Game Theory & Joint-Venture Negotiation
Kalai-Smorodinsky vs Nash Bargaining Split Formula
Allocate joint-venture surplus via Ehud Kalai and Meir Smorodinsky’s (1975) proportional ideal-point rule (u_1 - d_1)/(b_1 - d_1) = (u_2 - d_2)/(b_2 - d_2) when individual payoff caps bind.
Why Nash Bargaining ignores unilateral payoff caps — and how Kalai-Smorodinsky restores monotonicity!
Suppose two joint-venture partners divide Total Pie M > d_1 + d_2 over disagreement payoffs (d_1, d_2), EXCEPT Partner 1 faces a legal, regulatory, or capacity payoff cap C_1 < M - d_2 (so Partner 1’s Utopia / Ideal Point is b_1 = MIN(M - d_2, C_1), while uncapped Partner 2’s Utopia Point is b_2 = M - d_1)! Symmetric Nash Bargaining maximizes (u_1 - d_1)*(u_2 - d_2) subject to u_1 + u_2 = M and u_1 <= b_1, giving u_1_Nash = MIN(b_1, d_1 + 0.5*(M - d_1 - d_2)) — which gives Partner 1 ZERO credit versus Partner 2 until the cap literally truncates 50% of the pie! Ehud Kalai and Meir Smorodinsky (1975) replaced Nash’s Independence of Irrelevant Alternatives with Individual Monotonicity, equating fractional utopia gains (u_1 - d_1) / (b_1 - d_1) = (u_2 - d_2) / (b_2 - d_2) along Pareto frontier u_1 + u_2 = M (when b_1 + d_2 <= M): exact Kalai-Smorodinsky Payoff is u_1_KS = d_1 + ((b_1 - d_1) / ((b_1 - d_1) + (b_2 - d_2))) * (M - d_1 - d_2)!
Verify on Total_Pie M = 100 (B2), Disagreement_Payoffs d_1 = 0 (B3) and d_2 = 0 (B4), and Partner_1_Cap C_1 = 60 (B5) vs Uncapped Partner_2_Cap C_2 = 100 (B6): Utopia Gains are b_1 - d_1 = 60 and b_2 - d_2 = 100. Symmetric Nash ignores Partner 1’s tighter ceiling completely and awards 50.00 / 50.00! Kalai-Smorodinsky awards proportional utopia ratio 60 : 100 -> Partner 1 gets (60 / 160) * 100 = 37.50 and Partner 2 gets 62.50 (both achieving exact 37.50 / 60 = 62.50 / 100 = 0.6250 = 62.50% of their maximum possible aspiration!)!
| Bargaining Metric (M = 100, d = [0, 0], Caps = [60, 100]) | Formula | Verified Output |
|---|---|---|
| Partner 1 Utopia Aspiration (b_1) | MIN(M - d_2, C_1) | 60.00 (vs b_2 = 100.00) |
| Symmetric Nash Payoff (u_1_Nash) | MIN(b_1, d_1 + 0.5*(M - d_1 - d_2)) | 50.00 (Ignores Cap Asymmetry) |
| Kalai-Smorodinsky Payoff (u_1_KS) | d_1 + (g_1 / (g_1 + g_2)) * Surplus | 37.50 (u_2_KS = 62.50) |
| Equalized Utopia Achievement Ratio | (u_1_KS - d_1) / (b_1 - d_1) | 0.6250 (62.50% for Both!) |
Kalai-Smorodinsky Proportional Ideal-Point Payoff u_1_KS Formula
Validate positive cooperative surplus M > d_1 + d_2, non-binding joint feasibility C_1 + C_2 >= M, and strictly positive individual utopia gains b_i > d_i.
=LET(m_pi, B2, d_1, B3, d_2, B4, c_1, B5, c_2, B6, IF(OR(m_pi <= d_1 + d_2, c_1 <= d_1, c_2 <= d_2, c_1 + c_2 < m_pi), "Invalid Bargaining Set", LET(g_1, MIN(m_pi - d_2, c_1) - d_1, g_2, MIN(m_pi - d_1, c_2) - d_2, ROUND(d_1 + (g_1 / (g_1 + g_2)) * (m_pi - d_1 - d_2), 2))))Constrained Nash Bargaining Benchmark Payoff u_1_Nash Formula
When both caps are non-binding (C_1 = 100, C_2 = 100), Kalai-Smorodinsky and Nash coincide identically at 50.00!
=LET(m_pi, B2, d_1, B3, d_2, B4, c_1, B5, c_2, B6, IF(OR(m_pi <= d_1 + d_2, c_1 <= d_1, c_2 <= d_2, c_1 + c_2 < m_pi), "Invalid Bargaining Set", ROUND(MAX(m_pi - c_2, MIN(c_1, d_1 + 0.5 * (m_pi - d_1 - d_2))), 2)))Try the example
Use the original values or change them to test your understanding.