1. Desired Properties
- invariant to coordinate-wise affine transformation
2. Problem Statement
We consider a two-person bargaining problem formulated as follows
- : to every two-person game we associated a pair , where is a point in the plane and is a subset of the plane.
- The pair has the following intuitive interpretation: where is the level of utility that player receives if the two players do not cooperate with each other.
- Every point represents the level of utility for players 1 and 2 that can be reached by an outcome of the game which is feasible for the two players when they do cooperate.
- Assumption 1: There is at least one point such that . In other words, bargaining may prove worthwhile for both players.
- Assumption 2: is convex. This is justified under the assumption if two outcomes of the game give raise to points and in , then randomization of these two outcomes give raise to all convex combinations of and .
- Assumption 3: is compact.
- Assumption 4: for every . If this is not the case, we can disregard all the points of that fail to satisfy this condition because it is impossible that both players will agree to such a solution.
We let denote the set of pairs that satisfying these four conditions, and we call an element in a bargaining pair.
- Axiom 1: Pareto Optimality
- For every there is no such that and .
- Axiom 2: Symmetry
- We let be defined by and we require that for every , .
- Axiom 3: Invariance with Respect to Affine Transformation of Utility
- A is an afine transformation of utility if , , and the maps are of the form for some positive constant and some constant . We require that for such a transformation , .
- Axiom of Independence of Irrelavant Alternatives
- If and are bargaining pairs such that and , then .
- Interpretation: given a bargaining pair , for every point , consider the product . Then is the unique point in that maximizes this product.
- Many objectives are raised to Nash’s axiom of independence of irrelevant alternatives.
- Let be a function defined for in the following way
- if is the Pareto of .
- if there is no such .
- thus is the maximum player 2 can get if player 1 get at least x.
- By assumption 1 in the definition of a bargaining pair .
- By the compactness of , and are finite and are attained by points in .
- A pair will be called normalized if and . Clearly every game can be normalized by a unique affine transformation of the utilities.
Example objective to Nash’s Solution: consider the following two normalized pair and where
- = convex hull, and
- = convex hull,
- Nash’s solution for is , and for .
- Limitations pf Nash’s solution: Player 2 has good reasons to demand that he get more in the bargaining pair than he does in .
- This axiom states that if, for every utility level that player 1 may demand, the maximum feasible utility level that player 2 can simultaneously reach is increased, then the utility level assigned to player 2 according to the solution should also be increased.
- to normalize the utility function of each agent in such a way that it is worth zero at the status quo and one at this agent’s best outcome — given that all others get at least their status quo utility level
- to sharing equally the benefits of cooperation. In other words, this solution equalizes the relative benefit from status quo or equivalently the relative frustration until the shadow optimum.
- Independence of irrelevant alternatives can be substituted with a monotonicity condition. It is the point which maintains the ratios of maximal gains. In other words, if player 1 could receive a maximum of with player 2’s help (and vice versa for ), then the bargaining solution would yield the point on the Pareto frontier such that .