By John Nunn
Following his winning volumes on rook endings and pawnless endings, John Nunn and his desktop database flip their cognizance to the intricacies of endings with simply knights, bishops, and pawns -- tough endings of serious functional value.
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5. Thus, 1 cannot be an optimal pure strategy. o 6. Computational Methods: In this section we shall describe four programs used to compute strategies for the DSG and give examples of their application to specific games. These programs have been combined into an integrated Pascal program called SEARPAK which is listed in Clemson U. Tech Report 521 dated 1986. The routine SEARCH1 determines an optimal search sequence in response to finding probabilities Pll P2,' .. ,Pn and a fixed mixed strategy for the Hider.
In [Ruckle, 1983] this game is treated on p. 162 ff where it is called "Hide and Seek on a Complete Graph" (HSC). The dissertation of Norris cited above considers variations of this game such as the case when the Hider can move but at a cost. The main purpose of this paper is to establish a sufficient theoretical structure permitting the effective approximation of the solution of the DSG in the sense of Game Theory. That is, we want to approximate the value of the game as well as optimal strategies for Hider and Seeker.
23 Models for the Game of Liar's Dice If II challenges y, then P(I winslY = y) = g(y). If II accepts y, then P(I winslY = y) = g(y)(l- ip(O)) + (1- g(y))(I- ip((y - x)/(I- x))). Hence it is optimal for II to challenge y if and only if g(y) ::; g(y)(l- ip(O)) + (1- g(y))(I- ip((y - x)/(I- x))) or, equivalently, ip(O)f'(x) ::; 1- ip((y - x)/(I- x)). Suppose ip(O) < V. Then, using j'(x) = (j(x) - x)/((I- xlV) for 0 ::; x::; W, ip(O)j'(x) = ip(O)(y - x)/((I- xlV) < (y - x)/(1 - x) ::; 1- ip((y - x)/(I- x)) from Lemma 1.