Loss aversion, sunk cost and regression — what it costs you

PASSENGERS, RATE, ACC = 50_000_000, 1 / 10_000_000, 0.999
wanted = round(PASSENGERS * RATE)
ordinary = PASSENGERS - wanted
caught = round(wanted * ACC)
false_alarms = round(ordinary * (1 - ACC))
alarms = caught + false_alarms

record('air_passengers', f'{PASSENGERS:,}')
record('air_wanted', wanted)
record('air_accuracy', f'{ACC:.1%}')
record('air_caught', caught)
record('air_false', f'{false_alarms:,}')
record('air_alarms', f'{alarms:,}')
record('air_worth', f'1 in {round(alarms / caught):,}')
record('air_per_day', f'{round(false_alarms / 365):,}')
record('air_columns', ['', 'System raises an alarm', 'It does not', 'All'])
record('air_rows', [
    ['Actually wanted', f'{caught}', f'{wanted - caught}', f'{wanted}'],
    ['An ordinary passenger', f'{false_alarms:,}', f'{ordinary - false_alarms:,}', f'{ordinary:,}'],
    ['All', f'{alarms:,}', f'{PASSENGERS - alarms:,}', f'{PASSENGERS:,}'],
])
print(f'{caught} real of {alarms:,} alarms — about 1 in {round(alarms/caught):,}')
print(f'{round(false_alarms/365):,} innocent people stopped every day')

SINGLE = 1 / 8_500          # roughly the rate the court was given
record('clark_single', '1 in 8,500')
record('clark_squared', f'1 in {round(1 / SINGLE**2):,}')

# If a first cot death makes a second more likely — as later evidence
# suggested — the honest figure moves by orders of magnitude.
record('clark_columns', ['If a first death makes a second…', 'Chance of two'])
record('clark_rows', [
    ['no more likely (what the court was told)', f'1 in {round(1 / SINGLE**2):,}'],
    ['5 times more likely', f'1 in {round(1 / (SINGLE * SINGLE * 5)):,}'],
    ['10 times more likely', f'1 in {round(1 / (SINGLE * SINGLE * 10)):,}'],
])
for mult in (1, 5, 10):
    print(f'{mult:>3}x -> 1 in {round(1 / (SINGLE * SINGLE * mult)):,}')

n = 200
skill = rng.normal(100, 12, n)
q1 = skill + rng.normal(0, 18, n)
q2 = skill + rng.normal(0, 18, n)

top, bot = np.argsort(q1)[-20:], np.argsort(q1)[:20]
record('rep_n', n)
record('top_q1', f'{q1[top].mean():.0f}')
record('top_q2', f'{q2[top].mean():.0f}')
record('top_got_worse', f'{(q2[top] < q1[top]).mean():.0%}')
record('bot_q1', f'{q1[bot].mean():.0f}')
record('bot_q2', f'{q2[bot].mean():.0f}')
record('bot_improved', f'{(q2[bot] > q1[bot]).mean():.0%}')

def plot_reps(ax):
    for grp, colour, label in ((top, '#e2574c', 'best 10% of quarter one'),
                               (bot, '#3b6fd4', 'worst 10% of quarter one')):
        for i in grp:
            ax.plot([0, 1], [q1[i], q2[i]], color=colour, alpha=0.16, lw=1)
        ax.plot([0, 1], [q1[grp].mean(), q2[grp].mean()], color=colour, lw=2.6, label=label)
    ax.axhline(skill.mean(), color='#9aa0aa', ls='--', lw=1)
    ax.annotate('everyone\'s average', (1, skill.mean()), textcoords='offset points',
                xytext=(-6, 6), ha='right', fontsize=9, color='#77777f')
    ax.set_xticks([0, 1]); ax.set_xticklabels(['Quarter one', 'Quarter two'])
    ax.set_ylabel('% of target')
    ax.legend(frameon=False, loc='upper center')
    ax.grid(axis='x', visible=False)

save_fig('nobody-was-praised', plot_reps, figsize=(6.8, 4.0))
print(f'best 10%: {q1[top].mean():.0f} -> {q2[top].mean():.0f}, {(q2[top]<q1[top]).mean():.0%} got worse')
print(f'worst 10%: {q1[bot].mean():.0f} -> {q2[bot].mean():.0f}, {(q2[bot]>q1[bot]).mean():.0%} improved')

HABIT, WOBBLE, M = 6.0, 11.0, 500
truth = rng.normal(50, 15, M)
judge1 = truth + HABIT + rng.normal(0, WOBBLE, M)
judge2 = truth + HABIT + rng.normal(0, WOBBLE, M)
rule = truth + HABIT

err = lambda p: float(np.sqrt(np.mean((p - truth) ** 2)))
e1, e2, er = err(judge1), err(judge2), err(rule)
disagree = float(np.mean(np.abs(judge1 - judge2)))

record('judge_habit', f'{HABIT:.0f}')
record('judge_cases', M)
record('judge1_error', f'{e1:.1f}')
record('judge2_error', f'{e2:.1f}')
record('rule_error', f'{er:.1f}')
record('judges_disagree_by', f'{disagree:.1f}')
record('wobble_share', f'{(e1 - er) / e1:.0%}')
record('judge_columns', ['', 'Habit', 'Wobble', 'Typical error'])
record('judge_rows', [
    ['Underwriter one', f'+{HABIT:.0f}', 'yes', f'{e1:.1f}'],
    ['Underwriter two', f'+{HABIT:.0f}', 'yes', f'{e2:.1f}'],
    ['A written-down rule', f'+{HABIT:.0f}', 'none', f'{er:.1f}'],
])

def plot_judges(ax):
    ax.bar(['Underwriter\none', 'Underwriter\ntwo', 'A written-down\nrule'],
           [e1, e2, er], color=['#9aa0aa', '#9aa0aa', '#e2574c'])
    for i, v in enumerate([e1, e2, er]):
        ax.text(i, v + 0.25, f'{v:.1f}', ha='center', fontsize=11)
    ax.axhline(HABIT, color='#3b6fd4', ls='--', lw=1.2)
    ax.annotate('the habit they all share', (2.4, HABIT), textcoords='offset points',
                xytext=(0, 6), ha='right', fontsize=9, color='#3b6fd4')
    ax.set_ylabel('Typical error')
    ax.grid(axis='x', visible=False)

save_fig('habit-and-wobble', plot_judges, figsize=(6.6, 3.8))
print(f'judges {e1:.1f} and {e2:.1f}; rule {er:.1f}; they disagree by {disagree:.1f}')
print(f'{(e1-er)/e1:.0%} of the error was wobble')