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The Bargaining Problem
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The Bargaining Problem is a concept in game theory, a field of mathematics and economics that examines how individuals make decisions when their outcomes depend on the actions of others. Imagine two people trying to divide a pie. The challenge is how to split it fairly, considering each person's preferences.John Nash, a renowned mathematician, proposed a solution to this problem. He suggested a set of criteria to ensure a fair and mutually beneficial agreement. These criteria include:Efficiency: The solution should make the most of the available resources. In our pie example, this means no part of the pie should go to waste.Symmetry: If both people have the same preferences, they should get equal shares. So, if both love the pie equally, they should get equal slices.Independence of Irrelevant Alternatives: The final agreement should depend only on each person's preferences, not on unrelated options. If someone introduces another food option, it shouldn't change how the pie is split.Utility Maximization: Each person should get the most satisfaction (utility) possible from their share of the pie, considering their preferences.In real-world terms, Nash's Bargaining Solution applies to various scenarios where individuals or groups need to reach an agreement that benefits everyone involved, like business negotiations, diplomatic discussions, or even deciding what movie to watch with friends. It's a way to ensure fairness and cooperation in situations where interests and preferences might conflict.

This still leaves each individual's utility function determined only up to multiplication by a positive real number. Henceforth any utility functions used shall be understood to be so chosen. We may produce a graphical representation of the situation facing the two by choosing utility functions for them and plotting the utilities
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It is necessary to introduce assumptions about the nature of the set of points thus obtained. We wish to assume that this set of points is compact and convex, in the mathematical senses. It should be convex since an anticipation which will graph into any point on a straight line segment between two points of the set can always be obtained by the appropriate probability combination of two anticipations which graph into the two points. The condition of compactness implies, for one thing, that the set of points must be bounded, that is, that they can all be inclosed in a sufficiently large square in the plane. It also implies that any continuous function of the utilities assumes a maximum value for
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We shall regard two anticipations which have the same utility for any utility function corresponding to either individual as equivalent so that the graph becomes a complete representation of the essential features of the situation. Of course, the graph is only determined up to changes of scale since the utility functions are not completely determined. Now since our solution should consist of rational expectations of gain by the two bargainers, these expectations should be realizable by an appropriate agreement between the two. Hence, there should be an available anticipation which gives each the amount of satisfaction he should expect to get. It is reasonable to assume that the two, being rational, would simply agree to that anticipation, or to an equivalent one. Hence, we may think of one point in the set of the graph as representing the solution, and also representing all anticipations that the two
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We shall develop the theory by giving conditions which should hold for the relationship between this solution point and the set, and from these deduce a simple condition determining the solution point. We shall consider only those cases in which there is a possibility that both individuals could gain from the situation. (This does not exclude cases where, in the end, only one individual could have benefited because the "fair bargain" might consist of an agreement to use a probability method to decide who is to gain in the end. Any probability combination of available anticipations is an available anticipation.) This content downloaded from 165.82.98.155 on Tue, 25 Sep 2018 16:50:16 UTC All use subject to
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