 |
|
|
| | | | | |
|
|
|
|
|
Two-player Knock 'em Down
|
James Allen Fill, The Johns Hopkins University
David B Wilson, Microsoft
|
Abstract
We analyze the two-player game of Knock 'em Down, asymptotically as
the number of tokens to be knocked down becomes large. Optimal play
requires mixed strategies with deviations of order √n
from the naïve law-of-large numbers allocation. Upon rescaling
by √n and sending n→∞, we show
that optimal play's random deviations always have bounded support
and have marginal distributions that are absolutely continuous with
respect to Lebesgue measure.
|
Full text: PDF
Pages: 198-212
Published on: February 14, 2008
|
Bibliography
-
Arthur T. Benjamin and Matthew T. Fluet.
The best way to Knock 'm Down.
The UMAP Journal, 20(1):11--20, 1999.
-
Arthur T. Benjamin and Matthew T. Fluet.
What's best?
Amer. Math. Monthly, 107(6):560--562, 2000.
MR1767066
-
Arthur T. Benjamin, Matthew T. Fluet, and Mark L. Huber.
Optimal token allocations in Solitaire Knock 'm Down.
Electron. J. Combin., 8(2):Research Paper 2, 8 pp., 2001.
In honor of Aviezri Fraenkel on the occasion of his 70th birthday.
MR1853253
-
Ky Fan.
Minimax theorems.
Proc. Nat. Acad. Sci. U.S.A., 39:42--47, 1953.
MR0055678
-
Matthew T. Fluet.
Searching for optimal strategies in Knock 'm Down, 1999.
Senior thesis, Harvey Mudd College, Claremont, CA.
-
James A. Fill and David B. Wilson.
Solitaire Knock 'em Down, 2006.
Manuscript.
-
Gordon Hunt.
Knock 'm down.
Teaching Stat., 20(2):59--62, 1998.
-
Pertti Mattila.
Geometry of Sets and Measures in Euclidean Spaces, fractals
and rectifiability, volume 44 of Cambridge Studies in Advanced
Mathematics.
Cambridge University Press, Cambridge, 1995.
MR1333890
-
Daniel W. Stroock.
Probability Theory, an analytic view.
Cambridge University Press, Cambridge, 1993.
MR1267569
-
Maurice Sion and Philip Wolfe.
On a game without a value.
In Contributions to the theory of games, vol. 3, Annals of
Mathematics Studies, no. 39, pages 299--306. Princeton University Press,
Princeton, N. J., 1957.
MR0093742
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context