# Who should own the asset? The Grossman-Hart-Moore property-rights model, numerically.
# Two parties invest before they bargain: H (the foreign headquarters: product, brand, technology,
# investment h) and L (the local partner: distribution, customer and regulator relationships, l).
# Investment costs h^2/2 and l^2/2. Inside the relationship the venture is worth
#     V = a*h + b*l.
# If bargaining fails, each party has an outside option that depends on who owns the key asset
# (for example the local operating company holding licences and the customer base). The owner can
# still use part of its own investment with the asset (share 0.5); a non-owner keeps only 0.2 of
# its investment's value. Ex post they split the gain over outside options 50/50 (Nash bargaining),
# so each invests where marginal cost = 1/2 (marginal value inside + marginal outside option).
# Under joint ownership either party can block use of the asset, so neither has an outside option.
def outcome(owner, a, b):
    oh = 0.5 if owner == "H" else 0.2       # share of own investment usable outside the relationship
    ol = 0.5 if owner == "L" else 0.2
    if owner == "joint":                    # joint ownership: either party can veto use of the asset
        oh = ol = 0.0
    h = 0.5 * (a + oh * a)                  # first-order conditions: h = (a + oh*a)/2
    l = 0.5 * (b + ol * b)
    surplus = a * h + b * l - h * h / 2 - l * l / 2
    return h, l, surplus

for a, b, story in ((2.0, 1.0, "HQ's investment matters more (technology-led product)"),
                    (1.0, 2.0, "local investment matters more (distribution- and licence-led market)")):
    first_best = (a * a + b * b) / 2
    print(f"{story}: first-best surplus {first_best:.2f}")
    for owner in ("H", "L", "joint"):
        h, l, s = outcome(owner, a, b)
        print(f"   owner {owner:5s}: investments h = {h:.2f}, l = {l:.2f}, surplus {s:.2f} ({s / first_best:.0%} of first best)")
# HQ's investment matters more (technology-led product): first-best surplus 2.50
#    owner H    : investments h = 1.50, l = 0.60, surplus 2.29 (92% of first best)
#    owner L    : investments h = 1.20, l = 0.75, surplus 2.15 (86% of first best)
#    owner joint: investments h = 1.00, l = 0.50, surplus 1.88 (75% of first best)
# local investment matters more (distribution- and licence-led market): first-best surplus 2.50
#    owner H    : investments h = 0.75, l = 1.20, surplus 2.15 (86% of first best)
#    owner L    : investments h = 0.60, l = 1.50, surplus 2.29 (92% of first best)
#    owner joint: investments h = 0.50, l = 1.00, surplus 1.88 (75% of first best)

# Formal, relational or hybrid governance when the environment moves faster than contracts can be
# written. A stylised model in the spirit of Baker, Gibbons & Murphy (2002); numbers illustrative.
# Each period the venture creates surplus S = 10, and a shock of size z arrives, z ~ exponential with
# mean sigma (the speed of environmental change). Discount factor delta = 0.8.
#  Formal only: written clauses cover shocks up to Zw (drafting costs 0.15 * Zw per period). A larger
#    shock is a gap: the parties renegotiate at arm's length and lose 6 of that period's surplus.
#  Relational only: every shock is handled by an informal promise to adjust fairly. The promise holds
#    while z <= cap = delta / (1 - delta) * (S - fallback); a larger z is worth reneging over, and the
#    relationship ends. Even when nobody reneges, a partner may MISREAD a fair adjustment as a breach
#    (probability m per period); that also ends the relationship. m is higher across cultures.
#  Hybrid: written state-contingent milestones handle shocks up to Zw with no room for misreading;
#    the promise handles the rest. Breaking the promise now leaves the formal contract (worth 2 per
#    period), so the fallback rises and the cap shrinks: the substitution effect in Baker et al. (1994).
# Values are per-period equivalents, (1 - delta) times the discounted value of the relationship.
import math

S, LOSS, DRAFT, DELTA = 10.0, 6.0, 0.15, 0.8

def tail(a, sigma):
    return math.exp(-a / sigma)

def per_period(flow, p_end):
    return flow * (1 - DELTA) / (1 - DELTA * (1 - p_end))

def formal(sigma, m):
    return max((S - DRAFT * zw - LOSS * tail(zw, sigma), tail(zw, sigma))
               for zw in (k * 0.25 for k in range(401)))

def relational(sigma, m):
    cap = DELTA / (1 - DELTA) * S
    p_end = tail(cap, sigma) + (1 - tail(cap, sigma)) * m
    return per_period(S, p_end), p_end

def hybrid(sigma, m):
    cap = DELTA / (1 - DELTA) * (S - 2.0)
    best = None
    for zw in (k * 0.25 for k in range(401)):
        covered = 1 - tail(zw, sigma)
        reneg = tail(max(zw, cap), sigma)
        p_end = reneg + (1 - covered - reneg) * m          # misreading only where nothing is written
        v = per_period(S - DRAFT * zw, p_end)
        if best is None or v > best[0]: best = (v, p_end)
    return best

for m, who in ((0.005, "same-culture pair, misreading 0.5 %"), (0.04, "cross-cultural pair, misreading 4 %")):
    print(f"{who}:")
    print("  speed of change | formal: value (gaps) | relational: value (ends) | hybrid: value (ends) | best")
    for sigma in (1, 3, 6, 12, 24):
        fv, fp = formal(sigma, m); rv, rp = relational(sigma, m); hv, hp = hybrid(sigma, m)
        best = max((("formal", fv), ("relational", rv), ("hybrid", hv)), key=lambda x: x[1])[0]
        print(f"  {sigma:15d} | {fv:6.2f} ({fp:5.1%})       | {rv:6.2f} ({rp:5.1%})           |"
              f" {hv:6.2f} ({hp:5.1%})       | {best}")
# same-culture pair, misreading 0.5 %:
#   speed of change | formal: value (gaps) | relational: value (ends) | hybrid: value (ends) | best
#                 1 |   9.30 ( 2.4%)       |   9.80 ( 0.5%)           |   9.81 ( 0.4%)       | hybrid
#                 3 |   8.38 ( 7.6%)       |   9.80 ( 0.5%)           |   9.80 ( 0.5%)       | relational
#                 6 |   7.39 (14.7%)       |   9.76 ( 0.6%)           |   9.62 ( 1.0%)       | relational
#                12 |   6.03 (29.9%)       |   8.61 ( 4.0%)           |   7.71 ( 7.4%)       | relational
#                24 |   4.56 (60.0%)       |   5.64 (19.3%)           |   4.83 (26.7%)       | relational
# cross-cultural pair, misreading 4 %:
#   speed of change | formal: value (gaps) | relational: value (ends) | hybrid: value (ends) | best
#                 1 |   9.30 ( 2.4%)       |   8.62 ( 4.0%)           |   9.50 ( 0.4%)       | hybrid
#                 3 |   8.38 ( 7.6%)       |   8.62 ( 4.0%)           |   9.02 ( 1.2%)       | hybrid
#                 6 |   7.39 (14.7%)       |   8.58 ( 4.1%)           |   8.56 ( 3.1%)       | relational
#                12 |   6.03 (29.9%)       |   7.71 ( 7.4%)           |   7.01 (10.7%)       | relational
#                24 |   4.56 (60.0%)       |   5.30 (22.1%)           |   4.60 (29.3%)       | relational