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Simulation.Rmd
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---
title: "Simulation"
author: "Hamed Nilforoshan"
date: "12/29/2021"
output:
github_document:
pandoc_args: --webtex
---
```{r setup, include=FALSE}
library(sigmoid)
library(tidyverse)
library(furrr)
library(BBmisc)
library(latex2exp)
# Set number of workers.
plan(multicore, workers = 4)
# Set the ggplot theme.
theme_set(theme_bw(base_size = 20))
```
## Overview
We construct a stylized example of college admissions to analyze the downstream consequences
of implementing many causal fairness definitions as decision-making constraints
## Setup
A "book of life" is a draw of each potential outcome for a population of a given
size. We'll define a function that generates a book of life for a given set of
parameters.
```{r book_of_life}
gen_book_of_life <- function(
pop_size,
frac_minority,
discrete,
e_0,
e_r,
e_noise,
m_0,
m_e,
m_noise,
t_0,
t_m,
t_e,
t_e_m,
t_noise,
y_reject_noise,
y_reject_noise_mean,
...
) {
# Define the structural equations.
f_E <- function(r_i, noise){
return(e_0 + e_r*r_i + noise)
}
f_M <- function(e_i, noise){
return(m_0 + m_e*e_i + noise)
}
f_T <- function(m_i, e_i, noise){
return(round(t_0 + t_m*m_i + t_e*e_i + t_e_m*e_i*m_i + noise ,digits=ifelse(discrete,0,3)))
}
# Generate the book of life.
book_of_life <- tibble(
# Generate the exogenous variables.
R = as.numeric(as.integer(rbernoulli(pop_size, frac_minority))), # Race (minority=1, majority=0)
E_noise = rnorm(pop_size, mean = 0, sd = e_noise), #Educational opportunities
E_minority = f_E(1,noise = E_noise),
E_majority = f_E(0,noise = E_noise),
M_noise = rnorm(pop_size, mean = 0, sd = m_noise), #Preparation
M_minority = f_M(E_minority, noise=M_noise),
M_majority = f_M(E_majority, noise=M_noise),
# Generate the endogenous variables.
T_noise = rnorm(pop_size, mean = 0, sd = t_noise), #Test score
T_minority = f_T(M_minority, E_minority, noise=T_noise),
T_majority = f_T(M_majority, E_majority, noise=T_noise),
T_majority_star = f_T(M_minority, E_majority, noise=T_noise),
T_minority_star = f_T(M_majority, E_minority, noise=T_noise),
E = if_else(R==1,E_minority,E_majority),
M = if_else(R==1,M_minority,M_majority),
T = if_else(R==1,T_minority,T_majority),
# Generate potential outcomes
Y_unif = runif(pop_size,min=0,max=1),
Y_reject_noise = rnorm(pop_size, mean = y_reject_noise_mean, sd = y_reject_noise),
Y = as.integer(sigmoid(M) >=Y_unif) ,
Y_raw = sigmoid(M),
Y_reject = as.integer(sigmoid(M-0.5) >=Y_unif),
Y_reject_raw = sigmoid(M-0.5),
)
}
```
## Initialize parameters and books of life
We initialize the training population size for the book of life, the fraction of the population that is a minority, and all structural equation model parameters.
We then sample two books of life. One is for estimation, and the second is for testing what happens when the fairness definitions are applied.
```{r init}
set.seed(1)
POP_SIZE_TRAIN = 10000000
POP_SIZE_TEST = 100000
FRAC_minority = 0.33
DISCRETE = TRUE
E_0 = 1
E_R = -1
E_NOISE = 1
M_0 = -1.0
M_E = 1.0
M_NOISE = 1.0
T_0 = 50
T_M = 4
T_E = 4
T_E_M = 1.0
T_NOISE = 7
Y_REJECT_NOISE_SD = 0.3
Y_REJECT_NOISE_MEAN = -0.5
df_train <-gen_book_of_life(pop_size=POP_SIZE_TRAIN,
frac_minority=FRAC_minority,
discrete=DISCRETE,
e_0=E_0,
e_r=E_R,
e_noise=E_NOISE,
m_0=M_0,
m_e=M_E,
m_noise=M_NOISE,
t_0=T_0,
t_m=T_M,
t_e=T_E,
t_e_m=T_E_M,
t_noise=T_NOISE,
y_reject_noise=Y_REJECT_NOISE_SD,
y_reject_noise_mean=Y_REJECT_NOISE_MEAN,)
df_test <-gen_book_of_life(pop_size=POP_SIZE_TEST,
frac_minority=FRAC_minority,
discrete=DISCRETE,
e_0=E_0,
e_r=E_R,
e_noise=E_NOISE,
m_0=M_0,
m_e=M_E,
m_noise=M_NOISE,
t_0=T_0,
t_m=T_M,
t_e=T_E,
t_e_m=T_E_M,
t_noise=T_NOISE,
y_reject_noise=Y_REJECT_NOISE_SD,
y_reject_noise_mean=Y_REJECT_NOISE_MEAN,)
DIVERSITY_UTILITY = 0.25
FRAC_ADMIT = 0.50
POP_SIZE_TEST_ADMIT = FRAC_ADMIT * POP_SIZE_TEST
write.csv(df_train,'./df_train.csv')
write.csv(df_test,'./df_test.csv')
```
## Run linear programs in .ipynb and get output
```{r run_commands}
system(sprintf('export PATH="/Users/hamedn/opt/anaconda3/bin:$PATH";
export DIVERSITY_UTILITY=%s;
export FRAC_ADMIT=%s;
jupyter nbconvert --execute --to notebook --inplace 00-Maximize-Utility.ipynb;
jupyter nbconvert --execute --to notebook --inplace 01-Counterfactual-Fairness.ipynb;
jupyter nbconvert --execute --to notebook --inplace 02-Path-Specific-Fairness.ipynb;
jupyter nbconvert --execute --to notebook --inplace 03-Equalized_Odds.ipynb;
jupyter nbconvert --execute --to notebook --inplace 04-Principal-Fairness.ipynb;
jupyter nbconvert --execute --to notebook --inplace 05-Counterfactual-Predictive-Parity.ipynb;', DIVERSITY_UTILITY, FRAC_ADMIT))
```
```{r load_lp_results}
lp_results <- read.csv("lp_results.csv", sep = "\t")
PC_FAIR = c('Counterfactual Fairness\n& Path-Specific Fairness\n(Random)', 0, lp_results[lp_results$Policy.Name=='Counterfactual Fairness',]$Frac.Diverse.Candidates * POP_SIZE_TEST_ADMIT, lp_results[lp_results$Policy.Name=='Counterfactual Fairness',]$Total.Graduated)
EO_FAIR = c('Counterfactual Equalized Odds', 0, lp_results[lp_results$Policy.Name=='Counterfactual Equalized Odds',]$Frac.Diverse.Candidates *POP_SIZE_TEST_ADMIT , lp_results[lp_results$Policy.Name=='Counterfactual Equalized Odds',]$Total.Graduated)
PRINCIPAL_FAIR = c('Principal Fairness', 0, lp_results[lp_results$Policy.Name=='Principal Fairness',]$Frac.Diverse.Candidates *POP_SIZE_TEST_ADMIT , lp_results[lp_results$Policy.Name=='Principal Fairness',]$Total.Graduated)
ML_PREDICT_PAR = c('Counterfactual Pred. Parity', 0, lp_results[lp_results$Policy.Name=='Counterfactual Pred. Parity',]$Frac.Diverse.Candidates *POP_SIZE_TEST_ADMIT, lp_results[lp_results$Policy.Name=='Counterfactual Pred. Parity',]$Total.Graduated)
MAX_UTILITY = c(expression("Maximum utility (when " * lambda ~ " = 0.01)"), 0, lp_results[lp_results$Policy.Name=='Maximize Utility',]$Frac.Diverse.Candidates *POP_SIZE_TEST_ADMIT, lp_results[lp_results$Policy.Name=='Maximize Utility',]$Total.Graduated)
```
## Draw the pareto curve
```{r pareto_curve_02}
N=round(POP_SIZE_TEST_ADMIT) * FRAC_minority * 2.0
MAX_minority_ADMIT = POP_SIZE_TEST_ADMIT * FRAC_minority* 2.0
pareto_ideal = NULL
for (num_minority_admit in seq(from=1,to=FRAC_minority * POP_SIZE_TEST + 1000 ,by=as.integer(POP_SIZE_TEST_ADMIT/1000))){
cat(num_minority_admit)
frac_minority = num_minority_admit
df_test_minority<- df_test[order(df_test$T,decreasing=TRUE),] %>% filter( R == 1 )
df_test_majority<- df_test[order(df_test$T,decreasing=TRUE),] %>% filter( R == 0 )
lives_saved_outcomes = (sum(head(df_test_minority,n=num_minority_admit)$Y) + sum(head(df_test_majority,n=POP_SIZE_TEST_ADMIT-num_minority_admit)$Y))
pareto_ideal = rbind(pareto_ideal, data.frame(frac_minority ,lives_saved_outcomes))
}
# ggplot colors
peach <- "#F8766D"
purple <- "#C77CFF"
blue <- "#00BFC4"
green <- "#00BE6C"
browngreen <- "#A3A500"
darkgreen <- "#DEAA88"
navy <- "#055C9D"
darkpurple <- "#431C53"
transparency <- 0.4
labels = c("1", "2", "3", "4", "5")
random_x <- 0.1
random_y <- 280000
max_graduation <- pareto_ideal[which.max(pareto_ideal$lives_saved_outcomes),]
MAX_BOARDS_PASSAGE = c("Max Graduation", 0, max_graduation$frac_minority, max_graduation$lives_saved_outcomes)
pareto_ideal$cutoff = pareto_ideal$frac_minority >= as.double(MAX_BOARDS_PASSAGE[[3]])
lo = loess(lives_saved_outcomes ~ frac_minority, pareto_ideal,span = 0.5 )
pareto_ideal$smooth_y = predict(lo, pareto_ideal$frac_minority, se = FALSE)
p <- ggplot() +
geom_line(data= filter(pareto_ideal, cutoff),aes(x=frac_minority, y=smooth_y, color='ML Outcomes (Given T, R)'),size=1.5) +
geom_line(data= filter(pareto_ideal, !cutoff),aes(x=frac_minority, y=smooth_y, color='ML Outcomes (Given T, R)'), linetype = "dashed",size=1.5) +
scale_x_continuous("Admitted Applicants from Target Group", labels = scales::comma) +
scale_y_continuous("College Degree Attainment", labels = scales::comma) +
coord_cartesian(xlim = c(0.0, 30000),ylim = c(21800, 31500),expand = FALSE) +
geom_point(aes(x = as.double(MAX_BOARDS_PASSAGE[[3]]), y = as.double(MAX_BOARDS_PASSAGE[[4]])), color = peach, size = 4) +
annotate("text",size=5, x = as.double(MAX_BOARDS_PASSAGE[[3]]), y = as.double(MAX_BOARDS_PASSAGE[[4]]), label = MAX_BOARDS_PASSAGE[[1]]) +
geom_point(aes(x = as.double(MAX_UTILITY[[3]]), y = as.double(MAX_UTILITY[[4]])), color = blue, size = 4) +
annotate("text",size=5, x = as.double(MAX_UTILITY[[3]]), y = as.double(MAX_UTILITY[[4]]), label = expression("Max Utility (for " * lambda ~ " = " * frac(1, 4) *")")) +
geom_point(aes(x = as.double(PC_FAIR[[3]]), y = as.double(PC_FAIR[[4]])), color = purple, size = 4) +
annotate("text",size=5,lineheight = 0.85,hjust = 1, x = as.double(PC_FAIR[[3]]), y = as.double(PC_FAIR[[4]]), label = PC_FAIR[[1]]) +
geom_point(aes(x = as.double(PRINCIPAL_FAIR[[3]]), y = as.double(PRINCIPAL_FAIR[[4]])), color = darkgreen, size = 4) +
annotate("text",size=5, x = as.double(PRINCIPAL_FAIR[[3]]), y = as.double(PRINCIPAL_FAIR[[4]]) , label = PRINCIPAL_FAIR[[1]]) +
geom_point(aes(x = as.double(ML_PREDICT_PAR[[3]]), y = as.double(ML_PREDICT_PAR[[4]])), color = green, size = 4) +
annotate("text",size=5, x = as.double(ML_PREDICT_PAR[[3]]), y = as.double(ML_PREDICT_PAR[[4]]), label = ML_PREDICT_PAR[[1]]) +
geom_point(aes(x = as.double(EO_FAIR[[3]]), y = as.double(EO_FAIR[[4]])), color = browngreen, size = 4) +
annotate("text",size=5,lineheight = 0.85,hjust = 1, x = as.double(EO_FAIR[[3]]), y = as.double(EO_FAIR[[4]]), label = EO_FAIR[[1]]) +
scale_color_manual(
values = c('ML Outcomes (Given T, R)'="violet" )) +
theme(legend.title = element_blank(),legend.position="none", axis.text=element_text(size=15), axis.title=element_text(size=18), panel.grid = element_line(color = rgb(235, 235, 235, 100, maxColorValue = 255)))
ggsave(plot=p, filename='./frontier.pdf', height=6, width=7)
p
```