The nonsense that is the Nash Bargaining Solution.

John Nash mathematically resolved the problem of the "indeterminate contract curve" by introducing a unique, axiomatic framework known as the Nash Bargaining Solution. [1, 2]
Before Nash's breakthrough, classical economists (following Francis Ysidro Edgeworth) established that when two parties engage in a bilateral monopoly or barter exchange, the resulting allocations form a contract curve. While every point on this curve is Pareto efficient, economic theory considered the exact final division of gains indeterminate—meaning traditional supply-and-demand mechanics could not predict exactly where the two parties would agree to settle along that line. [1, 2, 3] 

How Nash Solved the Indeterminacy
Instead of trying to simulate the unpredictable psychological back-and-forth of human negotiation, Nash approached the problem backwards. He asked: What reasonable mathematical properties must a final, "fair" agreement satisfy?

In other words, Nash's solution is that if there is a fair agreement because fair agreement is militted for then there will be fair agreement though sadly nobody could ever say any agreement whatsoever wasn't fair or unfair or the product of sodomy with Satan 

He proposed four fundamental axioms that any rational bargaining solution should follow: [1, 2]
Pareto Efficiency: The parties will not settle for an agreement if another exists that leaves at least one person better off without harming the other. (The solution must lie on the contract curve). [1, 2]

This requires omniscience on both sides. Nobody would ever reach any agreement if this were a condition.  

Symmetry: If both players have identical preferences and strategic positions,

they must also have identical endowments (otherwise they would have different strategies to acquire what they lack and sell what they have too much off) and thus would not want to trade 

they will receive identical payoffs. 

Zero, because they won't want to trade. 

Invariance to Coordinate Transformations (Scale Invariance): Changing the units used to measure utility (e.g., switching from Fahrenheit to Celsius equivalent) will not change the physical outcome of the deal. [1, 2]

 i.e. utility isn't derived from what it is measured in. Sadly, this is not the case if it is measured in a transferable form- e.g. money.

Independence of Irrelevant Alternatives (IIA): If the parties prefer option A over option B out of a large set of choices, removing other unchosen options from the table shouldn't suddenly make them pick option B. [1, 2]

This means those unchosen options 

1) don't convey any information relevant to the decision. 

2) have no 'external' effects. In particular, there is no 'supermodularity' (complementarity)

3) don't alter risk or Knightian uncertainty. 

When it comes to economic transactions, market-makers exist. There is derived or speculative demand. Whether two peopl decide to do a deal rather than go through an arbitrageur has a lot to do what other options exist. When some of them disappear, their behaviour is likely to change for prudential reasons. 

    The Unique Mathematical Proof

Nash proved that if these four rules hold true, there is only one mathematically viable point on the entire contract curve.

There would be no contract curve. There would only be an omnescient, ubitquitous, information source, directing all actions without any need for communication or coordination. 

To find it, you establish a disagreement point (or threat point), which represents the payoff each party gets if negotiations completely collapse.

Both sides bluff. If an agent gets a reputation for being obdurate or extreme in their demands, they may not be able to transact business. The same may happen to people considered mendacious or of weak character.  

The unique Nash Bargaining Solution is the exact allocation that maximizes the product of the parties' utility gains above that disagreement point. 

This is the idea that an outsider can always improve deadlocked negotiations. It is false.Why? Because, in a repeated game, it leads to resources being diverted to improving the threat point.  

Where U represents the utility of the players and d represents their respective fallback positions. By tethering the final outcome directly to each party's "outside options" and risk preferences, Nash turned a vague, indeterminate range of options into a single, predictable mathematical equilibrium.

Nash published his result 75 years ago. Did indeterminacy disappear? Nope. It increased. Why? Mathsy masturbation has nothing to do with reality.