Celebratio Mathematica — Blackwell

Game theory

Works connected to Harold William Kuhn

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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

Harold William Kuhn	Related
Albert William Tucker	Related

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

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 \( p \), \( B = A = \overline{B} \).

Albert William Tucker	Related
Kenneth J. Arrow	Related
Edward William Barankin	Related
Harold William Kuhn	Related