# Deci, Koestner & Ryan (1999; interesting vs dull tasks: the 11 free-choice studies of its
# supplemental analysis, also reported in their 2001 review): free-choice behaviour, Cohen's d.
# Common-language effect size: P(rewarded person persists less than a control) = Phi(-d / sqrt(2)),
# assuming normal distributions with equal variance.
from statistics import NormalDist
PHI = NormalDist().cdf
ROWS = [
    ("all rewards", -0.24),
    ("verbal rewards (positive feedback)", 0.33),
    ("  ... children", 0.11),
    ("  ... college students", 0.43),
    ("tangible rewards", -0.34),
    ("  unexpected tangible", 0.01),
    ("  expected tangible", -0.36),
    ("    task-noncontingent", -0.14),
    ("    engagement-contingent", -0.40),
    ("    completion-contingent", -0.36),
    ("    performance-contingent", -0.28),
    ("tangible, interesting tasks (11 studies)", -0.68),
    ("tangible, dull tasks (11 studies)", 0.18),
]
print("reward type                               |     d | P(rewarded person persists less)")
for name, d in ROWS:
    print(f"{name:41s} | {d:+5.2f} | {PHI(-d / 2 ** 0.5):6.0%}")
print("50 % = no difference. Several small-k rows (unexpected, task-noncontingent, children's")
print("verbal rewards, dull tasks) were not statistically significant in the source.")
# reward type                               |     d | P(rewarded person persists less)
# all rewards                               | -0.24 |    57%
# verbal rewards (positive feedback)        | +0.33 |    41%
#   ... children                            | +0.11 |    47%
#   ... college students                    | +0.43 |    38%
# tangible rewards                          | -0.34 |    59%
#   unexpected tangible                     | +0.01 |    50%
#   expected tangible                       | -0.36 |    60%
#     task-noncontingent                    | -0.14 |    54%
#     engagement-contingent                 | -0.40 |    61%
#     completion-contingent                 | -0.36 |    60%
#     performance-contingent                | -0.28 |    58%
# tangible, interesting tasks (11 studies)  | -0.68 |    68%
# tangible, dull tasks (11 studies)         | +0.18 |    45%
# 50 % = no difference. Several small-k rows (unexpected, task-noncontingent, children's
# verbal rewards, dull tasks) were not statistically significant in the source.

# A reward as a signal, a stripped-down reading of Benabou & Tirole (2003); values illustrative.
# Task value HIGH (1.0) or LOW (0.2), prior 0.5; the principal knows the type and sets the reward,
# reward ~ Normal(mu_type, 0.15). The agent acts today if E[v | reward] + reward > cost and,
# with the reward withdrawn, tomorrow if E[v | reward] > cost (cost 0.55).
import random
from statistics import NormalDist
random.seed(3)
V = {"HIGH": 1.0, "LOW": 0.2}; COST = 0.55; SD = 0.15

def posterior_high(r, mu):
    lh = NormalDist(mu["HIGH"], SD).pdf(r) * 0.5
    ll = NormalDist(mu["LOW"], SD).pdf(r) * 0.5
    return lh / (lh + ll)

def world(gen, inf=None, announced=True, n=20000):
    # gen: how rewards are actually set; inf: how the agent believes they are set (None = ignores)
    today = tomorrow = highs = 0
    for _ in range(n):
        t = "HIGH" if random.random() < 0.5 else "LOW"
        r = max(0.0, random.gauss(gen[t], SD)) if gen else 0.0
        p = posterior_high(r, inf) if inf else 0.5
        ev = p * V["HIGH"] + (1 - p) * V["LOW"]
        today += ev + (r if announced else 0.0) > COST
        highs += t == "HIGH"
        tomorrow += t == "HIGH" and ev > COST
    return today / n, tomorrow / highs

TRACK = {"HIGH": 0.1, "LOW": 0.5}; FLAT = {"HIGH": 0.3, "LOW": 0.3}
RICH = {"HIGH": 0.5, "LOW": 0.5}
print("tomorrow's free choice is counted for tasks that are in fact HIGH-value")
print("condition                                          | engage today | HIGH tasks tomorrow")
for label, gen, inf, ann in (
        ("no reward (prior only: E[v] = 0.60)", None, None, False),
        ("reward tracks the principal's knowledge", TRACK, TRACK, True),
        ("same reward whatever the type, agent knows it", FLAT, FLAT, True),
        ("surprise reward to everyone, after the choice", FLAT, None, False),
        ("generous pay for every task, agent reads it as", RICH, TRACK, True),
        ("  a signal that the task needs inducement", None, None, None)):
    if ann is None:
        print(f"{label:50s} |"); continue
    a, b = world(gen, inf, ann)
    print(f"{label:50s} | {a:12.0%} | {b:19.0%}")
p = posterior_high(0.5, TRACK)
print(f"a reward of 0.5 moves P(task is worth doing for its own sake) from 50 % to {p:.0%}")
# tomorrow's free choice is counted for tasks that are in fact HIGH-value
# condition                                          | engage today | HIGH tasks tomorrow
# no reward (prior only: E[v] = 0.60)                |         100% |                100%
# reward tracks the principal's knowledge            |         100% |                 92%
# same reward whatever the type, agent knows it      |         100% |                100%
# surprise reward to everyone, after the choice      |         100% |                100%
# generous pay for every task, agent reads it as     |         100% |                 11%
#   a signal that the task needs inducement          |
# a reward of 0.5 moves P(task is worth doing for its own sake) from 50 % to 3%