Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T22:01:04.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Block Design Games

Published online by Cambridge University Press:  20 November 2018

A. J. Hoffman  and
Moses Richardson
A. J. Hoffman
Affiliation:
General Electric Company
Moses Richardson
Affiliation:
Brooklyn College and Princeton University
Rights & Permissions [Opens in a new window]

Extract

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 paper, we define and begin the study of an extensive family of simple n-person games based in a natural way on block designs, and hitherto for the most part unexplored except for the finite projective games (13). They should serve at least as a proving ground for conjectures about simple games. It is shown that many of these games are not strong and that many do not possess main simple solutions. In other cases, it is shown that they have no equitable main simple solution, that is, one in which the main simple vector has equal components. On the other hand, the even-dimensional finite projective games PG(2s, pn) with s > 1 possess equitable main simple solutions, although they are not strong either. These results are obtained by means of the study of the possible blocking coalitions. Interpretations in terms of graph theory, network flows, and linear programming are discusssed, as well as k-stability, automorphism groups, and some unsolved problems.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 110 - 128
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Bose, R. C., A note on the resolvability of balanced incomplete block designs, Sankhyâ, 6 (1942), 105–110.Google Scholar
2. Bose, R. C., Clatworthy, W. H., Shrikhande, S. S., Tables of partially balanced designs with two associate classes, Tech. Bull. No. 107, North Carolina Agricultural Experiment Station (1954).Google Scholar
3. Bose, R. C. and Nair, K. R., Partially balanced incomplete block designs, Sankhyâ, 4 (1939), 337–372.Google Scholar
4. Carmichael, R. D., Introduction to the theory of groups of finite order (Ginn, 1937).Google Scholar
5. Connor, W. S., Some relations among the blocks of symmetrical group divisible designs, Ann. Math. Stat., 23 (1952), 602–609.Google Scholar
6. Goldman, A. J., Tucker, A. W., Theory of linear programming. In H. W. Kuhn and A. W. Tucker, Linear inequalities and related systems (Princeton, 1956).Google Scholar
7. Gurk, H. M. and Isbell, J. R., Simple solutions. In A. W. Tucker and R. D. Luce, Contributions to the theory of games IV (Princeton, 1959).Google Scholar
8. Hall, M. Jr., A survey of combinatorial analysis. In I. Kaplansky, E. Hewitt, M. Hall, Jr., and R. Fortet, Some aspects of analysis and probability (Wiley, 1958).Google Scholar
9. Luce, R. D., A definition of k-stability for n-person games, Ann. Math., 59 (1954), 357–366.Google Scholar
10. Moore, E. H., Tactical memoranda, Amer. J. Math., 18 (1896), 264–303.Google Scholar
11. Netto, E., Lehrbuch der Combinatorik (Chelsea reprint of 1927 edition).Google Scholar
12. von Neumann, J. and Morgenstern, O., Theory of games and economic behavior (2nd ed.; Princeton, 1947).Google Scholar
13. Richardson, M., On finite projective games, Proc. Amer. Math. Soc, 7 (1956), 458–465.Google Scholar
14. Shapley, L. S., Lectures on n-person games (Princeton University notes, unpublished).Google Scholar
15. Shrikhande, S. S., On the dual of some balanced incomplete block designs, Biometrica, 8 (1952), 66–72.Google Scholar