Fibonacci-Type Sequences and Minimal Solutions of Discrete Silverman Games
Gerald A. Heuer, Ulrike Leopold‐Wildburger · The Fibonacci Quarterly · 1994
While the principal results of this paper seem to us to be of interest in their own right, and can be understood with no reference to game theory, the problems addressed arose in a game theory setting, and their solution has important consequences for the analysis of Silverman games. It seems appropriate therefore to sketch briefly the game theory background. Silverman games are two-person, zero-sum games in which, roughly speaking, the higher bid wins, unless it is too much higher than the other, in which case it loses. More precisely, let Sl and Su be sets of positive real numbers, and Tand v be parameters with T> \\ and v> 0. The sets Sl andSn are the pure strategy sets for Players I and II, respectively. Each player chooses a number from his strategy set, and the higher number wins 1, unless it is at least T times as large as the other, in which case it loses v. The parameters T and v are referred to as the threshold and the penalty, respectively. If SY = Slh the game is symmetric, and in this case, if optimal strategies exist they are the same for both players, and the game value is 0. The prototype games are attributed to David Silverman, although the earliest published mention of such a game of which we are aware is by Herstein and Kaplansky ([3], p. 212). The symmetric game on an open interval was analyzed by R. J. Evans [1] for arbitrary T and v, and the