Full Description
The structure of a Silverman game can be explained very quickly: Each of two players independently selects a number out of a prede termined set, not necessarily the same one for both of them. The higher number wins unless it is at least k times as high as the other one; if this is the case the lower number wins. The game ends in a draw if both numbers are equal. k is a constant greater than 1. The simplicity of the rules stimulates the curiosity of the the orist. Admittedly, Silverman games do not seem to have a direct applied significance, but nevertheless much can be learnt from their study. This book succeeds to give an almost complete overview over the structure of optimal strategies and it reveals a surprising wealth of interesting detail. A field like game theory does not only need research on broad questions and fundamental issues, but also specialized work on re stricted topics. Even if not many readers are interested in the subject matter, those who are will appreciate this monograph.
Contents
1. Introduction.- Survey of prior work.- The payoff function and expected payoffs.- The sequences {pk} and {vk}.- The sequences {Vk} and {Uk}.- Equivalent variations.- 2. Silverman's game on intervals: preliminaries.- The key mixed strategies.- 3. Intervals with equal left endpoints or equal right endpoints.- The regions LAn, and equal right endpoints.- Case 1. [(1, B)] x [(1, B)].- Case 2. [(1, B)] x [(1, D)],1 < B < D.- Case 3. [(1, B)] x [(A, B)], 1 < A < B.- 4. Intervals with no common endpoints.- Case 4. [(1, B)] x [(A, D)], 1 < A < B < D.- Case 5. [(1, D)] x [(A, B)], 1 < A < B < D.- Case 6. [(1, B)] x [(A, D)], 1 < B ? A < D.- Appendix. Multisimilar distributions.- 5. Reduction by dominance.- Type A dominance.- Type B dominance.- Type C dominance.- Type D dominance.- Semi-reduced games.- 6. The further reduction of semi-reduced games.- Games with |M| = 1. (Reduction to 2 x 2.).- Games with M = 0 which reduce to odd order.- Games with M = 0 which reduce to even order.- 7. The symmetric discrete game.- The symmetric game with v ? 1.- The symmetric game with v < v(n).- 8. The disjoint discrete game.- The disjoint game with v ? 1.- The disjoint game with v < 1.- 9. Irreducibility and solutions of the odd-order reduced games.- The reduced game matrix A and the associated matrix B.- The polynomial sequences.- The odd-order game of type (i).- The odd-order game of type (ii).- The odd-order game of type (iii).- The odd-order game of type (iv).- 10. Irreducibility and solutions of the even-order reduced games.- The reduced game matrix A and the associated matrix B.- Further polynomial identities.- The even-order game of type (i).- The even-order games of types (ii) and (iii).- The even-order game of type (iv).- 11. Explicit solutions.- The game onintervals.- The symmetric discrete game.- The disjoint discrete game.- The reduced discrete game.- Semi-reduced balanced discrete games with no changes of sign on the diagonal.- Maximally eccentric games.- References.
