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Finding saddle points on polyhedra: Solving certain continuous minimax problems

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  • George E. Monahan

Abstract

This article reviews procedures for computing saddle points of certain continuous concave‐convex functions defined on polyhedra and investigates how certain parameters and payoff functions influence equilibrium solutions. The discussion centers on two widely studied applications: missile defense and market‐share attraction games. In both settings, each player allocates a limited resource, called effort, among a finite number of alternatives. Equilibrium solutions to these two‐person games are particularly easy to compute under a proportional effectiveness hypothesis, either in closed form or in a finite number of steps. One of the more interesting qualitative properties we establish is the identification of conditions under which the maximizing player can ignore the values of the alternatives in determining allocation decisions. © 1996 John Wiley & Sons, Inc.

Suggested Citation

  • George E. Monahan, 1996. "Finding saddle points on polyhedra: Solving certain continuous minimax problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(6), pages 821-837, September.
    • Handle: RePEc:wly:navres:v:43:y:1996:i:6:p:821-837
    DOI: 10.1002/(SICI)1520-6750(199609)43:63.0.CO;2-6
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    References listed on IDEAS

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    1. Edward S. Pearsall, 1976. "A Lagrange Multiplier Method for Certain Constrained Min-Max Problems," Operations Research, INFORMS, vol. 24(1), pages 70-91, February.
    2. John S. Croucher, 1975. "Application of the fundamental theorem of games to an example concerning antiballistic missile defense," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 22(1), pages 197-203, March.
    3. Richard Schmalensee, 1976. "A Model of Promotional Competition in Oligopoly," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 43(3), pages 493-507.
    4. Jerome Bracken & James T. McGill, 1974. "Defense Applications of Mathematical Programs with Optimization Problems in the Constraints," Operations Research, INFORMS, vol. 22(5), pages 1086-1096, October.
    5. Shapley, Lloyd S & Shubik, Martin, 1977. "Trade Using One Commodity as a Means of Payment," Journal of Political Economy, University of Chicago Press, vol. 85(5), pages 937-968, October.
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    8. Jerome Bracken & James T. McGill, 1974. "Optimization of strategic defenses to provide specified post‐attack production capacities," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 21(4), pages 663-672, December.
    9. George E. Monahan, 1987. "The Structure of Equilibria in Market Share Attraction Models," Management Science, INFORMS, vol. 33(2), pages 228-243, February.
    10. Jerome Bracken & James T. McGill, 1974. "Technical Note—A Method for Solving Mathematical Programs with Nonlinear Programs in the Constraints," Operations Research, INFORMS, vol. 22(5), pages 1097-1101, October.
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