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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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