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The bargaining problem. (English) Zbl 1202.91122
The author considers the situations where two individuals may achieve any of a set of situations by agreement (e.g., transfer of a given item in exchange for varying possible amounts of money) but can only retain the status quo if they do not agree. Using the von Neumann-Morgenstern theory of utility, a situation may be represented by a point in a plane whose coordinates are the utilities of the two players when that situation obtains. If random combinations of situations are permitted, then the linearity properties of von Neumann-Morgenstern utility insure the convexity of the set. The utility scales may be so chosen that the utility of the status quo is zero to both players; then the origin belongs to the set $$S$$ of situations possible. If $$c(S)$$ is the point in $$S$$ finally chosen by agreement, the following assumptions are made: (1) There is no point in $$S$$ better than $$c(S)$$ for both players; (2) if $$c(T)$$ is chosen from $$T$$ and $$S$$ is a subset of $$T$$ which contains $$c(T)$$, then $$c(S)=c(T)$$; (3) if $$S$$ is symmetric about a $$45^\circ$$ line through the origin, then $$c(S)$$ lies on that line (i.e., bargaining skills are equal, so that symmetric conditions lead to symmetric solutions). It is then shown to follow that if $$S$$ is compact, then $$c(S)$$ is that point in the first quadrant which maximizes the product of the utilities of the two players among all points in $$S$$.

##### MSC:
 91B26 Auctions, bargaining, bidding and selling, and other market models 91A05 2-person games 91A30 Utility theory for games
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