe 
SimulationLoop <- function (runs, N, t.max, u, p, w, b, c, s, mu) {
  
  # initialise output data frame
  output <- InitialiseOutput(t.max)
  
  # set up runs loop, each run produces output. Add to output dataframe if it already exists
  for (i in 1:runs) {
    
    # for first run, create output data frame
    if (i == 1) {
    output <- RunSimulation(N, t.max, u, p, w, b, c, s, mu, output)
    } else {
      # otherwise add it on to the end of output
      output <- rbind(output, RunSimulation(N, t.max, u, p, w, b, c, s, mu, output))
    }
  }  # end of runs-loop
  
  # draw plots
  DrawPlots(output, N, w, b, p, c, u, s, mu, runs)

}

GetOutputSummary <- function (output) {
  # take the full multiple-runs output and generate means, maxs and mins across timesteps
  mean <- aggregate(output[, 2:4], by = list(output$timestep), mean)  # gives means
  max <- aggregate(output[, 2:4], by = list(output$timestep), max)  # gives maximums
  min <- aggregate(output[, 2:4], by = list(output$timestep), min)  # gives minimums
  
  # return output.summary
  data.frame(mean$n.SL, min$n.SL, max$n.SL, mean$w.SL, min$w.SL, max$w.SL, mean$W, min$W, max$W)
}

DrawPlots <- function (output, N, w, b, p, c, u, s, mu, runs) {

  # first need to get means and maximums/minimums from raw output
  output.summary <- GetOutputSummary(output)
  
  # create the graph title from parameter values
  graph.title <- paste("t.max = " ,nrow(output.summary), ", ", "N = ", N, ", ", "w = ", w, ", ", "b = ", b, ", ", "c = ", c, ", ", "s = ", s, ", ", "u = ", u, ", ", "mu = ", mu, ", ", "p = ", p, ", ", "runs = ", runs, sep = "")

  # plot number of social learners
  plot1 <- ggplot(data = output.summary, aes(x = 1:nrow(output.summary), y = mean.n.SL)) + geom_line() + geom_ribbon(aes(ymin = min.n.SL, ymax = max.n.SL), alpha = 0.2) + labs(x = "Generation", y = "Freq of social learners") + scale_y_continuous(limits = c(0,1)) + theme_classic(base_size = 14) + ggtitle(graph.title) + theme(plot.title = element_text(size = 12), axis.line.x = element_line(colour = "black", size = 0.5), axis.line.y = element_line(colour = "black", size = 0.5))
  
  # plot mean population fitness, W, with a dotted line showing expected W in a population of all individual learners
  plot2 <- ggplot(data = output.summary, aes(x = 1:nrow(output.summary), y = mean.W)) + geom_line() + geom_ribbon(aes(ymin = min.W, ymax = max.W), alpha = 0.2) + geom_hline(yintercept = w + b*(2*p - c - 1), linetype = 2) + labs(x = "Generation", y = "Population fitness") + scale_y_continuous(limits = c(0,NA)) + theme_classic(base_size = 14) + theme(axis.line.x = element_line(colour = "black", size = 0.5), axis.line.y = element_line(colour = "black", size = 0.5))
  
  # optionally: save to png image in folder 'graphs' with filename indicating parameters
  #graph.name <- paste("graphs/", "N" ,nrow(output.summary),"_w", w, "_b", b, "_c", c, "_s", s, "_u", u, "_mu", mu, "_p", p, "_runs", runs, ".png", sep = "")
  
  #png(file = graph.name, width = 20, height = 20, units = "cm", res = 300)
  #multiplot(plot1, plot2)
  #dev.off()

  # display 3 plots using multiplot from Rmisc
  multiplot(plot1, plot2)
}

# main function, with defaults
RogersModel <- function (runs = 10, N = 1000, t.max = 250, u = 0.2, p = 1, w = 1, b = 0.5, c = 0.9, s = 0.0, mu = 0.01, runin = 100) {
  
  # run loops with defaults
  SimulationLoop(runs, N, t.max, u, p, w, b, c, s, mu)

}

# run simulation here (all default values)
RogersModel()

```

## Environmental change

Removing environmental change, i.e. setting $u=0$, causes social learning to entirely replace individual learning (Figure 2). The initial individual learners do all the work of identifying the correct behaviour, and then social learners take over due to their lower learning costs. They never suffer from copying out-of-date behaviour as they do in Figure 1, because behaviour never becomes out-of-date. Here Rogers' paradox no longer applies, as the population of social learners does better than a population of individual learners (bottom graph of Figure 2).

In contrast, when the environment changes very rapidly $(u=0.8)$, then individual learning becomes more common (Figure 3). Individual learners can track the environmental change, unlike social learners.

```{r echo=FALSE, message=FALSE, fig.cap = "Unchanging environments favour social learning"}
RogersModel(u = 0)

```

```{r echo=FALSE, message=FALSE, fig.cap = "Rapidly changing environments favour individual learning"}
RogersModel(u = 0.8)

```


## Cost and accuracy of individual learning

Making individual learning less costly (reducing $c$) reduces the frequency of social learning (Figure 4). Making individual learning less accurate (reducing $p$) increases the frequency of social learning (Figure 5). Note that in the latter case, Rogers' paradox still holds. The heatmap in Figure 6 shows how social learning evolves when environments are stable and individual learning is costly. Figure 7 shows the same, but for inaccurate individual learning.


```{r echo=FALSE, message=FALSE, fig.cap = "Reducing the cost of individual learning favours individual learning"}
RogersModel(c = 0.4)

```

```{r echo=FALSE, message=FALSE, fig.cap = "Reducing the accuracy of individual learning favours social learning"}
RogersModel(p = 0.7)

```

```{r echo=FALSE, message=FALSE, fig.cap = "Social learning evolves when environments are stable and individual learning is costly"}

# main function but without graphs and things, much quicker, returns SL frequency
RogersModelShort <- function (runs = 5, N = 1000, t.max = 250, u = 0.2, p = 1, w = 1, b = 0.5, c = 0.9, s = 0.0, mu = 0.01, runin = 100) {
  
  # initialise output data frame
  output.SLfreq <- 0
  
  # set up runs loop, each run produces output. Add to output dataframe if it already exists
  for (i in 1:runs) {
    output <- InitialiseOutput(t.max)
    output <- RunSimulation(N, t.max, u, p, w, b, c, s, mu, output)
    output.SLfreq <- output.SLfreq + mean(output$n.SL[output$timestep > runin])
  }  # end of runs-loop
  
  output.SLfreq / runs # get mean SLfreq from all runs and return
}

# higher-level loop plotting frequency of SL for ranges of u and c
DrawHeatMap_uc <- function(runs, p) {

  # initialise 11x11 dataframe 'heat'
  c.heat <- rep(0:10, 11) # for now these are 0-10 to use in loops, convert to 0-1 later
  u.heat <- rep(0:10, each = 11)
  value.heat <- rep(0,121)
  heat <- data.frame(c.heat, u.heat, value.heat)  

# add values to heat matrix
for (u.loop in 0:10)
{
  for (c.loop in 0:10)
  {
    heat$value.heat[heat$c.heat == c.loop & heat$u.heat == u.loop] <- RogersModelShort(u = u.loop/10, c = c.loop/10, runs = runs, p = p)
  }
}

# convert c.heat and u.heat values
heat$c.heat <- heat$c.heat / 10
heat$u.heat <- heat$u.heat / 10

# draw heatmap
names(heat)[3] <- "Freq of social learners"  # easier than changing legend title via ggplot!
ggplot(data = heat, aes(x = c.heat, y = u.heat)) + geom_tile(aes(fill = `Freq of social learners`)) + labs(x = "c, cost of individual learning", y = "u, prob of environmental change") + scale_y_continuous(breaks = seq(0,1, by=0.1)) + scale_x_continuous(breaks = seq(0,1, by=0.1)) + theme_bw(base_size = 20) + scale_fill_gradient(low = "yellow", high = "red3", name="freq(SL)")
}

# higher-level loop plotting frequency of SL for ranges of u and p
DrawHeatMap_up <- function(runs, c) {
  
  # initialise 11x11 dataframe 'heat'
  p.heat <- rep(0:10, 11) # for now these are 0-10, convert to 0-1 later
  u.heat <- rep(0:10, each = 11)
  value.heat <- rep(0,121)
  heat <- data.frame(p.heat, u.heat, value.heat)  
  
  # add values to heat matrix
  for (u.loop in 0:10)
  {
    for (p.loop in 0:10)
    {
      heat$value.heat[heat$p.heat == p.loop & heat$u.heat == u.loop] <- RogersModelShort(u = u.loop/10, p = p.loop/10, runs = runs, c = c)
    }
  }
  
  # convert c.heat and u.heat values
  heat$p.heat <- heat$p.heat / 10
  heat$u.heat <- heat$u.heat / 10
  
  # draw heatmap
  names(heat)[3] <- "Freq of social learners"  # easier than changing legend title via ggplot!
  ggplot(data = heat, aes(x = p.heat, y = u.heat)) + geom_tile(aes(fill = `Freq of social learners`)) + labs(x = "p, prob of successful individual learning", y = "u, prob of environmental change") + scale_y_continuous(breaks = seq(0,1, by=0.1)) + scale_x_continuous(breaks = seq(0,1, by=0.1)) + theme_bw(base_size = 20) + scale_fill_gradient(low = "yellow", high = "red3", name="freq(SL)")
  }

DrawHeatMap_uc(runs = 1, p = 0.5)

```

```{r echo=FALSE, message=FALSE, fig.cap = "Social learning evolves when environments are stable and individual learning is inaccurate"}

DrawHeatMap_up(runs = 1, c = 0.9)

```

## Critical social learning

Enquist, Erikkson & Ghirlanda (2007) proposed a 'solution' to Rogers' paradox by relaxing the unrealistic assumption that organisms can only do either individual learning or social learning. In reality organisms typically do both, or at least have the capacity to do both. Enquist et al. (2007) proposed a third heritable strategy, *critical social learning*, in which individuals first try social learning, and if the result is unsatisfactory, then they try individual learning. As shown in Figure 8, critical social learners outperform pure individual learners and pure social learners (top panel). Moreover, they do not suffer from Rogers' paradox (bottom panel), clearly exceeding the fitness of pure individual learners.

```{r echo=FALSE, message=FALSE, fig.cap = "Critical social learning outperforms both social learning and individual learning"}

# load packages---------------------------
library(reshape2)

#  define functions----------------------------------

# initialise output data frame, to store results
InitialiseOutput.crit <- function(t.max) {
  timestep <- vector("numeric",t.max)  # timestep
  n.SL <- vector("numeric",t.max)  # number of social learners
  n.IL <- vector("numeric",t.max)  # number of individual learners
  n.CL <- vector("numeric",t.max)  # number of critical learners
  w.SL <- vector("numeric",t.max)  # fitness of social learners
  w.IL <- vector("numeric",t.max)  # fitness of individual learners
  w.CL <- vector("numeric",t.max)  # fitness of critical learners
  W <- vector("numeric",t.max)  # total absolute fitness of population
  data.frame(timestep, n.SL, n.IL, n.CL, w.SL, w.IL, w.CL, W)  # return this as 'output'
}

# learning function - now incorporates fitness calculations
LearningStage.crit <- function (agent, p, E, agent.previous, w, b, c, s) {

  # start each agent with baseline fitness w
  agent$fitness <- w
  
  # individual learners
  individual.learners <- agent$learning == "individual"  # vector of individual learners
  success <- runif(nrow(agent))  # vector with random numbers each between 0 and 1
  agent$behaviour[individual.learners & success < p] <- E  # if agent is an individual learner, and p exceeds success, then they learn the correct behaviour
  agent$behaviour[individual.learners & success >= p] <- 1 - E  # otherwise, they learn the incorrect behaviour
  # subtract bc from fitness of ILers
  agent$fitness[individual.learners] <- agent$fitness[individual.learners] - b*c
  
  # social learners
  social.learners <- agent$learning == "social"  # vector of social learners
  if (sum(social.learners) > 0) {  # do this only if there are social learners
    demonstrators <- sample(1:nrow(agent), sum(social.learners), replace = TRUE)  # a vector containing randomly chosen demonstrators, one for each social learner. With replacement, so that demonstrators can be picked more than once
    agent$behaviour[social.learners] <- agent.previous$behaviour[demonstrators]  # copy the behaviour of demonstrators into behaviour of social learners
    # subtract bs from fitness of SLers
    agent$fitness[social.learners] <- agent$fitness[social.learners] - b*s
    }
  
  # critical learners
  critical.learners <- agent$learning == "critical"  # vector of critical learners
  if (sum(critical.learners) > 0) {  # do this only if there are critical learners
    demonstrators <- sample(1:nrow(agent), sum(critical.learners), replace = TRUE)  # a vector containing randomly chosen demonstrators, one for each critical learner. With replacement, so that demonstrators can be picked more than once
    agent$behaviour[critical.learners] <- agent.previous$behaviour[demonstrators]  # copy the behaviour of demonstrators into behaviour of critical learners
    # subtract bs from fitness of CLers
    agent$fitness[critical.learners] <- agent$fitness[critical.learners] - b*s
    
    # get vector of critical learners who learned the wrong behaviour
    critical.wrong <- agent$learning == "critical" & agent$behaviour != E
    
    if (sum(critical.wrong) > 0) {  # do this only if there are wrong critical learners
    # repeat individual learning routine on these critical learners
    success <- runif(nrow(agent))  # vector with random numbers each between 0 and 1
    agent$behaviour[critical.wrong & success < p] <- E  # if agent is an individual learner, and p exceeds success, then they learn the correct behaviour
    agent$behaviour[critical.wrong & success >= p] <- 1 - E  # otherwise, they learn the incorrect behaviour
    # subtract bc from fitness of ILing CLers
    agent$fitness[critical.wrong] <- agent$fitness[critical.wrong] - b*c
    
    }
  }
  
  # for all agents:
  # for those agents with behaviour matched to the environment, add b
  agent$fitness[agent$behaviour == E] <- agent$fitness[agent$behaviour == E] + b  
  # for those agents with behaviour not matched to the environment, subtract b
  agent$fitness[agent$behaviour != E] <- agent$fitness[agent$behaviour != E] - b  
    
  agent  # return agent data frame
}

# new functions for getting fitness and number of ind and critical learners
GetILnumber <- function (agent) {
  sum(agent$learning == "individual") / nrow(agent)  # return n.IL as a proportion of N
}
GetCLnumber <- function (agent) {
  sum(agent$learning == "critical") / nrow(agent)  # return n.CL as a proportion of N
}
GetILfitness <- function (agent) {
  if (sum(agent$learning == "individual") > 0) {  # if there is at least one individual learner
    w.IL <- mean(agent$fitness[agent$learning == "individual"])  # get their mean fitness
  } else {
    w.IL <- 0  # otherwise return zero
  }
  w.IL
}
GetCLfitness <- function (agent) {
  if (sum(agent$learning == "critical") > 0) {  # if there is at least one critical learner
    w.CL <- mean(agent$fitness[agent$learning == "critical"])  # get their mean fitness
  } else {
    w.CL <- 0  # otherwise return zero
  }
  w.CL
}

#  produce new generation from previous
ReproductionStage.crit <- function (agent, agent.previous, mu) {
  
  # from now on, agent denotes new generation, agent.previous denotes previous
  
  agent$behaviour <- NA  # reset agent's behaviour and fitness
  agent$fitness <- NA
  
  # to determine learning, need to get probability of individual learner being a parent, weighted by relative fitness
  W <- GetW(agent.previous)  # get overall total fitness of previous generation
  
  # get probability of new agent being an IL, weighted by fitness
  if (sum(agent.previous$learning == "individual") > 0) {
    prob.IL <- sum(agent.previous$fitness[agent.previous$learning == "individual"]) / W 
  } else {
    prob.IL <- 0
  }
  
  # get probability of new agent being an SL, weighted by fitness
  if (sum(agent.previous$learning == "social") > 0) {
    prob.SL <- sum(agent.previous$fitness[agent.previous$learning == "social"]) / W 
  } else {
    prob.SL <- 0
  }
  
  # vector with random numbers each between 0 and 1, to determine prob of being ind learner
  success.IL <- runif(nrow(agent.previous))  
  # second vector to give prob of SL
  success.SL <- runif(nrow(agent.previous))
 
  # new agent is IL with probability prob.IL
  agent$learning[success.IL < prob.IL] <- "individual" 
  # otherwise SL, with prob prob.SL
  agent$learning[success.IL >= prob.IL & success.SL < prob.SL] <- "social"
  # otherwise CL
  agent$learning[success.IL >= prob.IL & success.SL >= prob.SL] <- "critical"
  
  # now mutation: 
  # vector with random numbers each between 0 and 1, to determine probability of mutation for each agent
  success.mu <- runif(nrow(agent.previous))  
  # random number 0 or 1, to determine which of the other two learning types to switch to
  mutant <- sample(0:1,nrow(agent.previous), replace = TRUE)  
  
  # individual learners: switch to either social or critical
  agent$learning[success.mu < mu & agent.previous$learning == "individual" & mutant == 0] <- "social"
  agent$learning[success.mu < mu & agent.previous$learning == "individual" & mutant == 1] <- "critical"
  
  # social learners: switch to either individual or critical
  agent$learning[success.mu < mu & agent.previous$learning == "social" & mutant == 0] <- "individual"
  agent$learning[success.mu < mu & agent.previous$learning == "social" & mutant == 1] <- "critical"
  
  # critical learners: switch to either social or indiviudal
  agent$learning[success.mu < mu & agent.previous$learning == "critical" & mutant == 0] <- "social"
  agent$learning[success.mu < mu & agent.previous$learning == "critical" & mutant == 1] <- "individual"
  
  agent  # return agent
}

GetOutput.crit <- function(output, t, agent) {
  output$timestep[t] <- t
  output$n.SL[t] <- GetSLnumber(agent)
  output$n.IL[t] <- GetILnumber(agent)
  output$n.CL[t] <- GetCLnumber(agent)
  output$w.SL[t] <- GetSLfitness(agent)
  output$w.IL[t] <- GetILfitness(agent)
  output$w.CL[t] <- GetCLfitness(agent)
  output$W[t] <- GetW(agent) / nrow(agent)
  output
}

#  run the simulation once
RunSimulation.crit <- function (N, t.max, u, p, w, b, c, s, mu, output) {
  
  # check parameters
  if (b*(1+c) > 1 || b*(1+s) > 1) {
    stop("Invalid parameter values: ensure b*(1+c) < 1 and b*(1+s) < 1")
  }
  
  #initialise agent data frame
  agent <- InitialiseAgent(N)
  
  #  initialise environment E
  E <- 0
  
  for (t in 1:t.max) { # cycle thru timesteps
    
    #  update agent as a result of learning  
    agent <- LearningStage.crit(agent, p, E, agent.previous, w, b, c, s)
    
    #  get variables needed for the plots later on
    output <- GetOutput.crit(output, t, agent)
    
    # copy this generation into agent.previous, to serve as demonstrators to next gen's social learners
    agent.previous <- agent
    
    #  create new generation of agents from agent.previous
    agent <- ReproductionStage.crit(agent, agent.previous, mu)
    
    #  possible environmental change
    E <- EnvironmentalChange(E, u)
    
  }  # end of timestep for
  
  output  # return output data frame
}

# simulation loop, repeat runs times and append output to output dataframe 
SimulationLoop.crit <- function (runs, N, t.max, u, p, w, b, c, s, mu) {
  
  # initialise output data frame
  output <- InitialiseOutput.crit(t.max)
  
  # set up s loop, each run produces output. Add to output dataframe if it already exists
  for (i in 1:runs) {
    
    # for first run, create output data frame
    if (i == 1) {
      output <- RunSimulation.crit(N, t.max, u, p, w, b, c, s, mu, output)
    } else {
      output <- rbind(output, RunSimulation.crit(N, t.max, u, p, w, b, c, s, mu, output))  # otherwise add it on
    }
  }  # end of runs loop
  
  # draw plots
  DrawPlots.crit(output, w, b, p, c, u, s, mu, runs)

}

GetOutputSummary.crit <- function (output) {
  mean <- aggregate(output[, 2:8], by = list(output$timestep), mean)  # gives means
  max <- aggregate(output[, 2:8], by = list(output$timestep), max)  # gives maximums
  min <- aggregate(output[, 2:8], by = list(output$timestep), min)  # gives minimums
  
  data.frame(mean$n.SL, min$n.SL, max$n.SL, mean$n.IL, min$n.IL, max$n.IL, mean$n.CL, min$n.CL, max$n.CL, mean$w.SL, min$w.SL, max$w.SL, mean$w.IL, min$w.IL, max$w.IL, mean$w.CL, min$w.CL, max$w.CL, mean$W, min$W, max$W)
}

DrawPlots.crit <- function (output, w, b, p, c, u, s, mu, runs) {
  # with ggplot (and multiplot from Rmisc)
  
  # first need to get means and maximums/minimums from raw output
  output.summary <- GetOutputSummary.crit(output)
  
  # create the graph title from parameter values
  graph.title <- paste("N = " ,nrow(output.summary), ", ", "w = ", w, ", ", "b = ", b, ", ", "c = ", c, ", ", "s = ", s, ", ", "u = ", u, ", ", "mu = ", mu, ", ", "p = ", p, ", ", "runs = ", runs, ", ", sep = "")
  
  # plot number of different learners
  # need to melt output into long format first, to plot three lines
  output.long <- output.summary[, c("mean.n.IL", "mean.n.SL","mean.n.CL")]
  names(output.long) <- c("Individual", "Social","Critical")
  output.long$Timestep <- c(1:nrow(output.summary))
  output.long <- melt(output.long, id = "Timestep", value.name = "Mean", variable.name = "Learning")
  
  plot1 <- ggplot(data = output.long, aes(x = Timestep, y = Mean, colour = Learning)) + geom_line() + labs(x = "Generation", y = "Frequency") + scale_y_continuous(limits = c(0,NA)) + theme_classic(base_size = 14) + ggtitle(graph.title) + theme(plot.title = element_text(size = 12), axis.line.x = element_line(colour = "black", size = 0.5), axis.line.y = element_line(colour = "black", size = 0.5))
  
  # plot mean fitness of individual, social and critical learners as three lines
  # need to melt output into long format first, to plot three lines
  output.long <- output.summary[, c("mean.w.IL", "mean.w.SL","mean.w.CL")]
  names(output.long) <- c("Individual", "Social", "Critical")
  output.long$Timestep <- c(1:nrow(output.summary))
  output.long <- melt(output.long, id = "Timestep", value.name = "Mean", variable.name = "Learning")
  
  plot2 <- ggplot(data = output.long, aes(x = Timestep, y = Mean, colour = Learning)) + geom_line() + labs(x = "Generation", y = "Fitness") + scale_y_continuous(limits = c(0,NA)) + theme_classic(base_size = 14) + theme(axis.line.x = element_line(colour = "black", size = 0.5), axis.line.y = element_line(colour = "black", size = 0.5))
  
  # plot mean population fitness, W, with a line showing expected W in a population of all individual learners
  plot3 <- ggplot(data = output.summary, aes(x = 1:nrow(output.summary), y = mean.W)) + geom_line() + geom_ribbon(aes(ymin = min.W, ymax = max.W), alpha = 0.2) + geom_hline(yintercept = w + b*(2*p - c - 1), linetype = 2) + labs(x = "Generation", y = "Population fitness") + scale_y_continuous(limits = c(0,NA)) + theme_classic(base_size = 14) + theme(axis.line.x = element_line(colour = "black", size = 0.5), axis.line.y = element_line(colour = "black", size = 0.5))
  
  # optionally, save to png image in folder 'graphs' with filename indicating parameters
  #graph.name <- paste("graphs/", "N" ,nrow(output.summary),"_w", w, "_b", b, "_c", c, "_s", s, "_u", u, "_mu", mu, "_p", p, "_runs", runs, ".png", sep = "")
  
  #png(file = graph.name, width = 20, height = 20, units = "cm", res = 300)
  #multiplot(plot1, plot2, plot3)
  #dev.off()
  
  # draw the plot again to leave it visible when programme finishes
  multiplot(plot1, plot2, plot3)
}

# main function, with defaults
RogersModel.crit <- function (runs = 10, N = 1000, t.max = 250, u = 0.1, p = 1, w = 1, b = 0.5, c = 0.9, s = 0, mu = 0.01, runin = 100) {
  
  SimulationLoop.crit(runs, N, t.max, u, p, w, b, c, s, mu)

}

# run simulation here
RogersModel.crit()

```

\newpage
\pagebreak

# References

Enquist, M., Eriksson, K., & Ghirlanda, S. (2007). Critical social learning: A solution to Rogers’ paradox of nonadaptive culture. American Anthropologist, 109(4), 727–734.

Rogers, A. (1988). Does biology constrain culture? American Anthropologist, 90(4), 819–831.