@article{oai:hama-med.repo.nii.ac.jp:00000213,
 author = {Noda, Akio},
 journal = {浜松医科大学紀要. 一般教育},
 month = {Mar},
 note = {application/pdf, This is a continuation of the author's previous papers [1] and [2], and is devoted to the study of optimal strategies arising in the card game of concentration. Suppose that two players A and B have both perfect memory and a set of cards consisting of all twin cards is used in their concentration game. In the first paper [1], we started to investigate two basic strategies, the strategies I and II. Note that the changing situation of this game is characterized by a vector (m,m - r), where the total number of the remaining twin cards is 2m , and the both players keep in memory m-r different cards including no twin cards. Now, the player having the move have to make a choice just before he turns up the second card. Indeed, he can select it randomly, either from the set of m+r-1 unknown cards or from the set of m-r memorized cards for 1≦m-r≦m-2, the former selection forms the strategy I and the latter does the strategy II. We are ready to explain our optimal strategy M. Before coming to a decision concerning the second card, the player calculate the expectation e1(m,m-r) of his points obtained if he adopts the strategy I in the present situation (m,m-r) and he also gets another expectation e2(m,m-r) based on the strategy II. Then he prefers the one giving rise to the maximum value max{e1(m,m-r),e2(m,m-r)}, which turns out to depend on the changing situation (m,m-r) in a rather complicated way. Such a strategy of selecting, in every situation of his move, the optimal one in a family of basic strategies that produces the maximum value of expectation is called the strategy M. The purpose of this paper is to give a full account of the strategy M and then to make a comparison of various strategies such as M, I , II and others. In order to exhibit an advantage of the strategy M over the other, we discuss several cases of the concentration game in which A constantly adopts the strategy M but B changes his strategy within the above-mentioned strategies. On the same line as in [1] and [2], we investigate the expectation of A's points in the situation (m,m-r) of, his own move, and succeed in proving its asmptotic behavior of the form αrm+δr+O(1/m), where m goes to ∞ for each fixed r. The detailed study of these coefficients αr and δr as sequences of r = 1,2,3, … enables us to state the following: The leading term αrm is independent of B's strategy, but the constant term δr does depend on it; if B changes his strategy from the strategy M to any one of other non-optimal strategies, then δr increases. The key point in discusions that follow in sections 2 and 3 is to show that A should select the strategy I when r is odd, and the strategy II when r is even, for sufficiently large values of m. For each value of m , on the other hand, the problem of solving the inequality e1(m,m-r) > e2(m,m-r) seems to be difficult, which one can see by checking several tables listed in the final section 4.},
 pages = {1--23},
 title = {A Mathematical Note of Card Games ,III; on Optimal Strategies in the Concentration Game},
 volume = {17},
 year = {2003}
}