Abstract
If two or more players agree to cooperate while playing a game, they help one another to minimize their respective costs as long as it is not to their individual disadvantages. This leads at once to the concept of undominated solutions to a game. Anundominated orPareto-optimal solution has the property that, compared to any other solution, at least one playerdoes worse or alldo the same if they use a solution other than the Pareto-optimal one.
Closely related to the concept of a Pareto-optimal solution is theabsolutely cooperative solution. Such a solution has the property that, compared to any other permissible solution,every playerdoes no better if a solution other than the absolutely cooperative one is employed.
This paper deals with control-space properties of Pareto-optimal and absolutely cooperative solutions for both static, continuous games and differential games. Conditions are given for cases in which solutions to the Pareto-optimal and absolutely cooperative games lie in the interior or on the boundary of the control set.
The solution of a Pareto-optimal or absolutely cooperative game is related to the solution of a minimization problem with avector cost criterion. The question of whether or not a problem with a vector cost criterion can be reduced to a family of minimization problems with ascalar cost criterion is also discussed.
An example is given to illustrate the theory.
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This research was supported in part by NASA Grant No. NGR-03-002-011 and ONR Contract No. N00014-69-A-0200-1020.
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Vincent, T.L., Leitmann, G. Control-space properties of cooperative games. J Optim Theory Appl 6, 91–113 (1970). https://doi.org/10.1007/BF00927045
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DOI: https://doi.org/10.1007/BF00927045