Axiomatic Bargaining Theory

Bargaining is uses when agents could make a mutually beneficial agreement but have a conflict of interest about which one to make. Like in Battle of the Sexes.
Axiomatic Bargaining Theory assumes no equilibrium. Instead it states the desired properties of a solution (axioms of bargaining solution).
The Nash bargaining solution is an early solution concept for a game with two agents trying to decide between a set of outcomes (o) and a fallback outcome (o_fallback). The set of axioms are:
- Invariance: u_i is relative, not absolute. We can multiply it by two the results should still be the same.
- Anonymity: switching the labels on the agents does not affect the outcome.
- Independence of irrelevant alternatives: if o is removed, the solution should not change.
- Pareto efficiency: it is not possible to give both players more utility.
- The best outcome that satisfies them is
- o^* = arg max_o(u₁(o) - u₁(o_fallback))(u₂(o) - u₂(o_fallback))
- Also nice because it's an unique solution
- It can be extended to more than two agents.
For example, say two agents are deciding how to split a dollar with a fallback value of 0. In this case all splits are in the Nash equilibrium (picture the matrix). However, the Nash bargaining solution gives us a .5/.5 split (why?).