Game Theory and Business Decisions
Unit 9 Overview: Bargaining and Negotiation in Game Theory
9.1 Cooperative and Non-cooperative Bargaining Models
9.2 The Nash Bargaining Solution
9.3 Multi-party Negotiations and Coalition Formation

The Nash Bargaining Solution is a powerful tool in cooperative game theory, providing a unique outcome for bargaining situations. It balances efficiency and fairness by maximizing the product of players' utility gains, ensuring a win-win solution on the Pareto frontier.

This solution stands out for satisfying key axioms like Pareto optimality, symmetry, scale invariance, and independence of irrelevant alternatives. Compared to other bargaining solutions, Nash's approach offers a comprehensive framework for analyzing and resolving cooperative bargaining scenarios.

Nash Bargaining Solution

Nash bargaining solution properties

  • Selects a unique outcome based on a set of axioms in cooperative bargaining
  • Pareto optimality ensures the solution is on the Pareto frontier, where no player can be made better off without making another player worse off (win-win outcome)
  • Symmetry treats players equally if they have indistinguishable preferences and disagreement points (identical starting positions)
  • Scale invariance maintains the solution even when utility functions undergo affine transformations (changes in scale or origin)
  • Independence of irrelevant alternatives (IIA) guarantees the solution remains the same if the set of alternatives is reduced, as long as the original outcome is still available (consistency in decision-making)

Calculation of Nash bargaining solution

  • Maximizes the product of the players' utility gains over their disagreement points (starting positions)
  • Formula for two-player bargaining problem: max⁡(u1,u2)∈S(u1−d1)(u2−d2)\max_{(u_1, u_2) \in S} (u_1 - d_1)(u_2 - d_2)
    • SS represents the set of feasible utility allocations (possible outcomes)
    • uiu_i denotes the utility of player ii in the bargaining solution (outcome)
    • did_i represents the disagreement point utility of player ii (starting position)
  • Steps to find the solution:
    1. Identify the feasible set of utility allocations and the disagreement point (possible outcomes and starting positions)
    2. Calculate the product of utility gains for each point in the feasible set (measure of improvement)
    3. Select the point that maximizes the product of utility gains (optimal outcome)

Efficiency and fairness in Nash bargaining

  • Efficiency achieved through Pareto optimality, ensuring no resources are wasted (maximizing total utility gains)
    • Selects an outcome on the Pareto frontier (best possible outcome)
  • Fairness addressed through the symmetry property, ensuring equal utility gains for players with identical preferences and disagreement points (equal treatment)
    • May not always align with other fairness criteria (equality or need-based distribution)
  • Balances efficiency and fairness by maximizing the product of utility gains, favoring outcomes that significantly improve both players' positions (win-win solution)

Nash vs other bargaining solutions

  • Kalai-Smorodinsky bargaining solution selects an outcome based on the ratio of maximum utility gains (relative gains)
    • Satisfies individual monotonicity but not independence of irrelevant alternatives (sensitive to changes in alternatives)
  • Egalitarian bargaining solution maximizes the minimum utility gain among players (focus on equality)
    • Prioritizes equality over efficiency (may sacrifice total gains for equal distribution)
  • Utilitarian bargaining solution maximizes the sum of utility gains for all players (focus on efficiency)
    • Prioritizes efficiency over fairness (may lead to unequal distribution of gains)
  • Nash bargaining solution is unique in satisfying all four key axioms (Pareto optimality, symmetry, scale invariance, and IIA), making it widely used and studied in bargaining theory (comprehensive approach)