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Block Design Games

  • A. J. Hoffman (a1) and Moses Richardson (a2)

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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.

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References

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1. Bose, R. C., A note on the resolvability of balanced incomplete block designs, Sankhyâ, 6 (1942), 105110.
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).
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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).
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15. Shrikhande, S. S., On the dual of some balanced incomplete block designs, Biometrica, 8 (1952), 6672.
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Block Design Games

  • A. J. Hoffman (a1) and Moses Richardson (a2)

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