Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T07:28:02.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Extended optima and equilibria for continuous games. I. General results

Published online by Cambridge University Press:  17 February 2009

D. J. Gates  and
M. Westcott
D. J. Gates
Affiliation:
C. S. I. R. O. Division of Mathematics and Statistics, P. O. Box 1965, Canberra, ACT 2601
M. Westcott
Affiliation:
C. S. I. R. O. Division of Mathematics and Statistics, P. O. Box 1965, Canberra, ACT 2601
Rights & Permissions [Opens in a new window]

Absrtact

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this, the first of three papers, we examine conditions, derived previously, which specify the equilibrium solutions of an adjustment process for N players engaged in a game with continuous (in fact, continuously differentiable) payoff functions, where each player's strategy is to choose a single real number. It is equivalent to the basic form of quantity-variation competition between N firms. The conditions are related to a new optimum which takes account of the ability of firms, or coalitions of firms, to discipline another firm that tries to increase its own profit. Closely related optima are also introduced and analysed. The new optima occupy N-dimensional regions in the strategy space, and contain the optima of Cournot, Pareto, von-Neumann and Morgenstern, and Nash as special cases.

Type
Research Article
Information
The ANZIAM Journal , Volume 22 , Issue 3 , January 1981 , pp. 291 - 307
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Day, R. H., “Adaptive processes and economic theory”, in Adaptive economic models (eds. Day, R. H. and Groves, T.), (Academic Press, New York, 1975), 1–38.Google Scholar
[2]Gates, D. J. and Rickard, J. A., “Pareto optimum by independent trials”, Bull. Austral. Math. Soc. 12 (1975), 259–265.CrossRefGoogle Scholar
[3]Gates, D. J., Rickard, J. A. and Wilson, D. J., “A convergent adjustment process for firms in competition”, Econometrica, 45 (1977), 1349–1364.CrossRefGoogle Scholar
[4]Gates, D. J., Rickard, J. A. and Wilson, D. J., “Convergence of a market related game strategy”, J. Math. Economics, 5 (1978), 97–109.CrossRefGoogle Scholar
[5]Gates, D. J., Rickard, J. A. and Westcott, M., “Exact cooperative solutions of a duopoly model without cooperation”, J. Math. Economics (to appear).Google Scholar
[6]Gates, D. J. and Westcott, M., “Extended optima and equilibria for continuous games. II. A class of economic models”, preprint, second paper of this series (1979).Google Scholar
[7]Gates, D. J. and Westcott, M., “Extended optima and equilibria for continuous games. III. Comparison with experiment”, preprint, third paper of this series (1979).Google Scholar
[8]Kirman, A. P., “Learning by firms about demand conditions”, in R. H. Day and T. Groves, loc.cit. 137–156.CrossRefGoogle Scholar
[9]Mayberry, J. P., Nash, J. F. and Shubik, M., “A comparison of treatments of a duopoly situation”, Econometrica, 21 (1953), 141–154.CrossRefGoogle Scholar
[10]von Neumann, J. and Morgenstern, O., Theory of games and economic behaviour, 2nd ed. (Princeton University Press, 1947).Google Scholar
[11]Seidmann, D. J., “The rationality of expections equilibria”, preprint, London School of Economics, 1979.Google Scholar