[1] R. Bellman and D. Blackwell :
“Some two-person games involving bluffing Proc. Nat. Acad. Sci. U. S. A.
35 : 10
(October 1949 ),
pp. 600–605 .
MR
0031700 Zbl
0041.44805 article
People
BibTeX
@article {key0031700m,
AUTHOR = {Bellman, Richard and Blackwell, David},
TITLE = {Some two-person games involving bluffing},
JOURNAL = {Proc. Nat. Acad. Sci. U. S. A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {35},
NUMBER = {10},
MONTH = {October},
YEAR = {1949},
PAGES = {600--605},
DOI = {10.1073/pnas.35.10.600},
NOTE = {MR:0031700. Zbl:0041.44805.},
ISSN = {0027-8424},
}
[2] D. Blackwell :
“On randomization in statistical games with \( k \) terminal actions ,”
pp. 183–187
in
Contributions to the theory of games ,
vol. II .
Edited by H. W. Kuhn and A. W. Tucker .
Annals of Mathematics Studies 28 .
Princeton University Press ,
1953 .
MR
0054923 Zbl
0050.14801 incollection
Abstract
People
BibTeX
In a two-person zero-sum game in which a player receives partial, i.e., statistical, information about his opponent’s strategy after which he takes one of \( k \) terminal actions, any randomized strategy is equivalent to a mixture of a countable number of pure strategies in the proportions
\[ \lambda_n = \Bigl(\frac{k-1}{k}\Bigr)^{n-1}\frac{1}{k}, \quad n = 1,2,\dots \]
The proof uses the fact that for any \( k \) non-negative numbers \( z_1,\dots \) , \( z_k \) with
\[ \sum_1^k z_i = 1 \]
there is a partition of the set of positive integers into disjoint sets \( S_1,\dots \) , \( S_k \) such that
\[ \sum_{n\in S_j}\lambda_n = z_j .\]
@incollection {key0054923m,
AUTHOR = {Blackwell, David},
TITLE = {On randomization in statistical games
with \$k\$ terminal actions},
BOOKTITLE = {Contributions to the theory of games},
EDITOR = {Kuhn, Harold William and Tucker, Albert
William},
VOLUME = {II},
SERIES = {Annals of Mathematics Studies},
NUMBER = {28},
PUBLISHER = {Princeton University Press},
YEAR = {1953},
PAGES = {183--187},
NOTE = {MR:0054923. Zbl:0050.14801.},
ISSN = {0066-2313},
ISBN = {9780691079356},
}
[3] K. J. Arrow, E. W. Barankin, and D. Blackwell :
“Admissible points of convex sets ,”
pp. 87–91
in
Contributions to the theory of games ,
vol. II .
Edited by H. W. Kuhn and A. W. Tucker .
Annals of Mathematics Studies 28 .
Princeton University Press ,
1953 .
MR
0054919 Zbl
0050.14203 incollection
Abstract
People
BibTeX
A point \( s \) of a closed convex subset \( S \) of \( k \) -space is admissible if there is no \( t\in S \) with \( t_i\leq s_i \) for all \( i=1,\dots \) , \( k \) , \( t\neq s \) . An example is given in which the set \( A \) of admissible points is not closed.
Let \( P \) be the set of vectors \( p=(p_1,\dots \) , \( p_k) \) with \( p_i > 0 \) and \( \sum_1^k p_i=1 \) , let \( B(p) \) be the set of \( s\in S \) with
\[ (p,s)=\min_{t\in S}(p,t) ,\]
and let \( B=\sum B(p) \) , so that \( B \) consists of exactly those points of \( S \) at which there is a supporting hyperplane whose normal has positive components.
\( B\subset A\subset \overline{B} \) . If \( S \) is determined by a finite set, there is a finite set \( p_1,\dots \) , \( p_N \) , with \( p_j\in P \) , such that \( B=\sum_{j=1}^N B(p_j) \) , so that, since \( B(p) \) is closed for fixed \( p \) , \( B = A = \overline{B} \) .
@incollection {key0054919m,
AUTHOR = {Arrow, K. J. and Barankin, E. W. and
Blackwell, D.},
TITLE = {Admissible points of convex sets},
BOOKTITLE = {Contributions to the theory of games},
EDITOR = {Kuhn, Harold William and Tucker, Albert
William},
VOLUME = {II},
SERIES = {Annals of Mathematics Studies},
NUMBER = {28},
PUBLISHER = {Princeton University Press},
YEAR = {1953},
PAGES = {87--91},
NOTE = {MR:0054919. Zbl:0050.14203.},
ISSN = {0066-2313},
ISBN = {9780691079356},
}
[4] N. M. Smith, Jr., S. S. Walters, F. C. Brooks, and D. H. Blackwell :
“The theory of value and the science of decision: A summary J. Operations Res. Soc. Amer.
1 : 3
(May 1953 ),
pp. 103–113 .
MR
0053466 article
People
BibTeX
@article {key0053466m,
AUTHOR = {Smith, Jr., Nicholas M. and Walters,
Stanley S. and Brooks, Franklin C. and
Blackwell, David H.},
TITLE = {The theory of value and the science
of decision: {A} summary},
JOURNAL = {J. Operations Res. Soc. Amer.},
FJOURNAL = {Operational Research Society Journal},
VOLUME = {1},
NUMBER = {3},
MONTH = {May},
YEAR = {1953},
PAGES = {103--113},
URL = {http://www.jstor.org/stable/166628},
NOTE = {MR:0053466.},
ISSN = {0160-5682},
}
[5] D. Blackwell :
“On multi-component attrition games Naval Res. Logist. Quart.
1 : 3
(1954 ),
pp. 210–216 .
MR
0068195 article
Abstract
BibTeX
A model is described for military games consisting of a large number of identical successive engagements, without resupply, with a player being defeated when his supply of any resource is exhausted. A method is given for determining which player should win.
@article {key0068195m,
AUTHOR = {Blackwell, David},
TITLE = {On multi-component attrition games},
JOURNAL = {Naval Res. Logist. Quart.},
FJOURNAL = {Naval Research Logistics Quarterly},
VOLUME = {1},
NUMBER = {3},
YEAR = {1954},
PAGES = {210--216},
DOI = {10.1002/nav.3800010308},
NOTE = {MR:0068195.},
ISSN = {0028-1441},
}
[6] D. Blackwell :
“Controlled random walks ,”
pp. 336–338
in
Proceedings of the International Congress of Mathematicians
(Amsterdam, 2 September–9 September 1954 ),
vol. III .
Wiskundig Genootschap .
E. P. Noordhoff (Groningen ),
1956 .
MR
0085141 Zbl
0073.13204 incollection
BibTeX
@incollection {key0085141m,
AUTHOR = {Blackwell, David},
TITLE = {Controlled random walks},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress
of {M}athematicians},
VOLUME = {III},
ORGANIZATION = {Wiskundig Genootschap},
PUBLISHER = {E. P. Noordhoff},
ADDRESS = {Groningen},
YEAR = {1956},
PAGES = {336--338},
NOTE = {(Amsterdam, 2 September--9 September
1954). MR:0085141. Zbl:0073.13204.},
}
[7] D. Blackwell :
“An analog of the minimax theorem for vector payoffs Pacific J. Math.
6 : 1
(1956 ),
pp. 1–8 .
MR
0081804 Zbl
0074.34403 article
BibTeX
@article {key0081804m,
AUTHOR = {Blackwell, David},
TITLE = {An analog of the minimax theorem for
vector payoffs},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {6},
NUMBER = {1},
YEAR = {1956},
PAGES = {1--8},
URL = {http://projecteuclid.org/euclid.pjm/1103044235},
NOTE = {MR:0081804. Zbl:0074.34403.},
ISSN = {0030-8730},
}
[8] D. Blackwell :
“Minimax and irreducible matrices J. Math. Anal. Appl.
3 : 1
(August 1961 ),
pp. 37–39 .
MR
0139495 Zbl
0111.33404 article
Abstract
BibTeX
Call an \( n{\times}n \) matrix \( A = \|A(i,j)\| \) with nonnegative elements irreducible if for every \( i,j \) there is an \( N \) for which \( A^{(N)}(i,j) > 0 \) , where
\[ A^{(N)} = \|A^{(N)}(i,j)\| \]
is the \( N \) -th power of \( A \) . A useful result of Frobenius [Karlin 1959] is: Every irreducible \( A \) has a positive eigenvalue \( \lambda_0 \) which is at least as large as the absolute value of any other eigenvalue. This eigenvalue is simple and its eigenvector has all coordinates strictly of the same sign.
We shall obtain this result from the minimax theorem of game theory. Our main result is the
If \( A \) is irreducible, there is exactly one number \( \lambda_0 \) for which the value of the game with matrix \( A - \lambda_0 I \) is zero (\( I \) is the \( n{\times}n \) identity matrix). \( \lambda_0 > 0 \) . Each player has a unique good strategy, and all coordinates of each player’s good strategy are positive.
We shall see that this \( \lambda_0 \) is the \( \lambda_0 \) of Frobenius’ result, and that the strategy vectors are the left, right eigenvectors for \( \lambda_0 \) .
@article {key0139495m,
AUTHOR = {Blackwell, David},
TITLE = {Minimax and irreducible matrices},
JOURNAL = {J. Math. Anal. Appl.},
FJOURNAL = {Journal of Mathematical Analysis and
Applications},
VOLUME = {3},
NUMBER = {1},
MONTH = {August},
YEAR = {1961},
PAGES = {37--39},
DOI = {10.1016/0022-247X(61)90005-1},
NOTE = {MR:0139495. Zbl:0111.33404.},
ISSN = {0022-247x},
}
[9] D. Blackwell :
“Infinite games and analytic sets Proc. Nat. Acad. Sci. U.S.A.
58 : 5
(November 1967 ),
pp. 1836–1837 .
MR
0221466 Zbl
0224.90077 article
Abstract
BibTeX
@article {key0221466m,
AUTHOR = {Blackwell, David},
TITLE = {Infinite games and analytic sets},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy
of Sciences of the United States of
America},
VOLUME = {58},
NUMBER = {5},
MONTH = {November},
YEAR = {1967},
PAGES = {1836--1837},
DOI = {10.1073/pnas.58.5.1836},
NOTE = {MR:0221466. Zbl:0224.90077.},
ISSN = {0027-8424},
}
[10] D. Blackwell and T. S. Ferguson :
“The big match Ann. Math. Stat.
39
(1968 ),
pp. 159–163 .
MR
0223162 Zbl
0164.50305 article
Abstract
People
BibTeX
@article {key0223162m,
AUTHOR = {Blackwell, David and Ferguson, T. S.},
TITLE = {The big match},
JOURNAL = {Ann. Math. Stat.},
FJOURNAL = {Annals of Mathematical Statistics},
VOLUME = {39},
YEAR = {1968},
PAGES = {159--163},
DOI = {10.1214/aoms/1177698513},
NOTE = {MR:0223162. Zbl:0164.50305.},
ISSN = {0003-4851},
}
[11] D. Blackwell :
“Infinite \( G_{\delta} \) -games with imperfect information ,”
Zastos. Mat.
10
(1969 ),
pp. 99–101 .
MR
0243837 Zbl
0232.90068 article
BibTeX
@article {key0243837m,
AUTHOR = {Blackwell, D.},
TITLE = {Infinite \$G_{\delta}\$-games with imperfect
information},
JOURNAL = {Zastos. Mat.},
FJOURNAL = {Zastosowania Matematyki},
VOLUME = {10},
YEAR = {1969},
PAGES = {99--101},
NOTE = {MR:0243837. Zbl:0232.90068.},
ISSN = {0044-1899},
}
[12] D. Blackwell :
“Borel sets via games Ann. Probab.
9 : 2
(1981 ),
pp. 321–322 .
MR
606995 Zbl
0455.28002 article
Abstract
BibTeX
A family of games \( G = G(\sigma, u) \) is defined such that (a) for each \( \sigma \) the set of all \( u \) for which Player I can force a win in \( G(\sigma, u) \) is a Borel set \( B(u) \) and (b) every Borel set is a \( B(u) \) for some \( u \) .
@article {key606995m,
AUTHOR = {Blackwell, D.},
TITLE = {Borel sets via games},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {9},
NUMBER = {2},
YEAR = {1981},
PAGES = {321--322},
DOI = {10.1214/aop/1176994474},
NOTE = {MR:606995. Zbl:0455.28002.},
ISSN = {0091-1798},
CODEN = {APBYAE},
}
[13] D. Blackwell :
“Operator solution of infinite \( G_\delta \) games of imperfect information ,”
pp. 83–87
in
Probability, statistics, and mathematics: Papers in honor of Samuel Karlin .
Edited by T. W. Anderson, K. B. Athreya, and D. L. Iglehart .
Academic Press (Boston, MA ),
1989 .
MR
1031279 Zbl
0696.90094 incollection
People
BibTeX
@incollection {key1031279m,
AUTHOR = {Blackwell, David},
TITLE = {Operator solution of infinite \$G_\delta\$
games of imperfect information},
BOOKTITLE = {Probability, statistics, and mathematics:
{P}apers in honor of {S}amuel {K}arlin},
EDITOR = {Anderson, Theodore Wilbur and Athreya,
Krishna B. and Iglehart, Donald L.},
PUBLISHER = {Academic Press},
ADDRESS = {Boston, MA},
YEAR = {1989},
PAGES = {83--87},
NOTE = {MR:1031279. Zbl:0696.90094.},
ISBN = {9780120584703},
}
[14] D. Blackwell :
“Games with infinitely many moves and slightly imperfect information ,”
pp. 407–408
in
Games of no chance
(MSRI, Berkeley, CA 11–21 July 1994 ).
Edited by R. J. Nowakowski .
MSRI Publications 29 .
Cambridge University Press ,
1996 .
Zbl
0873.90115 incollection
People
BibTeX
@incollection {key0873.90115z,
AUTHOR = {Blackwell, David},
TITLE = {Games with infinitely many moves and
slightly imperfect information},
BOOKTITLE = {Games of no chance},
EDITOR = {Nowakowski, Richard J.},
SERIES = {MSRI Publications},
NUMBER = {29},
PUBLISHER = {Cambridge University Press},
YEAR = {1996},
PAGES = {407--408},
NOTE = {(MSRI, Berkeley, CA 11--21 July 1994).
Zbl:0873.90115.},
ISBN = {9780521574112},
}
[15] D. Blackwell :
“The square-root game 35–37
in
Game theory, optimal stopping, probability and statistics: Papers in honor of Thomas S. Ferguson .
Edited by F. T. Bruss and L. Le Cam .
IMS Lecture Notes–Monograph Series 35 .
Institute of Mathematical Statististis (Beachwood, OH ),
2000 .
MR
1833849 Zbl
0988.91004 incollection
Abstract
People
BibTeX
@incollection {key1833849m,
AUTHOR = {Blackwell, David},
TITLE = {The square-root game},
BOOKTITLE = {Game theory, optimal stopping, probability
and statistics: {P}apers in honor of
{T}homas~{S}. {F}erguson},
EDITOR = {Bruss, F. Thomas and Le Cam, Lucien},
SERIES = {IMS Lecture Notes -- Monograph Series},
NUMBER = {35},
PUBLISHER = {Institute of Mathematical Statististis},
ADDRESS = {Beachwood, OH},
YEAR = {2000},
PAGES = {35--37},
DOI = {10.1214/lnms/1215089742},
NOTE = {MR:1833849. Zbl:0988.91004.},
ISBN = {9780940600485},
}
[16] D. Blackwell :
“The prediction of sequences Internat. J. Game Theory
31 : 2
(2002 ),
pp. 245–251 .
Special anniversary issue.
MR
1968990 Zbl
1082.91036 article
Abstract
BibTeX
The problem of predicting the short-term future behavior of a sequence, after observing it as long as we please, so as to achieve a specified reliability against all possible sequences is considered. For a particular problem, namely, predicting when in a sequence of 0s and 1s the pair \( (1,0) \) in that order is not coming next, a reliability of \( 3/4 \) can be approximated as closely as we please, but not achieved.
@article {key1968990m,
AUTHOR = {Blackwell, David},
TITLE = {The prediction of sequences},
JOURNAL = {Internat. J. Game Theory},
FJOURNAL = {International Journal of Game Theory},
VOLUME = {31},
NUMBER = {2},
YEAR = {2002},
PAGES = {245--251},
DOI = {10.1007/s001820200114},
NOTE = {Special anniversary issue. MR:1968990.
Zbl:1082.91036.},
ISSN = {0020-7276},
CODEN = {IJGTA2},
}
