Week 11: Fairness vs. Social Welfare

DSAN 5450: Data Ethics and Policy
Spring 2025, Georgetown University

Author

Affiliation

Jeff Jacobs

jj1088@georgetown.edu

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Utility \(\rightarrow\) Social Welfare

# For slides
library(ggplot2)
cbPalette <- c("#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7")
options(ggplot2.discrete.colour = cbPalette)
# Theme generator, for given sizes
theme_dsan <- function(plot_type = "full") {
    if (plot_type == "full") {
        custom_base_size <- 16
    } else if (plot_type == "half") {
        custom_base_size <- 22
    } else if (plot_type == "quarter") {
        custom_base_size <- 28
    } else {
        # plot_type == "col"
        custom_base_size <- 22
    }
    theme <- theme_classic(base_size = custom_base_size) +
        theme(
            plot.title = element_text(hjust = 0.5),
            plot.subtitle = element_text(hjust = 0.5),
            legend.title = element_text(hjust = 0.5),
            legend.box.background = element_rect(colour = "black")
        )
    return(theme)
}

knitr::opts_chunk$set(fig.align = "center")
g_pointsize <- 5
g_linesize <- 1
# Technically it should always be linewidth
g_linewidth <- 1
g_textsize <- 14

remove_legend_title <- function() {
    return(theme(
        legend.title = element_blank(),
        legend.spacing.y = unit(0, "mm")
    ))
}

Externalities

• Jeef and Keef are roommates: Jeef loves listening to Tony Danza Tapdance Extravaganza, but Keef is normal and slowly dies inside with each additional song

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ lubridate 1.9.3     ✔ tibble    3.2.1
✔ purrr     1.0.2     ✔ tidyr     1.3.1
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

music_df <- tribble(
  ~Songs, ~Jeef, ~Keef,
  0, 0, 0,
  1, 13, -2,
  2, 18, -6,
  3, 24, -13,
  4, 28, -20,
  5, 30, -42
)
music_df <- music_df |>
  mutate(Total = Jeef + Keef)
music_df

Songs	Jeef	Keef	Total
0	0	0	0
1	13	-2	11
2	18	-6	12
3	24	-13	11
4	28	-20	8
5	30	-42	-12

long_df <- music_df |>
  pivot_longer(!Songs, names_to="Roommate", values_to="Utility")
util_df <- long_df |>
  filter(Roommate != "Total")
ggplot(util_df, aes(x=Songs, y=Utility, color=Roommate)) +
  geom_line(linewidth=g_linewidth) +
  geom_point(size=g_pointsize) +
  labs(
    title="Individual Utility: Jeef vs. Keef",
    x="Number of Songs Played",
    y="Utility"
  ) +
  theme_dsan("quarter")

welfare_df <- long_df |>
  filter(Roommate == "Total")
ggplot(welfare_df, aes(x=Songs, y=Utility, color=Roommate)) +
  geom_line(linewidth=g_linewidth) +
  geom_point(size=g_pointsize) +
  labs(
    title="Social Welfare: Jeef and Keef",
    x="Number of Songs Played",
    y="Social Welfare"
  ) +
  scale_color_manual(values=c(cbPalette[3]), labels=c("Total      ")) +
  theme_dsan("quarter") +
  remove_legend_title()

So What’s the Issue?

• These utility values are not observed
• If we try to elicit them, both Jeef and Keef have strategic incentives to lie (over-exaggerate)
• Jeef maximizes own utility by reporting \(u_j(s) = \infty\)
• Keef maximizes own utility by reporting \(u_k(s) = -\infty\)
• (…Second price auctions tho)

Now with Scarce Resources

• In a given week, Jeef and Keef have 14 meals and 7 aux hours to divide amongst them

\[ \begin{align*} \max_{m_1,m_2,a_1,a_2}& W(u_1(m_1,a_1),u_2(m_2,a_2)) \\ \text{s.t. }& m_1 + m_2 \leq 14 \\ \phantom{\text{s.t. }} & ~ \, a_1 + a_2 \; \leq 7 \end{align*} \]

• Let’s assume \(u_i(m_i, a_i) = m_i + a_i\) for both
• \(\Rightarrow\) One solution: \(m_1 = 14, m_2 = 0, a_1 = 7, a_2 = 0\)…
• \(\Rightarrow\) Another: \(m_1 = 0, m_2 = 14, a_1 = 0, a_2 = 7\)…
• Who decides? Any decision implies \(\omega_1, \omega_2\) (\(\omega_1 + \omega_2 = 1\))

The Dark Secret Behind Fairness in AI

The Conveniently-Left-Out Detail

• Recall predictive parity:

\[ \mathbb{E}[Y \mid D = 1, A = 1] = \mathbb{E}[Y \mid D = 1, A = 0] \]

• Who decides which \(Y\) to pick?
• Answer: Whoever picks the objective function!
• For profit-maximizing firm: \(\mathbb{E}[D (Y - c)]\)
• For welfare-maximizing society: \(W(u_1(D), \ldots, u_n(D))\)
• Do these align? Sometimes yes, often no (affirmative action!)

Remaining (Challenging) Details

• Who gets included in the SWF?
• People in one community? One state? One country?
• People in the future?
• Animals?

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