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Balanced Silverman Games on General Discrete Sets

  • Book
  • © 1991

Overview

Authors:
  1. Gerald A. Heuer

    1. Concordia College, Moorhead, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Ulrike Leopold-Wildburger

    1. University of Graz, Graz, Austria

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Lecture Notes in Economics and Mathematical Systems (LNE, volume 365)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-v
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  2. Introduction

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 1-4

  3. Games with saddle points

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 5-5

  4. The 2 by 2 games

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 6-7

  5. Some games which reduce to 2 by 2 when v ≥ 1

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 8-18

  6. Reduction by dominance

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 19-28

  7. Balanced 3 by 3 games

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 29-31

  8. Balanced 5 by 5 games

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 32-33

  9. Reduction of balanced games to odd order

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 34-71

  10. Reduction of balanced games to even order

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 72-113

  11. Games with ±1 as central diagonal element

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 114-118

  12. Further reduction to 2 by 2 when v = 1

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 119-125

  13. Explicit solutions for certain classes

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 126-137

  14. Concluding remarks on irreducibility

    • Gerald A. Heuer, Ulrike Leopold-Wildburger
    Pages 138-139

  15. Back Matter

    Pages 140-142
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Keywords

About this book

A Silverman game is a two-person zero-sum game defined in terms of two sets S I and S II of positive numbers, and two parameters, the threshold T > 1 and the penalty v > 0. Players I and II independently choose numbers from S I and S II, respectively. The higher number wins 1, unless it is at least T times as large as the other, in which case it loses v. Equal numbers tie. Such a game might be used to model various bidding or spending situations in which within some bounds the higher bidder or bigger spender wins, but loses if it is overdone. Such situations may include spending on armaments, advertising spending or sealed bids in an auction. Previous work has dealt mainly with special cases. In this work recent progress for arbitrary discrete sets S I and S II is presented. Under quite general conditions, these games reduce to finite matrix games. A large class of games are completely determined by the diagonal of the matrix, and it is shown how the great majority of these appear to have unique optimal strategies. The work is accessible to all who are familiar with basic noncooperative game theory.

Authors and Affiliations

  • Concordia College, Moorhead, USA

    Gerald A. Heuer

  • University of Graz, Graz, Austria

    Ulrike Leopold-Wildburger

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