9.2 Nash bargaining solution

The Nash bargaining solution is a key concept in cooperative game theory. It provides a unique outcome for two-player bargaining problems, satisfying four important axioms: Pareto optimality, symmetry, scale invariance, and independence of irrelevant alternatives.

This solution maximizes the product of players' utility gains relative to their disagreement point. It's widely applied in various scenarios, from labor negotiations to resource allocation, offering a balanced approach to efficiency and fairness in bargaining situations.

Nash Bargaining Solution

Definition and Properties

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• The Nash bargaining solution is a cooperative game theory concept providing a unique solution to a two-player bargaining problem, assuming players can communicate and make binding agreements
• Satisfies four axioms:
  • Pareto optimality: No other outcome can make one player better off without making the other worse off
  • Symmetry: If players are indistinguishable, the solution should provide equal payoffs
  • Scale invariance: The solution is unaffected by linear transformations of utility functions
  • Independence of irrelevant alternatives: If the solution is chosen from a set of alternatives, it should still be chosen if the set is reduced, as long as the original solution remains available

Mathematical Formulation

• The Nash bargaining solution $(x^*, y^*)$ maximizes the Nash product: $(x^* - d_1)(y^* - d_2)$
  • $d_1$ and $d_2$ are the disagreement payoffs for players 1 and 2, respectively
  • The disagreement point represents the payoffs players would receive if no agreement is reached
• The feasible set is the set of all possible payoff combinations
• The solution selects the point in the feasible set that maximizes the product of players' utility gains relative to the disagreement point

Applying the Nash Bargaining Solution

Two-Player Bargaining Situations

• Can be applied to various two-player bargaining situations (labor negotiations, resource allocation, international disputes)
• To find the Nash bargaining solution:
  1. Determine the disagreement point (payoffs if no agreement is reached)
  2. Determine the feasible set (all possible payoff combinations)
  3. Find the point in the feasible set that maximizes the Nash product
• Example: In a labor negotiation, the disagreement point could be the current wages and benefits, while the feasible set includes all possible combinations of wage increases and benefit improvements

Step-by-Step Application

1. Identify the players involved in the bargaining situation
2. Determine each player's utility function, mapping outcomes to numerical values representing their preferences
3. Establish the disagreement point $(d_1, d_2)$, the payoffs players would receive if no agreement is reached
4. Define the feasible set of all possible payoff combinations, considering any constraints or limitations
5. Find the point $(x^*, y^*)$ in the feasible set that maximizes the Nash product: $(x^* - d_1)(y^* - d_2)$
6. The point $(x^*, y^*)$ represents the Nash bargaining solution, the unique outcome satisfying the four axioms

Efficiency and Fairness of Nash Bargaining

Pareto Efficiency

• The Nash bargaining solution is Pareto efficient, always selecting an outcome on the Pareto frontier
• On the Pareto frontier, no player can be made better off without making the other worse off
• Ensures that the solution maximizes the total welfare of the players given the available options
• Example: In a resource allocation problem, the Nash bargaining solution will allocate resources such that no reallocation can improve one player's outcome without harming the other

Fairness Considerations

• Satisfies the symmetry axiom, ensuring equal payoffs for indistinguishable players
• Fairness may be questioned when players have unequal bargaining power or the disagreement point is not equitable
• Does not account for the bargaining process or potential strategic behavior by players
• Example: If one player has a significantly better disagreement point, the Nash bargaining solution may favor that player, even if the outcome is not perceived as fair

Nash Bargaining vs Other Solutions

Kalai-Smorodinsky Bargaining Solution

• An alternative satisfying different axioms, including individual monotonicity (which Nash does not satisfy)
• Selects the Pareto efficient outcome maintaining the ratio of players' maximum possible gains
• Focuses on maintaining the relative gains of players from the disagreement point
• Example: If player 1's maximum possible gain is twice that of player 2's, the Kalai-Smorodinsky solution will select an outcome where player 1's gain is twice player 2's gain

Egalitarian and Utilitarian Solutions

• Egalitarian bargaining solution focuses on equalizing players' gains from the disagreement point, prioritizing equality over efficiency
• Utilitarian bargaining solution maximizes the sum of players' utilities, emphasizing total welfare rather than individual gains
• These solutions prioritize different aspects of fairness and efficiency compared to the Nash bargaining solution
• Example: In a resource allocation problem, the egalitarian solution would allocate resources to equalize gains, while the utilitarian solution would maximize the total utility of all players